FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY NOTIONS (F.U.N.)

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FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY

NOTIONS (F.U.N.)

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FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY

NOTIONS (F.U.N.)

JOHN MICHAEL RASSIAS EDITOR

Nova Science Publishers, Inc. New York

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Published by Nova Science Publishers, Inc. New York

CONTENTS

Preface vii

Chapter 1 Ulam's Stability of a Class of Linear Cauchy Functional Equations 1 Ahmed Charifi, Mohamed Ait Sibaha and Belaid Bouikhalene

Chapter 2 Sequential Antagonistic Games with an Auxiliary Initial Phase 15 Jewgeni H. Dshalalow and Weijun Huang

Chapter 3 Some Stability Results for Equations and Inequalities Connected with the Exponential Function

37

Włodzimierz Fechner and Roman Ger

Chapter 4 On a Problem of John M. Rassias Concerning the Stability in Ulam Sense of Euler–Lagrange Equation

47

L. Găvruţa and P. Găvruţa

Chapter 5 Hyers–Ulam–Aoki–Rassias Sstability and Ulam–Gavruta–Rassias Stability of Quadratic Homomorphisms and Quadratic Derivations on Banach Algebras

55

M. Eshaghi Gordji and N. Ghobadipour

Chapter 6 Fundamental Solutions for the Generalized Elliptic Gellerstedt Equation

73

Anvar Hasanov, John Michael Rassias and Mamasali Turaev

Chapter 7 Pointwise Superstability and Superstability of the Jordan Equation 85 Ji-Rong Lv, Huai-Xin Cao and J.M. Rassias

Chapter 8 A Problem with Non-Local Conditions on the Line of Degeneracy and Parallel Characteristics for a Mixed Type Equation with Singular Coefficient

95

M. Mirsaburov and M. Kh. Ruziev

Chapter 9 On the Stability of AN Additive Functional Inequality in Normed Modules

107

Choonkil Park

Contents vi

Chapter 10 Cubic Derivations and Quartic Derivations on Banach Modules 119 Choonkil Park and John Michael Rassias

Chapter 11 Tetrahedron Isometry Ulam Stability Problem 131 John Michael Rassias

Chapter 12 Hyers–Ulam Stability of Cauchy Type Additive Functional Equations

143

Matina J. Rassias

Chapter 13 Solution and Ulam Stability of a Mixed Type Cubic and Additive Functional Equation

149

J.M. Rassias, K. Ravi, M. Arunkumar and B.V. Senth. Kumar

Chapter 14 Stability of Mappings Approximately Preserving Orthogonality and Related Topics

177

Aleksej Turnšek

Chapter 15 The Frankl Problem for Second Order Nonlinear Equations of Mixed Type with Non-smooth Degenerate Curve

191

Guo Chun Wen

Index 209

PREFACE 1. Ulam’s Biography: S. M. ULAM was born in Lwow, Poland on April 3, 1909 and died in Santa Fe, U.S.A.

on May 13, 1984. He graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwow in 1933. Ulam worked at: The Institute for Advanced Study, Princeton (1936), Harvard University (1939-40), University of Wisconsin (1941-43), Los Alamos Scientific Laboratory (1943-65), University of Colorado (1965-76), and University of Florida (1974-). He was a member of the American Academy of Arts and Sciences and the National Academy of Sciences. He made fundamental contributions in mathematics, physics, biology, computer science, and the design of nuclear weapons. His early mathematical work was in set theory, topology, group theory, and measure theory. While still a schoolboy in Lwow, Ulam signed his notebook "S. Ulam, astronomer, physicist and mathematician". As Ulam notes, "the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation." Ulam worked with Stefan Banach, Kazimir Kuratowski, Karol Borsuk, Stanislaw Mazur, Hugo Steinhaus, John von Neumann, Garrett Birkhoff, Cornelius Everett, Dan Mauldin, D. H. Hyers, Mark Kac, P. R. Stein, Enrico Fermi, John Pasta, Richard Feynman, Ernest Lawrence, J. Robert Oppenheimer, Teller, and many other people of applied and exact sciences. Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. Ulam's most remarkable achievement at Los Alamos was his contribution to the postwar development of the thermonuclear or hydrogen (H-) bomb in which nuclear energy is released when two hydrogen or deuterium nuclei fuse together. One of Ulam's early insights was to use the fast computers at Los Alamos to solve a wide variety of problems in a statistical manner using random numbers. This method has become appropriately known as the Monte Carlo method. One example that may have biological relevance is the subfield of cellular automata founded by Ulam and von Neumann. Finally Ulam had a unique ability to raise important unsolved problems. One of these problems was solved by the editor of this F.U.N. volume (J. Approx. Th., Vol. 57, 268-273, 1989).

John Michael Rassias viii

2. Ulam’s volume F. U. N. : Functional Equations and Difference Inequalities and Ulam Stability Notions, is a

forum for exchanging ideas among eminent mathematicians and physicists, from many parts of the world, as a tribute to the first centennial birthday anniversary of Stanislaw Marcin ULAM.

This collection is composed of outstanding contributions in mathematical and physical equations and inequalities and other fields of mathematical and physical sciences. It is intended to boost the cooperation among mathematicians and physicists working on a broad variety of pure and applied mathematical areas. This transatlantic collection of mathematical ideas and methods comprises a wide area of applications in which equations, inequalities and computational techniques pertinent to their solutions play a core role.

Ulam's influence has been tremendous on our everyday life, because new tools have been developed, and revolutionary research results have been achieved , bringing scientists of exact sciences even closer, by fostering the emergence of new approaches, techniques and perspectives.

The central scope of this commemorating 100birthday anniversary volume is broad, by deeper looking at the impact and the ultimate role of mathematical and physical challenges, both inside and outside research institutes, scientific foundations and organizations.

We have recently observed a more rapid development in the areas of research of Ulam worldwide.

This F.U.N. volume is suitable for graduate students and researchers interested in functional equations, and differential equations and would make an ideal supplementary reading or independent study research text.

This item will also be of interest to those working in other areas of mathematics and

physics. It is a work of great interest and enjoyable read as well as unique in market. This Ulam's volume (F.U.N.) consists of research papers containing various parts of

contemporary pure and applied mathematics with emphasis to Ulam's mathematics. It contains various parts of Functional Equations and Difference Inequalities as well as

related topics in Mathematical Analysis, namely: Ulam’s stability of a class of linear Cauchy functional equations; Sequential antagonistic games with an auxiliary initial phase; Some stability results for

equations and inequalities connected with the exponential function; On a problem of John M. Rassias concerning the stability in Ulam sense of Euler-Lagrange equation; Hyers-Ulam-Aoki-Rassias stability and Ulam-Gavruta-Rassias stability of quadratic homomorphisms and quadratic derivations on Banach algebras; Fundamental solutions for the generalized elliptic Gellerstedt equation; Pointwise superstability and superstability of the Jordan equation; A problem with non-local conditions on the line of degeneracy and parallel characteristics for a mixed type equation with singular coefficient; On the stability of an additive functional inequality in normed modules; Cubic derivations and quartic derivations on Banach modules; Tetrahedron isometry Ulam stability problem; Hyers-Ulam stability of Cauchy type additive functional equations; Solution and Ulam stability of a mixed type cubic and additive

Preface ix

functional equation; Stability of mappings approximately preserving orthogonality and related topics; The Frank problem for second order nonlinear equations of mixed type with non-smooth degenerate curve.

John Michael RASSIAS Editor of Ulam’ s volume F. U. N. National and Capodistrian University of Athens 4, Agamemnonos Str., Aghia Paraskevi, Attikis 15342 Athens, GREECE E-mail: [email protected] Web: http://www.primedu.uoa.gr/~jrassias

In: Functional Equations, Difference Inequalities...Editor: John Michael Rassias, pp. 1-14

ISBN 978-1-60876-461-7c© 2010 Nova Science Publishers, Inc.

Chapter 1

ULAM ’ S STABILITY OF A CLASS OF L INEAR

CAUCHY FUNCTIONAL EQUATIONS

Ahmed Charifia, Mohamed Ait Sibahaa, Belaid Bouikhaleneb,∗

and Samir Kabbajaa Laboratory: L. A. M. A., Department of MathematicsUniversity of Ibn Tofail, Faculty of Sciences BP 133

Kenitra 14000, Moroccob University Sultan Moulay Slimane

Polydisciplinaire Faculty Beni MellalMorocco

Abstract

In this work, we describe the solution of (1.1) connected with additive functionsand we study the Ulam’s problem of this equation. Some applications deal with newequations of type linear Cauchy in Banach spaces, are given.

2000 Mathematics Subject Classifications:39B32, 39B42, 39B72.

Key words: Functional equation, Ulam problem, stability.

1. Introduction

LetE andF be Banach spaces with norms‖·‖ and‖·‖ respectively. A mappingf : E −→F is called, additive function, if it satisfies the Cauchy functional equationf(x + y) =f(x) + f(y) for all x, y ∈ E.

In 1940, S. M. Ulam (see [22]) raised the question concerning the stability of grouphomomorphisms: “when is it true that the solution of an equation differing slightly froma given one, must of necessity be close to the solution of the given equation?”. The firstanswer to Ulam’s question, concerning the Cauchy equation, was given by D. H. Hyers[8]. Thus we speak about the Hyers–Ulam stability. This terminology is also applied tothe case of other functional equations. Th. M. Rassias [19] generalized the theorem of

∗E-mail address: [email protected]

2 A. Charifi, M. Ait Sibaha, B. Bouikhalene et al.

Hyers for approximately linear mappings [19]. The stability phenomena that wasprovedby Th. M. Rassias [19] is called the Hyers–Ulam–Rassias stability. The modified Ulam’sstability problem with the generalization control function was proved by P. Gavruta [6] inthe following way

Theorem 1.1.LetE be a vector space,E be a Banach space and letϕ : E×E −→ [0,+∞[be a function satisfying

ψ(x, y) =+∞∑

k=0

1

2k+1ϕ(2kx, 2ky) < +∞

for all x, y ∈ E. If a functionf : E −→ F satisfies the functional inequality

‖f(x+ y) − f(x) − f(y)‖ < ϕ(x, y)

for all x, y ∈ E. Then there exists a unique additive functionT : E −→ F which satisfies‖f(x) − T (y)‖ ≤ ψ(x, y) for all x ∈ E.

J. M. Rassias [14]–[18] solved the Ulam’s problem for different mappings, in the fol-lowing way

Theorem 1.2. Let X be a real normed linear space and letY be a real normed linearspace. Assume in addition that iff : X −→ Y is a mapping for which there exist constantδ > 0 andp, q ∈ R such thatr = p+ q 6= 1 andf satisfies the inequality

‖f(x+ y) − f(x) − f(y)‖ ≤ δ‖x‖p‖y‖q

for all x, y ∈ X. Then there exists a unique additive mappingL : X −→ Y satisfying

‖f(x) − L(x)‖ ≤δ

‖2r − 1‖‖x‖r

for all x ∈ X.

As the words “differing slightly” and “be close” in Ulam’s problem have various mean-ings, different kinds of stability can be dealt with. It may happen that all approximatesolution are in fact exact solutions. Then we speak about superstability. To get acquaintedwith the theory of the stability of functional equation we refer to papers [8]–[15].

In this paper, we introduce the new general linear Cauchy type functional equation ofthe form

M∑

j=1

(n∑

i=1

xi + aj

)= M

n∑

i=1

f(xi) for all x1, . . . , xn ∈ E (1.1)

and for any arbitrary fixed elementsa1, . . . , aM inE.Whenaj = 0 for all j ∈ 1, . . . ,M,the equation (1.1) reduces to the equation

f

(n∑

i=1

xi

)=

n∑

i=1

f(xi) for all x1, . . . , xn ∈ E. (1.2)

Ulam’s Stability 3

The Ulam’s problem of the question (1.2) was studied by J. M. Rassias in [18 Theorem2].If n = 2 andM = 2, the equation (1.1) becomes

f(x+ y + a) + f(x+ y + b) = 2f(x) + 2f(y). (1.3)

The Ulam’s problem of this equation was studied by authors in [1]. Following this investi-gation, we study here the Ulam’s problem of the functional equation (1.1) and a bouquet ofspecial cases, namely

f(x+ y + z + a) + f(x+ y + z + b) = 2f(x) + 2f(y) + 2f(z) (1.4)

andf(x+ y + a) + f(x+ y + b) + f(x+ y + c) = 3f(x) + 3f(y), (1.5)

for all x, y, z ∈ E and any arbitrarya, b, c ∈ E. This paper is organized as follows: inthe first section, after this introduction we gave the general solution of (1.1). In the secondsection we investigate the Ulam’s problem for the general linear Cauchy equation (1.1). InCorrollaries (5.5) and (5.6) we deduce the Ulam’s stability for equation (1.4) and (1.5).

2. Solution of (1.1)

In this section we give the general solution of the functional equation (1.1).

Theorem 2.1. LetM , N be integers,M > 0 andN > 1. LetE andF be vectors spaces.A functionf : E −→ F satisfies the functional equation (1.1) if and only if there exists anadditive functiong : E −→ F such that

f(x) =M∑

j=1

g((N − 1)x+ aj)

M(N − 1)for all x ∈ E.

Proof. If f is a solution of (1.1), then by substitutingx1, . . . , xN−1 by 0 andxN by x1 +· · · + xN in (1.1) we have

M∑

j=1

f

(N∑

i=1

xi + aj

)= M(N−1)f(0)+Mf

(N∑

i=1

xi

)for all x1, . . . , xN ∈ E. (2.6)

By takingx1 = · · · = xN = 0 in (1.1) we get

M∑

j=1

(f − f(0))(aj) = M(N − 1)f(0).

From (1.1) and (2.1) we deduce that

N∑

i=1

f(xi) = (N − 1)f(0) + f

(N∑

i=1

xi

)


soN∑

i=1

(f − f(0))(xi) = (f − f(0))

(N∑

i=1

xi

).

We poseg = f − f(0), then

f(x) = g(x) +

M∑j=1

g(aj)

M(N − 1)=

M∑j=1

g((N − 1)x+ aj)

M(N − 1).

Conversely, letg : E −→ F be an additive function. It’s elementary to verify thatf(x) =M∑

j=1

g((N−1)x+aj)

M(N−1) is a solution of (1.1). This ends the proof.

3. Ulam’s Stability for the Functional Equation (1.1)

In this section we establish the Ulam’s stability for equation (1.1).

Theorem 3.1. LetE be a vector space,F a Banach space andδ > 0. Suppose that thefunctionf : E −→ F satisfies the inequality

∥∥∥∥∥∥

M∑

j=1

f

(N∑

i=1

xi + aj

)−M

N∑

i=1

f(xi)

∥∥∥∥∥∥< δ for all x1, . . . , xN ∈ E. (3.7)

Then, there exists a unique functionS : E −→ F solution of (1.1) such that

‖f(x) − S(x)‖ ≤δ


Proof. Assume thatf : E −→ F satisfies the inequality (3.1), we use induction onn to

prove that the sequence functionsf0(x) = f(x) andfn(x) =M∑

j=1fn−1(Nx + aj) for all

x ∈ E, n ∈ N∗ satisfy the following statements

‖fn(x) −MNfn−1(x)‖ < Mn−1δ for all x ∈ E, n ∈ N∗, (3.8)

‖fn(x) −MnNnf0(x)‖ < Mn−1Nn − 1

N − 1δ for all x ∈ E, n ∈ N

∗, (3.9)∥∥∥∥∥∥

M∑

j=1

fn

(N∑

i=1

xi + aj

)−M

N∑

i=1

fn(xi)

∥∥∥∥∥∥< Mnδ for all x1, . . . , xN ∈ E, n ∈ N

∗.

(3.10)

By takingx1 = · · · = xN = x in (3.1), we get that

‖f1(x) −MNf0(x)‖ < δ.


We have for allx1, . . . , xn ∈ E andn ∈ N∗

∥∥∥∥∥∥

M∑

j=1

f1

(N∑

i=1

xi + aj

)−M

N∑

i=1

f1(xi)

∥∥∥∥∥∥

=

∥∥∥∥∥∥

M∑

j=1

M∑

l=1

f0

(N

N∑

i=1

xi +Naj + al

)−

M∑

j=1

M

N∑

i=1

f0(Nxi + aj)

∥∥∥∥∥∥

≤M∑

j=1

∥∥∥∥∥

M∑

l=1

f0

(N∑

i=1

(Nxi + aj) + al

)−M

N∑

i=1

f0(Nxi + aj)

∥∥∥∥∥ < Mδ.

Consequently, the assertions (3.2), (3.3) and (3.4) are trues forn = 1. Assuming that theassertions are trues for all integersi, 1 ≤ i ≤ n. It follows from the induction assumptionthat

‖fn+1(x) −MNfn(x)‖ =

∥∥∥∥∥∥

M∑

j=1

fn(Nx+ aj) −MN

M∑

j=1

fn−1(Nx+ aj)

∥∥∥∥∥∥

≤

M∑

j=1

‖fn(Nx+ aj) −MNfn−1(Nx+ aj)‖ < M(Mn−1δ) = Mnδ

for all x ∈ E. We have for allx ∈ E that

∥∥fn+1(x) −Mn+1Nn+1f0(x)∥∥ =

∥∥∥∥∥

n∑

i=0

M iN ifn+1−i(x) −M i+1N i+1fn−i(x)

∥∥∥∥∥

≤n∑

i=0

M iN i ‖fn+1−i(x) −MNfn−i(x)‖ ≤n∑

i=0

M iN iMn−iδ = MnNn+1 − 1

N − 1δ.

Now, for allxi, 1 ≤ i ≤ n, we get∥∥∥∥∥∥

M∑

j=1

fn+1

(N∑

i=1

xi + aj

)−M

N∑

i=1

fn+1(xi)

∥∥∥∥∥∥

=

∥∥∥∥∥∥

M∑

j=1

M∑

l=1

fn

(N

N∑

i=1

xi +Naj + al

)−

M∑

j=1

M

N∑

i=1

fn(Nxi + aj)

∥∥∥∥∥∥

≤

M∑

j=1

∥∥∥∥∥

M∑

l=1

fn

(N∑

i=1

(Nxi + aj) + al

)−M

N∑

i=1

fn(Nxi + aj)

∥∥∥∥∥ < M(Mnδ) = Mn+1δ,

(3.11)

which gives the sought results. From (3.2), it follows that the sequence functions( fn(x)

MnNn

)n

is a Cauchy sequence. SinceF is complete, the above sequence has a limit inF. We defineS : E −→ F by

S(x) = limn→+∞

fn(x)

MnNn.


From (3.3), it follows that

‖f(x) − S(x)‖ ≤δ


From (3.4), we deduce thatS : E −→ F satisfies the functional equation (1.1).In the next we will show the uniqueness ofS. LetS′ : E −→ F be another solution of

the functional equation (1.1) which satisfies

‖f(x) − S′(x)‖ ≤δ


We will prove by induction that

∥∥fn(x) −MnNnS′(x)∥∥ ≤

Mn−1

N − 1δ for all x ∈ E. (3.12)

SinceS′ satisfies (1.1), we get

∥∥f1(x) −MNS′(x)∥∥ =

∥∥∥∥∥∥

M∑

j=1

f0(Nx+ aj) −M∑

j=1

S′(Nx+ aj)

∥∥∥∥∥∥

≤M∑

j=1

∥∥f(Nx+ aj) − S′(Nx+ aj)∥∥ ≤

Mδ

M(N − 1)=

δ

N − 1.

Assumingthat (3.6) is true for all integersi, 1 ≤ i ≤ n, hence we have

∥∥fn+1(x) −Mn+1Nn+1S′(x)∥∥ =

∥∥∥∥∥∥

M∑

j=1

fn(Nx+ aj) −MnNnM∑

j=1

S′(Nx+ aj)

∥∥∥∥∥∥

≤M∑

j=1

∥∥fn(Nx+ aj) −MnNnS′(Nx+ aj)∥∥ ≤

Mn

N − 1δ.

By lettingn −→ +∞ we get from inequality∥∥∥∥fn(x)

MnNn− S′(x)

∥∥∥∥ ≤δ

M(N − 1)Nn

thatS = S′. Thisends the proof of theorem (3.1).

Theorem 3.2. Let E be a vector space, letF be a Banach space and letϕ : EN −→[0,+∞[ be a function. We defineϕ0 = ϕ and

ϕn(x1, . . . , xN ) =M∑

j=1

ϕn−1(Nx1+aj , . . . , NxN+aj) for all x1, . . . , xN ∈ E, n ∈ N∗.

(3.13)


Suppose that

ϕ(x1, . . . , xN ) =+∞∑

n=0

1

Mn+1Nn+1ϕn(x1, . . . , xN ) < +∞ for all x1, . . . , xN ∈ E.

(3.14)Assume that a functionf : E −→ F satisfies the inequality

‖M∑

j=1

f(N∑

i=1

xi + aj) −M

N∑

i=1

f(xi)‖ < ϕ(x1, . . . , xN ) for all x1, . . . , xN ∈ E. (3.15)

Then, there exists a unique functionS : E −→ F solution of (1.1) such that

‖f(x) − S(x)‖ ≤ ϕ(x, . . . , x) for all x ∈ E. (3.16)

Proof. We use induction onn to prove that the sequence functionsf0(x) = f(x) and

fn(x) =M∑

j=1fn−1(Nx+ aj) for all x ∈ E, n ∈ N

∗, satisfy the following statements

‖fn(x) −MNfn−1(x)‖ < ϕn−1(x, . . . , x) for all x ∈ E, n ∈ N∗, (3.17)

‖fn(x) −MnNnf0(x)‖ <

n−1∑

i=0

M iN iϕn−1−i(x, . . . , x) for all x ∈ E, n ∈ N∗,

(3.18)∥∥∥∥∥∥

M∑

j=1

fn

(N∑

i=1

xi+aj

)−M

N∑

i=1

fn(xi)

∥∥∥∥∥∥<ϕn(x1, . . . , xN ) for all x1, . . . , xN ∈E, n∈N.

(3.19)

By using the definition offn and the inequality (3.9) we have

‖f1(x) −MNf0(x)‖ =

∥∥∥∥∥∥

M∑

j=1

f(Nx+ aj) −MNf(x)

∥∥∥∥∥∥< ϕ(x, . . . , x).

We have from (3.5)

∥∥∥∥∥∥

M∑

j=1

f1

(N∑

i=1

xi + aj

)−M

N∑

i=1

f1(xi)

∥∥∥∥∥∥

≤M∑

j=1

∥∥∥∥∥

M∑

l=1

f0

(N∑

i=1

(Nxi + aj) + al

)−M

N∑

i=1

f0(Nxi + aj)

∥∥∥∥∥

<

M∑

j=1

ϕ(Nx1 + aj , . . . , Nxn + aj) = ϕ1(x1, . . . , xN ).


The assertions (3.11), (3.12) and (3.13) are now trues forn = 1. Assuming that the as-sertions are trues for all integersi, 1 ≤ i ≤ n. It follows from (3.9) and the inductionassumption that

‖fn+1(x) −MNfn(x)‖ =

∥∥∥∥∥∥

M∑

j=1

fn(Nx+ aj) −MN

M∑

j=1

fn−1(Nx+ aj)

∥∥∥∥∥∥

≤M∑

j=1

‖fn(Nx+ aj) −MNfn−1(Nx+ aj)‖

<

M∑

j=1

ϕn−1(Nx+ aj , . . . , Nx+ aj) = ϕn(x, . . . , x).

We have from (3.5)

∥∥∥∥∥∥

M∑

j=1

fn+1

(N∑

i=1

xi + aj

)−M

N∑

i=1

fn+1(xi)

∥∥∥∥∥∥

≤M∑

j=1

∥∥∥∥∥

M∑

l=1

fn

(N∑

i=1

(Nxi + aj) + al

)−M

M∑

l=1

fn(Nxi + al)

∥∥∥∥∥

<

M∑

j=1

ϕn(Nx1 + aj , . . . , NxN + aj) = ϕn+1(x1, . . . , xN ).

Also we have

∥∥fn+1(x) −Mn+1Nn+1f0(x)∥∥ =

∥∥∥∥∥

n∑

i=0

M iN i [fn+1−i(x) −MNfn−i(x)]

∥∥∥∥∥

<

n∑

i=0

M iN iϕn−i(x, . . . , x).

This gives the sought results. It follows from (3.12) that the sequence( fn(x)

MnNn

)n

is a Cauchysequence. SinceF is complete, the sequence has a limit inF . We defineS : E −→ F , by

S(x) = limn→+∞

fn(x)

MnNn.

It follows from (3.12) that

‖f(x) − S(x)‖ ≤ ϕ(x, . . . , x) for all x ∈ E.

From (3.13), it follows that the functionS : E −→ F satisfies the functional equation (1.1).In the next we will show the uniqueness ofS. LetS′ : E −→ F be another solution of

the functional equation (1.1) which satisfies

‖f(x) − S′(x)‖ ≤ ϕ(x, . . . , x) for all x ∈ E.


We will prove by induction that

∥∥fn(x) −MnNnS′(x)∥∥ < Mn−1Nn−1

+∞∑

i=n

1

M iN iϕi(x, . . . , x) for all x ∈ E, n ∈ N

∗.

(3.20)By using (3.5), (3.7), (3.8) and (3.10), we get

∥∥f1(x) −MNS′(x)∥∥ =

∥∥∥∥∥∥

M∑

j=1

f0(Nx+ aj) −M∑

j=1

S′(Nx+ aj)

∥∥∥∥∥∥

≤

M∑

j=1

∥∥f0(Nx+ aj) − S′(Nx+ aj)∥∥ ≤

M∑

j=1

ϕ(Nx+ aj , . . . , Nx+ aj)

=+∞∑

n=0

M∑

j=1

1

Mn+1Nn+1ϕn(Nx+ aj , . . . , Nx+ aj)

=+∞∑

n=0

1

Mn+1Nn+1ϕn+1(x, . . . , x) =

+∞∑

i=1

1

M iN iϕi(x, . . . , x).

Assuming that the assertion (3.14) is true for all integersi, 1 ≤ i ≤ n, hence we have

∥∥fn+1(x) −Mn+1Nn+1S′(x)∥∥ =

∥∥∥∥∥∥

M∑

j=1

fn(Nx+ aj) −MnNnM∑

j=1

S′(Nx+ aj)

∥∥∥∥∥∥

≤M∑

j=1

∥∥fn(Nx+ aj) −MnNnS′(Nx+ aj)∥∥

<

M∑

j=1

Mn−1Nn−1+∞∑

i=n

1

M iN iϕi(Nx+ aj , . . . , Nx+ aj)

= Mn−1Nn−1+∞∑

i=n

1

M iN i

M∑

j=1

ϕi(Nx+ aj , . . . , Nx+ aj)

= Mn−1Nn−1+∞∑

i=n

1

M iN iϕi+1(x, . . . , x)

= MnNn+∞∑

i=n+1

1

M iN iϕi(x, . . . , x).

This gives (3.14). Consequently, by lettingn −→ +∞ we obtain thatS = S′. This endsthe proof of theorem (3.2).

Corollary 3.3. Let δ > 0 andp < 1. Letf : E −→ F be a function from a normed vector


spaceE into a Banach spaceF, which satisfies∥∥∥∥∥∥

M∑

j=1

f(N∑

i=1

xi + aj) −M

N∑

i=1

f(xi)

∥∥∥∥∥∥< δ

N∑

i=1

‖xi‖p for all x1, . . . , xN ∈ E, N ≥ 2.

(3.21)Then, there exists a unique mappingS : E −→ F solution of (1.1) such that

‖f(x) − S(x)‖ ≤δ‖x‖p

M+ δ

Np

M(N −Np)

(‖x‖ +

a

N − 1

)p

for all x ∈ E, (3.22)

wherea = max(‖a1‖, . . . , ‖aM‖).

Proof. We poseϕ(x1, . . . , xN ) = δN∑

i=1‖xi‖

p, x1, . . . , xN ∈ E. By induction we will prove

that

ϕn(x1, . . . , xN ) ≤MnNnpδ

N∑

i=1

(‖xi‖ + a

n∑

k=1

1

Nk

)p

for all x1, . . . , xN ∈ E, n ∈ N∗.

(3.23)Forn = 1 we have

ϕ1(x1, . . . , xN ) =M∑

j=1

δ

N∑

i=1

‖Nxi + aj‖p ≤MNpδ

N∑

i=1

(‖xi‖ +

a

N

)p

.

Assumingthat (3.17) is true for all integers i,1 ≤ i ≤ n, hence we have

ϕn+1(x1, . . . , xN ) =M∑

j=1

ϕn(Nx1 + aj , . . . , NxN + aj)

≤MnNnpδ

M∑

j=1

N∑

i=1

(‖Nxi + aj‖ + a

n∑

k=1

1

Nk

)p

≤MnNnpδ

M∑

j=1

N∑

i=1

(‖Nxi‖ + ‖aj‖ + a

n∑

k=1

1

Nk

)p

≤MnN (n+1)pδ

M∑

j=1

N∑

i=1

(‖xi‖ +

a

N+ a

n∑

k=1

1

Nk+1

)p

≤Mn+1N (n+1)pδ

N∑

i=1

(‖xi‖ + a

n+1∑

k=1

1

Nk

)p

for all x1, . . . , xN ∈ E, n ∈ N∗. So that gives

ϕ(x1, . . . , xN ) =1

MN

+∞∑

n=0

1

MnNnϕn(x1, . . . , xN )


≤δ

MN

N∑

i=1

‖xi‖p +

δ

MN

+∞∑

n=1

(Np−1)nN∑

i=1

(‖xi‖ + a

n∑

k=1

1

Nk

)p

≤δ

MN

N∑

i=1

‖xi‖p +

δ

M

Np−1

N −Np

N∑

i=1

(‖xi‖ +

a

N − 1

)p

for all x1, . . . , xN ∈ E.

From Theorem 3.2 we deduce that there exists a unique functionS : E −→ F solution of(1.1) such that

‖f(x) − S(x)‖ ≤ ϕ(x, . . . , x) ≤δ‖x‖p

M+

Npδ

M(N −Np)

(‖x‖ +

a

N − 1

)p

for all x ∈ E.

Corollary 3.4. Let δ > 0 and p < 1. Suppose thatf : E −→ F be a function from anormed vector spaceE into a Banach spaceF such that

∥∥∥∥∥f(

N∑

i=1

xi

)−

N∑

i=1

f(xi)

∥∥∥∥∥ < δ

N∑

i=1

‖xi‖p for all x1, . . . , xN ∈ E, N ≥ 2.

Then, there exists a unique additive mappingS : E −→ F which satisfies

‖f(x) − S(x)‖ ≤δ‖x‖p

1 −Np−1for all ∈ E.

Corollary 3.5. Letf : E −→ F be a function from a normed vector spaceE into a Banachspace F such that∥∥∥∥∥∥

M∑

j=1

f

(N∑

i=1

xi + aj

)−M

N∑

i=1

f(xi)

∥∥∥∥∥∥< α+ δ

N∏

i=1

‖xi‖pi for all x1, . . . , xN ∈ E,

(3.24)

whereα, δ are a positive numbers andp1, . . . , pN ∈ R are such thatp =N∑

i=1pi < 1. Then,

there exists a unique functionS : E −→ F solution of (1.1) such that

‖f(x)−S(x)‖≤α

M(N−1)+δ‖x‖p

MN+

δNp−1

M(N−Np)

(‖x‖+

a

N−1

)p

for all x∈E,

(3.25)wherea = max(‖a1‖, . . . , ‖aM‖).

Proof. We poseϕ(x1, . . . , xN ) = α+ δN∏

i=1‖xi‖

pi , x1, . . . , xN ∈ E. By induction we will

prove that

ϕn(x1, . . . , xN ) ≤Mnα

+MnNnpδ

N∏

i=1

(‖xi‖ + a

n∑

k=1

1

Nk

)pi

for all x1, . . . , xN ∈ E, n ∈ N∗. (3.26)


Forn = 1 we have

ϕ1(x1, . . . , xN ) = Mα+

M∑

j=1

δ

N∏

i=1

‖Nxi + aj‖pi ≤Mα+MNpδ

N∏

i=1

(‖xi‖ +

a

N

)pi

.

Assumingthat (3.20) is true for all integers i,1 ≤ i ≤ n, hence we have

ϕn+1(x1, . . . , xN ) =M∑

j=1

ϕn(Nx1 + aj , . . . , NxN + aj)

≤ αMn+1 +MnNnpδ

M∑

j=1

N∏

i=1

(‖Nxi + aj‖ + a

n∑

k=1

1

Nk

)pi

≤ αMn+1 +MnNnpδ

M∑

j=1

N∏

i=1

(‖Nxi‖ + ‖aj‖ + a

n∑

k=1

1

Nk

)pi

≤ αMn+1 +MnN (n+1)pδ

M∑

j=1

N∏

i=1

(‖xi‖ +

a

N+ a

n∑

k=1

1

Nk+1

)pi

≤ αMn+1 +Mn+1N (n+1)pδ

N∏

i=1

(‖xi‖ + a

n+1∑

k=1

1

Nk

)pi

for all x1, . . . , xN ∈ E, n ∈ N∗. So that gives

ϕ(x1, . . . , xN ) =1

MN

+∞∑

n=0

1

MnNnϕn(x1, . . . , xN )

≤α

M(N − 1)+

δ

MN

N∏

i=1

‖xi‖pi +

1

MN

+∞∑

n=1

(Np−1)nδ

N∏

i=1

(‖xi‖ + a

n∑

k=1

1

Nk

)pi

≤α

M(N − 1)+

δ

MN

N∏

i=1

‖x‖pi +δNp−1

M(N −Np)

N∏

i=1

(‖xi‖ +

a

N − 1

)pi

for all x1, . . . , xN ∈ E.

FromTheorem 3.2, we deduce that there exists a unique functionS : E −→ F solution of(1.1) such that

‖f(x)−S(x)‖ ≤ ϕ(x, . . . , x) ≤α

M(N − 1)+

δ

MN‖x‖p+

δNp−1

M(N −Np)

(‖x‖ +

a

N − 1

)p

for all x ∈ E. In the case wherep = p1 + p2 < 1, we get the following corollary whichcompletes Theorem 1.2.

Corollary 3.6. Letf : E −→ F be a function from a normed vector spaceE into a Banachspace F such that

∥∥∥∥∥f(

N∑

i=1

xi

)−

N∑

i=1

f(xi)

∥∥∥∥∥ < α+ δ

N∏

i=1

‖xi‖pi for all x1, . . . , xN ∈ E, (3.27)


whereα, δ are a positive numbers andp1, . . . , pN ∈ R are such thatp =N∑

i=1pi < 1. Then,

there exists a unique additive mappingS : E −→ F such that

‖f(x) − S(x)‖ ≤α

M(N − 1)+

δ‖x‖p

(N −Np)for all x ∈ E. (3.28)

Corollary 3.7. Let δ > 0, letE be a vector space andF a Banach space. Assume that afunctionf : E −→ F satisfies the functional inequality∥∥∥f(x+y+z+a)+f(x+y+z+b)−2f(x)−2f(y)−2f(z)

∥∥∥<δ for all x, y, z∈E, (3.29)

wherea and b are two arbitrary elements ofE. Then, there exists a unique functionS :E −→ F solution of the functional equation

f(x+y+z+a)+f(x+y+z+b) = 2f(x)+2f(y)+2f(z) for all x, y, z ∈ E, (3.30)

such that‖f(x) − S(x)‖ ≤ δ4 for all x ∈ E.

Corollary 3.8. Let δ > 0, letE be a vector space andF a Banach space. Assume that afunctionf : E −→ F satisfies the functional inequality∥∥∥f(x+y+a)+f(x+y+b)+f(x+y+c)−3f(x)−3f(y)

∥∥∥ < δ for all x, y, z ∈ E, (3.31)

wherea, b and c are three arbitrary elements ofE. Then, there exists a unique functionS : E −→ F solution of the functional equation

f(x+y+a)+f(x+y+ b)+f(x+y+ c) = 3f(x)+3f(y) for all x, y, z ∈ E, (3.32)

such that‖f(x) − S(x)‖ ≤ δ3 for all x ∈ E.

References

[1] Ait Sibaha M., Bouikhalene B. and Elqorachi E., 2007, Ulam–Gavruta–Rassias Sta-bility of a linear functional equation.International Journal of Applied Mathematicsand Statistics.Vol. 7, No. Fe07, 157–166.

[2] Aoki T., 1950, On the Stability of the linear transformation in Banach spaces,J. Math.Sc. Japan2, 64–66.

[3] Bouikhalene B., 2004, The stability of some linear functional equations,JIMPA, J.Inequal. Pure Appli.Vol. 5, Issue 2, Article 49.

[4] Bouikhalene B. and Elqorachi E, 2007, On the Hyers–Ulam–Rassias stability of linearCauchy functional equation,TOJMS, Vol.23No. 4, 449–459.

[5] Gajda Z., 1991, On stability of additive mappings,Internat J. Math. Sci.14, 431–434.


[6] GavrutaP., 1994, A generalization of the Hyers–Ulam–Rassias stability of approxi-mately additive mappings,Journal of Math. Analysis and Application.184, 431–436.

[7] Gavruta P., 1999, An answer to a question of John M. Rassias concerning the stabilityof Cauchy equation, in:Advences in Equations and Inequalities, in: Hadronic Math.Ser., 67–71.

[8] Hyers D. H., 1941, On the stability of the linear functional equation,Proc. Nat. Acad.Sci. U. S. A.27, 222–224.

[9] Hyers D. H., 1983, The stability of homomorphisms and related topics, in GlobalAnalysis-Analysis on Manifolds, in:Teubner–Texte Math.75, 140–153.

[10] Kuczma M., 1985, “An introduction to the theory of functional equations and inequal-ities”, Panstwowe Wydawnictwo Naukowe, Uniwersyttet Slaski, Warszawa, Karkow,Katowice.

[11] Jung S. -M, 1998, On the Hyers-Ulam stability of the functional equations that havethe quadratic property,J. Math. Anal. Appl.222, 126–137.

[12] Park C., 2005, Homomorphisms between Lie JC*-algebras and Cauchy–Rassias sta-bility of Lie JC*-algebra derivations,J. Lie Theory15, 393–414.

[13] Rassias J. M. 1997,Mathematical Equations and Inequalities,Volume 1, Athens,Greece.

[14] Rassias J. M., 1982, On approximation of approximately linear mappings,J. Funct.Anal.46, 126–130.

[15] Rassias J. M., 1984, On approximation of approximately linear mappings,Bull. Sci.Math.108, 445–446.

[16] Rassias J. M., 1989, Solution of a problem of Ulam,J. Approx. Theory57, 268–273.

[17] Rassias J. M., 1992, Solution of a stability problem of Ulam,Discuss, Math.12, 95–103.

[18] Rassias J. M., 1994, complete solution of the multi-dimentional problem of Ulam,Discuss. Math.14, 101–107.

[19] Rassias Th. M., 1978, On the stability of linear mappings in Banach spaces,Proc.Amer. Math. Soc.72, 297–300.

[20] Rassias Th. M., 1991, Approximate homomorphisms,Aequationes Mathematicae,44,125–153.

[21] Ravi K. and Arunkumar M., 2007, On the Ulam–Gavruta–Rassias stability of the or-thoganally Euler–Lagrange type functional equation,Euler’s Tri-centennial BirthdayAnniversary Issue in FIDA, Internat. J. Appl. Math. Stat.7, 143–156.

[22] Ulam S. M., 1964, A collection of mathematical problems,Interscience Publ. NewYork, 1961. Problems in Modern Mathematics, Wiley, New York.



Chapter 2

SEQUENTIAL ANTAGONISTIC GAMES

WITH AN AUXILIARY I NITIAL PHASE∗

Jewgeni H. Dshalalow† and Weijun HuangDepartment of Mathematical Sciences

College of Science, Florida Institute of TechnologyMelbourne, FL 32901, USA

Abstract

We use random walk processes and tools of fluctuation theory to analyze stochasticgames of two players running two phases. The casualties between the players aremodeled by antagonistic random walks. The game is observed by a point processforming a bivariate random walk process that runs within a fixed rectangle R1 andregisters the moves of the game at the observation epochs. Phase 1 of the game endswhen the process crosses R1 while being contained within a larger rectangle R2. Thatis, Phase 1 ends when a players sustains serious but restricted damages. Phase 2 beginsthereafter and ends with the ruin of one of the players, which occurs when the randomwalk process leaves the area of a rectangle R3, containing R2. Unlike the assumptionsmade in our recent work, the observation process is no longer independent of theantagonistic walks, which suggests an analytical treatment of yet another auxiliaryphase (called initial phase). Three functionals are evaluated on each of the three phasesthat are merged together to form the entire game utilizing not only boundary valuesbut also some key past parameters. The results give predictions of the exit times fromphases 1 and 2 in the form of analytically tractable functionals.

2000 Mathematics Subject Classifications: 82B41, 60G51, 60G55, 60G57,91A10, 91A05, 91A60, 60K05, 60K10.

Key words: Antagonistic stochastic games, sequential games, fluctuation theory, markedpoint processes, Poisson process, ruin time, exit time.

∗This research is supported by the US Army Grant No. W911NF-07-1-0121.†E-mail address: [email protected]

16 Jewgeni H. Dshalalow and Weijun Huang

1. Introduction

The Game. The game presented in this article models a long term conflict between play-ers A and B that exchange hostile actions at random times and exert damages of randommagnitudes. Each player can sustain a limited amount of damage and once this thresholdis reached or exceeded, the player is ruined. A simple game would be over when one ofthe players is ruined. Of course, any real conflict continues until the loser is completelydestroyed or becomes aware of his defeat and his official surrender takes place. In the gamepresented here, the conflict continues over several phases.

The Terminology. Cooperation between players is often assumed to occur even if agame is, loosely speaking, noncooperative. However, there is little agreement in the litera-ture on whether the terms “noncooperative” and “antagonistic” are completely synonymousbut we have used these terms interchangably in some of our past work. In the game pre-sented here, we assume that the game is totallyantagonisticand there is no cooperationwhatsoever between the players. We also assume that the end of the entire game orexittime, and the ruin time of a player are the same. We would not have a problem letting thefinal phase of the game go past the exit time until the defeated player surrenders, but doingso would not yield any interesting analytical results and only make the final formulas morecomplex.

Another feature of the game presented in this paper is itsstochasticity. That is, thegame is modeled by stochastic processes and unlike most traditional game-theoretical work,this game does not offer an optimal strategy for winning or reaching equilibrium. In ouranalysis, we strive to predict major events of the game (such as defeat of a losing player),the status of casualties upon the end of the game, and to impose some reasonable control.

The game presented here runssequentially. That is, the game consists of two separatephases (or games) where the first phase has limited damage inflicted to the players allowingthem to continue to the second phase which runs differently.

Modeling. The antagonistic game in this paper is a model of a conflict but in turn, thegame is modeled by a multivariate random walk or multivariate marked point process. Theconflict starts with a hostile action by one of the players followed by hostile responses fromthe second player. At some time, the first phase of the conflict ends when one of the playerssustains a certain amount of damage. At this time however, no player is ruined and thegame simply morphs into a second phase which is more intense. At this point and over theentire second phase of the game, the thresholds are higher than in the first phase.

To make the model more realistic, we allow the status of the conflict to be updated onlyat certain epochs of time (anobservation process) which causes certain delays. The statusupdate information can be arbitrarily crude or fine and thus, we have a full control over it.The observation process is loosely specified and very general thereby leaving enough spaceto manage it.

The Initial Phase. In our past and recent work [5], [6], [8] on antagonistic games, wealways assumed that the hostile actions between players A and B start at some point of timeminr1, w1 (wherer1 ≥ 0 is the time of the first attack on player A andw1 ≥ 0 is thetime of the first attack on player B) and that the game will be observed by a third-partystochastic process beginning with a random timet0 (or t1), a nonnegative r.v. (random

Sequential Antagonistic Games with an Auxiliary Initial Phase 17

variable). Under these assumptions, the initial observation time, sayt0, can begin in anyinterval: [0, minr1, w1), or [minr1, w1, maxr1, w1), or [maxr1, w1,∞). Thistype of uncertainty was motivated by analytical complexities where our prior investigationshowed that specifying the initial observation time would form a peliminary phase attachedto the forthcoming phases using a “doubly-formed” boundary condition. However, in thispaper we assume the initial observationt0 starts aftermaxr1, w1, i.e. at random epocht0 = maxr1, w1 + ∆0, where∆0 is an independent r.v.. The evolution of the game thatgoes on beyondt0 gets scrutinized all the way from0 to t0 to make an attachment that isanalytically compatible. With this assumption, some preliminary analysis was required thatis separate from the rest of the game.

Attaching the Phases.The game studied here has three separate phases. The initialphase begins with time zero and the inception of the conflict atminr1, w1, and lastsuntil t0, when the first instant of the observation actually takes place. Phase 1 (Game 1)occurs while the actions of the players and the damage is relatively limited. Phase 2 (Game2) occurs when more intense actions are taking place and lasts until one of the players iscompletely ruined. For an illustration, see Figure 1 below:

1 1 2

0 3

t t

t t

ρ

ρ

τ

τ− −= =

= =2

3

µ

ν

τ τ

τ τ

=

=

t

t

t

2M

M

1M

2N

N

1N

1t− 0t 1t 2t 3t

Figure 1.


It may appear that to merge all three phases (being distinct in nature but linked throughinput parameters), we need the exit times from each phase and the values of the damagesat the exits. However, it turns out that not only the exit times and exit values are required,but also, information from the preceding observation in order to attach the phases. The firsteffort to accomplish such a merge was made in our recent work [8]. In this paper we includean initial phase with outputs that impact the entire game.

A Formal Description. Let (Ω,F(Ω), Ft, P ) be a filtered probability space and letFA1 ,FB1,FS ⊆ F (Ω) be independentσ-subalgebras. We suppose that

A1 :=∑

j≥1

djεrj and B1 :=∑

k≥1

zkεwk(1.1)

areFA1 -measurable andFB1-measurable marked Poisson random measures (εa is a pointmass ata), with respective intensitiesλA andλB and position independent marking. Therandom measures are specified by their transforms

Ee−uA1(·) = eλA|·|[hA(u)−1], hA(u) = Ee−ud1 , Re(u) ≥ 0, (1.2)

Ee−uB1(·) = eλB|·|[hB(u)−1], hB(u) = Ee−uz1 , Re(u) ≥ 0, (1.3)

where| · | is the Borel–Lebesgue measure anddj andzk are nonnegative r.v.’s.Phase 1 (or Game 1) starts with hostile actions initiated by one of the players A or B

at r1 or w1, and with respective strikes of magnitudesd1 andz1 respectively. The playerscan exchange with several more strikes before the initial information is picked up by anobserver at timet0. We therefore assume that

t0 ≥ maxr1, w1. (1.4)

The initial observation timet0 will be formalized below. All forthcoming observations willbe rendered in accordance with a point process

T0 =∑

i≥0

εti = εt0 + S, with S =∑

i≥1

εti ,

0 < t0 < t1 < · · · < tn < · · · (tn → ∞, with n → ∞),(1.5)

and its extensionT := εt−1 + T0, with t−1 := minr1, w1, (1.6)

such that the tailS =∑

i≥1 εti of T0 is FS-measurable. The increments∆1 := t1 −t0, ∆2 := t2 − t1, ∆3 := t3 − t2, . . . are all independent and identically distributed, and allbelong to the equivalence class[∆] of r.v.’s with the common transform

δ(θ) := Ee−θ∆. (1.7)

Define the initial observation as

t0 = maxr1, w1 + ∆0, (1.8)


where∆0 ∈ [∆] and∆0 are independent from the rest of the∆’s. t0 is included inT0 ofequation (1.5) and because it contains some of theA1 andB1, T0 is notFS -measurable.However,T0 is a delayed renewal process.

We assignt−1 to the genuine start of the game at time minr1, w1 mentioned in (1.6).That is,

t−1 = minr1, w1. (1.9)

Now, sincet−1 andt0 − t−1 are dependent (throughr1 andw1), the extended processT of (1.6) is not a renewal process, and not even a delayed renewal, as it was in [5], [6], [8].

It should be clear thatt0 depends uponr1 andw1 and thus onA1 andB1, which makesT0 depend on the namedσ-algebras.

Define the continuous time parameter process

(α(t), β(t)) := A1 ⊗ B1([0, t]), t ≥ 0, (1.10)

to be adapted to the filtration(Ft)t≥0. Also introduce its embedding overT0 :

(αj , βj) := (α(tj), β(tj)) = A1 ⊗ B1([0, tj]), j = 0, 1, . . . , (1.11)

which form observations ofA1 ⊗ B1 overT0, with respective increments

(ξj , ηj) := A1 ⊗ B1((tj−1, tj ]), j = 1, . . . . (1.12)

In addition, let(ξ0, η0) := A1 ⊗ B1((maxr1, w1, t0]) (1.13)

to be used later on.Introduce the embedded bivariate marked point process

AT0 ⊗ BT0 := (α0, β0)εt0 +∑

j≥1

(ξj , ηj)εtj , (1.14)

where the marginal marked point processes

AT0 = α0εt0 +∑

i≥1

ξiεti and BT0 = β0εt0 +∑

i≥1

ηiεti (1.15)

are with position dependent marking and withξj andηj being dependent. For the forth-coming sections we introduce the Laplace–Stieltjes transform

g(u, v, θ) := Ee−uξj−vηj−θ∆j , Re(u) ≥ 0, Re(v) ≥ 0, Re(θ) ≥ 0, j ≥ 0, (1.16)

which will be evaluated as the following:

E[e−uξj−vηj−θ∆j

]= E

[e−θ∆j E

[e−uξj−vηj

∣∣∆j

]]

= E[e−θ∆j · E

[e−uA1((tj−1,tj])

∣∣∆j

]· E

[e−vB1((tj−1,tj])

∣∣∆j

]]

= E[e−θ∆j · eλA∆j(hA(u)−1) · eλB∆j(hB(v)−1)

]

= E[e−

θ+λA(1−hA(u))+λB(1−hB(v))

∆j

]

= δ(θ∗), j = 1, 2, . . . , (1.17)

θ∗ : = θ + λA(1− hA(u)) + λB(1− hB(v)), (1.18)


where the functionalδ has been defined in (1.7).

In Section 2 we will refine the embedded bivariate random walk process of (1.14) toinclude the information aboutt−1 and the corresponding values of the process att−1,in notationA1

T ⊗ B1T . In the upcoming sections, we further extendA1

T ⊗ B1T to games 1

and 2. Game 1 (Sections 3 and 4) will continue until one of the players sustains damagesexceeding either two fixed thresholds,M1 andN1, respectively, but limited to two largerthersholdsM (> M1) andN (> N1). In other words, the random walk will end game 1when it enters the area R2−R1, where R1= [0, M1] × [0, N1] and R2= [0, M ] × [0, N ].After game 1 ends, the extension ofA1

T ⊗B1T drifts in R2−R1 initiating game 2 (Section 5),

which terminates when the damage to one of the players exceedsM2 (> M ) or N2 (> N ).The random walk may drift further in the area R3−R2, where R3= [0, M2] × [0, N2], butonce it leaves R3−R2, the entire game is over.

Related Literature. The modeling and analysis of the game can be classified in twodifferent ways. Topically, our model belongs to the game-theoretical literature [1]–[3], [5]–[12], [14]–[16],[18]–[23], and more particularly to sequential games [3], [7], [8], [10], [11],[14], [18]–[20], [23]. Furthermore, the game falls into the subcategory of stochastic games[1], [5], [6], [8], [15], [16]. It also overlaps with the area of stochastic hybrids [1], [4],[5], [6], [8], of which [1] and [5] are true hybrid stochastic games. The antagonistic natureof our modeling suggests yet another category of games, which are purely antagonistic ornoncooperative, and are widely used in economics with highly competitive parties [2], [5],[6], [8], [9], [12], [16], [18], [20], [21] and warfare [5], [6], [8], [22]. Methodologically,the paper falls into the area of fluctuations of random walk processes [5], [6], [8], [17]. Theliterature on this topic is very rich and we cite only a few pertinent articles.

The Layout of the Paper. The present article generalizes our past and recent workon hybrid and sequential antagonistic games [5], [6], [8] in which the initial observationepoch could take place at any time independently of the inception of the conflict. The latterwould make it possible for the initial observation epoch, and even some of its followingones to take place before the conflict begins. This assumption offered tame analytics butturned out to be less realistic. In this paper, we form a strict chronology of the events so thatthe first observation does not take place before two sides exchange with mutual hostilities.The resulting dependence between all processes (which was not assumed in [5], [6], [8])yielded analytical challenges that gave rise to this article. We therefore divided the wholegame into three separate phases, of which the first phase takes place in the interval[0, t0](t0 =maxr1, w1 + ∆0) and is referred to as the initial phase. The details of the initialphase are developed in Section 2 along with all other formalities of the game. Game 1,which continues fromt0 until one of the players ends up suffering some moderate andlimited losses, is treated in Section 3 and results in an explicit functional of the end of game1 (total exit time), the damages to both players, and other important reference points. InSection 4 we impose restrictions on how much damage each player can sustain and furthermodify the “truncated” functional obtained in Section 4. This completes the first phaseof the conflict. In Section 5, we work on game 2, which begins on the heels of game 1,but under different conditions and under the control of different processes. At the end ofSection 5 we calculate the functional that includes only the paths of the game when playerA is defeated. All results are given in analytically tractable forms.


2. The Initial Phase of the Game

Extended Game 1 will include the recording of the conflict between players A and B knownto an observer upon processT (informally, t−1, t0, t1, . . .) from its inception upont−1

followed by the initial observation at timet0. Extended Game 1 is defined below. Theactual start of the game att−1 is unknown to the observer, as this moment takes place priorto t0. From the construction of the extended game, the point processT is obviously “doublydelayed” (in light of its attachmentt−1). The information ont−1 will be presented in theupcoming sections.

The initial phase of the game is specified as follows. Define the respective damages tothe players att−1 as

(ξ−1, η−1) := (α−1, β−1) := (α(t−1), β(t−1)) = (d11r1≤w1, z11r1≥w1). (2.1)

Therefore, the embedded process∑

k≥−1

εtk(αk, βk) obeys the extended initial conditions

A1t−1

⊗ B1t−1

= (α−1, β−1) = (d1, 0), on traceσ-algebraF(Ω)∩ r1 < w1, (2.2)

A1t−1

⊗ B1t−1

= (α−1, β−1) = (0, z1), onF(Ω)∩ r1 > w1, (2.3)

A1t−1

⊗ B1t−1

= (α−1, β−1) = (d1, z1), onF(Ω)∩ r1 = w1. (2.4)

Observe that none of the relations below is correct:

α0 = ξ−1 + ξ0 and β0 = η−1 + η0.

The extended form of game 1 is formally defined as the bivariate marked point process

A1T ⊗ B1

T := (ξ−1, η−1)εt−1 + (α0 − ξ−1, β0 − η−1)εt0 +∑

j≥1

(ξj , ηj)εtj (2.5)

which is embedded overT .Becauser1 andw1 are continuous r.v.’s,r1 = w1 is a P -null set. Hence, the as-

sociated traceσ-algebraF(Ω) ∩ r1 = w1 contains only a.s. negligible paths of gameA1

T ⊗ B1T , which will have no impact on the upcoming functionals.

As we will see it in the next section, game 1 will require knowledge ofA1T ⊗B1

T at t−1

andt0. Consequently, we begin to work on the functional

φ0 := φ0(a0, b0, ϑ0, u0, v0, θ0) = E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0t0

](2.6)

that describes what we call, the initial phase of the game.

Theorem 1. The functionalφ0 of the initial phase of the game satisfies the following for-mula:

φ0 =λAλBδ(θ* )

ϑ0 + θ0 + λA + λB

(1

θA + λBhA(a0 + u0)hB(v0) +

1θB + λA

hA(u0)hB(b0 + v0))

,

(2.7)whereθ∗ is defined in(1.18) and

θA := θ0 − λA(hA(u0) − 1), (2.8)

θB := θ0 − λB(hB(v0) − 1), (2.9)

δ(θ) := E[e−θ∆0

]. (2.10)


Proof. Recall thatt−1 = minr1, w1 andt0 = maxr1, w1 + ∆0. Then, the functionalφ0 may be rewritten as

φ0 = E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0 t0

]

= E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0t01r1<w1

]

+ E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0t01r1>w1

]. (2.11)

Sincer1 = w1 is aP -null set,

E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0t01r1=w1

]= 0.

Case1. r1 < w1. This corresponds to

t−1 = r1 and t0 = w1 + ∆0. (2.12)

Letξ′ = A1((r1, w1]). (2.13)

To keep up with the abundance of notation, the initial phase is depicted in Figure 2:

1d

2d

'ξ

0ξ1ξ

1w

0η

1η

Player A

Player B

0

0

0

1r 2r

1t− 0t 1t

1α−

0α 1α

1β−

0β

1β

1z

3d

3r

2z

2w

Figure 2.

The first term ofφ0 in (2.11) can be evaluated as follows:

E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0t01r1<w1

]

= E[e−a0d1−u0(d1+ξ′+ξ0)−b0·0−v0(0+z1+η0)−ϑ0r1−θ0(r1+(w1−r1)+∆0)1r1<w1

]

(due to assumed independent marking and independence of Poisson processesA1 andB1)

= Ee−(a0+u0)d1Ee−v0z1E[e−u0ξ′−(ϑ0+θ0)r1−θ0(w1−r1)1r1<w1

]E

[e−u0ξ0−v0η0−θ0∆0

].


Thus, we have


]

= hA(a0 + u0)hB(v0)E[e−u0ξ′−(ϑ0+θ0)r1−θ0(w1−r1)1r1<w1

]E

[e−u0ξ0−v0η0−θ0∆0

]

(2.14)

With notation (2.13), the third factor of (2.14) reads

E[e−u0ξ′−(ϑ0+θ0)r1−θ0(w1−r1)1r1<w1

]

= E[e−(ϑ0+θ0)r1−θ0(w1−r1)1r1<w1E

[e−u0ξ′ |r1, w1

]]

= E[e−(ϑ0+θ0)r1−θ0(w1−r1)1r1<w1 · e

λA(w1−r1)(hA(u0)−1)]

= E[e−[ϑ0+λA(hA(u0)−1)]r1−[θ0−λA(hA(u0)−1)]w11r1<w1

]

= E[e−ϑAr1−θAw11r1<w1

], (2.15)

whereϑA = ϑ0 + λA(hA(u0) − 1) andθA = θ0 − λA(hA(u0) − 1). Becauser1 andw1

are independent and exponentially distributed,

E[e−ϑAr1−θAw11r1<w1

]=

∞∫

x=0

∞∫

y=x

e−ϑAx−θAyλAe−λAxλBe−λBy dy dx

=

∞∫

x=0

λAe−(ϑA+λA)x[ −λB

θA + λBe−(θA+λB)y

]∞y=x

dx

=

∞∫

x=0

λAe−(ϑA+λA)x λB

θA + λBe−(θA+λB)x dx

=λAλB

θA + λB

∞∫

x=0

e−(ϑA+θA+λA+λB)x dx

=λAλB

(θA + λB)(ϑA + θA + λA + λB)=

1θA + λB

· λAλB

ϑ0 + θ0 + λA + λB. (2.16)

Since∆0 belongs to the equivalence class[∆] with the common transformδ(θ), the lastfactor of (2.14) is

E[e−u0ξ0−v0η0−θ0∆0

]= δ(θ∗), (2.17)

as per (1.17) withθ∗ := θ0 + λA(1 − hA(u0)) + λB(1 − hB(v0)) defined so in (1.18).In summary,


]

=1

θA + λB· λAλBδ(θ∗)ϑ0 + θ0 + λA + λB

hA(a0 + u0)hB(v0). (2.18)

Case2. r1 > w1 corresponds to

t−1 = w1 and t0 = r1 + ∆0. (2.19)


Denoteη′ = B1((t−1, r1]). (2.20)

Then by interchanging the roles ofr1 andw1 we have from (2.18),

E[e−a0α−1−u0α0−b0β−1−v0β0−ϑ0t−1−θ0t01r1>w1

]

=1

θB + λA· λAλBδ(θ∗)ϑ0 + θ0 + λA + λB

hA(u0)hB(b0 + v0). (2.21)

Summing up (2.18) and (2.21) yields (2.7).

3. The Development of Game 1 aftert0

After passing the initial phase, game 1 continues with its status registered at epochsT .Game 1 ends when at least one of the players sustains damages in excess of thresholdsM1

or N1. To further formalize game 1 pastt0 we introduce the following random exit indices

ν1 := infj ≥ 0 : αj = α0 + ξ1 + · · ·+ ξj > M1

, (3.1)

ν2 := infk ≥ 0 : βk = β0 + η1 + · · ·+ ηk > N1

. (3.2)

Game 1 is thus assumed to be over attρ (first passage time or exit from game 1) where

ρ := minν1, ν2. (3.3)

Note that while one of the players will be defeated, it will not be explicitly revealedwhich of the two is defeated without going over particular paths of the game. The latter isnot our objective during this phase however, because neither player will be ruined as perSection 4.

Definition 1. The Terminated Game 1 is the random measure

[A1 ⊗ B1]ρ = (ξ−1, η−1)εt−1 + (α0 − ξ−1, β0 − η−1)εt0 +ρ∑

j=1

(ξj , ηj)εtj (3.4)

on(Ω,F(Ω), (Ft)tρt=0, P ) (with the first two terms incorporating the pair of the initial condi-

tions), wheretρ is the end of game 1. Here the process is adapted to thetρ-head of filtration(Ft)t≥0.

For this phase of the game, we consider

φρ := φρ(a1, b1, ϑ1, u1, v1, θ1) = E[e−a1αρ−1−u1αρ−b1βρ−1−v1βρ−ϑ1tρ−1−θ1tρ

]. (3.5)

To evaluate this functional we introduce the Laplace–Carson transform

Lp1q1(·)(x1, y1) := x1y1

∞∫

p1=0

∞∫

q1=0

e−x1p1−y1q1(·) d(p1, q1), Re(x1) > 0, Re(y1) > 0,

(3.6)


with the inverse

L−1x1y1

(·)(p1, q1) = L−1(· 1x1y1

)(p1, q1), (3.7)

whereL−1 is the inverse of the bivariate Laplace transform.Theorem 2 (below) establishes an explicit formula forφρ. We use the following abbre-

viations based on (1.16):

g := g(a1 + u1 + x1, b1 + v1 + y1, ϑ1 + θ1), (3.8)

G := g(u1 + x1, v1 + y1, θ1), (3.9)

G1 := g(u1, v1 + y1, θ1), (3.10)

G2 := g(u1 + x1, v1, θ1), (3.11)

G12 := g(u1, v1, θ1), (3.12)

Φ∗0 := φ0(0, 0, 0, a1 + u1 + x1, b1 + v1 + y1, ϑ1 + θ1), (3.13)

Φ0 := φ0(a1, b1, ϑ1, u1 + x1, v1 + y1, θ1), (3.14)

Φ10 := φ0(a1 + x1, b1, ϑ1, u1, v1 + y1, θ1), (3.15)

Φ20 := φ0(a1, b1 + y1, ϑ1, u1 + x1, v1, θ1), (3.16)

Φ120 := φ0(a1 + x1, b1 + y1, ϑ1, u1, v1, θ1). (3.17)

Theorem 2. The functionalφρ of game1 satisfies the following formula:

φρ = L−1x1y1

(Φ12

0 − Φ0 +Φ∗

0

1 − g(G12 − G)

)(M1, N1), (3.18)

provided that Re(a1 + u1 + x1) > 0, Re(b1 + v1 + y1) > 0, Re(ϑ1 + θ1) > 0,

(3.18a)

with any two of the three strict inequalities relaxed with≥.

Proof. We first extend the random indicesν1 andν2 to the families of indices

ν1(p1) := inf

j ≥ 0 : αj = α0 + ξ1 + · · ·+ ξj > p1

, p1 ≥ 0

(3.19)

andν2(q1) := inf

k ≥ 0 : βk = β0 + η1 + · · ·+ ηk > q1

, q1 ≥ 0

. (3.20)

The parametric analog ofρ is then

ρ(p1, q1) := min

ν1(p1), ν2(q1)

, p1 ≥ 0, q1 ≥ 0

. (3.21)

Next, introduce the following parametric families of measurable sets:

H1,2=ν1(p1)<ν2(q1)

, H12 =

ν1(p1)=ν2(q1)

, H2,1=

ν1(p1)>ν2(q1)

.

(3.22)


The corresponding parametric extension of the primary functionalφρ can be decomposedin accordance with (3.22) as follows:

φρ(p1,q1) : = φρ(p1,q1)(a1, b1, ϑ1, u1 , v1, θ1)

= E[e−a1αρ(p1,q1)−1−u1αρ(p1,q1)−b1βρ(p1 ,q1)−1−v1βρ(p1,q1)−ϑ1tρ(p1,q1)−1−θ1tρ(p1,q1)

]

= E[e−a1αρ(p1,q1)−1−u1αρ(p1,q1)−b1βρ(p1 ,q1)−1−v1βρ(p1,q1)−ϑ1tρ(p1,q1)−1−θ1tρ(p1,q1)1H1,2

]

+ E[e−a1αρ(p1,q1)−1−u1αρ(p1,q1)−b1βρ(p1 ,q1)−1−v1βρ(p1,q1)−ϑ1tρ(p1,q1)−1−θ1tρ(p1,q1)1H12

]

+ E[e−a1αρ(p1,q1)−1−u1αρ(p1,q1)−b1βρ(p1 ,q1)−1−v1βρ(p1,q1)−ϑ1tρ(p1,q1)−1−θ1tρ(p1,q1)1H2,1

],

(3.23)

or in notation, =F1,2 + F12 + F2,1,

Below we will be concerned with transformations ofF1,2, F12 andF2,1 under the operatorLp1q1 to be applied toφρ(p1,q1).

Case1. ν1(p1) < ν2(q1). This will follow the paths of game 1 on the traceσ-algebraF(Ω)∩ ν1(p1) < ν2(q1) and yieldρ(p1, q1) = ν1(p1):

F1,2 =∑

j≥0

∑

k>j

E[e−a1αj−1−u1αj−b1βj−1−v1βj−ϑ1tj−1−θ1 tj1ν1(p1)=j,ν2(q1)=k

]. (3.24)

By Fubini’s theorem,

Lp1q1(F1,2)(x1, y1)

=∑

j≥0

∑

k>j

E[e−a1αj−1−u1αj−b1βj−1−v1βj−ϑ1tj−1−θ1tj(e−x1αj−1−e−x1αj)(e−y1βk−1−e−y1βk)

].

(3.25)

Case:j = 0. This case will include the entire information on the initial phase observed att0 and prior tot0, includingt−1. In a few lines below, we are going to implement the resultof Theorem 1 and utilize all necessary versions of the functionalφ0 :

∑

k>0

E[e−a1α−1−u1α0−b1β−1−v1β0−ϑ1t−1−θ1t0(e−x1α−1 − e−x1α0)(e−y1βk−1 − e−y1βk)

]

=∑

k>0

E[e−a1α−1−u1α0−b1β−1−v1β0−ϑ1t−1−θ1t0(e−x1α−1 − e−x1α0)

×e−y1β0e−y1(η1+ ...+ηk−1)(1 − e−y1ηk)]

=E

[e−(a1+x1)α−1−u1α0−b1β−1−(v1+y1)β0−ϑ1t−1−θ1t0

]

−E[e−a1α−1−(u1+x1)α0−b1β−1−(v1+y1)β0−ϑ1t−1−θ1t0

]

×∑

k>0

E[e−y1(η1+···+ηk−1)(1 − e−y1ηk)

]


=φ0(a1 + x1, b1, ϑ1, u1, v1 + y1, θ1)− φ0(a1, b1, ϑ1, u1 + x1, v1 + y1, θ1)

×∑

k>0

[g(0, y1, 0)

]k−1(1 − g(0, y1, 0))

= Φ10 − Φ0, (3.26)

where the summation overk > 0 converges to 1 as per Lemma 1 of [5]: the associatedconvergence of

∑k>0

[g(0, y1, 0)]k−1 is guaranteed provided thatRe(y1) > 0. The last line in

(3.26) is due to notation (3.14)–(3.15).

Case:j > 0. This case also contains parts of functionalφ0 in the information related to thereference pointt0.

Transformation (3.25) for this case is∑

j>0

∑

k>j

E[e−a1αj−1−u1αj−b1βj−1−v1βj−ϑ1tj−1−θ1tj(e−x1αj−1 − e−x1αj )(e−y1βk−1 − e−y1βk)

]

=∑

j>0

∑

k>j

E

[e−(a1+u1+x1)αj−1−(b1+v1+y1)βj−1−(ϑ1+θ1)tj−1

]

×E[e−u1ξj(1 − e−x1ξj )e−(v1+y1)ηj−θ1∆j

]E

[e−y1(ηj+1+···+ηk−1)(1 − e−y1ηk)

]

=∑

j>0

E

[e−(a1+u1+x1)α0−(b1+v1+y1)β0−(ϑ1+θ1)t0

]

×E[e−(a1+u1+x1)(ξ1+···+ξj−1 )−(b1+v1+y1)(η1+···+ηj−1 )−(ϑ1+θ1)(∆1+···+∆j−1 )

]

×E[e−u1ξj (1 − e−x1ξj )e−(v1+y1)ηj−θ1∆j

] ∑

k>j

E[e−y1(ηj+1+···+ηk−1 )(1 − e−y1ηk)

], (3.27)

where the third factor can be written as

E[e−u1ξj−(v1+y1)ηj−θ1∆j

]− E

[e−(u1+x1)ξj−(v1+y1)ηj−θ1∆j

]= G1 − G

(as per notation (3.9)–(3.10)) and the summation overk > j converges to1, for Re(y1) > 0,as per Lemma 1 of [5]. Then, after some algebra in (3.27) and the use of notation (3.8)–(3.10) and (3.13), we arrive at

φ0(0, 0, 0, a1 + u1 + x1, b1 + v1 + y1, ϑ1 + θ1) ·∑

j>0

gj−1 · (G1 − G)

= Φ∗0 ·

∑

j>0

gj−1 · (G1 − G) =Φ∗

0

1 − g(G1 − G), (3.28)

with the convergence of∑j>0

gj−1 under the condition that the parameters ofg satisfy:

Re(a1 + u1 + x1) > 0, Re(b1 + v1 + y1) > 0, Re(ϑ1 + θ1) > 0,

with any two of the three strict inequalities relaxed with≥.With the casesj = 0 andj > 0 combined together, we will arrive at

Lp1q1(F1,2)(x1, y1) = (Φ10 − Φ0) +

Φ∗0

1 − g(G1 − G). (3.29)


Case2. ν1(p1) > ν2(q1). This will follow the paths of game 1 on the traceσ-algebraF(Ω)∩ ν1(p1) > ν2(q1) and yieldingρ(p1, q1) = ν2(q1).

With the roles ofx1 andy1 interchanged, we find that

Lp1q1(F2,1)(x1, y1)

=φ0(a1, b1 + y1, ϑ1, u1 + x1, v1, θ1)− φ0(a1, b1, ϑ1, u1 + x1, v1 + y1, θ1)

+φ0(0, 0, 0, a1 + u1 + x1, b1 + v1 + y1, ϑ1 + θ1) ·∑

j>0

gj−1 · (G2 − G)

= (Φ20 − Φ0) +

Φ∗0

1 − g(G2 − G). (3.30)

Case3. ν1(p1) = ν2(q1). This implies

ρ(p1, q1) = ν1(p1) = ν2(q1).

The corresponding transformation is

Lp1q1(F12)(x1, y1)

=∑

j≥0

E[e−a1αj−1−u1αj−b1βj−1−v1βj−ϑ1tj−1−θ1tj(e−x1αj−1−e−x1αj)(e−y1βj−1−e−y1βj)

].

(3.31)

Case:j = 0.

(e−x1αj−1 − e−x1αj)(e−y1βj−1 − e−y1βj)

for j = 0 give

= e−x1α−1−y1β−1 − e−x1α−1−y1β0 − e−x1α0−y1β−1 + e−x1α0−y1β0 (3.32)

and thus the transformation (3.31) can be written as

φ0(a1 + x1, b1 + y1, ϑ1, u1, v1, θ1)− φ0(a1 + x1, b1, ϑ1, u1, v1 + y1, θ1)−φ0(a1, b1 + y1, ϑ1, u1 + x1, v1, θ1) + φ0(a1, b1, ϑ1, u1 + x1, v1 + y1, θ1)

= Φ120 − Φ1

0 − Φ20 + Φ0 (3.33)

Case:j > 0.Transformation (3.31) reads∑

j>0

E[e−a1αj−1−u1αj−b1βj−1−v1βj−ϑ1tj−1−θ1tj(e−x1αj−1 − e−x1αj)(e−y1βj−1 − e−y1βj)

]

=∑

j>0

E[e−(a1+u1+x1)αj−1−(b1+v1+y1)βj−1−(ϑ1+θ1)tj−1

]

×E[e−u1ξj−v1ηj−θ1∆j (1 − e−x1ξj)(1− e−y1ηj)

]


= Φ∗0 ·

∑

j>0

gj−1 · (G12 − G1 − G2 + G) =Φ∗

0

1 − g(G12 − G1 − G2 + G). (3.34)

Thus,

Lp1q1(F12)(x1, y1) = (Φ120 − Φ1

0 − Φ20 + Φ0) +

Φ∗0

1 − g(G12 − G1 − G2 + G). (3.35)

Finally after simple algebra, the sum all of three cases is

Lp1q1 (φρ(p1,q1))(x1, y1) = Lp1q1(F1,2)(x1, y1) + Lp1q1(F12)(x1, y1) + Lp1q1(F2,1)(x1, y1)

= Φ120 − Φ0 +

Φ∗0

1 − g(G12 − G).

4. The Restricted Random Walk

In this section we form a bridge from the first phase to the second phase (game 2). Becauseat the end of game 1, each player is supposed to have sustained only limited damage not inexcess ofM or N , and because the winner of game 1 is not specified, we need to reducethe damages to their maximal values ofM or N in the event excesses take place. A similarprocedure was rendered in [8]. Let us define

αρ = minαρ, M (4.1)

and

βρ = minβρ, N. (4.2)

The corresponding functional to be worked on is

φρ := φρ(a2, b2, ϑ2, u2, v2, θ2) = E[e−a2αρ−1−u2αρ−b2βρ−1−v2 βρ−ϑ2tρ−1−θ2tρ

]. (4.3)

Theorem 3 (below) is similar to that of [8] (applied to a different functional) but for the sakeof consistency we give a proof.

Theorem 3. The functionalφρ of the tandem game upon the beginning of phase2 satisfiesthe following formula:

φρ = L−1x2y2

[φρ(a2, b2, ϑ2, u2 + x2, v2 + y2, θ2) − e−u2Mφρ(a2, b2, ϑ2, x2, v2 + y2, θ2)

− e−v2Nφρ(a2, b2, ϑ2, u2 + x2, y2, θ2) + e−u2M−v2N φρ(a2, b2, ϑ2, x2, y2, θ2)](M, N )

+ e−v2NL−1x2

[φρ(a2, b2, ϑ2, u2 + x2, 0, θ2) − e−u2Mφρ(a2, b2, ϑ2, x2, 0, θ2)

](M )

+ e−u2ML−1y2

[φρ(a2, b2, ϑ2, 0, v2 + y2, θ2) − e−v2Nφρ(a2, b2, ϑ2, 0, y2, θ2)

](N )

+ e−u2M−v2Nφρ(a2, b2, ϑ2, 0, 0, θ2), Re(x2) > 0, Re(y2) > 0. (4.4)

HereL−1xy is the inverse of the Laplace–Carson transform introduced in the earlier sections.


Proof. Let

φ1(a2, b2, ϑ2, u2, v2, θ2) = E[e−a2αρ−1−u2 αρ−b2βρ−1−v2βρ−ϑ2tρ−1−θ2 tρ

], (4.5)

which is the “truncated” functionalφρ(a2, b2, ϑ2, u2, v2, θ2) only w.r.t. the first componentαρ (but notβρ). That is, in the event the total damage to player A upon exit from game1 exceedsM (while surely crossingM1 which is greater thanM ), it will be reduced toM , because of our assumption on the maximum casualty to player A. Analogously, weintroduce the truncated functional w.r.t. the second componentβρ:

φ2(a2, b2, ϑ2, u2, v2, θ2) = E[e−a2αρ−1−u2αρ−b2βρ−1−v2 βρ−ϑ2tρ−1−θ2 tρ

], (4.6)

which will represent the joint functional of the damages to players A and B and exit timefrom game 1, with restricted casulaties to player B, but not to player A.

Due to Theorem 2 [3], we have

T1φρ(a2, b2, ϑ2, u2, v2, θ2) := φ1(a2, b2, ϑ2, u2, v2, θ2)

= L−1x2

[φρ(a2, b2, ϑ2, u2 + x2, v2, θ2) − e−u2Mφρ(a2, b2, ϑ2, x2, v2, θ2)

](M)

+e−u2Mφρ(a2, b2, ϑ2, 0, v2, θ2), (4.7)

expressed through operatorT1 acting on variableu w.r.t. a fixed parameterM . Define op-eratorT2 which is similar toT1, only acting on variablev w.r.t. another fixed parameterN :

T2φρ(a2, b2, ϑ2, u2, v2, θ2) := φ2(a2, b2, ϑ2, u2, v2, θ2)

= L−1y2

[φρ(a2, b2, ϑ2, u2, v2 + y2, θ2)− e−v2Nφρ(a2, b2, ϑ2, u2, y2, θ2)

](N)

+e−v2Nφρ(a2, b2, ϑ2, u2, 0, θ2). (4.8)

Thus, merging operatorsT1 andT2 makes

φρ(a2, b2, ϑ2, u2, v2, θ2) = T2 T1φρ(a2, b2, ϑ2, u2, v2, θ2)

= L−1y2

[φ1(a2, b2, ϑ2, u2, v2 + y2, θ2)− e−v2N φ1(a2, b2, ϑ2, u2, y2, θ2)

](N)

+e−v2N φ1(a2, b2, ϑ2, u2, 0, θ2). (4.9)

Then, in light of (4.7), the first term of (4.9) can be rewritten as

L−1y2

[φ1(a2, b2, ϑ2, u2, v2 + y2, θ2) − e−v2N φ1(a2, b2, ϑ2, u2, y2, θ2)

](N )

=L−1

x2y2


](M, N )

+e−u2ML−1y2

[φρ(a2, b2, ϑ2, 0, v2 + y2, θ2)

](N )

−e−v2NL−1

x2y2

[φρ(a2, b2, ϑ2, u2 + x2, y2, θ2) − e−u2Mφρ(a2, b2, ϑ2, x2, y2, θ2)

](M, N )

+e−u2ML−1y2

[φρ(a2, b2, ϑ2, 0, y2, θ2)

](N )

= L−1x2y2


−e−v2Nφρ(a2, b2, ϑ2, u2 + x2, y2, θ2) + e−u2M−v2Nφρ(a2, b2, ϑ2, x2, y2, θ2)](M, N )


+e−u2ML−1y2


](N ) (4.10)

and the second term

e−v2N φ1(a2, b2, ϑ2, u2, 0, θ2)

= e−v2NL−1

x2


](M)

+e−u2Mφρ(a2, b2, ϑ2, 0, 0, θ2)

= e−v2NL−1x2


](M)

+e−u2M−v2Nφρ(a2, b2, ϑ2, 0, 0, θ2). (4.11)

Hence,

T2 T1φρ(a2, b2, ϑ2, u2, v2, θ2)

= L−1x2y2

[φρ(a2, b2, ϑ2, u2 + x2, v2 + y2, θ2)− e−u2Mφρ(a2, b2, ϑ2, x2, v2 + y2, θ2)

−e−v2Nφρ(a2, b2, ϑ2, u2 + x2, y2, θ2) + e−u2M−v2Nφρ(a2, b2, ϑ2, x2, y2, θ2)](M, N)

+e−v2NL−1x2


](M)

+e−u2ML−1y2


](N)

+e−u2M−v2Nφρ(a2, b2, ϑ2, 0, 0, θ2), (4.12)

which is a closed form ofφρ(a2, b2, ϑ2, u2, v2, θ2).

5. The Final Phase

During the final phase of the game, the conflict may intensify. To realistically model thispart of the conflict we assume the presence of different parameters beginningtρ, which isthe exit time from phase 1. Perhaps there is a time of truce between the players lasting fromtρ and ending at some epoch when game 2 starts, but this would be analytically insignifi-cant and we bypass this time as nonexistent. We therefore allow game 2 to develop underdifferent parameters, but inheriting the values of the process attρ andtρ−1. As we will see,merging the two phases will automatically require us to bypass this time on attaching theinitial phase to phase 1.

Now, the rest of the procedure is very similar to the development in our last paper [8],but to make this paper self-contained we include some details.

We assume that the processes describing the casualties to player A and B, as well asobservation process, will be different.

We start again with some independentσ-algebrasFA,FB,FT ⊆ F(Ω). We also as-sume that they are independent from the previously introducedσ-subalgebrasFA1, FB1 ,FT .

We need two auxiliary marked Poisson random measures

PA :=∑

j≥1

πAj εϕj and PB :=

∑

k≥1

πBk εζk

(5.1)


with respective intensitiesΛA andΛB and position independent marking, so thatPA andPB areFA- andFB-measurable, respectively. Now based on (5.1), we form the damageprocesses:

A :=∑

j≥0

πAj εδj and B :=

∑

k≥0

πBk εςk , (5.2)

where

δ0 = tρ, δj = tρ + ϕj , j = 1, 2, . . . , πA0 = αρ, (5.3)

ς0 = tρ, ςk = tρ + ζk , k = 1, 2, . . . , πB0 = βρ. (5.4)

To attach game 2 to game 1, we use the delayed components ofA andB in (5.3-5.4).While the increments of the associated point processes

∑j≥0

εδj and∑k≥0

εςk are independent

and exponentially distributed with respective parametersΛA and ΛB as per (5.1)–(5.4),the associated marks (being a.s. nonnegative) counting from the first one are iid with thetransforms

HA(u) = Ee−uπA1 , Re(u) ≥ 0, (5.5)

HB(u) = Ee−uπB1 , Re(u) ≥ 0. (5.6)

The start of game 2 will be at pointsδ0 andς0 (actually, atminδ0, ς0) and their initialpositionsπA

0 andπB0 of processes (5.2). They are to be known from the functional of

φρ(a2, b2, ϑ2, u2, v2, θ2) = E[e−a2αρ−1−u2 αρ−b2βρ−1−v2 βρ−ϑ2tρ−1−θ2tρ

]

derived in Theorem 3. According to (5.3) and (5.4), the corresponding marginal transfor-mations are

Ee−θδ0 = Ee−θς0 = φρ(0, 0, 0, 0, 0, θ), (5.7)

Ee−uαρ = φρ(0, 0, 0, u, 0, 0), Re(u) ≥ 0, (5.8)

Ee−vβρ = φρ(0, 0, 0, 0, v, 0), Re(v) ≥ 0. (5.9)

Notice that these two delayed Poisson marked processes describe the conflict between play-ers A and B in the second phase, and they start in accordance with the truncated terminalconditions of the two-variate random walk process from the previous phase.

Furthermore, as in game 1, we introduce anFT -measurable point process

T :=∑

i≥0

ετi , τ0 := tρ, (5.10)

which is a delayed renewal process aimed to oversee the second phase of the game.If

(A(t), B(t)) := A ⊗ B([0, t]), t ≥ 0, (5.11)

then(Aj , Bj) := (A(τj), B(τj)) = A ⊗ B([0, τj]), j = 0, 1, . . . , (5.12)


form observations ofA⊗ B embedded uponT , with respective increments

(Xj, Yj) = A ⊗ B((τj−1, τj]), j = 1, . . . , (5.13)

X0 = A0, Y0 = B0, τ−1 := tρ−1, A−1 := αρ−1, B−1 := βρ−1. (5.14)

Note that due to the formation of sequential games and the upcoming analysis, not only dothe initial values of game 2 absorb the terminal values of game 1, but they will also need“pre-exit” values of game 1, which we then include in (5.14).

Obviously, the bivariate marked point measure

AT ⊗ BT :=∑

j≥0

(Xj , Yj)ετj , (5.15)

with marginal measures

AT =∑

i≥0

Xiετi and BT =∑

i≥0

Yiετi , (5.16)

is with position dependent marking thus makingXj andYj dependent. With the notation

∆j := τj − τj−1, j = 1, 2, . . . , ∆0 := τ0, (5.17)

we can evaluate the functional

γ(u, v, θ) = Ee−uXj−vYj−θ∆j (5.18)

using straightforward probabilistic arguments (cf. [5], formula (3.19)),

γ(u, v, θ) = γθ + ΛA(1− HA(u)) + ΛB(1 − HB(v))

, j = 1, 2, . . . , (5.19)

whereγ(θ) = Ee−θ∆j (5.20)

is the marginal Laplace–Stieltjes transform of∆1, ∆2, . . . .Now we introduce the exit indices of game 2:

µ = infm ≥ 0 : A0 + X1 + · · ·+ Xm = Am > M2; A0 = αρ

, (5.21)

ν = infn ≥ 0 : B0 + Y1 + · · ·+ Yn = Bn > N2; B0 = βρ

. (5.22)

Since in game 2, and thus the game as the whole, we are interested in the paths that lead tothe defeat of player A, the main functional of the game will be

Φµν := Φµν(a, a′, b, b′, h, h′) = E[e−aAµ−1−a′Aµ−bBµ−1−b′Bµ−hτµ−1−h′τµ1µ<ν

]

(5.23)

Definition 2. Game 2 will be defined as the random measure

[AT ⊗ BT ]µ = [A1 ⊗ B1]ρ +µ∑

j=0

(Xj, Yj)ετj


on (Ω,F(Ω), (Ft)τµ

t=0, P ), with the history[A1 ⊗ B1]ρ of the conflict and the pair of initialconditions:

(A−1 = A(τ−1) = αρ−1, B−1 = B(τ−1) = βρ−1), (5.24)

(A0 = A(τ0) = αρ, B0 = B(τ0) = βρ). (5.25)

The moment of timeτµ is called the end of game 2 and this will also be the end of the entiregame. Game 2 will be observed along the paths from the traceσ-algebraF(Ω)∩ µ < ν,which all terminate atτµ. The game process will be adapted to the head-filtration(Ft)

τµ

t=0.

Below Theorem 4 establishes an explicit formula forΦµν . We use the following abbre-viations based on (5.18) with all involved variables being fixed:

γ := γ(a + a′ + c, b + b′ + s, h + h′), (5.26)

Γ := γ(a′ + c, b′ + s, h′), (5.27)

Γ 1 := γ(a′, b′ + s, h′). (5.28)

Theorem 4. The tandem game of two players A and B is with the both players initiating theconflict and with player A losing the entire game; the functionalΦµν describing the gamesatisfies the following formula:

Φµν(a, a′, b, b′, h, h′)

= L−1cs

[φρ(a + c, b, h, a′, b′ + s, h′) − φρ(a, b, h, a′ + c, b′ + s, h′)

+ φρ(0, 0, 0, a+ a′ + c, b + b′ + s, h + h′) · 11 − γ

(Γ 1 − Γ )](M2, N2)

under the conditions that

Re(c) > 0, Re(s) > 0, Re(a + a′ + c) > 0, Re(b + b′ + s) > 0, e(h + h′) > 0,(5.29)

with any two strict inequalities from the latter three being replaced with≥.

The proof of Theorem 4 is identical to that of Theorem 3 of [8] and thus will be omitted.

Acknowledgment

The authors are very grateful to the referees who offered a wealth of constructive sugges-tions which we were happy to implement and that greatly improved the presentation of thepaper.

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Chapter 3

SOME STABILITY RESULTS FOR EQUATIONS

AND I NEQUALITIES CONNECTED

WITH THE EXPONENTIAL FUNCTION

Włodzimierz Fechner and Roman Ger∗

Institute of Mathematics, Silesian UniversityBankowa 14, 40-007 Katowice, Poland

Abstract

We generalize some earlier results connected with the Hyers-Ulam stability offunctional equations and inequalities related to the exponential function. In our firsttheorem we prove the stability of equationf = f ′ in reflexive normed spaces. Fur-ther, we apply this result jointly with some other facts to prove the stability of severalrelated functional equations.

2000 Mathematics Subject Classifications:39B62, 39B82.

Key words: exponential function, functional inequality, Hyers–Ulam stability.

1. Introduction

Throughout the paperI stands for an open real interval,R denotes the set of all real numbersandN = 1, 2, . . ..

C. Alsina and J. L. Garcia-Roig [1] investigated the following functional inequalities:

f(y) − f(x)

y − x≤

f(x) + f(y)

2, x, y ∈ R, x 6= y, (1)

and

0 ≤f(y) − f(x)

y − x≤

f(x) + f(y)

2, x, y ∈ R, x 6= y. (2)

They have proved that a functionf : R → R satisfies (1) if and only if there exists anonincreasing functiond : R → R such thatf(x) = d(x)ex for x ∈ R [1, Theorem 1].

∗E-mail addresses: [email protected], [email protected]

38 Włodzimierz Fechner and Roman Ger

Further,f : R → R is a solution of (2) if and only if there exists a continuous nonincreasingfunctiond : R → R such thatf(x) = d(x)ex for x ∈ R andd(x + t) ≥ e−td(x) for x ∈ R

andt > 0 [1, Theorem 2]. Moreover, their results remain true (with the same proofs) if thereal lineR is replaced by an open intervalI.

C. Alsina and R. Ger [2] and later W. Fechner [3] have dealt with the following func-tional inequality

f

(x + y

2

)≤

f(y) − f(x)

y − x, x, y ∈ I, x < y. (3)

In particular, it has been proved that if a functionf : I → R satisfies (3) jointly with thefollowing condition:

lim suph→0+

f(x + h) ≥ f(x), x ∈ I. (4)

then there exists a nondecreasing nonnegative functioni : I → R such thatf(x) = i(x)ex

for x ∈ I [3, Theorem 2].In [4] the following inequality is examined:

6f(y) − f(x)

y − x≤ 4f

(x + y

2

)+ f(x) + f(y), x, y ∈ I, x 6= y, (5)

as well as a more general functional inequality:

f(y) − f(x)

y − x≤ N (g(M1(x, y)), g(M2(x, y))) , x, y ∈ I, x 6= y, (6)

whereM1, M2 andN stand for arbitrary means andN is continuous. It was proved thatsolutions of (5) which satisfy (4) are of the formf(x) = i(x)ex for x ∈ I, wherei is anondecreasing function. Further, continuous solutions of (6) satisfy inequality

f(y) − f(x) ≤

y∫

x

g(t) dt, x, y ∈ I, x ≤ y. (7)

Remark 1. The representation of functionf : I → R in the form f(x) = d(x)ex orf(x) = i(x)ex with d andi being arbitrary nonincreasing and a nondecreasing function,respectively, is equivalent to the validity of the following respective estimations:

f(y) ≥ f(x) · ey−x, x, y ∈ I, x < y;

f(y) ≤ f(x) · ey−x, x, y ∈ I, x < y.

It seems that C. Alsina and R. Ger [2] were the first authors who investigated the Hyers–Ulam stability of differential equations. In particular, they have shown [2, Theorem 1 andRemark] that the equationf ′ = f is stable; more precisely, they have proved that given anε ≥ 0 if f : I → R is a differentiable mapping such that

|f ′(x) − f(x)| ≤ ε, x ∈ I,

Exponential Function 39

then there exists a constantc0 ∈ R suchthat

|f(x) − c0 · ex| ≤ 3ε, x ∈ I.

The purpose of the present paper is to investigate the Hyers–Ulam stability of the above-mentioned functional inequalities and of related functional equations. Moreover, we gener-alize the above-mentioned result of C. Alsina and R. Ger from [2] for mappings with valuesin a reflexive normed linear space.

2. Results

Our first result is the following theorem.

Theorem 1. Given anε ≥ 0 and a reflexive real normed linear space(X, ‖ · ‖) assumethatf : (0, +∞) → X is a differentiable mapping such that

‖f ′(x) − f(x)‖ ≤ ε, x ∈ (0,+∞).

Then there exists a vectorc0 ∈ X such that

‖f(x) − exc0‖ ≤ 3ε, x ∈ (0,+∞). (8)

Proof. For every memberx∗ of the closed unit ballS∗ in the dual spaceX∗ one has∣∣x∗(f ′(x) − f(x))

∣∣ ≤ ε, x ∈ (0,+∞),

stating that the mapx∗ f satisfies the assumptions of the above-mentioned result of C.Alsina and R. Ger from [2]. Therefore, there exists a real constantc = c(x∗) such that

∣∣(x∗ f)(x) − c(x∗)ex∣∣ ≤ 3ε, x ∈ (0,+∞). (9)

Clearly, we may assume that the assignment

S∗ ∋ x∗ 7→ c(x∗) ∈ R

yields a function fromS∗ into R. Moreover, for every other elementy∗ ∈ S∗ such that‖x∗ + y∗‖ ≤ 1 one has also

∣∣(y∗ f)(x) − c(y∗)ex∣∣ ≤ 3ε, x ∈ (0,+∞),

and ∣∣((x∗ + y∗) f)(x) − c(x∗ + y∗)ex∣∣ ≤ 3ε, x ∈ (0,+∞).

Consequently,∣∣[c(x∗ + y∗) − c(x∗) − c(y∗)]ex

∣∣ ≤ 9ε, x ∈ (0,+∞),

whence, by passing here to the infinity withx, we deduce that

x∗, y∗, x∗ + y∗ ∈ S∗ implies c(x∗ + y∗) = c(x∗) + c(y∗). (10)


Likewise, if α ∈ [−1, 1] andx∗ ∈ S∗ weget∣∣[((αx∗) f)(x) − c(αx∗)

]ex

∣∣ ≤ 3ε, x ∈ (0,+∞),

as well as ∣∣[((αx∗) f)(x) − αc(x∗)]ex

∣∣ ≤ 3|α|ε, x ∈ (0,+∞),

whence ∣∣[c(αx∗) − αc(x∗)]ex

∣∣ ≤ 3(1 + |α|)ε, x ∈ (0,+∞),

which forces the equalityc(αx∗) = αc(x∗) (11)

for everyα ∈ [−1, 1] and everyx∗ ∈ S∗.Now, we may extendc onto the whole dual spaceX∗ by putting

c(x∗) := nc

(1

nx∗

), x∗ ∈ X∗,

wheren ∈ N is large enough to have1n‖x∗‖ ≤ 1. To see that such an extension is well

defined (does not depend on the choice ofn) we apply a standard reasoning based on (10)and (11). For, observe first that (10) implies (induction)

nc

(1

ny∗

)= c(y∗) provided that y∗ ∈ S∗.

Therefore, fixing arbitrarily a memberx∗ of the dual spaceX∗ and takingn, m ∈ N largeenough to have1

nx∗, 1

mx∗ in S∗ we infer that

nc

(1

nmx∗

)= c

(1

mx∗

)

aswell as

mc

(1

nmx∗

)= c

(1

nx∗

).

Consequently,1

nc

(1

mx∗

)= c

(1

nmx∗

)=

1

mc

(1

nx∗

),

whence

mc

(1

mx∗

)= nc

(1

nx∗

).

Now, a simple calculation shows thatc : X∗ → R is both additive and homogeneous,i.e. linear. On the other hand, in virtue of (9) applied forx = 1, we get

sup‖x∗‖≤1

|c(x∗)| ≤ sup‖x∗‖≤1

1

e

(3ε +

∣∣(x∗ f)(1)∣∣)

≤3

eε + sup

‖x∗‖≤1

1

e

∣∣(x∗ f)(1)∣∣ ≤ 3

eε +

1

e‖f(1)‖ < +∞


stating thatc is continuous. Thus

c ∈ X∗∗

and, due to the reflexivity ofX, there exists ac0 ∈ X such that

c(x∗) = x∗(c0)

for all x∗ ∈ X∗. Consequently, relation (9) assumes the form

∣∣x∗(f(x) − exc0)∣∣ ≤ 3ε, x ∈ (0,+∞), x∗ ∈ S∗,

which immediately implies the assertion claimed.

Remark 2. Generalizationsto semireflexive locally convex linear topological spaces in thespirit of L. Szekelyhidi (see [8]) as well as to sequentially complete locally convex lineartopological spaces (see Z. Gajda [5] and [6]) are also possible. We have omitted the detailsof such approach to keep greater readibility of the statements.

In what follows we will deal with the stability of functional inequalities (1) and (3) andwe use the results obtained to prove stability of the corresponding functional equations.Then, we derive similar results for (5). Finally, we will focus on the stability of (6) and therelated equality.

Remark 3. Using a result of J. Ger [7, Theorem 1] concerning the Sahoo–Riedel equationson an interval one can easily check that the only solution of each of the following functionalequations

f(x) + f(y)

2=

f(x) − f(y)

x − y, x, y ∈ I, x < y;

f

(x + y

2

)=

f(x) − f(y)

x − y, x, y ∈ I, x < y; (12)

and

f(x) + f(y) + 4f

(x + y

2

)= 6

f(x) − f(y)

x − y, x, y ∈ I, x < y, (13)

is the zero functionf = 0. Therefore, in order to prove that these equations are stable in thesense of Hyers–Ulam, we need to show that each solution of the corresponding “approxi-mate equation” is in a sense close to the zero function.

Proposition 1. Assume thatε ≥ 0 andf0 : I → R satisfies

f0(y) − f0(x)

y − x≤

f0(x) + f0(y)

2+ ε, x, y ∈ R, x 6= y.

Thenf0(x) = f(x) − ε for x ∈ I, wheref is a solution of(1).

Proof. It is enough to putf(x) := f0(x) + ε for x ∈ I and check thatf satisfies (1).


Now, we are going to discuss the following stability problem:∥∥∥∥

1

y − x[f(y) − f(x)] −

f(x) + f(y)

2

∥∥∥∥ ≤ ε, x, y ∈ I, x < y, (14)

wheref mapsI into a normed linear space (not necessarily complete). Our result reads asfollows.

Theorem 2. Given anε ≥ 0, let(X,‖ · ‖) be a real normed linear space and letf : I → X

satisfy(14). Then the estimate

∥∥f(y) − ey−xf(x)∥∥ ≤ [ey−x − 1]ε, x, y ∈ I, x < y, (15)

holds true.

Proof. First, we derive (15) from Proposition 1 and C. Alsina and J. L. Garcia–Roig [1,Theorem 1] in a special case, where(X, ‖·‖) = (R, | · |). In this situation (14) is equivalentto the following system of two functional inequalities:

f(x) + f(y)

2≤

f(y) − f(x)

y − x+ ε, x, y ∈ I, x < y;

f(x) + f(y)

2≥

f(y) − f(x)

y − x− ε, x, y ∈ I, x < y.

Now, apply Proposition 1 jointly with [1, Theorem 1] and Remark 1 twice, forf and for−f , to get

f(y) + ε ≤ [f(x) + ε]ey−x, x, y ∈ I, x < y;

−f(y) + ε ≤ [−f(x) + ε]ey−x, x, y ∈ I, x < y,

which leads to

∣∣f(y) − ey−xf(x)∣∣ ≤ [ey−x − 1]ε, x, y ∈ I, x < y.

Now, the general case can be derived from the real one. Indeed, iff satisfies (14), thenfor each linear and continuous functionalx∗ the mapx∗ f satisfies the real version of thisinequality and (15) follows from the Hahn–Banach Theorem.

In a similar way we may proceed with the inequality (3). In order to make [3, Theorem2] applicable we will be assuming that the functions in question are continuous (condition(4) needs to be satisfied for both mappingsf and−f ).

Proposition 2. Given anε ≥ 0, let f0 : I → R satisfy

f0

(x + y

2

)≤

f0(y) − f0(x)

y − x+ ε, x, y ∈ I, x < y.

Thenf0(x) = f(x) + ε for x ∈ I, wheref is a solution of(3).


Theorem 3. Givenanε ≥ 0, let (X, ‖ ·‖) be a real normed linear space and letf : I → X

be a continuous solution to the functional inequality∥∥∥∥f

(x + y

2

)−

1

y − x[f(y) − f(x)]

∥∥∥∥ ≤ ε, x, y ∈ I, x < y. (16)

Then the estimate(15)holds true.

Proof. It suffices to apply Proposition 2, then [3, Theorem 2], Remark 1 and the Hahn–Banach Theorem.

One may obtain analogous results for equation (13). It suffices to apply [4, Theorem 7]and repeat the reasoning used previously.

Proposition 3. Given anε ≥ 0, let f0 : I → R satisfy

6f0(y) − f0(x)

y − x≤ 4f0

(x + y

2

)+ f0(x) + f0(y) + ε,

for eachx, y ∈ I such thatx < y. Thenf0(x) = f(x)− 16ε for x ∈ I, wheref is a solution

of (5).

Theorem 4. Given anε ≥ 0, let (X, ‖ · ‖) be a real normed linear space and letf : I → X

be continuous and satisfies∥∥∥∥2

3f

(x + y

2

)+

f(x) + f(y)

6−

1

y − x[f(y) − f(x)]

∥∥∥∥ ≤ ε, (17)

for eachx, y ∈ I such thatx < y. Then the estimate(15)holds true.

Remark 4. In view of Remark 3, it is reasonable to expect that each continuous solution of(14), (16) and (17) has to be in a sense close to the zero function. That is really the case. Wewill derive this fact from the previous theorems. For (16) letf : I → X satisfy assumptionsof Theorem 3. Fixx ∈ I andh > 0 such thatx + 2h ∈ I. By Theorem 3 the estimate (15)is valid whence

‖f(x + h) − ehf(x)‖ ≤ [eh − 1]ε, (18)

‖f(x + 2h) − e2hf(x)‖ ≤ [e2h − 1]ε. (19)

On the other hand, (16) applied fory = x + 2h implies that∥∥∥∥f(x + h) −

1

2hf(x + 2h) +

1

2hf(x)

∥∥∥∥ ≤ ε.

Let us rewrite these three estimates in the following form:

‖2hf(x + h) − 2hehf(x)‖ ≤ [2heh − 2h]ε,

‖ − f(x + 2h) + e2hf(x)‖ ≤ [e2h − 1]ε,

‖ − f(x) − 2hf(x + h) + f(x + 2h)‖ ≤ 2hε.


Adding these three inequalities side-by-side we finally arrive at

‖f(x)‖ ≤e2h + 2heh − 1

e2h − 2heh − 1· ε.

Notethat the right-hand side of this estimation does not depend uponx. Therefore, we mayconclude that in the class of continuous mappings the functional equation (12) is stable inthe sense of Hyers-Ulam. Moreover, if the intervalI is unbounded from the right, then wemay pass withh to +∞, to obtain the estimation

‖f(x)‖ ≤ ε, x ∈ I. (20)

An analogous reasoning can be applied for (14) (without assuming the continuity off ).Indeed, with the aid of this estimation fory = x + 2h we get

∥∥∥∥f(x) + f(x + 2h) −1

h[f(x + 2h) − f(x)]

∥∥∥∥ ≤ 2ε,

which jointly with (19) (which is valid by Theorem 2) gives us

‖f(x)‖ ≤

∣∣∣∣(h − 1)(e2h − 1) + 2h

(h − 1)e2h + h + 1

∣∣∣∣ · ε, x ∈ I, h > 0, x + h ∈ I,

stating that (14) is stable. Again, if additionallyI is unbounded from the right, then bylettingh tend to+∞ we can see that (20) holds.

Finally, (17) applied fory = x + 2h implies that∥∥4hf(x + h) + (h + 3)f(x) + (h − 3)f(x + 2h)

∥∥ ≤ 6hε,

which jointly with (18) and (19) (which is valid by Theorem 4) leads to the estimation

‖f(x)‖ ≤

∣∣∣∣he2h + 4heh + h − 1

(h − 3)e2h + 4heh + h + 3

∣∣∣∣ · ε, x ∈ I, h > 0, x + h ∈ I,

stating that (14) is stable in the sense of Hyers–Ulam. Again, if additionallyI is unboundedfrom the right, then (20) holds.

Now, we will discuss the stability of a more general equality:

f(y) − f(x)

y − x= N (g(M1(x, y)), g(M2(x, y))) , x, y ∈ I, x 6= y, (21)

under an additional assumption that there exist an injective functionϕ : I → R such that

N(u + h, v + h) = N(u, v) + ϕ(h), (22)

for eachu, v ∈ I andh ∈ R such thatu + h, v + h ∈ I.

Proposition 4. Given anε ≥ 0, let M1, M2 andN be arbitrary means and assume thatN

satisfies(22)with an injective mappingϕ : I → R andf0 : I → R andg0 : I → R satisfy

f0(y) − f0(x)

y − x≤ N (g0(M1(x, y)), g0(M2(x, y))) + ε,

for eachx, y ∈ I such thatx < y. Thenf0 = f andg0(x) = g(x)−ϕ−1(ε) for x ∈ I,where(f, g) is a solution of(6).


By a suitable modification of some previously used arguments, we obtain the followingresult,which states that in the stability inequality the general mean appearing in the right-hand side of (21) can be replaced by the integral mean.

Theorem 5. Given anε ≥ 0, let M1, M2 andN be arbitrary means and assume thatN

is continuous and satisfies(22) with an injective mappingϕ : I → R andf : I → R andg : I → R satisfy

∣∣∣∣f(y) − f(x)

y − x− N (g(M1(x, y)), g(M2(x, y)))

∣∣∣∣ ≤ ε,

for eachx, y ∈ I such thatx < y. Then∣∣∣∣∣∣f(y) − f(x)

y − x−

1

y − x

y∫

x

g(t) dt

∣∣∣∣∣∣≤ ϕ−1(ε), x, y ∈ I.

We terminate the paper with a stability result for (7).

Proposition 5. Assume thatε ≥ 0 andf0 : I → R is a continuous function satisfying

1

y − x

y∫

x

f0(t) dt ≤f0(y) − f0(x)

y − x+ ε, x, y ∈ I, x < y.

Thenf0(x) = f(x) + ε for x ∈ I, wheref is a solution of(7).

Proof. Putf(x) := f0(x) − ε for x ∈ I and check thatf satisfies (7).

Corollary 1. Givenan ε ≥ 0 and a reflexive real normed linear space(X, ‖ · ‖) assumethatf : (0, +∞) → X is a differentiable mapping such that

∥∥∥∥∥∥f(y) − f(x)

y − x−

1

y − x

y∫

x

f(t) dt

∥∥∥∥∥∥≤ ε, x, y ∈ I, x < y.

Then there exists a vectorc0 ∈ X such that the estimation(8) holds true.

Proof. Clearly, a differentiable function is Pettis (Bochner) integrable. Consequently, theintegral mean

1

y − x

y∫

x

f(t) dt

will tend to the valuef(x) whenevery tends tox. Therefore, it suffices to passy → x andto apply Theorem 1.

Corollary 2. Givenan ε ≥ 0, let M1, M2 andN be arbitrary means and assume thatN

is continuous and satisfies(22) with an injective mappingϕ : I → R whereasf : I → R

satisfies ∣∣∣∣f(y) − f(x)

y − x− N (f(M1(x, y)), f(M2(x, y)))

∣∣∣∣ ≤ ε,


for eachx, y ∈ I such thatx < y. Then, there exists a constantc ∈ R such that

|f(x) − cex| ≤ 3ϕ−1(ε), x ∈ I.

Added in proof. On January 20, 2009 some of the above results were presented at theSeminar on Functional Equations held at the Silesian University of Katowice (Poland). Dur-ing the discussion Peter Volkmann [9] remarked that the re exivity assumption in Theorem1 (and a fortiori in Corollary 1) may be dropped.

References

[1] C. Alsina, J. L. Garcia-Roig, On some inequalities characterizing the exponentialfunction,Arch. Math. (Brno)26 (1990), No. 2-3, 67–71.

[2] C. Alsina, R. Ger, On some inequalities and stability results related to the exponentialfunction,J. Inequal. Appl.2 (1998), 373–380.

[3] W. Fechner, On some inequalities connected with the exponential function,Arch.Math. (Brno)44 (2008), No. 3, 217–222.

[4] W. Fechner,A functional characterization of two inequalities between means, (sub-mitted).

[5] Z. Gajda, On stability of the Cauchy equation on semigroups,Aequationes Math.36(1988), No. 1, 76–79.

[6] Z. Gajda, Invariant means and representations of semigroups in the theory of func-tional equations,Prace Naukowe UniwersytetuSlaskiego w Katowicach [ScientificPublications of the University of Silesia],1273, UniwersytetSlaski, Katowice, 1992.

[7] J. Ger, On Sahoo-Riedel equations on a real interval,Aequationes Math.63 (2002),No. 1-2, 168–179.

[8] L. Szekelyhidi, Note on Hyers’s theorem,C. R. Math. Rep. Acad. Sci. Canada8(1986), No. 2, 127–129.

[9] P. Volkmann, Oral communication.



Chapter 4

ON A PROBLEM OF JOHN M. R ASSIAS

CONCERNING THE STABILITY IN ULAM SENSE

OF EULER –LAGRANGE EQUATION

L. Gavruta and P. Gavruta∗

Department of Mathematics, University “Politehnica”Timisoara, Piata Victoriei no. 2, 300006, Romania

Abstract

In this paper we solve a problem posed by John M. Rassias in 1992, concerningthe stability of Euler-Lagrange equation in the Ulam sense.


Key words: Ulam stability, Euler–Lagrange mapping.

1. Introduction

The study of stability problems for functional equations originated from a talk of S. Ulamin 1940 (see [19]) when he proposed the following problem:

LetG be a group endowed with a metricd. Givenε > 0, does there exist ak > 0 suchthat for every functionf : G→ G satisfying the inequality

d(f(xy), f(x)f(y)) < ε, ∀x, y ∈ G,

there exists an automorphisma of G with

d(f(x), a(x)) < kε, ∀x ∈ G ?

For results concerning this area see the papers [1]–[18]. In the following theorem John M.Rassias [12] proved the stability of Euler–Lagrange equation in the Ulam sense:

∗E-mail addresses: [email protected], [email protected]

48 L. Gavruta and P. Gavruta

Theorem 1.1. LetX bea normed linear space,Y be a Banach space, andf : X → Y. Ifthere existsa ≥ 0, b ≥ 0 such thata+ b < 2 , andc2 ≥ 0 such that:

‖f(x+ y) + f(x− y) − 2 · [f(x) + f(y)]‖ ≤ c2 · ‖x‖a · ‖y‖b

for all x, y ∈ X, there exists a unique non-linear mappingN : X → Y such that:

‖f(x) −N(x)‖ ≤ c · ‖x‖a+b

andN(x+ y) +N(x− y) = 2 · [N(x) +N(y)]

for all x, y ∈ X, wherec = c2/(4 − 2a+b)

In the same paper he puts the following question: “What is the situation in the abovetheorem in the casea+ b = 2?” In the present paper we give an answer to this problem.

A similar result for Cauchy functional equation was obtained in 1999 by the secondauthor of this paper[4].

2. The Result

Theorem 2.1. Let be0 < a < 2. Then, there exists a mappingf : R → R so that:

|f(x+ y) + f(x− y) − 2f(x) − 2f(y)| ≤ k|x|a|y|2−a, (2.1)

wherek does not depend onx, y, for all x, y ∈ R and for any quadratic mappingQ : R →R and everyα ∈ R we have:

supx6=0

|f(x) −Q(x)|

|x|α= ∞ (2.2)

Proof. We take the functionf : R → R,

f(x) =

x2 ln |x|, if x 6= 0

0, if x = 0

Step (I). We verify (2.1) for allx ≥ y > 0. Forx = y > 0:∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∣∣ = 4x2 ln 2x− 4x2 lnx = (4 ln 2)x2.

If x > y > 0 we have:

f(x+ y) + f(x− y) − 2f(x) − 2f(y)

xay2−a

=(x+ y)2 ln(x+ y) + (x− y)2 ln(x− y) − 2x2 lnx− 2y2 ln y

xay2−a

=(x2 + y2) ln(x2 − y2) + 2xy[ln(x+ y) − ln(x− y)] − x2 lnx2 − y2 ln y2

xay2−a

On a Problem of John M. Rassias Concerning the Stability... 49

=x2 ln x2

−y2

x2 + y2 ln x2−y2

y2 + 2xy ln x+yx−y

xay2−a

=

(x

y

)2−a

ln

(1 −

y2

x2

)+

(y

x

)a

ln

(x2

y2− 1

)+ 2

(x

y

)1−a

ln

xy

+ 1xy− 1

= t2−a ln

(1 −

1

t2

)+

1

taln

(t2 − 1

)+ 2t1−a ln

t+ 1

t− 1,

wheret := xy> 1.

We prove that the functionFa : (1,∞) → R,

Fa(t) = t2−a ln

(1 −

1

t2

)+

1

taln

(t2 − 1

)+ 2t1−a ln

t+ 1

t− 1

is bounded.We prove that

limt→∞

Fa(t) = 0. (2.3)

First term:

limt→∞

t2−a ln

(1 −

1

t2

)=

1

t=u

limu→0u>0

ua−2 ln(1 − u2)

= limu→0u>0

ln(1 − u2)

u2−a

= limu→0u>0

−2u1−u2

(2 − a)u1−a

= −2

2 − alimu→0u>0

ua

1 − u2.

Secondterm:

limt→∞

ln(t2 − 1)

ta= lim

t→∞

2tt2−1

ata−1

=2

alimt→∞

t2−a

t2 − 1

= 0,

since2 − a < 2.The last term:

limt→∞

t1−a lnt+ 1

t− 1= lim

t→∞

1

taln

(t+ 1

t− 1

)t

= 0 · 2 = 0

We prove that:limt→1t>1

Fa(t) (2.4)


exists and it is finite. We write:

Fa(t) = t2−a ln(t− 1) +1

taln(t− 1) − 2t1−a ln(t− 1)

+ t2−a ln(t+ 1) +1

taln(t+ 1) + 2t1−a ln(t+ 1)

− 2t2−a ln t

hence:

Fa(t) =(t− 1)2

taln(t− 1) +

(t+ 1)2 ln(t+ 1)

ta− 2t2−a ln t.

We havelimt→1t>1

Fa(t) = 4 ln 2

SinceFa is continuous on(1,∞), it follows that it is a bounded function.

Step (II). We prove (2.1) for allx, y ∈ R using the following lemma:

Lemma 2.2. Let bef : R → R an even function so thatf(0) = 0 and for every0 < a < 2there existsc = c(a) > 0 so that:

∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)∣∣ ≤ cxay2−a for all x ≥ y > 0 (2.5)

Then, for every0 < a < 2 there isk = k(a) such that:∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∣∣ ≤ k|x|a|y|2−a for all x, y ∈ R. (2.6)

Proof of Lemma.Case 1)0 < x < y. Then:∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∣∣=

∣∣f(x+ y) + f(y − x) − 2f(y) − 2f(x)∣∣

≤ c(2 − a)y2−axa.

If k = maxc(a), c(2 − a) it follows∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∣∣ ≤ kxay2−a, x, y > 0

Case 2)x = 0 or y = 0. Clear.Case 3)x > 0, y < 0. If u = −y > 0, then

∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)∣∣

=∣∣f(x− u) + f(x+ u) − 2f(x) − 2f(u)

∣∣

≤ kxau2−a = kxa|y|2−a

Case 4)x < 0, y > 0. Putv = −x

∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)∣∣

=∣∣f(−v + y) + f(−v − y) − 2f(−v) − 2f(y)

∣∣


=∣∣f(v + y) + f(y − v) − 2f(v) − 2f(y)

∣∣

≤ kvay2−a.

Case 5)x < 0, y < 0. If u = −y, v = −x we have:

∣∣f(x+ y) + f(x− y) − 2f(x) − 2f(y)∣∣

=∣∣f(u+ v) + f(v − u) − 2f(v) − 2f(u)

∣∣

≤ kvau2−a = k|x|a|y|2−a.

Step (III). We prove (2.2). First we prove that ifQ : R → R is a quadratic mapping then

Q(2n) = 4nQ(1), n ∈ Z.

FromQ(x+ y) +Q(x− y) = 2Q(x) + 2Q(y) for x = y we obtain

Q(2x) +Q(0) = 4Q(x)

Forx = 0 it follows 2Q(0) = 4Q(0) =⇒ Q(0) = 0.HenceQ(2x) = 4Q(x). We obtain:Q(2nx) = 4nQ(x), n ∈ N.

That implies

Q(2n) = 4nQ(1), n ∈ N

and forx = 12n wehave:

Q(1) = 4nQ(2−n) ⇐⇒ Q(2−n) = 4−nQ(1).

Forα ≤ 2 :

supx6=0

|f(x) −Q(x)|

|x|α≥ sup

n∈N

|f(2n) −Q(2n)|

2nα

= supn∈N

|4nn ln 2 − 4nQ(1)|

2nα

= supn∈N

2n(2−α) |n ln 2 −Q(1)| = ∞.

Forα > 2 :

supx6=0

|f(x) −Q(x)|

|x|α≥ sup

n∈N

|f(2−n) −Q(2−n)|

2−nα

= supn∈N

|4−nn ln 2 − 4−nQ(1)|

2−nα

= supn∈N

2n(α−2) |n ln 2 −Q(1)| = ∞.


References

[1] T. Aoki, On the stability of the linear transformation in Banach spaces,J. Math. Soc.Japan2 (1950), 64–66.

[2] S. Czerwik, Functional Equations and Inequalities in Several Variables,World Scien-tific, New Jersey, London, Singapore, Hong Kong, 2002.

[3] P. Gavruta, A generalization of the Hyers–Ulam–Rassias stability of approximatelyadditive mappings,J. Math. Anal. Appl.184(1994), 431–436.

[4] P. Gavruta, An answer to a question of John M. Rassias concerning the stability ofCauchy functional equation,Advances in Equations and Inequalities, Hadronic Math.Ser.(1999), 67–71.

[5] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in SeveralVariables,Birkhauser, Basel,1998.

[6] D. H. Hyers, On the stability of the linear functional equation,Proc. Natl. Acad. Sci.27 (1941), 222–224.

[7] G. Isac and Th. M. Rassias, On the Hyers–Ulam stability ofψ-additive mappings,J.Approx. Theory72 (1993), 131–137.

[8] S-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in MathematicalAnalysis,Hadronic Press, Palm Harbor, 2001.

[9] M. S. Moslehian, Ternary derivations, stability and physical aspects,Acta Appl. Math.100(2008), No. 2, 187–199.

[10] P. Nakmahachalasint, On the generalized Ulam–Gavruta–Rassias stability of mixed-type linear and Euler–Lagrange–Rassias functional equations,International Journalof Mathematics and Mathematical Sciences,vol. 2007, Article ID 63239, 10 pages,2007.

[11] C. G. Park, Stability of an Euler–Lagrange–Rassias type additive mapping,Intern. J.Appl. Math. Stat.7(Fe07)(2007), 101–111.

[12] J. M. Rassias, On the stability of the Euler–Lagrange functional equation,ChineseJournal of Mathematics20 (1992), No. 2, 185–190.

[13] J. M. Rassias and M. J. Rassias, Refined Ulam stability for Euler–Lagrange type map-pings in in Hilbert spaces,Intern. J. Appl. Math. Stat.7(Fe07)(2007), 126–132.

[14] J. M. Rassias, On approximation of approximately linear mappings by linear map-pings,Bull. Sci. Math. (2)108(1984), No. 4, 445–446.

[15] J. M. Rassias, On approximation of approximately linear mappings by linear map-pings,Journal of Functional Analysis46 (1982), No. 1, 126–130.


[16] J. M. Rassias, Solution of a problem of Ulam,Journal of Approximation Theory57(1989), No. 3, 268–273.

[17] Th. M. Rassias, On the stability of the linear mapping in Banach spaces,Proc. Amer.Math. Soc.72 (1978), 297–300.

[18] Th. M. Rassias, On the stability of the quadratic functional equation and its applica-tions,Studia Univ. Babes-Bolyai Math.43(3)(1998), 89–124.

[19] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, science ed.,Wiley, NewYork,1940.



Chapter 5

HYERS–ULAM –AOKI –RASSIAS SSTABILITY

AND ULAM –GAVRUTA –RASSIAS STABILITY

OF QUADRATIC HOMOMORPHISMS AND

QUADRATIC DERIVATIONS ON BANACH ALGEBRAS

M. Eshaghi Gordji∗ and N. Ghobadipour†

Department of Mathematics, Semnan UniversityP. O. Box 35195-363, Semnan

Iran

Abstract

In this paper, we establish the Hyers–Ulam–Aoki–Rassias stability and Ulam–Gavruta–Rassias stability of the quadratic homomorphisms and quadratic derivationson Banach algebras.


Key words: Hyers–Ulam–Aoki–Rassias stability, Ulam–Gavruta–Rassias stability-Homomorphism- Derivation- Quadratic function.

1. Introduction

Throughout this paper we suppose thatA is a Banach algebra andX is a BanachA-module.Quadratic functional equation was used characterize inner product spaces [2,3,18]. Severalother functional equations were also to characterize inner product spaces. A square normon an inner product space satisfies the important parallelogram equality

‖x+ y‖2 + ‖x− y‖2 = 2(‖x‖2 + ‖y‖2).

The functional equation

f(x+ y) + f(x− y) = 2f(x) + 2f(y) (1.1)

∗E-mail addresses: [email protected]†E-mail addresses: [email protected]

56 M. Eshaghi Gordji and N. Ghobadipour

is related to symmetric bi-additive function [2], [18]. It is natural that this equationis calleda quadratic functional equation. In particular, every solution of the quadratic equation (1.3)is said to be a quadratic function. It is well known that a functionf between real vectorspaces is quadratic if and only if there exits a unique symmetric bi-additive function B suchthatf(x) = B(x, x) for all x (see [2], [18]). The bi-additive functionB is given by

B(x, y) =1

4(f(x+ y) − f(x− y)). (1.2)

A mappingf : A −→ A is called a quadratic homomorphism iff is a quadratic functionsatisfiesf(ab) = f(a)f(b) for all a, b ∈ A. For instance, letA be commutative, then themappingf : A −→ A defined byf(a) = a2(a ∈ A), is a quadratic homomorphism.

A mappingf : A −→ X is called a quadratic derivation iff is a quadratic functionsatisfiesf(ab) = f(a)b+ af(b) for all a, b ∈ A. The following example is a slight modifi-cation of an example due to [11].

Example. LetA be a Banach algebra. Then we take

T =

0 A A0 0 A0 0 0

,

T is a Banach algebra equipped with the usual matrix-like operations and the followingnorm: ∥∥∥∥∥∥

0 a b

0 0 c

0 0 0

∥∥∥∥∥∥= ‖a‖ + ‖b‖ + ‖c‖ (a, b, c ∈ A).

It is known that

T ∗ =

0 A∗ A∗

0 0 A∗

0 0 0

,

is the dual ofT under the following norm∥∥∥∥∥∥

0 f g

0 0 h

0 0 0

∥∥∥∥∥∥= max

‖f‖, ‖g‖, ‖h‖

(f, g, h ∈ A∗).

Let the left module action ofT onT ∗ be trivial and let the right module action ofT onT ∗

is defined as follows.⟨

0 f g

0 0 h

0 0 0

0 a b

0 0 c

0 0 0

,

0 x y

0 0 z

0 0 0

⟩= f(ax) + g(by) + h(cz),

for all f, g, h ∈ A∗, a, b, c, x, y, z ∈ A. Then T ∗ is a BanachT -module. Let

0 k g

0 0 h

0 0 0

∈ T ∗. We defineD : T −→ T ∗ by

D

0 a b

0 0 c

0 0 0

=

0 k g

0 0 h

0 0 0

0 0 ac

0 0 00 0 0

(a, b, c ∈ A).

Hyers–Ulam–Aoki–Rassias Stability. . . 57

Then we can see thatD is a quadratic derivation fromT into T ∗ [11].

It is easy to see that a quadratic homomorphism (derivation) is a linear homomorphism(derivation) if and only if it is zero function.

The stability problem of functional equations originated from a question of Ulam [67]in 1940, concerning the stability of group homomorphisms. Let(G1, .) be a group and let(G2, ∗) be a metric group with the metricd(., .).Givenǫ > 0, dose there exist aδ > 0, suchthat if a mappingh : G1 −→ G2 satisfies the inequalityd(h(x.y), h(x) ∗ h(y)) < δ for allx, y ∈ G1, then there exists a homomorphismH : G1 −→ G2 with d(h(x), H(x)) < ǫ forall x ∈ G1? In the other words, Under what condition dose there exists a homomorphismnear an approximate homomorphism? The concept of stability for functional equation ariseswhen we replace the functional equation by an inequality which acts as a perturbation ofthe equation. In 1941, D. H. Hyers [19] gave a first affirmative answer to the question ofUlam for Banach spaces. Letf : E −→ E′ be a mapping between Banach spaces such that

∥∥f(x+ y) − f(x) − f(y)∥∥ ≤ δ

for all x, y ∈ E, and for someδ > 0. Then there exists a unique additive mappingT :E −→ E′ such that

‖f(x) − T (x)‖ ≤ δ

for all x ∈ E. Moreover iff(tx) is continuous in t for each fixedx ∈ E, thenT is linear.In 1950, T. Aoki [3] was the second author to treat this problem for additive mappings.

Finally in 1978, Th. M. Rassias [61] proved the following Theorem.

Theorem 1.1. Letf : E −→ E′ be a mapping from a norm vector spaceE into a BanachspaceE′ subject to the inequality

∥∥f(x+ y) − f(x) − f(y)∥∥ ≤ ǫ

(‖x‖p + ‖y‖p

)(1.3)

for all x, y ∈ E, whereǫ and p are constants withǫ > 0 andp < 1. Then there exists aunique additive mappingT : E −→ E′ such that

‖f(x) − T (x)‖ ≤2ǫ

2 − 2p‖x‖p, (1.4)

for all x ∈ E. If p < 0 then inequality (1.3) holds for allx, y 6= 0, and (1.4) forx 6= 0.Also, if the functiont 7→ f(tx) fromR intoE′ is continuous for each fixedx ∈ E, then T islinear.

In 1991, Z. Gajda [14] answered the question for the casep > 1, which was rased byRassias. On the other hand J. M. Rassias [53]–[55], generalized the Hyers stability resultby presenting a weaker condition controlled by a product of different powers of norms.According to J. M. Rassias Theorem:

Theorem 1.2. LetΘ ≥ 0 and letp1, p2 ∈ R with p = p1 + p2 6= 1. Supposef : E −→ E′

is a mapping from a norm spaceE into a Banach spaceE′ such that the inequality

∥∥f(x+ y) − f(x) − f(y)∥∥ ≤ ǫ‖x‖p1‖y‖p2 ,


for all x, y ∈ E, then there exists a unique additive mappingT : E −→ E′ such that

‖f(x) − T (x)‖ ≤Θ

2 − 2p‖x‖p,

for all x ∈ E. If in addition for everyx ∈ E, f(tx) is continuous, thenT is linear.

Following the techniques of the proof of the corollary of D. H. Hyers [19] it is observedthat D. H. Hyers introduced (in 1941) the following Hyers continuity condition: about thecontinuity of the mappingf(tx) in realt for each fixedx, and then he proved homogenouityof degree one and therefore the famous linearity. This condition has been assumed furthertill now, through the complete Hyers direct method, in order to prove linearity for general-ized Hyers–Ulam stability problem forms. A number of mathematicians were attracted tothe pertinent stability results of T. Aoki [3], Th. M. Rassias [61] and J. M. Rassias [53]–[55],and stimulated to investigate the stability problems of functional equations. The stabilityphenomenon that was introduced and proved by T. Aoki, Th. M. Rassias and J. M. Ras-sias is called Hyers–Ulam–Aoki–Rassias stability for the sum and Ulam–Gavruta–Rassiasstability for the product of powers of norms. And then the stability problems of severalfunctional equations have been extensively investigated by a number of authors and thereare many interesting results concerning this problem [12], [13] (see also [1], [4], [6], [15],[16], [18]–[22] and [56]–[64]). A Ulam–Gavruta–Rassias stability Theorem for quadraticfunctional equation (1.1) was proved by J. M. Rassias [53]–[55] (see also [10]). During the34th International Symposium on Functional Equations, Gy. Maksa [27] posed the problemconcerning the Hyers–Ulam stability of the functional equation

f(xy) = f(x)y + xf(y), (1.5)

on the interval(0, 1] and J. Tabor gave an answer to the equation of Maksa in [66]. On theother hand, Zs. Pales [47] remarked that the functional equation (1.1) for real-valued func-tions has a superstability on the interval[1,∞). In 1997, C. Borelli [7] demonstrated theHyers–Ulam stability of functional equation (1.5) on restricted domain ofR. Jung and Park[23] have solved the functional equationf(x+y+xy) = f(x)+f(y)+f(x)y+xf(y) mo-tivated by the equation (1.5), and then investigated the Hyers–Ulam–Aoki–Rassias stabilityproblem on the interval(−1, 0] and the superstability on[0,∞) (see also [16], [20]–[22]).

Hyers–Ulam–Aoki–Rassias stability and problem for the quadratic functional equation(1.1) was proved by Skof for functionsf : A −→ B, were A is normed space and B Banachspace(see [65]). Cholewa [9] noticed that the Theorem of Skof is still true if relevant domainA is replaced an abelian group. In the paper [10], Czerwik proved the Hyers–Ulam–Aoki–Rassias stability of the equation (1.3). Grabiec [17] has generalized these result mentionedabove (see [2], [10], [24], [25], [46] and [48]–[51]). For the stability of linear homomor-phisms and derivations we refer the reader to [4], [5], [8], [26] and [28]–[45]. In sectiontwo we investigate the situation that the generalized Hyers–Ulam–Aoki–Rassias stabilityand Ulam–Gavruta–Rassias stability for quadratic homomorphisms on Banach algebras.In section three we study the generalized Hyers–Ulam–Aoki–Rassias stability and Ulam-Gavruta-Rassias stability of quadratic derivations from a Banach algebras into its Banachmodules.


2. Quadratic Homomorphisms

In this section we study the stability of quadratic homomorphisms on Banach algebras.

Theorem 2.1. Let A, B be Banach algebras. Suppose functionsψ,Q : A × A → [0,∞)satisfying

Q(x) :=

∞∑

i=0

1

4iQ(2ix, 2ix) <∞, (2.6)

lim1

16iψ(2ix, 2iy) = 0, (2.7)

and

limQ(2ix, 2iy) = 0 (2.8)

for all x, y ∈ A. If f : A→ B is a mapping such that

∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)∥∥ ≤ Q(x, y), (2.9)

and that

‖f(xy) − f(x)f(y)‖ ≤ ψ(x, y), (2.10)

for all x, y ∈ A, then there exists a unique quadratic homomorphismH : A→ B such that

‖f(x) −H(x)‖ ≤1

4Q(x) (2.11)

for all x ∈ A.

Proof. Letting y=x in (2.4), we get

∥∥∥∥1

4f(2x) − f(x)

∥∥∥∥ ≤1

4Q(x, x). (2.12)

Replacex by 2x in (2.7) and result divide by 4 to obtain

∥∥∥∥1

42f(4x) −

1

4f(2x)

∥∥∥∥ ≤1

42Q(2x, 2x). (2.13)

Now, combine (2.7) and (2.8) by use of the triangle inequality to get

∥∥∥∥1

42f(4x) − f(x)

∥∥∥∥ ≤1

4Q(x, x) +

1

42Q(2x, 2x).

Now, proceed in this way to prove by induction that

∥∥∥∥1

4nf(2nx) − f(x)

∥∥∥∥ ≤n−1∑

i=0

1

4i+1Q(2ix, 2ix). (2.14)


In order to show that functionshn(x) = 14n f(2nx) form a convergent sequence, we used

Cauchy convergence criterion. Indeed, replace x by2mx in (2.9) and result divide by4m,wherem is an arbitrary positive integer. We find that,

∥∥∥∥1

4n+mf(2n+mx) −

1

4mf(2mx)

∥∥∥∥ ≤1

4

m+n−1∑

i=m

1

4iQ(2ix, 2ix). (2.15)

By (2.1) and since B is complete, it follows thatlimn→∞

Hn(x) exists for allx ∈ A.

Let m=0 andn→ ∞ in (2.10), we have

‖H(x) − f(x)‖ ≤1

4

∞∑

i=0

1

4iQ(2ix, 2ix) =

1

4Q(x)

suchthat H defined byH : A → X, H(x) = lim 14n f(2nx). On the other hand, by using

(2.1), for allx, y ∈ A, we have

∥∥H(x+ y) +H(x− y) − 2H(x) − 2H(y)∥∥

= lim1

4n

∥∥f(2n(x+ y)) + f(2n(x− y)) − 2f(2nx) − 2f(2ny)∥∥

≤ lim1

4nQ(2nx, 2ny) = 0.

This means that H is quadratic. Using (2.2) to obtain

∥∥H(xy) −H(x)H(y)∥∥ =

∥∥∥∥lim1

4nf(2nxy) − lim

1

4nf(2nx). lim

1

4nf(2ny)

∥∥∥∥

=

∥∥∥∥lim1

16nf(2nx2ny) − lim

1

16nf(2nx)f(2ny)

∥∥∥∥

≤ lim1

16nψ(2nx, 2ny) = 0.

Hence,H(xy) = H(x)H(y).

Now, suppose there is another such functionH : A → B satisfiesH(x+ y) + H(x−y) = 2H(x) + 2H(y) and‖H(x) − f(x)‖ ≤ 1

4Q(x). Then for allx ∈ A, we have

∥∥H(x) − H(x)∥∥ =

1

4n

∥∥H(2nx) − H(2nx)∥∥

≤1

4n

(‖H(2nx) − f(2nx)‖ + ‖H(2nx) − f(2nx)‖

)

≤2

4n+1Q(2nx) =

1

2

∞∑

i=0

1

4i+nQ(2i+nx, 2i+nx)

=1

2

∞∑

i=n

1

4iQ(2ix, 2ix).

By n→ ∞ we get,H(x) = H(x).


Corollary 2.2. Let P < 2, Θ > 0, and letA, B be Banach algebras. Suppose mappingψ : A×A→ [0,∞) satisfies

lim1

16iψ(2ix, 2iy) = 0,

for all x, y ∈ A, moreover, suppose mappingf : A→ B satisfies∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∥∥ ≤ Θ(‖x‖p + ‖y‖p),

and‖f(xy) + f(x)f(y)‖ ≤ ψ(x, y),

for all x, y ∈ A. Then there exists a unique quadratic homomorphismH : A → B suchthat for all x ∈ A,

‖H(x) − f(x)‖ ≤Θ‖x‖p

2· · ·

1

1 − 2p−2.

Proof. It follows from above Theorem by takingQ(x, y) := θ(‖x‖p + ‖y‖p).

In the following Corollary, we show that the superstability for the inequality (2.4)isvalid whenf is a quadratic function.

Corollary 2.3. Let letA andB be Banach algebras. Suppose a mappingψ : A × A →[0,∞) satisfies

lim1

16iψ(2ix, 2iy) = 0,

moreover, suppose a quadratic mappingf : A→ B satisfies

‖f(xy) + f(x)f(y)‖ ≤ ψ(x, y),

for all x, y ∈ A. Thenf is a homomorphism.

Proof. It follows from above Theorem by takingQ(x, y) := 0.

Theorem 2.4. LetA, B beBanach algebras. Suppose functionsψ,Q : A × A → [0,∞)satisfying

Q(x) :=∞∑

i=1

4iQ( x

2i,x

2i

)<∞, (2.16)

limi→∞

16iψ( x

2i,y

2i

)= 0, (2.17)

andlimi→∞

Q( x

2i,y

2i

)= 0, (2.18)

for all x, y ∈ A. Moreover, iff : A→ B is a mapping such that∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∥∥ ≤ Q(x, y), (2.19)

and that‖f(xy) − f(x)f(y)‖ ≤ ψ(x, y), (2.20)


for all x, y ∈ A, then there exists a unique quadratic homomorphismH : A→ B such that

‖f(x) −H(x)‖ ≤1

4Q(x), (2.21)

for all x ∈ A.

Proof. Letting y=x in (2.14) we get

‖f(2x) − 4f(x)‖ ≤ Q(x, x), (2.22)

replacex by x2 in (2.17), to obtain

∥∥∥f(x) − 4f(x

2

)∥∥∥ ≤ Q(x

2,x

2

). (2.23)

Replacingx by x2 in (2.18) and result divide by14 , we lead to

∥∥∥4F(x

2

)− 16f

(x4

)∥∥∥ ≤ 4Q(x

4,x

4

). (2.24)

Combine(2.18) and (2.19), to get∥∥∥f(x) − 16f

(x4

)∥∥∥ ≤ Q(x

2,x

2

)+ 4Q

( x22,x

22

). (2.25)

Now, Proceed in this way to prove by induction that,

∥∥∥Hnf( x

22

)− f(x)

∥∥∥ ≤n∑

i=1

4i−1Q( x

2i,x

2i

). (2.26)

In order to show that functionsHn(x) = 4nf(

x2n

)form a convergent sequence, we used

Cauchy convergence criterion. Indeed, replace x byx2m in (2.21) and result divide by1

4m ,wherem is an arbitrary positive integer. we lead to

∥∥∥4n+mf( x

2n2m

)− 4mf

( x

2m

)∥∥∥ ≤m+n∑

i=1+m

4i−1Q( x

2i,x

2i

). (2.27)

SinceB is complete, then bylimn→∞

Hn(x) exists for allx ∈ A. Let m=0 andn → ∞

in (2.22), we have,‖H(x) − f(x)‖ ≤ 14 Q(x), whichH : A → B defined by,H(x) =

lim 4nf(

x2n

).

Now, for all x, y ∈ A, it follows that

∥∥H(x+ y) +H(x− y) − 2H(x) − 2H(y)∥∥

= lim4n∥∥∥f(2−n(x+ y)) + f(2−n(x− y)) − 2f(2−nx) − 2f(2−ny)

∥∥∥

≤ lim 4nQ(2−nx, 2−ny) = 0.

ThusH(x+ y) +H(x− y) = 2H(x) + 2H(y).


So we obtain

‖H(xy) −H(x)H(y)‖ =∥∥∥ lim 4nf(2−nxy) − lim 4nf(2−nx) lim 4nf(2−ny)

∥∥∥

≤ lim 16nψ( x

2n,y

2n

)= 0.

This means that,H(xy) = H(x)H(y).Now, suppose there is another quadratic homomorphismH : A → B satisfies (2.16).

Then we have

‖H(x) − H(x)‖ = 4n∥∥∥H

( x

2n

)− H

( x

2n

)∥∥∥

≤ 4n(∥∥∥H

( x

2n

)− f

( x

2n

)∥∥∥ +∥∥∥H

( x

2n

)− f

( x

2n

)∥∥∥)

≤ 4n

(1

4Q

( x

2n

)+

1

4Q

( x

2n

))

=1

2

∑4i+nQ

( x

2i+n,x

2i+n

)=

1

2

∞∑

i=1+n

4iQ( x

2i,x

2i

).

By (2.11) it follows thatH(x) = H(x), for all x ∈ A.

Corollary 2.5. LetP , Θ bepositive real numbers such thatp > 2, and let A,B be Banachalgebras. Suppose functionψ : A×A→ [0,∞) satisfies

lim 16iψ( x

2i,y

2i

)= 0,

for all x, y ∈ A. Moreover, iff : A→ B is a mapping such that∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∥∥ ≤ Θ(‖x‖p + ‖y‖p

),

and ∥∥f(xy) − f(x)f(y)∥∥ ≤ ψ(x, y),

then there exists a unique quadratic homomorphismH : A→ B such that

‖H(x) − f(x)‖ ≤ Θ1

2‖x‖p.

1

2P−2 − 1,

for all x ∈ A.

3. Quadratic Derivations

In this section we establish the stability of quadratic derivations.

Theorem 3.1. LetA be a Banach algebra andX be a BanachA-Module. Suppose mapsψ,Q : A×A→ [0,∞) satisfying

Q(x) :=∑ 1

4iQ(2ix, 2ix) <∞, (3.28)

lim1

8iψ(2ix, 2iy) = 0, (3.29)


andlimQ(2ix, 2iy) = 0. (3.30)

Moreover, iff : A→ X is a mapping such that for allx, y ∈ A,∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∥∥ ≤ Q(x, y), (3.31)

and that ∥∥2−nf(4nxy) + 2nxf(2ny) − 2nf(2nx).y∥∥ ≤ ψ(2nx, 2ny), (3.32)

for all n ∈ N, then there exists a unique quadratic derivationD : A→ X such that for allx ∈ A

‖f(x) −D(x)‖ ≤1

4Q(x). (3.33)

Proof. By Theorem 2.1, the limitD(x) := lim 14n f(2nx) exists for everyx ∈ A. Now for

all x, y ∈ A, we have∥∥D(x+ y) +D(x− y) − 2D(x) − 2D(y)

∥∥

=

∥∥∥∥lim1

4nf(2n(x+ y)) + lim

1

4nf(2n(x− y)) − lim

1

4n2f(2nx) − lim

1

4n2f(2ny)

∥∥∥∥

≤ lim1

4nQ(2nx, 2ny) = 0.

ThereforeD(x+ y) +D(x− y) = 2D(x) + 2D(y) andD is quadratic.On the other hand, we have

∥∥D(xy) − xD(y) −D(x)y∥∥

=

∥∥∥∥1

4nD(2nxy) − xD(y) −D(x)y

∥∥∥∥∥∥∥∥lim

1

16nf(4nxy) − x lim

1

4nf(2ny) − lim

1

4nf(2nx).y

∥∥∥∥

=

∥∥∥∥lim1

16nf(2nx.2ny) − lim

1

8n(2nx).f(2ny) − lim

1

8nf(2nx).(2ny)

∥∥∥∥

= lim1

8n

∥∥∥2−nf(2nx.2ny) − (2nx).f(2ny) − f(2nx).(2ny)∥∥∥

≤ lim1

8nψ(2nx, 2ny) = 0.

ThusD(xy) = xD(y) +D(x)y.Now, suppose there is another such functionD : A→ X with D(x+ y)+ D(x− y) =

2D(x) + 2D(y) and‖D(x) − f(x)‖ ≤ 14 Q(x) for all x ∈ A. So, for allx ∈ A we have

∥∥D(x) − D(x)∥∥ =

1

4n

∥∥D(2nx) − D(2nx)∥∥

=1

4n

(∥∥D(2nx) − f(2nx)∥∥ +

∥∥D(2nx) − f(2nx)∥∥)

≤1

2

∞∑

i=0

1

4i+nQ(2i+nx, 2i+nx) =

1

2

∞∑

i=n

1

4iQ(2ix, 2ix).

Using(3.1) and takingn→ ∞ we get,D(x) = D(x).


Corollary 3.2. Let P < 2 and Θ > 0, and letA be a Banach algebra,X be a BanachA-module. Suppose functionψ : A×A→ [0,∞) satisfies

lim1

8iψ(2ix, 2iy) = 0,

moreover, iff : A→ X is a mapping such that,

∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)∥∥ ≤ Θ

(‖x‖P + ‖y‖P

)

and ∥∥2−nf(4nxy) + 2nxf(2ny) − 2nf(2nx).y∥∥ ≤ ψ(2nx, 2ny),

then there exists a unique quadratic derivationD : A→ X such that

‖D(x) − f(x)‖ ≤θ‖x‖P

2.

1

1 − 2P−2.

Theorem 3.3.LetA be a Banach algebra andX be a BanachA-module. Suppose functionsQ,ψ : A×A→ [0,∞) satisfying

Q(x) :=∞∑

i=1

4iQ( x

2i,x

2i

)<∞, (3.34)

limi→∞

8iψ( x

2i,y

2i

)= 0, (3.35)

andlimi→∞

Q( x

2i,y

2i

)= 0. (3.36)

Moreover, suppose the mappingf : A→ X satisfies

∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)∥∥ ≤ Q(x, y), (3.37)

and

∥∥2nf(2−nx.2−ny) − 2−nxf(2−ny) − f(2−nx).(2−ny)∥∥ ≤ ψ(x, y), (3.38)

for all x, y ∈ A, and for all n ∈ N. Then there exists a unique quadratic derivationD : A→ X such that for allx ∈ A,

‖D(x) − f(x)‖ ≤1

4Q(x). (3.39)

Proof. By Theorem 2.3, the limitD(x) := lim 4nf(

x2n

)exists for allx ∈ A. Now, by

(3.7), for allx, y ∈ A, we have

∥∥D(x+ y) +D(x− y) − 2D(x) − 2D(y)∥∥

=∥∥∥ lim 4nf(2−n(x+ y)) + lim 4nf(2−n(x− y)) − 2 lim 4nf(2−nx) − 2 lim 4nf(2−ny)

∥∥∥

≤ lim 4nq(2−nx, 2−ny) = 0.


ThereforeD(x + y) +D(x − y) = 2D(x) + 2D(y). This means that D is quadratic. Onthe other hand by (3.8) and (3.11), for allx, y ∈ A, we have

∥∥D(xy) − xD(y) −D(x)y∥∥ =

∥∥4nD(2−nxy) − xD(y) −D(x)y∥∥

=∥∥∥ lim 16nf(2−nx2−ny) − x lim 4nf(2−ny) − lim 4nf(2−nx).y

∥∥∥

=∥∥∥ lim 16nf(2−nx.2−ny) − (2−nx) lim 8nf(2−ny) − lim 8nf(2−nx).(2−ny)

∥∥∥

= lim 8n∥∥∥2nf(2−nx.2−ny) − 2−nxf(2−ny) − f(2−nx).(2−ny)

∥∥∥

≤ lim 8nψ(2−nx, 2−ny) = 0.

On the other word, D is multiplicative derivation. Now, suppose there is another suchfunctionD : A→ X, with D(x+y)+D(x−y) = 2D(x)+2D(y) and‖D(x)−f(x)‖ ≤14 Q(x). It follows that

‖D(x) − D(x)‖ = 4n∥∥D(2−nx) − D(2−nx)

∥∥

= 4n(∥∥D(2−nx) − f(2−nx)

∥∥ +∥∥D(2−nx) − f(2−nx)

∥∥)

≤ 4n

(1

4Q(2−nx) +

1

4Q(2−nx)

)=

1

2

∞∑

i=1+n

4iQ(2−ix, 2−ix).

By (3.7), we getD(x) = D(x) for all x ∈ A by takingn −→ ∞.

Corollary 3.4. Let P > 2 and θ > 0, and letA be a Banach algebra,X be a BanachA-module. Suppose mappingψ : A×A −→ [0,∞) satisfies

limi→∞

8iψ( x

2i,y

2i

)= 0,

moreover, iff : A→ X is mapping such that,∥∥f(x+ y) + f(x− y) − 2f(x) − 2f(y)

∥∥ ≤ θ(‖x‖P + ‖y‖P

),

and that∥∥2nf(2−nx.2−ny) − 2−nxf(2−ny) − f(2−nx).(2−ny)

∥∥ ≤ ψ(x, y), (3.11)

for all x, y ∈ X and for all n ∈ N. Then there exists a unique quadratic derivationD :A→ X such that for allx ∈ A,

‖D(x) − f(x)‖ ≤θ

2‖x‖P 1

2P−2 − 1.

Proof. The proof follows from above Theorem by takingQ(x, y) := θ(‖x‖P +‖y‖P ).

Acknowledgement

The authors would like to express their sincere thanks to professor J. M. Rassias for hisinvaluable comments. Also, the second author would like to thank the office of giftedstudents at Semnan University for its financial support.


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In: Functional Equations, Difference Inequalities… ISBN: 978-1-60876-461-7Editor: John Michael Rassias, pp. 73-83 © 2010 Nova Science Publishers, Inc.

Chapter 6

FUNDAMENTAL SOLUTIONS FOR THE GENERALIZEDELLIPTIC GELLERSTEDT EQUATION

Anvar Hasanov1,a, John Michael Rassias2,b and Mamasali Turaev3,c

1 Uzbek Academy of Sciences, Institute of Mathematics29 F. Hodjaev Street, Tashkent 700125, Uzbekistan

2 National and Capodistrian University of Athens, Pedagogical DepartmentSection of Mathematics and Informatics

4, Agamemnonos Str., Aghia Paraskevi, Attikis 15342, Greece.3 Institute for Advanced Studies

16 Spitamen Street, Tashkent 100121, Uzbekistan

Abstract

In 2002, J. M. Rassias “Uniqueness of quasi-regular solutions for a bi-parabolic ellipticbi-hyperbolic Tricomi problem”, Complex Variables, 47(8) (2002), 707-718) imposed andinvestigated the bi-parabolic elliptic bi-hyperbolic mixed type partial differential equation ofsecond order. In this paper fundamental solutions for the generalized and degenerated ellipticGellerstedt equation are constructed in the first quadrant via Appell hypergeometric functionsof two variables. Besides employing the formula expansion of Appell hypergeometricfunctions, we prove that the four constructed fundamental solutions possess a certainlogarithmic singularity. Classical references in this field of mixed type partial differentialequations are given by: J. M. Rassias (Lecture Notes on Mixed Type Partial DifferentialEquations, World Scientific, 1990, pp. 1-144) and M. M. Smirnov (Equations of Mixed Type,Translations of Mathematical Monographies, 51, American Mathematical Society,Providence, R. I., 1978, pp. 1-232).

2000 Mathematics Subject Classification. Primary 35А08, 35J70; secondary 35M10.

a E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected].

Anvar Hasanov, John Michael Rassias and Mamasali Turaev74

Key words and phrases. Degenerated elliptic Gellerstedt equation; fundamental solutions;logarithmic singularity; hypergeometric Appell functions; expansion of hypergeometricfunction of two variables.

1. Introduction

The theory of the degenerated equations of elliptic type is one of the most importanttopics of the modern theory of partial differential equations. Degenerated elliptic equationsappear in the solution of many important applied problems. Determination of fundamentalsolutions of partial differential equations is one of the core targets. Both the theory ofpotentials and solution of boundary value problems are established by means of fundamentalsolutions. Well-known fundamental solutions exist today only for several degenerated ellipticpartial differential equations.

Example. Let us consider the classical degenerated elliptic Gellerstedt equation [1, 5]

0, 0,mxx yyy u u m const+ = = > (1.1)

\in the field of ( ) 1 , : , 0x y x y+ = −∞ < < ∞ > . Regarding this equation, we now know

pertinent fundamental solutions expressed via hypergeometric Gauss functions ([4], p. 41,(2.7)-(2.8), see also [5]):

( ) ( ) ( )21 0 0 1 1, ; , , ;2 ;1 ,q x y x y k r F

ββ β β σ

−= − (1.2)

( ) ( ) ( ) ( )1 222 0 0 2 1, ; , 1 1 ,1 ;2 2 ;1 ,q x y x y k r F

β βσ β β β σ− −= − − − − − (1.3)

where

( )( )

22 2 22 2 2

0 0221

4 2 2 ,2 22

m mrx x y y

m mr m

+ +⎫ ⎛ ⎞= − +⎬ ⎜ ⎟+ ++ ⎝ ⎠⎭

∓2

21

,rr

σ = (1.4)

( ),

2 2m

mβ =

+

( )( )

2 2

11 4 ,

4 2 2k

m

β βπ β

Γ⎛ ⎞= ⎜ ⎟+ Γ⎝ ⎠

( )( )

2 2 2

2

11 44 2 2 2

km

β βπ β

− Γ −⎛ ⎞= ⎜ ⎟+ Γ −⎝ ⎠, (1.5)

and hypergeometric Gauss function ( ); ; ;F a b c x [6], [7], as well as

( ) ( ) ( )( )0

, ; ; ,!

ii i

i i

a bF a b с x x

с i

∞

=

=∑ (1.6)

( ) ( )( ) ( ) ( ) ( )

111

0

, ; ; 1 1 ,c b abcF a b c x t t xt dt

b c b− − −−Γ

= − −Γ Γ − ∫ Re Re 0c b> > , (1.7)

Fundamental Solutions for the Generalized Elliptic Gellerstedt Equation 75

( ) ( ) ( )/μ

λ λ μ λ= Γ + Γ is a symbol of Pochhammer (or the shifted factorial).

In 2002, J. M. Rassias in [8]-[9] imposed the bi-parabolic elliptic bi-hyperbolic mixedtype partial differential equation of second order

( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 , , .x yx yK y M x u M x K y u r x y u f x y⎡ ⎤+ + =⎡ ⎤⎣ ⎦ ⎣ ⎦ (1.8)

which is parabolic on both lines 0, 0x y= = , elliptic in the first quadrant 0, 0x y> > and

hyperbolic in both quadrants 0, 0x y< > ; 0, 0x y> < and established the proof of quasi-regular solutions of the Tricomi problem (or Problem T) associated to this equation (1.8).

If we suppose equation (1.8) in the region ( ) 2 , : 0, 0R x y x y+ = > > , such that

( ) 11

nM x x= , ( ) 22

nM x x= , ( ) 11

mK y y= , ( ) 22

mK y y= , ( ) ( ), , 0,r x y f x y= =

then the following generalized elliptic Gellerstedt equation

( ) 1 2 1 2 0,m n n mx yx y

L u y x u x y u⎡ ⎤ ⎡ ⎤≡ + =⎣ ⎦ ⎣ ⎦ (1.9)

holds with conditions

1 2 0,n n+ > 20 1,n≤ < 1 2 0,m m+ > 20 1,m≤ < 1 2 1 2, , , .n n m m ∈ (1.10)

In this paper we explicitly construct four fundamental solutions for the afore-mentionedgeneralized elliptic Gellerstedt equation (1.9) in the field of ( ) 2 , : 0, 0x y x y+ = > > ,

associated to hypergeometric Appell functions in two variables. Furthermore and by means ofexpansions for hypergeometric function, we prove that constructed fundamental solutionspossess a logarithmic singularity as 0→r . Classical references in this field of mixed typepartial differential equations are by: J. M. Rassias (Lecture Notes on Mixed Type PartialDifferential Equations, World Scientific, 1990, pp. 1-144) and M. M. Smirnov (Equations ofMixed Type, Translations of Mathematical Monographies, 51, American MathematicalSociety, Providence, R. I., 1978, pp. 1-232).

2. Green's Formulas

Let us consider equation (1.9) in a bounded domain ( ) 2 , : 0, 0D x y x y+⊂ = > >

with smooth DΓ = ∂ its boundary. We can obviously prove the following differentialidentity


( ) ( ) ( ) ( )2 1 1 2 .n m n mx x y yuL v vL u x y uv vu x y uv vu

x y∂ ∂ ⎡ ⎤⎡ ⎤− = − + −⎣ ⎦ ⎣ ⎦∂ ∂

(2.1)

Integrating this identity, and applying Green's formula, we obtain the integral identity

( ) ( ) ( ) ( )1 2 2 1( ) ,n m n my y x x

D

uL v vL u dxdy x y uv vu dx x y uv vu dyΓ

− = − − + −⎡ ⎤⎣ ⎦∫∫ ∫ (2.2)

where DΓ = ∂ is the contour of domain D .We note that formula (2.2) holds under the following three assumptions:

i). Functions ( ),u x y , ( ),v x y and their partial derivatives of the first order are

continuous in the closure D D D D= ∪∂ = ∪Γ ;ii). Partial derivatives of the second order are continuous in the interior of D ; andiii). Integrals involving ( )L u and ( )L v exist.

Let ( ),u x y and ( ),v x y be solutions of the equation (1.9). From formula (2.2), we have

[ ] [ ]( ) 0,s suA v vA u dsΓ

− =∫ (2.3)

where s the arc length along the smooth boundary DΓ = ∂ and

[ ] 2 1 1 2n m n ms

dy dxA x y x yds x ds y

∂ ∂= −

∂ ∂. (2.4)

Besides from the following differential identity

( ) ( ) ( )2 1 1 2 2 1 1 22 2 0,n m n m n m n mx y x yx y

uL u x y uu x y uu x y u x y u≡ + − − = (2.5)

we establish the Green’s integral identity

[ ]2 1 1 22 2 ,n m n mx y s

D

x y u x y u dxdy uA u dsΓ

⎡ ⎤+ =⎣ ⎦∫∫ ∫ (2.6)

where ( ),u x y a solution of equation (1.9). From formula (2.3) and letting 1v = , we obtain

[ ] 0sA u dsΓ

=∫ . (2.7)


This relation (2.7) is a compatibility condition for the solution of the exterior Neumannproblem for (1.9).

3. Fundamental Solutions

A solution of equation (1.9) is established in ( ) 2 , : 0, 0x y x y+ = > > , in the form of

( ) ( )0 0, ; , , ,u P x y x y ω ξ η= (3.1)

where2 22

21 0 02

2

1 1 1 1 ,q q p p

rr x x y y

q q p pr

⎫ − −⎛ ⎞ ⎛ ⎞⎪ ⎜ ⎟ ⎜ ⎟= + + −⎬ ⎜ ⎟ ⎜ ⎟⎪ ⎜ ⎟ ⎜ ⎟− +⎝ ⎠ ⎝ ⎠⎭

( ) ( )20 0, ; ,P x y x y r

α β− −= , (3.2)

2 21

2 ,r rr

ξ −=

2 22

2 ,r rr

η −= 1 2 1 22 2, ,

2 2n n m mq p− + − +

= = (3.3)

( ) ( )1 2 1 2

1 2 1 2

, ,0 2 ,2 12 2 2 2

n n m mn n m m

α β α β+ += = < <

− + − +. (3.4)

In fact, substituting (3.1) into (1.9), we get

1 2 3 1 2 0,A A A B B Cξξ ξη ηη ξ ηω ω ω ω ω ω+ + + + + = (3.5)

where

1 2 1 22 21 ,m n n m

x yA P y x x yξ ξ⎡ ⎤= +⎣ ⎦1 2 1 2

2 2 ,m n n mx x y yA P y x x yξ η ξ η⎡ ⎤= +⎣ ⎦

1 2 1 22 23 ,m n n m

x yA P y x x yη η⎡ ⎤= +⎣ ⎦

( ) ( )( ),

21

21

2

1

2121

21212121

ymn

xnm

yymn

xxnm

yymn

xxnm

yxmxynP

xxxyPPxxPxyB

ξξ

ξξξξ−− ++

+++=

( ) ( )( ),`

21

21

2

2

2121

21212121

ymn

xnm

yymn

xxnm

yymn

xxnm

yxmxynP

xxxyPPxxPxyB

ηη

ηηηη−− ++

+++=

1 2 1 2 1 2 1 21 12 2 .m n n m m n n m

xx yy x yC y x P x y P n y x P m x y P− −= + + +


Therefore we determine

( )1 1

1 024 1 ,n m

q qx yA P x xr

ξ ξ−= − − (3.6)

1 1 1 1

2 0 02 24 4 ,n m n m

p p q qx y x yA P y y P x xr r

ξη ξη− −= + (3.7)

( )1 1

3 024 1 ,n m

p px yA P y yr

η η−= − − (3.8)

( )1 1 1 1

1 0 02 24 2 1 2 4 ,n m n m

q q p px y x yB P x x P y yr r

α α β ξ βξ− −= − − + + +⎡ ⎤⎣ ⎦ (3.9)

( )1 1 1 1

2 0 02 24 4 2 1 2 ,n m n m

q q p px y x yB P x x P y yr r

αη β α β η− −= − − + +⎡ ⎤⎣ ⎦ (3.10)

( ) ( )1 1 1 1

0 02 24 4 .n m n m

q q p px y x yC P x x P y yr r

α β α α β β− −= + + + (3.11)

Employing substitutions (3.6)-(3.11) in the equation (3.5), we determine the followingdifferential system of hypergeometric Appell functions ([6], p.44, ( 2F )), such that:

( ) ( ) ( )( ) ( ) ( )1 2 1 2 0,

1 2 1 2 0.ξξ ξη ξ η

ηη ξη η ξ

ξ ξ ω ξηω α α β ξ ω αηω α β αω

η η ω ξηω β α β η ω βξω α β βω

⎧ − − + − + + − − + =⎡ ⎤⎪ ⎣ ⎦⎨

− − + − + + − − + =⎡ ⎤⎪ ⎣ ⎦⎩ (3.12)

This system pertinent to the above equation (3.5) has the solutions ([6], p. 50, (11)):

( ) ( )1 2, ; , ;2 , 2 ; , ,Fω ξ η α β α β α β ξ η= + (3.13)

( ) ( )1 22 2, 1 ;1 , ;2 2 , 2 ; , ,Fαω ξ η ξ α β α β α β ξ η−= − + − − (3.14)

( ) ( )1 23 2, 1 ; ,1 ;2 ,2 2 ; , ,Fβω ξ η η α β α β α β ξ η−= + − − − (3.15)

( ) ( )1 2 1 24 2, 2 ;1 ,1 ;2 2 , 2 2 ; , ,Fα βω ξ η ξ η α β α β α β ξ η− −= − − − − − − (3.16)

where ( )2 ; , '; , '; ,F a b b c c x y hypergeometric Appell function ([6], p. 14 (12)):

( )( ) ( ) ( )( ) ( )

1 22 1 2 1 2

, 0 1 2

; , ; , ; , ,! !

i j i j i j

i j i j

a b bF a b b с с x y x y

с с i j

∞+

=

= ∑ (3.17)


having the following integral representation ([6], p. 28 (2)):

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )1 1 2 21 2

1 22 1 2 1 2

1 2 1 1 2 2

1 11 11 1

1 2 1 2 1 2 1 20 0

; , ; , ; ,

1 1 1 ,c b c b ab b

c cF a b b c c x y

b b c b c b

t t t t xt yt dt dt− − − − −− −

Γ Γ=Γ Γ Γ − Γ −

⋅ − − − −∫ ∫(3.18)

1 1 2 2Re Re 0, Re Re 0.c b c b> > > >

Therefore by substituting solutions (3.13) - (3.16) into the solution (3.1), we find foursolutions of the generalized elliptic Gellerstedt equation (1.9):

( ) ( ) ( )21 0 0 1 2, ; , ; , ;2 , 2 ; , ,q x y x y l r F

α βα β α β α β ξ η

− −= + (3.19)

( ) ( ) ( )( ) ( )

2 2

11 2 1 2

1 4 4 ;4 2 2 2 2

ln n m m

α βα β α β

π α βΓ Γ Γ +⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟− + − + Γ Γ⎝ ⎠ ⎝ ⎠(3.20)

( ) ( ) ( )2 21 1 12

2 0 0 2 0 2, ; , 1 ;1 , ;2 2 ,2 ; , ,n nq x y x y l r x x Fα β

α β α β α β ξ η− − − −= − + − − (3.21)

( ) ( ) ( )( ) ( )

2 2 2

21 2 1 2

1 11 4 4 ;4 2 2 2 2 2

ln n m m

α βα β α β

π α β

−Γ − Γ Γ − +⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟− + − + Γ − Γ⎝ ⎠ ⎝ ⎠ (3.22)

( ) ( ) ( )2 21 1 12

3 0 0 3 0 2, ; , 1 ; ,1 ;2 ,2 2 ; , ,m mq x y x y l r y y Fα β

α β α β α β ξ η− + − − −= + − − − (3.23)

( ) ( ) ( )( ) ( )

2 2 2

31 2 1 2

1 11 4 4 ;4 2 2 2 2 2

ln n m m

α βα β α β

π α β

−Γ Γ − Γ + −⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟− + − + Γ Γ −⎝ ⎠ ⎝ ⎠(3.24)

( )

( ) ( )2 2 2 2

4 0 0

2 1 1 1 124 0 0 2

, ; ,

2 ;1 ,1 ;2 2 ,2 2 ; , ,n m n m

q x y x y

l r x y x y Fα β

α β α β α β ξ η+ − − − − −= − − − − − −

(3.25)

( ) ( ) ( )( ) ( )

2 2 2 2

41 2 1 2

1 1 21 4 4 .4 2 2 2 2 2 2

ln n m m

α βα β α β

π α β

− −Γ − Γ − Γ − −⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟− + − + Γ − Γ −⎝ ⎠ ⎝ ⎠ (3.26)


It is easy to note that the constructed functions possesses the following eight properties:

( )21 0 0

0

, ; , 0,n

x

x q x y x yx =

∂=

∂( )2

1 0 00

, ; , 0,m

y

y q x y x yy =

∂=

∂(3.27)

( )2 0 0 0, ; , 0,

xq x y x y

== ( )2

2 0 00

, ; , 0,m

y

y q x y x yy =

∂=

∂(3.28)

( )23 0 0

0

, ; , 0,n

x

x q x y x yx =

∂=

∂ ( )3 0 0 0

, ; , 0,y

q x y x y== (3.29)

( )4 0 0 0, ; , 0,

xq x y x y

== ( )4 0 0 0

, ; , 0.y

q x y x y== (3.30)

Applying the formula of differentiation ([6], p. 19, (20)), from (3.19) we get

( )( ) ( ) ( )

( ) ( ) ( )1 22 1 2 1 2 2 1 2 1 2

1 2

; , ; , ; , ; , ; , ; , ,i j

i j i ji j

i j

a b bF a b b с с x y F a i j b i b j с i с j x y

x y c c

++∂

= + + + + + +∂ ∂

and

( )

( )( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2

1 2

1 2

1 2

1 0 0

12 21 0 2

12 21 0 2

12 21 0 2

, ; ,

1 12 ; , ;2 ,2 ; ,

12 1 ;1 , ;1 2 ,2 ; ,

1 12 1 ;1 , ;1 2 ,2 ; ,2

2

n

n nq q

n nq

n nq q

x q x y x yx

l r x x x Fq q

l r x x Fq

l r x x x Fq q

α β

α β

α β

α β α β α β α β ξ η


αα β ξ α β α β α β ξ ηα

β

+− − −

+− − −

+− − −

∂=

∂⎛ ⎞

− + − +⎜ ⎟⎝ ⎠

− + + + + +

⎛ ⎞⎡− + − + + + +⎜ ⎟⎢⎣⎝ ⎠

+ ( )2 1 ; ,1 ;2 ,1 2 ; , .Fη α β α β α β ξ ηβ

⎤+ + + + ⎥

⎦

(3.31)

By virtue of an adjacent relation for hypergeometric Appell functions ([6], p. 21), one finds


( ) ( )

( ) ( )

1 22 1 2 1 2 2 1 2 1 2

1 2

2 1 2 1 2 2 1 2 1 2

1 ;1 , ;1 , ; , 1 ; ,1 ; ,1 ; ,

1 ; , ; , ; , ; , ; , ; , ,

b bxF a b b c c x y yF a b b c c x yc c

F a b b c c x y F a b b c c x y

+ + + + + + +

= + −

and from (3.31), we establish

( )

( ) ( ) ( )

( ) ( ) ( )

2

1 2

1 2

1 0 0

12 21 0 2

12 21 0 2

, ; ,

12 1 ;1 , ;1 2 ,2 ; ,

1 12 1 ; , ;2 , 2 ; ,

n

n nq

n nq q

x q x y x yx

l r x x Fq

l r x x x Fq q

α β

α β



+− − −

+− − −

∂=

∂

− + + + + +

⎛ ⎞− + − + +⎜ ⎟

⎝ ⎠

(3.32)

Considering conditions (1.10) from (3.32), one finds (3.27). Properties (3.28)-(3.30) aresimilarly proved. We shall note that properties (3.27)-(3.30) will be used for the solution ofboundary value problems associated to the afore-mentioned generalized elliptic Gellerstedtequation (1.9).

4. Logarithmic Singularities of Fundamental Solutions

We claim that the constructed solutions of the equation (1.9), have logarithmic singularities as0→r . In fact, we first determine a fundamental solution ( )1 0 0, ; ,q x y x y :

By virtue of a well-known expansion ([10], p. 253 (26)), we find

( )( ) ( ) ( )( ) ( ) ( ) ( )

2 1 2 1 2

1 21 1 2 2

0 1 2

; , ; , ; ,

, ; ; , ; ; ,!

i ii i i

i i i

F a b b c c x y

a b bx y F a i b i c i x F a i b i c i y

c c i

∞

=

= + + + + + +∑ (4.1)

and thus for function ( )1 0 0, ; ,q x y x y , we have

( ) ( ) ( ) ( ) ( )( ) ( )

2 2 2 22 1 2

1 0 0 1 2 20

2 2 2 21 2

2 2

, ; ,2 2 !

, ;2 ; , ;2 ; ,

i i

i i i

i i i

r r r rq x y x y l ri r r

r r r rF i i i F i i ir r

α β α β α βα β

α β α α α β β β

∞− −

=

+ ⎛ ⎞ ⎛ ⎞− −= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− −× + + + + + + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

∑ (4.2)

Using the formula ( ) ( ) ( )( ), ; ; 1 , ; ; / 1bF a b c x x F c a b c x x−= − − − and (4.2), we get


( ) ( ) ( ) ( ) ( ) ( )( ) ( )

2 2 2 22 2 1 2

1 0 0 1 1 2 2 20 1 2

2 2 2 21 1

2 21 1

, ; ,2 2 !

, ;2 ; , ;2 ; .

i i

i i i

i i i

r r r rq x y x y l r ri r r

r r r rF i i F i ir r

α β α β α βα β

α β α α β α β β

∞− −

=

+ ⎛ ⎞ ⎛ ⎞− −= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− −× − + + − + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

∑ (4.3)

Therefore [7], we establish as 0r → , the following essential relations:

( ) ( ) ( )( ) ( )

, ; ;1 ,c c a b

F a b cc a c b

Γ Γ − −=Γ − Γ −

( )0, 1, 2,..., Re 0,c c a b≠ − − − − > (4.4)

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )( ) ( )

( )( )

2

21

2

22

22, ;2 ;1 , ;2 ;1 ,

22, ;2 ;1 , ;2 ; .

i

i

i

i

rF i i F i ir

rF i i F i ir

αα βα β α α α β α α

α β α α β

ββ αβ α β β β α β β

α β β α β

Γ Γ⎛ ⎞− + + − → − + + =⎜ ⎟ Γ + Γ +⎝ ⎠

Γ Γ⎛ ⎞− + + − → − + + =⎜ ⎟ Γ + Γ +⎝ ⎠

(4.5)

Hence, as 0r → , from expansion (4.3) we obtain

( ) ( ) ( ) ( ) ( )( )

4 2 22 2

1 0 0 1 1 2 2 2 2 2 21 2 1 2

2 2, ; , , ; ; 1 .r r rq x y x y l r r F

r r r rα β α β

α β α βα β

− − Γ Γ ⎛ ⎞→ + − − +⎜ ⎟Γ + ⎝ ⎠

(4.6)

Similarly, we get

( ) ( )( ) ( ) ( ) ( )

( )( ) ( )

( ) ( )( )

( ) ( ) ( ) ( )22 20

, ; ; , ;1;1 ln 1

2 1 1 ,!

j

j

a bF a b a b z F a b z z

a b

a b a j b jj a j b j z

a b jψ ψ ψ

∞

=

Γ ++ = − − −

Γ Γ

Γ + Γ + Γ ++ + − + − + −⎡ ⎤⎣ ⎦Γ Γ ∑

(4.7)

( )arg 1 , , 0, 1, 2,...z a bπ π− < − < ≠ − −

via the logarithmic derivative ( )zψ of ( )zΓ [7]:

( )0

1 1ln ln 1 , 0;n

z z zn z n z

ψ∞

=

⎡ ⎤⎛ ⎞= − − + >⎜ ⎟⎢ ⎥+ +⎝ ⎠⎣ ⎦∑

( )1 1

0 0

1ln , Re 01

zt tz e t dt dt z

tψ

∞ −− −

= + >−∫ ∫


Thus from relation (4.6) as 0r → , we establish

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

( )( )

2 21 0 0 1 1 2

2 22 2

2 2 2 2 2 2 2 21 2 1 2 1 2 1 2

22

2 2 2 2 20 1 2 1 2

2 2, ; ,

1 1 1 1, ;1; ln ln

1 1 1 1 ,!

j

jj

q x y x y l r r

r rF r rr r r r r r r r

j j rh r Or r r rj

α β α βα β α β

α β

α βα β

− −

∞

=

Γ Γ=

Γ + Γ Γ

⎧ ⎡ ⎤ ⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪ ⎪× − + − + + −⎨ ⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎪ ⎣ ⎦ ⎩ ⎭⎩

⎫⎡ ⎤Γ + Γ + ⎛ ⎞ ⎪+ + − +⎬⎢ ⎥⎜ ⎟Γ Γ ⎝ ⎠⎣ ⎦ ⎪⎭∑

(4.8)

( ) ( ) ( )2 1 .jh j j jψ ψ α ψ β= + − + − +⎡ ⎤⎣ ⎦

Formula (4.8) proves that the solution ( )1 0 0, ; ,q x y x y has a logarithmic singularity as

0→r . Hence ( )1 0 0, ; ,q x y x y is the first fundamental solution of the above-mentioned

equation (1.9). Similarly, we determine the other three fundamentalsolutions ( )2 0 0, ; ,q x y x y , ( )3 0 0, ; ,q x y x y , ( )4 0 0, ; ,q x y x y of the generalized and

degenerated Elliptic Gellerstedt equation (1.9).

References

[1] F. G. Tricomi, Sulle Equazioni Lineari alle derivate Parziali di 2º Ordine di Tipo Misto,Atti Accad. Naz. dei Lincei. 14 (5) 1923, 133-247.

[2] S. Gellerstedt, Sur un Probleme aux Limites pour une Equation Lineaire aux DeriveesPartielles du Second Ordre de Type Mixte. Doctoral Thesis, Uppsala, 1935, JbuchFortschritte Math. 61, 1259.

[3] F. I. Frankl, Selected Works in Gas Dynamics. Nauka, Moscow, 1973, p. 712 (inRussian).

[4] M. M. Smirnov, Degenerating elliptic and hyperbolic equations, Izdat. Nauka, Moscow,1966, pp. 1-292 (in Russian).

[5] M. M. Smirnov, Equations of Mixed Type, Translations of MathematicalMonographies, 51, American Mathematical Society, Providence, R. I., 1978, 1-232.

[6] P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques;Polynomes d'Hermite, Gauthier - Villars. Paris, 1926.

[7] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher TranscendentalFunctions, vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.

[8] J. M. Rassias, Lecture Notes on Mixed Type Partial Differential Equations, WorldScientific, 1990, 1-144.

[9] J. M. Rassias, Uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem, Complex Variables, 47 (8) 2002, 707-718.

[10] J. L. Burchnall and T. W. Chaundy, Expansions of Appell’s double HypergeometricFunctions. Quart. J. Math., Oxford, 1940, Ser. 12, 112-128.



Chapter 7

POINTWISE SUPERSTABILITY

AND SUPERSTABILITY OF THE JORDAN EQUATION

Ji-Rong Lva, Huai-Xin Caoa and J.M. Rassiasb,∗

aCollege of Mathematics and Information Science, Shaanxi Normal UniversityXi’an 710062, P. R. China

bPedagogical Department, Section of Mathematics and InformaticsNational and Capodistrian University of Athens

Athens 15342, Greece

Abstract

In this paper,ε-approximate Jordan mappings and strongε-approximate Jordanmappings are introduced,(A ,B)-pointwise superstability and(A ,B)-superstabilityof the Jordan equation are defined. It is proved that ifA andB are normed algebrassuch that the norm ofB is multiplicative, then the Jordan equation is both(A ,B)-pointwise superstable and(A ,B)-superstable.

2000 Mathematics Subject Classifications:39B82.

Key words: Jordan mapping, Jordan equation,ε-approximate Jordan mapping, pointwisesuperstability, superstability.

1. Introduction

In 1940 S. M. Ulam [32] gave a talk before the Mathematics Club of the University ofWisconsin in which he discussed a number of unsolved problems. Among these was thefollowing celebrated Ulam question concerning the stability of homomorphisms.

We are given a groupG and a metric groupG′ with metricρ. Given anε > 0, doesthere exist aδ > 0 such that iff : G → G′ satisfiesρ(f(xy), f(x)f(y)) < δ for all x, y inG, then a homomorphismg : G → G′ exists withρ(f(x), g(x)) < ε for all x in G?

By now an affirmative answer has been given in several cases, and some interestingvariations of the problem have also been investigated. We shall call such anf : G → G′ anapproximate homomorphism.

∗E-mail addresses: [email protected], [email protected] [email protected]

86 Ji-Rong Lv, Huai-Xin Cao and J. M. Rassias

In 1941 D. H. Hyers [10] considered the case of approximately additive mappingsf :E → E′ whereE andE′ areBanach spaces andf satisfies Hyers inequality

∥∥f(x + y) − f(x) − f(y)∥∥ ≤ ε (1.1)

for all x, y in E. It was shown that the limit

L(x) = limn→∞

2−nf(2nx) (1.2)

exists for allx ∈ E and thatL is the unique additive mapping satisfying‖f(x)−L(x)‖ ≤ ε

for all x in E, whereε is a positive constant. No continuity conditions are required forthis pioneering Hyers–Ulam stability result. However, if one applies the Hyers continuitycondition thatt 7→ f(tx) is continuous in the real variablet for each fixedx ∈ E, thenhe obtains thatL is real linear, and iff is continuous at a single point ofE, thenL is alsocontinuous.

Besides D. H. Hyers [10] studied the problem of knowing if in the case that a mapf

is “near” to hold the Cauchy additive functional equation, then there exists another mapL

acting in the same space “near” tof and satisfying this equation. Here “near” means thatf

andL are close in the sense of a metric structure inside the considered functional space.In 1982–1994, a generalization of Hyers stability result was proved by J. M. Rassias

([21]–[24], [26]). This author assumed the following generalized condition (or weakerinequality or Cauchy–Gavruta–Rassias inequality)

∥∥f(x + y) − f(x) − f(y)∥∥ ≤ θ‖x‖p‖y‖q (1.3)

for all x, y in E, controlled by (or involving) a product of different powers of norms, whereθ ≥ 0 and realp, q are real numbers such thatr = p + q 6= 1, and retained the condition ofcontinuity off(tx) in t for a any fixedx in E. Besides J. M. Rassias investigated that it ispossible to replaceε in the above Hyers inequality by a non-negative real-valued functionsuch that the pertinent series converges and other conditions hold and still obtain stabilityresults. In all the cases investigated in these results, the approach to the existence questionwas to prove asymptotic type formulas of the form:

L(x) = limn→∞

2−nf(2nx), or L(x) = limn→∞

2nf(2−nx). (1.4)

J. M. Rassias stability Theorem ([21]–[23], [26]) states: LetX be a real normed linearspace and letY be a real complete normed linear space. Assume in addition thatf : X → Y

is an approximately additive mapping for which there exist constantsθ ≥ 0 andp, q ∈ R

such thatr = p+ q 6= 1 and satisfies the inequality (1.3) for allx, y in X. Then there existsa unique additive mappingL : X → Y satisfying

‖f(x) − L(x)‖ ≤θ

|2r − 2|‖x‖r (1.5)

for all x ∈ X. If, in addition, the transformationt 7→ f(tx) is continuous int ∈ R fora any fixedx in X, then theL is aR-linear mapping. In 1999, P. Gavruta [7] gave a nicecounterexample to the Ulam–Gavruta–Rassias stability of this theorem in the singular case:r = 1.

Pointwise Superstability and Superstability of the Jordan Equation 87

The stability problems of several functional equations have been extensively investi-gated by a number of authors and there are many interesting research results concerningthe above Ulam stability problem ([1], [2], [4]–[9], [12], [21]–[24], [26], [29]). A largelist of references can be found therein. Interesting pertinent counterexamples can be foundin ([5], [7], [15]). Euler–Lagrange (E–L) mappings and Euler–Lagrange–Rassias (E–L–R) mappings were introduced and named by J. M. Rassias [25] (in 1992) and via ([27],[28]). Already several specialists on the Ulam stability problem have employed these newfunctional mappings ([11], [15], [17]–[20], [30]). These E–L–R quadratic functional equa-tions (and mappings) are important in analysis, because they marry functional equations(and mappings) with the probability theory and mathematical statistics (stochastic anal-ysis) through the weighted quadratic means and quadratic mean equations. Besides theUlam–Gavruta–Rassias (UGR) stability was introduced by J. M. Rassias ([21]–[23], [26])(in 1982–1989, 1994) for linear mappings via the above J. M. Rassias stability theorem,where the initial fixed bound proposed by D. H. Hyers [10] was changed in the sequel byanother condition involving a product of different powers of norms. Already several authorshave used this new stability ([18], [30]). Also the Ulam–Aoki–Rassias (UAR) stability orequivalently Hyers–Ulam–Rassias (HUR) or Hyers–Ulam–Aoki–Rassias (HUAR) stabilitywas introduced by T. Aoki [1] (in 1950) for additive mappings and by Th. M. Rassias [29](in 1978) for linear mappings, where the initial Hyers fixed bound [10] was changed by acondition involving a sum of powers of norms.

Young Whan Lee, Gwang Hui Kim in [33] discussed the superstability of Jordan func-tional on a normed algebra. In this paper, we will discuss the pointwise superstability andsuperstability of the Jordan equation. We first recall the definitions ofε-homomorphism,Jordan mapping, Jordan equation,ε-approximate Jordan mapping and strongε-approximateJordan mapping.

Definition 1.1. Let A , B be normed algebras, andf : A → B be a linear mapping, iffsatisfies the inequality ∥∥f(ab) − f(a)f(b)

∥∥ ≤ ε‖a‖ ‖b‖

for all a, b ∈ A , thenf is called anε-homomorphism.

Definition 1.2. Let A , B be Banach algebras,φ : A → B be a linear mapping. Then theequation

φ(a2) − (φ(a))2 = 0 (∀ a ∈ A ) (1.1)

is said to be theJordan equation.Moreover, a solution of the Jordan equation (1.1) is called a Jordan mapping. Especially,

a Jordan mapping fromA into C is called a Jordan functional onA .

Definition 1.3. Let A , B be normed algebras,f : A → B be a linear mapping satisfyingthe inequality ∥∥f(a2) − (f(a))2

∥∥ ≤ ε‖a‖2

for all a ∈ A . Thenf is called anε-approximate Jordan mapping. Especially, ifB is thecomplex fieldC, thenf is called anε-approximate Jordan functional onA .

Definition 1.4. Let A , B be normed algebras,f : A → B be a linear mapping satisfying


the inequality ∥∥f(a2) − (f(a))2∥∥ ≤ ε

for all a ∈ A . Thenf is called astrong ε-approximate Jordan mapping. Especially, ifBis the complex fieldC, thenf is called astrong ε-approximate Jordan functional onA .

2. The Pointwise Superstability and Superstabilityof the Jordan Equation

In this section, we will study the pointwise superstability and superstability of the Jordanequation.

Let f : A → B be anε-approximate Jordan mapping, define

Df =a ∈ A : f(a2) = (f(a))2

,

Eεf =

a ∈ A : ‖f(a)‖ ≤ C(ε)‖a‖

,

where

C(ε) =1 +

√1 + 4ε

2.

Definition 2.1. If everyε-approximate Jordan mappingf : A → B satisfies

A = Df ∪ Eεf ,

then the Jordan equation (1.1) is called(A , B)-pointwise superstable.

Definition 2.2. If a mappingf : A → B satisfies

inf‖a‖=1

‖f(a)‖ > C(ε),

then it is said to beε-lower bounded.

Definition 2.3. If every ε-lower boundedε-approximate Jordan mappingf : A → B is aJordan mapping, then (1.1) is called(A , B)-superstable.

Theorem 2.1. Let A be an algebra, B be a commutative normed algebra with the mul-tiplicative norm, and f : A → B be a strong ε-approximate Jordan mapping. Then foreach a ∈ A \ ker f, f(a2) = (f(a))2. In particular, if f is also an injection, then f is aJordan mapping on A .

Proof. Sincef is a strongε-approximate Jordan mapping, for everyx, y ∈ A \ ker f , wehave‖f((x + y)2) − (f(x + y))2‖ ≤ ε. Thus,

ε ≥∥∥f((x + y)2) − (f(x + y))2

∥∥

=∥∥∥f(x2 + xy + yx + y2) − (f(x))2 − (f(y))2 − 2f(x)f(y)

∥∥∥

=∥∥∥f(x2) − (f(x))2 + f(y2) − (f(y))2 + f(xy + yx) − 2f(x)f(y)

∥∥∥

≥∥∥f(xy + yx) − 2f(x)f(y)

∥∥ −∥∥f(x2) − (f(x))2

∥∥ −∥∥f(y2) − (f(y))2

∥∥≥

∥∥f(xy + yx) − 2f(x)f(y)∥∥ − 2ε.


Therefore,‖f(xy + yx) − 2f(x)f(y)‖ ≤ 3ε. This proves that

∥∥f(xy) − f(x)f(y)∥∥ ≤

3ε

2, ∀x ∈ y′. (2.1)

Clearly,C(ε)2−C(ε) = ε andC(ε) > 1. Leta ∈ A \ ker f , for f(a) 6= 0, we may assumethat‖f(a)‖ > C(ε) because‖f(ta)‖ = ‖tf(a)‖ > C(ε) for somet ∈ R andf((ta)2) =(f(ta))2 impliesf(a2) = (f(a))2. So there exists ap > 0 such that‖f(a)‖ = C(ε) + p.Then

‖f(a2)‖ =∥∥f(a2) − (f(a))2 + (f(a))2

∥∥≥ ‖(f(a))2‖ −

∥∥f(a2) − (f(a))2∥∥

= ‖f(a)‖2 − ‖f(a2) − (f(a))2‖

≥ (C(ε) + p)2 − ε

= 2pC(ε) + p2 + C(ε)

> 2pC(ε) + C(ε)

> C(ε) + 2p.

This proves that whenn = 1, inequality

‖f(a2n

)‖ > C(ε) + 2np (2.2)

holds. Suppose that whenn = k, (2.2) holds, that is‖f(a2k

)‖ > C(ε) + 2kp, then whenn = k + 1, we have

‖f(a2k+1

)‖ = ‖f((a2k

)2)‖

=∥∥f((a2k

)2) − (f(a2k

))2 + (f(a2k

))2∥∥

≥ ‖(f(a2k

))2‖ −∥∥f((a2k

)2) − (f(a2k

))2∥∥

≥ (C(ε) + 2kp)2 − ε

= C(ε) + 4kpC(ε) + 4k2p2

> C(ε) + 4kpC(ε)

> C(ε) + 4kp

≥ C(ε) + (2k + 2)p

= C(ε) + 2(k + 1)p.

So by induction, (2.2) holds for alln = 1, 2, 3, . . .. For all mutually commutative elementsx, y, z ∈ A \ ker f , we see from (2.1) that

∥∥f(xyz) − f(xy)f(z)∥∥ ≤

3ε

2,

∥∥f(xyz) − f(x)f(yz)∥∥ ≤

3ε

2.

Hence∥∥f(xy)f(z) − f(x)f(yz)

∥∥ =∥∥f(xy)f(z) − f(xyz) + f(xyz) − f(x)f(yz)

∥∥≤ ‖f(xy)f(z) − f(xyz)‖ + ‖f(xyz) − f(x)f(yz)‖

≤ 3ε.


So∥∥f(xy)f(z) − f(x)f(y)f(z)

∥∥ =∥∥f(xy)f(z) − f(x)f(yz) + f(x)f(yz) − f(x)f(y)f(z)

∥∥≤ ‖f(xy)f(z) − f(x)f(yz)‖ + ‖f(x)‖ · ‖f(yz) − f(y)f(z)‖

≤ 3ε + ‖f(x)‖ ·3ε

2.

In particular, lettingx = y = a, z = a2n

, by the above inequality and (2.2), we have

∥∥f(a2) − (f(a))2∥∥ =

‖f(a2)f(a2n

) − (f(a))2f(a2n

)‖

‖f(a2n)‖

≤3ε + ‖f(a)‖ · 3ε

2

‖f(a2n)‖

<3ε + ‖f(a)‖ · 3ε

2

C(ε) + 2np

→ 0 (n → ∞).

Thereforef(a2) = (f(a))2. This completes the proof.

Lemma 2.1. Let A be a normed algebra, B be a normed algebra with the multiplicativenorm, and f : A → B be an ε-approximate Jordan mapping. If a ∈ A with ‖f(a)‖ >

C(ε)‖a‖, then there exists a p > 0 such that

‖f(a2n

)‖ ≥ (C(ε) + 2np)‖a‖2n

(n = 1, 2, . . .). (2.3)

Proof. Clearly, a 6= 0. Puttingp = ‖f(a)‖‖a‖

− C(ε), we see thatp > 0 and‖f(a)‖ =

(C(ε) + p)‖a‖. Thus

‖f(a2)‖ =∥∥f(a2) − (f(a))2 + (f(a))2

∥∥≥ ‖(f(a))2‖ − ‖f(a2) − (f(a))2‖

= ‖f(a)‖2 − ‖f(a2) − (f(a))2‖

≥[(C(ε) + p)2 − ε

]‖a‖2

=[2pC(ε) + p2 + C(ε)

]‖a‖2

>[2pC(ε) + C(ε)

]‖a‖2

>[C(ε) + 2p

]‖a‖2.

This proves that whenn = 1, inequality

‖f(a2n

)‖ ≥ (C(ε) + 2np)‖a‖2n

holds. Suppose that whenn = k, (2.3) holds, that is‖f(a2k

)‖ ≥ [C(ε) + 2kp]‖a‖2k

, thenwhenn = k + 1, we have

‖f(a2k+1

)‖ =∥∥f((a2k

)2)∥∥

=∥∥f((a2k

)2) − (f(a2k

))2 + (f(a2k

))2∥∥


≥ ‖(f(a2k

))2‖ −∥∥f((a2k

)2) − (f(a2k

))2∥∥

≥[(C(ε) + 2kp)2 − ε

]‖a‖2k+1

=[C(ε) + 4kpC(ε) + 4k2p2

]‖a‖2k+1

>[C(ε) + 4kpC(ε)

]‖a‖2k+1

>[C(ε) + 4kp

]‖a‖2k+1

≥[C(ε) + (2k + 2)p

]‖a‖2k+1

=[C(ε) + 2(k + 1)p

]‖a‖2k+1

.

Therefore, (2.3) holds for alln = 1, 2, 3, . . ..

Theorem 2.2. Let A , B be normed algebras and B with the multiplicative norm, f :A → B be an ε-approximate Jordan mapping. If ‖f(a)‖ > C(ε)‖a‖ for some a ∈ A ,then f(ak) = (f(a))k(k = 1, 2, . . .).

Proof. Clearly,a 6= 0. Let c = a‖a‖

. Since‖f(a)‖ > C(ε)‖a‖, by Lemma 2.1, there existsa constant numberp > 0 such that

‖f(c2n

)‖ ≥ C(ε) + 2np (n = 1, 2, . . .). (2.4)

For allm, n ∈ N+, we get from the definition ofε-approximate Jordan mapping that

∥∥f(cncm) − f(cn)f(cm)∥∥

=

∥∥∥∥1

2

[f((cn+cm)2)−f((cn)2)−f((cm)2)

]−

1

2

[(f(cn+cm))2−(f(cn))2−(f(cm))2

]∥∥∥∥

=1

2

∥∥[f((cn+cm)2)−(f(cn+cm))2

]−

[f((cn)2)−(f(cn))2

]−

[f((cm)2)−(f(cm))2

]∥∥

≤1

2

(∥∥f((cn+cm)2)−(f(cn+cm))2∥∥+

∥∥f((cn)2)−(f(cn))2∥∥+

∥∥f((cm)2)−(f(cm))2∥∥)

≤ε

2

(‖cn + cm‖2 + ‖cn‖2 + ‖cm‖2

)

≤ 3ε.

Therefore ∥∥f(cncm) − f(cn)f(cm)∥∥ ≤ 3ε. (2.5)

We get from (2.4) and (2.5) that

∥∥f(ck) − (f(c))k∥∥ =

1

‖f(c2n)‖

∥∥∥∥[f(ck)f(c2n

) − f(ck · c2n

)]

−k−1∑

i=0

[(f(c))k−if(cic2n

) − (f(c))k−i−1 · f(ci+1 · c2n

)]∥∥∥∥

≤1

‖f(c2n)‖

(∥∥f(ck)f(c2n

) − f(ck · c2n

)∥∥


+k−1∑

i=0

∥∥∥(f(c))k−if(ci · c2n

) − (f(c))k−i−1 · f(ci+1 · c2n

)∥∥∥)

=1

‖f(c2n)‖

(∥∥f(ck)f(c2n

) − f(ck · c2n

)∥∥

+k−1∑

i=0

‖f(c)‖k−i−1 ·∥∥f(c) · f(ci · c2n

) − f(ci+1 · c2n

)∥∥)

≤1

‖f(c2n)‖

3ε + 3ε

k−1∑

i=0

‖f(c)‖k−i−1

≤1

C(ε) + 2np

3ε + 3ε

k−1∑

i=0

‖f(c)‖k−i−1

→ 0 (n → ∞).

Thusf(ck) = (f(c))k, that isf(ak) = (f(a))k. This completes the proof.

As an application of Theorem 2.2., we can get the following corollary.

Corollary 2.1. Let A , B be normed algebras and B have multiplicative norm. Then theJordan equation (1.1) is (A , B)-pointwise superstable.

Theorem 2.3. Let A , B be normed algebras and Jordan equation (1.1) be (A , B)-pointwise superstable. Then (1.1) is (A , B)-superstable.

Proof. Let f : A → B be anε-approximate Jordan mapping andε-lower bounded. Forall a ∈ A \ 0, we have‖f(a)‖ > C(ε)‖a‖. Since the Jordan equation (1.1) is(A , B)-pointwise superstable,f(a2) = (f(a))2. f is therefore a Jordan mapping. This shows thatthe Jordan equation (1.1) is(A , B)-superstable. This completes the proof.

Use Corollary 2.1 and Theorem 2.3, we obtain the following corollary.

Corollary 2.2. Let A , B be normed algebras, and B have multiplicative norm. Then theJordan equation (1.1) is (A , B)-superstable.

Acknowledgment

This work was supported by the NNSF of China (No: 10571113, 10871224).

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[8] Gilanyi, A., 2000, On the stability of monomial functional equations,Publ. Math.Debrecen, 56: 201–212.

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Chapter 8

A PROBLEM WITH NON-LOCAL CONDITIONS ON THELINE OF DEGENERACY AND PARALLEL

CHARACTERISTICS FOR A MIXED TYPE EQUATIONWITH SINGULAR COEFFICIENT

M. Mirsaburov1 and M. Kh. Ruziev2

1 Termez State University, Termez, Uzbekistan2 Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences,

F. Khodzhaev str., 29, Tashkent, Uzbekistan

Abstract

In the paper, we consider the problem for a mixed type equation with singular coefficientwith non-local conditions on the line of degeneracy and parallel characteristics. We prove theuniqueness of solution for the problem by an analogue of the Bitsadze extremume principle.And we use the method of integral equations to prove the existence of solution.

2000 Mathematics Subject Classification: 35M10.

Key words and phrases. Non-local condition, mixed type equation, parallel characteristics,singular coefficient, extremum principle.

Introduction

Some boundary value problems for partial differential equations were investigated in(Polosin, 1996; Rassias, 1999; Protter, 1951; Holmgren, 1926). In (Wen, Chen, Cheng, 2007),the general Tricomi-Rassias problem was investigated for generalized Chaplygin equation. Inthe paper, the representation of solution of the general Tricomi-Rassias problem is given forthe first time, as well as the uniqueness and the existence of solution for the problem is

M. Mirsaburov and M. Kh. Ruziev96

proved by a new method. In (Rassias, 2002), the general Tricomi problem was investigatedfor the new bi-parabolic elliptic bi-hyperbolic equation.

Unlike known problems for mixed type equations, in the present paper we investigatenew non-local problem with the Bitsadze-Samarskii condition on parallel characteristics andan analogue of the Frankl condition on a segment of the degeneracy line for a class of mixedtype equations.

Statement of the Problem

Consider the equation

( 2 ) 0mxx yy ysigny y u u m y u| | + − / = , (1)

where 0m const= > , in a finite one-connected domain Ω of the plane of independent

variables x, y bounded at 0y > by the normal curve 2 2 2 20 4( 2) 1mx m yσ − +: + + = , with

ends in points ( 1 0)A − , , (1 0)B , , and at 0y < – by characteristics AC and BC of theequation (1).

Denote the parts of Ω lying in the half-planes 0y > and 0y < as +Ω and −Ω ,respectively. Let 0C and 1C be points of intersection of characteristics AC and BC with

the characteristic starting from the point ( 0)E c, , respectively, here ( 1 1)c J∈ = − , is aninterval of the axis 0y = .

Let 2( ) [ 1 ]p x C c∈ − , be a diffeomorphism from the set of points of the segment[ 1 ]c− , into the set of points of the segment [ 1]c, , and also ( ) 0p x′ < , ( 1) 1p − = ,

( )p c c= . As an example of such function, one can mention a linear function( )p x kxδ= − where (1 ) (1 )k c c= − / + , 2 (1 )c cδ = / + .

Problem GF. To find the function ( )( )u x y C, ∈ Ω in Ω satisfying the following

conditions:

1. 2( ) ( )u x y C +, ∈ Ω and satisfies (1) in this domain;

2. ( )u x y, is a generalized solution of the class 1R (Smirnov, 1985) ( ( )xτ ′ , ( )x Hν ∈ ,

definition for ( )xτ and ( )xν is given below) in the domain ( )10\ ECEC ∪Ω− ;

3. On the degeneracy interval, the following conjugation condition

( ) cJxyuy

yuy m

y

m

y\,limlim 2

0

2

0∈

∂∂

=∂∂

− −

+→

−

−→(2)

A Problem with Non-Local Conditions on the Line of Degeneracy… 97

hold, moreover these limits can have at 1x = ± , x c= singularities of the order less thanunit;

4.

0( ) ( ) 1 1u x y x xσ ϕ, | = , − ≤ ≤ , (3)

[ ( )] [ ( ( ))] ( ) 1u x u p x x x cθ θ ψ∗− = , − ≤ ≤ , (4)

( 0) ( ( ) 0) ( ) 1u x u p x f x x c, − , = , − ≤ ≤ , (5)where

2 ( 2)0 0 0( ) ( 1) 2 [( 2)(1 ) 4] mx x i m xθ / += − / − + + / ,

2 ( 2)0 0 0( ( )) ( ( )) 2 [( 2) ( ( ) ) 4] mp x c p x i m p x cθ ∗ / += + / − + / − / ,

i.e. 0( )xθ ( 0( ( ))p xθ ∗ ) is the affix of the point of intersection of the characteristic 0AC( 1EC ) with the characteristic starting from the point 0( 0)x , , 0 [ 1 ]x c∈ − , ( 0( ( ) 0)p x , ,

0( ) [ 1]p x c∈ , ), given functions 1( ) ( ) [ 1 ] ( 1 )f x x C c C cαψ ,, ∈ − , ∩ − , , 0( ) [ 1 1]x C αϕ ,∈ − , ,

and also ( ) ( ) ( )xxx ϕϕ ~1 2−= , where ( ) [ ]1,1~ ,0 −∈ αϕ Cx , moreover, by virtue of thecoordination condition, we have ( 1) ( 1) (1) 0f ϕ ϕ− = − − = from (5) at 1x = − .

Conditions (4) and (5) are analogues of the Bitsadze-Samarskii (Bitsadze, Samarskii,1969) and Frankl (Morawetz, 1954; Frankl, 1956) conditions, respectively.

Uniqueness of the Solution of Problem GF

The following assertion takes place.

Theorem 1. Problem GF with homogeneous boundary conditions( ( ) ( ) ( ) 0x x f xϕ ψ≡ ≡ ≡ ) has only the trivial solution.

Proof. By the d’Alembert’s formula (Bitsadze, 1981) giving the solution of the modifiedCauchy problem in −Ω with data

2

0( ) ( 0) ( ) lim ( ) m

y

ux u x x J x y x Jy

τ ν − /

→−

∂= , , ∈ ; = − , ∈ ,

∂

we have from the boundary condition (4) the following:

( ) ( ) ( ( )) ( ) ( ( )) ( ) 2 ( ) [ 1 ]x x p x p x p x p x x x cτ ν τ ν ψ′ ′ ′ ′ ′− − + = , ∈ − , . (6)


Rewrite (5) in the form

( ) ( ( )) ( ) [ 1 ]x p x f x x cτ τ− = , ∈ − , . (7)

Transform (6) with the help of (7) to the form

( ) ( ( )) ( ) ( ) 2 ( ) [ 1 ]x p x p x f x x x cν ν ψ′ ′ ′− = − , ∈ − , . (8)

Let’s show that if ( ) 0xϕ ≡ , ( ) 0xψ ≡ , ( ) 0f x ≡ , then the solution of problem GF in

the domain Ω equals identically to zero. Suppose the opposite, let ( ) 0u x y, ≠ in the

domain +Ω , hence it has the positive maximum and negative minimum in +Ω . By theHopf’s principle (Bitsadze, 1981), the function ( )u x y, to be found does not attain its

positive maximum in interior points of +Ω . According to the corresponding homogeneouscondition (3) we have the same in 0σ .

Let ( )u x y, attain its positive maximum in an interior point of the segment EAB \ .Then by virtue of the corresponding homogeneous condition (5), this extremum is attained intwo points: 0( 0)x , and 0( ( ) 0)p x , . Therefore we have in these points 0( ) 0xν < ,

0( ( )) 0p xν < , (Volkodavov, 1970), what follows 0 0 0( ) ( ( )) ( ) 0x p x p xν ν ′− < . But this

contradicts to the corresponding homogeneous relation (8) at 0x x= , hence there is no

positive maximum on EAB \ .

Thus, the function ( )u x y, to be found attains its positive maximum in +Ω in the point

( 0)E c, . Analogously, one can prove that ( )u x y, attains its negative minimum in +Ω also

in ( 0)E c, . Obtained contradiction shows that ( ) 0u x y, ≡ in +Ω , hence ( ) 0u x y, ≡

in Ω.

Existence of the Solution of Problem GF

Theorem 2. Problem GF is uniquely solvable.

Proof. The following formula (Mirsaburov, Eshankulov, 2003):

12 2

21

1 4( ) ( ) ln ( )2 ( 2)

mu x y t x t ym

νπ

⎡ ⎤⎢ ⎥+⎢ ⎥⎢ ⎥⎣ ⎦−

⎧, = − + −⎨ +⎩

∫2

2 22

4ln (1 )( 2)

mtxt y dtm

⎫⎡ ⎤⎪⎢ ⎥+ ⎪⎬⎢ ⎥⎪⎢ ⎥⎪⎣ ⎦ ⎭

− − + ++


12( 2) 2 2 2

11

( 2)(1 ) ( )[ ( )] ( )4

mm R r r dϕ ξ η ξ ξπ

− + / − −

−

+ −+ + ,∫

where2 2 2 24( 2) mR x m y− += + + , 2 2 24( 2) 1mmξ η− ++ + = ,

2 22 2

2 22

221

4( )( 2)

m mrx y

mrξ η

+ +⎫⎪ ⎛ ⎞⎪ ⎜ ⎟⎬ ⎜ ⎟⎪ ⎝ ⎠⎪⎭

= − + ± ,+

gives in +Ω the solution of the modified problem N:

0

2

0( ) ( ) lim ( )m

y

uu x y x x J y x x Jyσ ϕ ν− /

→+

∂, | = , ∈ ; = , ∈ ,

∂

for the equation (1). By this formula,

1

1

1 1( ) ( ) ( ) [ 1 1]1

tx t dt F x xt x xt

τ νπ −

⎛ ⎞′ = − − + , ∈ − ,⎜ ⎟− −⎝ ⎠∫ , (9)

where1

2 ( 2) 2 2 1

1

2( ) (1 ) ( )[ ( )] [1 2 ]2

mm dF x x x x ddx

ϕ ξ η ξ ξ ξπ

− + / −

−

⎡ ⎤+= − − + ,⎢ ⎥

⎣ ⎦∫

( ) 1( ) ( )F x C J C J∈ ∩ .

By virtue of (9), we obtain from the condition (7) that

1 1

1 1

1 1( ) ( ) ( )1 ( ) 1 ( )

t tt dt t p x dtt x xt t p x p x t

ν ν− −

⎛ ⎞⎛ ⎞ ′− − − =⎜ ⎟⎜ ⎟− − − −⎝ ⎠ ⎝ ⎠∫ ∫

0 ( ) ( 1 )F x x c= , ∈ − , , (10)where

00 ( ) ( ( ) ( ( )) ( ) ( )) ( 1 )F x F x F p x p x f x C cαπ ,′ ′= − − ∈ − , .

Now decomposing each integral on the left side of (10) onto two ones with the intervals( 1 )c− , and ( 1)c, , substituting ( )t p s= into integrals with limits ( 1)c, , taking into account(8), we have

1 1

1 1 ( )( ) ( )1 ( ) 1 ( )

c ct p st dt s dst x xt p s x xp s

ν ν− −

⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎜ ⎟− − − −⎝ ⎠ ⎝ ⎠∫ ∫

1

1( ) ( )( ) 1 ( )

c tt p x dtt p x p x t

ν−

⎛ ⎞ ′− − +⎜ ⎟− −⎝ ⎠∫


11

1 ( )( ) ( ) ( ) ( 1 )( ) ( ) 1 ( ) ( )

c p ss p x ds F x x cp s p x p x p s

ν−

⎛ ⎞ ′+ − = , ∈ − ,⎜ ⎟− −⎝ ⎠∫ , (11)

where

1 01

1 ( )( ) ( ) (2 ( ) ( ))( ) 1 ( )

c p sF x F x s f s dsp s x xp s

ψ−

⎛ ⎞′ ′= + − − −⎜ ⎟− −⎝ ⎠∫

0

1

1 ( )(2 ( ) ( )) ( ) ( 1 )( ) ( ) 1 ( ) ( )

c p ss f s p x ds C cp s p x p x p s

αψ ,

−

⎛ ⎞′ ′ ′− − − ∈ − , .⎜ ⎟− −⎝ ⎠∫

Further we investigate problem GF in two cases: 0c = and 0c ≠ .

1. The case of 0c = , ( )p x x= − . In this case, (11) has the form

0

12 2 2 21

1 14 ( ) ( ) ( 1 0)1

t tdt F x xt x x t

ν−

⎛ ⎞− = , ∈ − , .⎜ ⎟− −⎝ ⎠∫

Substituting 2tξ = , t ξ= − ; 2y x= , x y= − , we obtain the following first ordersingular integral Tricomi equation:

1

20

1 1( ) ( ) (0 1)1

d F y yy y

ρ ξ ξξ ξ⎛ ⎞

− = , ∈ , ,⎜ ⎟− −⎝ ⎠∫ (12)

where

( )ρ ξ ν ξ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= − , 2 11( )2

F y F y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= − − .

We’ll look up the solution ( )yρ of the equation (12) in the Hölder class of functions

(0 1)H , bounded at the point 1y = and unbounded at the point 0y = , i.e. in the class(1)h . Applying the Carleman method (Smirnov, 1985), we obtain the inversion formula for

the singular integral equation (12):

1

220

1 1 1 1( ) ( )1 1

yy F dy y yξρ ξ ξ

π ξ ξ ξ⎛ ⎞+

= − − .⎜ ⎟+ − −⎝ ⎠∫

2. The case of 0c ≠ , ( )p x kxδ= − where 2 (1 )c cδ = / + , (1 ) (1 )k c c= − / + . In thiscase, introduce the function


( )( ) ( )( ) [ 1 ] [ 1 ]1 ( ) ( ) 1

p x p t tc x t C c cp x p t xt′

, = − ∈ − , × − ,− −

except for the point ( 1 1)− ,− where the function is bounded. In this case, we transform (11)into the form:

1 1

1 1 1 ( )( ) ( )1 2 ( ) ( )

c ct p xt dt t dtt x xt p t x t p x

ν ν− −

′⎛ ⎞⎛ ⎞− = + +⎜ ⎟⎜ ⎟− − − −⎝ ⎠ ⎝ ⎠∫ ∫

11[ ] ( ) ( 1 )2

R F x x cν+ + , ∈ − , , (13)

where

1

1 ( ) ( )[ ] ( ) ( )2 1 ( ) 1 ( )

c p t tp xR c x t t dtxp t p x t

ν ν−

′⎛ ⎞= − + − ,⎜ ⎟− −⎝ ⎠

∫ (14)

is a regular operator. The integrand expression 1 ( )

( ) ( )p x

p t x t p x′

+− −

of the right side of (13)

have at t c= , x c= the first order singularity by virtue of the equality ( )p c c= , thereforethis summand is selected separately (Polosin, 1996; Mirsaburov, 2001).

Rewrite (13) in the following form:

31

1 ( ) ( ) ( 1 )1

c t t dt F x x ct x xt

ν−

⎛ ⎞− = , ∈ − ,⎜ ⎟− −⎝ ⎠∫ , (15)

where

3 11

1 1 ( ) 1( ) ( ) [ ] ( )2 ( ) ( ) 2

c p xF x t dt R F xp t x t p x

ν ν−

′⎛ ⎞= + + + .⎜ ⎟− −⎝ ⎠∫ (16)

Solving (15) by the Carleman method, we have

321

1 ( )(1 ) 1 1( ) ( )( )(1 ) 1

c c t ctx F t dtc x cx t x xt

νπ −

− − ⎛ ⎞= − − .⎜ ⎟− − − −⎝ ⎠∫ (17)

Transform (17) with regard to (16) to the form

21 1

1 1 1( ) ( )2 1

c c c tx s dsc x t x xt

ν νπ − −

− ⎛ ⎞= − − ×⎜ ⎟− − −⎝ ⎠∫ ∫

1 41 ( ) [ ] ( ) ( 1 )

( ) ( )p t dt R F x x c

p s t s p tν

′⎛ ⎞× − + + , ∈ − ,⎜ ⎟− −⎝ ⎠

, (18)

where


1 21 1

1 1 1 1[ ] ( ) 12 1 1

c c ct c tR s dscx c x t x xt

ν νπ − −

⎛ ⎞− − ⎛ ⎞= − − − ×⎜ ⎟ ⎜ ⎟⎜ ⎟− − − −⎝ ⎠⎝ ⎠∫ ∫

21

1 ( ) 1 ( )(1 ) 1 1 [ ]( ) ( ) ( )(1 ) 1

cp t c t ctdt R dtp s t s p t c x cx t x xt

νπ −

′⎛ ⎞ − − ⎛ ⎞× − − − ,⎜ ⎟ ⎜ ⎟− − − − − −⎝ ⎠⎝ ⎠∫

04 12

1

1 ( )(1 ) 1 1( ) ( ) ( 1 )2 ( )(1 ) 1

c c t ctF x F t dt C cc x cx t x xt

α

π,

−

− − ⎛ ⎞= − − ∈ − , .⎜ ⎟− − − −⎝ ⎠∫

Calculating the interior integral of the right side in (18), we have

1 22

1

1 ( ) 1 1 1( ) 2(1 ) 12 2 2 1

c s ds c xx c Fks x cc x

ννπ δ

/

−

⎧ ⎡ −⎛ ⎞= − − + ,− , ; +⎨ ⎜ ⎟⎢− − +− ⎝ ⎠⎣⎩∫

3 21 22 (1 ) 5 11 1 [ 2(1 )

3 1 2 1c c kF c

ks ks s kxδ δ δ

//⎤+ +⎛ ⎞+ , , ; − − + ×⎜ ⎟⎥+ − + − − +⎝ ⎠⎦

3 21 1 2 (1 ) 5 (1 )1 1 12 2 1 3(1 ) 2 1

c x k c k cF Fc s s

/ ⎤− + +⎛ ⎞ ⎛ ⎞× ,− , ; + , , ; +⎜ ⎟ ⎜ ⎟⎥+ − −⎝ ⎠ ⎝ ⎠⎦3 2 3 21 2 (1 ) 3 5 (1 ) 2 (1 )1

1 3 1 2 2 1 3 1x c x c cF

x kxs xc xs ksδ δ

/ /⎡ + + +⎛ ⎞+ , , ;− − ×⎜ ⎟⎢− + − − + −⎝ ⎠⎣3 25 1 2 (1 ) 3 5 (1 )1 1 1

2 1 3(1 ) 2 2 1c k x c x cF F

ks k sx x xc xcδ δ

/⎡+ ⎤ + − +⎛ ⎞ ⎛ ⎞× , , ; + , , ; −⎜ ⎟ ⎜ ⎟⎢⎥+ − + − − −⎝ ⎠ ⎝ ⎠⎦ ⎣

1 42 5 (1 )1 1 [ ] ( ) ( 1 )

3(1 ) 2 1k k cF R F x x c

s sν

⎫⎤+⎛ ⎞− , , ; + + , ∈ − , .⎬⎜ ⎟⎥− −⎝ ⎠⎦⎭(19)

Introduce the notations:

1 2 1 20

1 1( ) 2(1 ) 1 2(1 )2 2 1

c xK x c F cc

/ /−⎛ ⎞= − + ,− , ; + + ,⎜ ⎟+⎝ ⎠ (20)

3 21 2 1 2 1 22 (1 ) 5 1( ) 1 1 2(1 ) ( )

3 1 2 1c cL s F c k c s

ks ksπ

δ δ

// / /+ +⎛ ⎞= , , ; − + + − ,⎜ ⎟+ − + −⎝ ⎠

(21)

3 21 2 1 2 1 22 (1 ) 5 (1 )( ) 1 1 2(1 ) ( )

3(1 ) 2 1k c k cM s F c k c s

s sπ

// − / /+ +⎛ ⎞= , , ; − + + − .⎜ ⎟− −⎝ ⎠

(22)


It is easy to verify that functions 0 ( )K x , ( )L s , ( )M s are infinitesimals of the orderless than 1/2 at the point c, i.e.

1 2 1 2 1 20 ( ) ( ) ( ) ( ) ( ) ( )K x o c x L s o c s M s o c s/ / /⎛ ⎞ ⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= − , = − , = − . (23)

We prove only the last relation in (23). Applying the L’Hospital rule for calculatinglimits, we have

1 12 21

2

( ) ( )lim lim( ) ( )s c s c

M s M sc s c s −→ →

′= =

− − −

12

2 (1 ) 5 (1 )2lim (1 ) 1 13 1 2 1s c

d k c k cc Fds s s→

⎧ ⎡ + + ⎤⎛ ⎞− + , , ; −⎨ ⎜ ⎟⎢ ⎥− −⎝ ⎠⎣ ⎦⎩12 1 1

2 2( ) ( )2k c s c sπ ⎫−

⎪− ⎪⎬⎪⎪⎭

− − − .

Applying the well-known formulas for the Gauss hypergeometric functions (Smirnov, 1985)

1( ) ( 1 )a ad x F a b c x ax F a b c xdx

−, , ; = + , , ; ,

( ) (1 ) ( )c a bF a b c x x F c a c b c x− −, , ; = − − , − , ; ,

2( ) ( ) 1( 1)( ) ( ) 2c c a bF a b cc a c b

πΓ Γ − − ⎛ ⎞, , ; = , Γ = ,⎜ ⎟Γ − Γ − ⎝ ⎠

it is not difficult to calculate that

1 12 2

12

( )lim 2 02 2( )s c

M s k kc s

π π− −

→

⎛ ⎞= − − = .⎜ ⎟⎜ ⎟− ⎝ ⎠

Hence, 12( ) ( )M s o c s⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

= − . Other two relations in (23) can be proved analogously.

Taking into consideration (23) and notations (20)-(22), we transform the equation (19) tothe form

1 2

1

1 1( ) ( )2

ck c sx s dsc x ks x s kx

ν νπ δ δ

/

−

− ⎛ ⎞= − +⎜ ⎟− − − − +⎝ ⎠∫2 4[ ] ( ) ( 1 )R F x x cν+ + , ∈ − , , (24)


where

0 02 1 2

1

( ) ( ) ( ( ) ( ))1 ( )[ ] [ ]2

c K x L s k K x M ss dsR Rks x s kxc x

νν νπ δ δ−

+ +⎧= − − +⎨ − − − +− ⎩∫3 2 3 21 2 (1 ) 3 5 (1 ) 2 (1 )1

1 3(1 ) 2 2 1 3 1x c x c cF

x kxs xc xc ksδ δ

/ /⎡ + + +⎛ ⎞+ , , ;− − ×⎜ ⎟⎢− + − − + −⎝ ⎠⎣3 25 1 2 (1 )1 1

2 1 3(1 )c k x cF

ks k sx x xcδ δ

/⎡+ ⎤ +⎛ ⎞× , , ; + ×⎜ ⎟ ⎢⎥+ − + − −⎝ ⎠⎦ ⎣

3 5 (1 ) 2 5 (1 )1 1 12 2 1 3(1 ) 2 1

x c k k cF Fxc s s

⎫⎤− + +⎛ ⎞ ⎛ ⎞× , , ; − , , ; ⎬⎜ ⎟ ⎜ ⎟⎥− − −⎝ ⎠ ⎝ ⎠⎦⎭(25)

is a regular operator.Rewrite (24) in the form

1 2

1

1 1( )2 ( ) ( )

ck c sxc x c s k c x c s

νπ

/

−

⎛−= +⎜− − + − / −⎝

∫

2 41 ( ) [ ] ( )

1 ( ) ( )s ds R F x

k c x c sν ν⎞

+ + + .⎟+ − / − ⎠(26)

If we substitute in (26) (1 ) ts c c e−= − + , (1 ) yx c c e−= − + (Polosin, 1996) and

introduce the new unknown function ( ) [ (1 ) ]y yy c c e eρ ν − −= − + , then (26) has the form

3 50

( ) ( ) ( ) [ ] ( )y K y t t dt R F yρ ρ ρ∞

= − + +∫ , (27)

where1 2

2 2 2 2

1 1( )2 x x x x

kK xke e e keπ

/

/ − / / − /

⎡ ⎤= + ,⎢ ⎥+ +⎣ ⎦

3 2 5 4[ ] [ ] ( ) [ (1 ) ]y y y yR R e e F y F c c e eρ ρ − − −= , = − + ,

3[ ]R ρ is a regular operator.The equation (27) is the Winner-Hopf equation (Gakhov, Cherskii, 1978). The index of

(27) will be the index of the expression 1 ( )K x∧− where

( ) ( )ixtK x e K t dt∞

∧

−∞

= .∫


Calculating the Fourier integral with the help of the residues theory, we obtain

1 2 ( )( ) 12

ixlnk ixlnkk e e cos xlnkK xch xkch x kch x

π ππ ππ π

/ −∧ ⎛ ⎞

= + = ≤ .⎜ ⎟⎝ ⎠

Thus, 1 ( ) 0K x∧− ≥ , hence , (1 ( )) 0Ind K x∧− = , i.e. changing of the argument of

1 ( )K x∧− on the real axis expressed in terms of total turns equals to zero (Gakhov, Cherskii,1978). This and the uniqueness of the solution of problem GF yield uniquely solvability of(27). Therefore problem GF is uniquely solvable.

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equation. Volzhskii matem. sb., Kuibyshev State Pedag. Inst. 1:55-65.Wen G., Chen D., Cheng X. (2007) General Tricomi-Rassias problem and oblique derivative

problem for generalized Chaplygin equations. J. Math. Appl.333:679-694.



Chapter 9

ON THE STABILITY OF AN ADDITIVE FUNCTIONAL

I NEQUALITY IN NORMED M ODULES

Choonkil Park∗

Department of Mathematics, Hanyang UniversitySeoul 133-791, Republic of Korea

To the memory of Professor Stanislaw Marcin Ulamon the occasion of his 100-th birthday anniversary

Abstract

In this paper, we investigate the following additive functional inequality‖f(x) +f(y)+f(z)+f(w)‖ ≤ ‖f(x)+f(y+z+w)‖ in normed modules over aC∗-algebra.This is applied to understand homomorphisms inC∗-algebras.

Moreover, we prove the generalized Hyers–Ulam stability of the following func-tional inequality‖f(x)+f(y)+f(z)+f(w)‖ ≤ ‖f(x)+f(y+ z+w)‖+θ(‖x‖p +‖y‖p +‖z‖p +‖w‖p) in real Banach spaces, whereθ, p are positive real numbers withp 6= 1.

Using fixed point methods, we prove the generalized Hyers–Ulam stability of theprevious functional inequality in real Banach spaces.

2000 Mathematics Subject Classifications: Primary 39B72, 39B62, 47H10,46L05.

Key words: Functional equation, fixed point, generalized Hyers–Ulam stability, functionalinequality, linear mapping in normed modules over aC∗-algebra.

1. Introduction

The stability problem of functional equations originated from a question of Ulam (33) con-cerning the stability of group homomorphisms. Hyers (13) gave a first affirmative partialanswer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized byAoki (1) for additive mappings and by Th. M. Rassias (28) for linear mappings by consid-ering an unbounded Cauchy difference. The paper of Th. M. Rassias (28) has provided a lot

∗E-mail address: baak@@hanyang.ac.kr

108 C. Park

of influence in the development of what we callgeneralized Hyers–Ulam stabilityof func-tional equations. A generalization of the Th.M. Rassias theorem was obtained by Gavruta(10) by replacing the unbounded Cauchy difference by a general control function in thespirit of Th. M. Rassias’ approach. In 1982, J. M. Rassias (24) followed the innovativeapproach of the Th. M. Rassias’ theorem (28) in which he replaced the factor‖x‖p + ‖y‖p

by ‖x‖p · ‖y‖q for p, q ∈ R with p+ q 6= 1.

Theorem 1.1. (J. M. Rassias: (24)–(27)). LetX be a real normed linear space andY areal Banach space. Assume thatf : X → Y is a mapping for which there exist constantsθ ≥ 0 andp, q ∈ R such thatr = p+ q 6= 1 andf satisfies the functional inequality

∥∥f(x+ y) − f(x) − f(y)∥∥ ≤ θ‖x‖p‖y‖q

for all x, y ∈ X. Then there exists a unique additive mappingL : X → Y satisfying

‖f(x) − L(x)‖ ≤θ

|2r − 2|‖x‖r

for all x ∈ X. If, in addition, f : X → Y is a mapping such that the transformationt→ f(tx) is continuous int ∈ R for each fixedx ∈ X, thenL is anR-linear mapping.

The stability problems of several functional equations have been extensively investi-gated by a number of authors and there are many interesting results concerning this problem(see (6), (9), (14), (21), (22), (29)–(31)).

We recall a fundamental result in fixed point theory.

Theorem 1.2((2),(7)). Let(X, d) be a complete generalized metric space and letJ : X →X be a strictly contractive mapping with Lipschitz constantL < 1. Then for each givenelementx ∈ X, either

d(Jnx, Jn+1x) = ∞

for all nonnegative integersn or there exists a positive integern0 such that(1) d(Jnx, Jn+1x) <∞, ∀n ≥ n0;(2) the sequenceJnx converges to a fixed pointy∗ of J ;(3) y∗ is the unique fixed point ofJ in the setY =

y ∈ X | d(Jn0x, y) <∞

;

(4) d(y, y∗) ≤ 11−L

d(y, Jy) for all y ∈ Y .

In 1996, G. Isac and Th. M. Rassias (15) were the first to provide applications of stabil-ity theory of functional equations for the proof of new fixed point theorems with applica-tions. By using fixed point methods, the stability problems of several functional equationshave been extensively investigated by a number of authors (see (4), (17), (18), (19), (23)).

Gilanyi (11) showed that iff satisfies the functional inequality

∥∥2f(x) + 2f(y) − f(x− y)∥∥ ≤ ‖f(x+ y)‖ (1.1)

thenf satisfies the Jordan-von Neumann functional equation

2f(x) + 2f(y) = f(x+ y) + f(x− y).

Stability of Additive Functional Inequality 109

See also (32). Fechner (8) and Gilanyi (12) proved the generalized Hyers–Ulam stabil-ity of the functional inequality (1.1). Park, Cho and Han (20) investigated the functionalinequality ∥∥f(x) + f(y) + f(z)

∥∥ ≤ ‖f(x+ y + z)‖ (1.2)

in Banach spaces, and proved the generalized Hyers–Ulam stability of the functional in-equality (1.2) in Banach spaces.

In this paper, we investigate a module linear mapping associated with the functionalinequality

∥∥f(x) + f(y) + f(z) + f(w)∥∥ ≤ ‖f(x) + f(y + z + w)‖. (1.3)

This is applied to understand homomorphisms inC∗-algebras. Moreover, we prove thegeneralized Hyers–Ulam stability of the functional inequality

∥∥f(x) + f(y) + f(z) + f(w)∥∥

≤ ‖f(x) + f(y + z + w)‖ + θ(‖x‖p + ‖y‖p + ‖z‖p + ‖w‖p

)(1.4)

in real Banach spaces. Using fixed point methods, we prove the generalized Hyers–Ulamstability of the functional inequality (1.4) in real Banach spaces.

2. Functional Inequalities in Normed Modules overa C

∗-Algebra

Throughout this section, letA be a unitalC∗-algebra with unitary groupU(A) and unite,and letB be aC∗-algebra. Assume thatX is a normedA-module with norm‖ · ‖ and thatY is a normedA-module with norm‖ · ‖.

In this section, we investigate anA-linear mapping associated with the functional in-equality (1.3).

Theorem 2.1. Letf : X → Y be a mapping such that∥∥f(x) + f(y) + f(z) + uf(w)

∥∥ ≤∥∥f(x) + f(y + z + uw)

∥∥ (2.5)

for all x, y, z, w ∈ X and allu ∈ U(A). Then the mappingf : X → Y isA-linear.

Proof. Lettingx = y = z = w = 0 andu = e ∈ U(A) in (2.1), we get

‖4f(0)‖ ≤ ‖2f(0)‖.

Sof(0) = 0.Lettingx = w = 0 in (2.1), we get

‖f(y) + f(z)‖ ≤ ‖f(y + z)‖ (2.6)

for all y, z ∈ X.Replacingy andz by x andy + z + w in (2.2), respectively, we get

∥∥f(x) + f(y + z + w)∥∥ ≤ ‖f(x+ y + z + w)‖

110 C. Park

for all x, y, z, w ∈ X. So∥∥f(x) + f(y) + f(z) + f(w)

∥∥ ≤ ‖f(x+ y + z + w)‖ (2.7)

for all x, y, z, w ∈ X.Lettingz = w = 0 andy = −x in (2.3), we get

‖f(x) + f(−x)‖ ≤ ‖f(0)‖ = 0

for all x ∈ X. Hencef(−x) = −f(x) for all x ∈ X.Lettingz = −x− y andw = 0 in (2.3), we get∥∥f(x) + f(y) − f(x+ y)

∥∥ =∥∥f(x) + f(y) + f(−x− y)

∥∥ ≤ ‖f(0)‖ = 0

for all x, y ∈ X. Thusf(x+ y) = f(x) + f(y)

for all x, y ∈ X.Lettingz = −uw andx = y = 0 in (2.1), we get

‖ − f(uw) + uf(w)‖ = ‖f(−uw) + uf(w)‖ ≤ ‖2f(0)‖ = 0

for all w ∈ X and allu ∈ U(A). Thus

f(uw) = uf(w) (2.8)

for all u ∈ U(A) and allw ∈ X.Now let a ∈ A (a 6= 0) andM an integer greater than4|a|. Then | a

M| < 1

4 <

1 − 23 = 1

3 . By Theorem 1 of (16), there exist three elementsu1, u2, u3 ∈ U(A) such that3 a

M= u1 + u2 + u3. So by (2.4)

f(ax) = f

(M

3· 3

a

Mx

)= M · f

(1

3· 3

a

Mx

)=M

3f

(3a

Mx)

=M

3h(u1x+ u2x+ u3x)

=M

3

(f(u1x) + f(u2x) + f(u3x)

)=M

3(u1 + u2 + u3)f(x) =

M

3· 3

a

Mf(x)

= af(x)

for all x ∈ X. Sof : X → Y isA-linear, as desired.

Corollary 2.2. Letf : A→ B bea multiplicative mapping such that

∥∥f(x) + f(y) + f(z) + µf(w)∥∥ ≤

∥∥f(x) + f(y + z + µw)∥∥ (2.9)

for all x, y, z, w ∈ A and allµ ∈ T :=λ ∈ C | |λ| = 1

. Then the mappingf : A → B

is aC∗-algebra homomorphism.

Proof. By Theorem 2.1, the multiplicative mappingf : A → B is C-linear, sinceC∗-algebras are normed modules overC. So the multiplicative mappingf : A → B is aC∗-algebra homomorphism, as desired.


3. Generalized Hyers–Ulam Stability of Functional Inequalities

Throughoutthis section, assume thatX is a real normed linear space and thatY is a realBanach space.

In this section, we prove the generalized Hyers–Ulam stability of the functional inequal-ity (1.4) in real Banach spaces.

Theorem 3.1. Assume thatf : X → Y is an odd mapping for which there exist constantsθ ≥ 0 andp ∈ R such thatp 6= 1 andf : X → Y satisfies the functional inequality

∥∥f(x) + f(y) + f(z) + f(w)∥∥

≤ ‖f(x) + f(y + z + w)‖ + θ(‖x‖p + ‖y‖p + ‖z‖p + ‖w‖p

)(3.10)

for all x, y, z, w ∈ X. Then there exists a unique Cauchy additive mappingA : X → Y

satisfying

‖f(x) −A(x)‖ ≤2p + 2

|2p − 2|θ‖x‖p (3.11)

for all x ∈ X. If, in addition, f : X → Y is a mapping such that the transformationt→ f(tx) is continuous int ∈ R for each fixedx ∈ X, thenA is anR-linear mapping.

Proof. Sincef is odd,f(0) = 0 andf(−x) = −f(x) for all x ∈ X.Lettingx = 0, z = y andw = −2y in (3.1), we get

‖2f(y) − f(2y)‖ ≤ (2 + 2p)θ‖y‖p (3.12)

for all y ∈ X. So

∥∥∥f(x) − 2f(x

2

)∥∥∥ ≤2 + 2p

2pθ‖x‖p

for all x ∈ X. Hence

∥∥∥2lf( x

2l

)− 2mf

( x

2m

)∥∥∥ ≤2 + 2p

2p

m−1∑

j=l

2j

2pjθ‖x‖p (3.13)

for all nonnegative integersm andl with m > l and allx ∈ X.Assume thatp > 1. It follows from (3.4) that the sequence

2kf

(x2k

)is Cauchy for

all x ∈ X. SinceY is complete, the sequence2kf

(x2k

)converges. So one can define the

mappingA : X → Y by

A(x) := limk→∞

2kf( x

2k

)

for all x ∈ X.Letting l = 0 andm→ ∞ in (3.4), we get

‖f(x) −A(x)‖ ≤2p + 2

2p − 2θ‖x‖p

for all x ∈ X.

112 C. Park

It follows from (3.1) that

∥∥∥2kf( x

2k

)+ 2kf

( y

2k

)+ 2kf

( z

2k

)+ 2kf

( w2k

)∥∥∥

≤

∥∥∥∥2kf( x

2k

)+ 2kf

(y + z + w

2k

)∥∥∥∥ +2kθ

2pk

(‖x‖p + ‖y‖p + ‖z‖p + ‖w‖p

)(3.14)

for all x, y, z, w ∈ X. Lettingk → ∞ in (3.5), we get∥∥A(x) +A(y) +A(z) +A(w)

∥∥ ≤ ‖A(x) +A(y + z + w)‖ (3.15)

for all x, y, z, w ∈ X. It is easy to show thatA : X → Y is odd. Lettingw = −y − z andx = 0 in (3.6), we getA(y + z) = A(y) +A(z) for all y, z ∈ X. So there exists a Cauchyadditive mappingA : X → Y satisfying (3.2) for the casep > 1.

Now, letT : X → Y be another Cauchy additive mapping satisfying (3.2). Then wehave

‖A(x) − T (x)‖ = 2q∥∥∥A

( x2q

)− T

( x2q

)∥∥∥

≤ 2q(∥∥∥L

( x2q

)− f

( x2q

)∥∥∥ +∥∥∥T

( x2q

)− f

( x2q

)∥∥∥)

≤2p + 2

2p − 2·2 · 2q

2pqθ‖x‖p,

which tends to zero asq → ∞ for all x ∈ X. So we can conclude thatA(x) = T (x) for allx ∈ X. This proves the uniqueness ofA.

Assume thatp < 1. It follows from (3.3) that∥∥∥∥f(x) −

1

2f(2x)

∥∥∥∥ ≤2p + 2

2θ‖x‖p

for all x ∈ X. Hence

∥∥∥∥1

2lf(2lx) −

1

2mf(2mx)

∥∥∥∥ ≤2p + 2

2

m−1∑

j=l

2pj

2jθ‖x‖p (3.16)

for all nonnegative integersm andl with m > l and allx ∈ X.It follows from (3.7) that the sequence

12k f(2kx)

is Cauchy for allx ∈ X. Since

Y is complete, the sequence

12k f(2kx)

converges. So one can define the mappingA :

X → Y by

A(x) := limk→∞

1

2kf

(2kx

)

for all x ∈ X.Letting l = 0 andm→ ∞ in (3.7), we get

‖f(x) −A(x)‖ ≤2p + 2

2 − 2pθ‖x‖p

for all x ∈ X.


The rest of the proof is similar to the above proof. So there exists a unique Cauchyadditive mappingA : X → Y satisfying

‖f(x) −A(x)‖ ≤2p + 2

|2p − 2|θ‖x‖p (3.17)

for all x ∈ X.Assume thatf : X → Y is a mapping such that the transformationt → f(tx) is

continuous int ∈ R for each fixedx ∈ X. By the same reasoning as in the proof ofTheorem 1.1, one can prove thatA is anR-linear mapping.

Using fixed point methods, we prove the generalized Hyers–Ulam stability of thefunc-tional inequality (1.4) in Banach spaces.

Theorem 3.2. Let f : X → Y be an odd mapping for which there exists a func-tion ϕ : X4 → [0,∞) such that there exists anL < 1 such thatϕ(x, y, z, w) ≤12 Lϕ(2x, 2y, 2z, 2w) for all x, y, z, w ∈ X, and

∥∥f(x) + f(y) + f(z) + f(w)∥∥ ≤ ‖f(x) + f(y + z + w)‖ + ϕ(x, y, z, w) (3.18)


satisfying

‖f(x) −A(x)‖ ≤L

2 − 2Lϕ(0, x, x,−2x) (3.19)

for all x ∈ X.

Proof. Consider the setS := g : X → Y

and introduce thegeneralized metriconS:

d(g, h) = infK ∈ R+ : ‖g(x) − h(x)‖ ≤ Kϕ(0, x, x,−2x), ∀x ∈ X

.

It is easy to show that(S, d) is complete. (See the proof of Theorem 2.5 of (3).)Now we consider the linear mappingJ : S → S such that

Jg(x) := 2g(x

2

)

for all x ∈ X.It follows from the proof of Theorem 3.1 of (2) that

d(Jg, Jh) ≤ Ld(g, h)

for all g, h ∈ S.Sincef : X → Y is odd,f(0) = 0 andf(−x) = −f(x) for all x ∈ X. Letting

z = y = x andw = −2x in (3.9), we get

‖2f(x) − f(2x)‖ = ‖2f(x) + f(−2x)‖ ≤ ϕ(0, x, x,−2x) (3.20)

114 C. Park

for all x ∈ X.It follows from (3.11) that

∥∥∥f(x) − 2f(x

2

)∥∥∥ ≤ ϕ(0,x

2,x

2,−x

)≤L

2ϕ(0, x, x,−2x)

for all x ∈ X. Henced(f, Jf) ≤ L2 .

By Theorem 1.2, there exists a mappingA : X → Y satisfying the following:(1)A is a fixed point ofJ , i.e.,

A(x

2

)=

1

2A(x) (3.21)

for all x ∈ X. ThenA : X → Y is an odd mapping. The mappingA is a unique fixed pointof J in the set

M =g ∈ S : d(f, g) <∞

.

This implies thatA is a unique mapping satisfying (3.12) such that there exists aK ∈(0,∞) satisfying

‖f(x) −A(x)‖ ≤ Kϕ(0, x, x,−2x)

for all x ∈ X;(2) d(Jnf,A) → 0 asn→ ∞. This implies the equality

limn→∞

2nf( x

2n

)= A(x) (3.22)

for all x ∈ X;(3) d(f,A) ≤ 1

1−Ld(f, Jf), which implies the inequality

d(f,A) ≤L

2 − 2L.

This implies that the inequality (3.10) holds.It follows from (3.9) and (3.13) that

∥∥A(x) +A(y) +A(z) +A(w)∥∥ ≤ ‖A(x) +A(y + z + w)‖

for all x, y, z, w ∈ X. By Theorem 2.1, the mappingA : X → Y is a Cauchy additivemapping.

Therefore, there exists a unique Cauchy additive mappingA : X → Y satisfying (3.11),as desired.

Corollary 3.3. Let r > 1 andθ be nonnegative real numbers, and letf : X → Y be anodd mapping such that

∥∥f(x) + f(y) + f(z) + f(w)∥∥

≤ ‖f(x) + f(y + z + w)‖ + θ(‖x‖r + ‖y‖r + ‖z‖r + ‖w‖r

)(3.23)


such that

‖f(x) −A(x)‖ ≤2r + 2

2r − 2θ‖x‖r

for all x ∈ X.


Proof. Theproof follows from Theorem 2.1 by taking

ϕ(x, y, z, w) := θ(‖x‖r + ‖y‖r + ‖z‖r + ‖w‖r

)

for all x, y, z, w ∈ X. Then we can chooseL = 21−r and we get the desired result.

Remark 3.4. Let f : X → Y be an odd mapping for which there exists a functionϕ :X4 → [0,∞) satisfying (3.9). By a similar method to the proof of Theorem 3.2, onecan show that if there exists anL < 1 such thatϕ(x, y, z, w) ≤ 2Lϕ

(x2 ,

y2 ,

z2 ,

w2

)for all

x, y, z, w ∈ X, then there exists a unique Cauchy additive mappingA : X → Y satisfying

‖f(x) −A(x)‖ ≤1

2 − 2Lϕ(0, x, x,−2x)

for all x ∈ X.For the case0 < r < 1, one can obtain a similar result to Corollary 3.3: Let0 < r < 1

andθ ≥ 0 be real numbers, and letf : X → Y be an odd mapping satisfying (3.14). Thenthere exists a unique Cauchy additive mappingA : X → Y satisfying

‖f(x) −A(x)‖ ≤2 + 2r

2 − 2rθ‖x‖r

for all x ∈ X.

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Chapter 10

CUBIC DERIVATIONS AND QUARTIC DERIVATIONS

ON BANACH M ODULES

Choonkil Parka and John Michael Rassiasb,∗

aDepartment of Mathematics, Hanyang UniversitySeoul 133-791, Republic of Korea

bPedagogical Department, E.E., National and CapodistrianUniversity of Athens, 4, Agamemnonos Str., Aghia Paraskevi

Athens 15342, Greece

To the memory of Professor Stanislaw Marcin Ulamon the occasion of his 100-th birthday anniversary

Abstract

In this paper, we define a cubic derivation and a quartic derivation on a Banachmodule over a normed algebra and prove the generalized Hyers–Ulam stability of thecubic derivation and the quartic derivation on a Banach module over a normed algebra.

2000 Mathematics Subject Classifications:Primary 39B82, 39B72.

Key words: Cubic functional equation, quartic functional equation, cubic derivation, quar-tic derivation, generalized Hyers–Ulam stability.

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam (23) con-cerning the stability of group homomorphisms. Hyers (5) gave a first affirmative partialanswer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized byAoki (1) for additive mappings and by Th. M. Rassias (15) for linear mappings by con-sidering an unbounded Cauchy difference. The paper of Th. M. Rassias (15) has provideda lot of influence in the development of what we callgeneralized Hyers–Ulam stabilityoffunctional equations. A generalization of the Th. M. Rassias theorem was obtained by

∗E-mail addresses: baak@@hanyang.ac.kr, jrassias@@primedu.uoa.gr

120 C. Park and J.M. Rassias

Gavruta(4) by replacing the unbounded Cauchy difference by a general control function inthe spirit of Th. M. Rassias’ approach.

A square norm on an inner product space satisfies the parallelogram equality

‖x+ y‖2 + ‖x− y‖2 = 2‖x‖2 + 2‖y‖2.


f(x+ y) + f(x− y) = 2f(x) + 2f(y)

is called aquadratic functional equation. In particular, every solution of the quadraticfunctional equation is said to be aquadratic mapping. A generalized Hyers–Ulam stabilityproblem for the quadratic functional equation was proved by Skof (22) for mappingsf :X → Y , whereX is a normed space andY is a Banach space. Cholewa (2) noticed thatthe theorem of Skof is still true if the relevant domainX is replaced by an Abelian group.In (3), Czerwik proved the generalized Hyers–Ulam stability of the quadratic functionalequation. Several functional equations have been investigated in (6), (7), (9) and (11)–(21).

Jun and Kim (8) introduced the following functional equation

f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 12f(x) (1.1)

and they established the general solution and the generalized Hyers–Ulam stability problemfor the functional equation(1.1).

It is easy to see that the functionf(x) = cx3 is a solution of the above functionalequation(1.1). Thus, it is natural that(1.1) is called acubic functional equationand everysolution of the cubic functional equation(1.1) is said to be acubic mapping.

In (10), S. Lee et al. considered the following quartic functional equation

f(2x+ y) + f(2x− y) = 4f(x+ y) + 4f(x− y) + 24f(x) − 6f(y). (1.2)

It is easy to show that the functionf(x) = cx4 satisfies the functional equation (1.2), whichis called aquartic functional equationand every solution of the quartic functional equationis said to be aquartic mapping.

Throughout this paper, we suppose thatA is a normed algebra and thatX is a BanachA-module.

Definition 1.1. A cubic mappingf : A → X is called acubic derivationif f satisfiesf(xy) = x3f(y) + f(x)y3 for all x, y ∈ A.

Definition 1.2. A quartic mappingf : A → X is called aquartic derivationif f satisfiesf(xy) = x4f(y) + f(x)y4 for all x, y ∈ A.

Example 1.3. Assume thatA is a commutative normed algebra. Letω ∈ X be fixed.(i) Definef : A → X by f(x) := x3ω − ωx3 for all x ∈ A. It is easy to show that

f : A→ X is a cubic derivation.(ii) Define f : A → X by f(x) := x4ω − ωx4 for all x ∈ A. It is easy to show that

f : A→ X is a quartic derivation.

In this paper, we prove the generalized Hyers-Ulam stability of the cubic derivation andof the quartic derivation on a Banach module over a normed algebra.

Cubic Derivations and Quartic Derivations on Banach Modules 121

2. On the Stability of Cubic Derivations on Banach Modules

In this section, we prove the generalized Hyers–Ulam stability of the cubic derivation on aBanach module over a normed algebra.

Theorem 2.1. Suppose that a functionψ : A×A→ [0,∞) satisfies

ψ(x, y) :=∞∑

i=0

1

8iψ(2ix, 2iy) <∞ (2.3)

for all x, y ∈ A. If f : A→ X is a mapping such that∥∥f(2x+ y) + f(2x− y) − 2f(x+ y) − 2f(x− y) − 12f(x)

∥∥ ≤ ψ(x, y), (2.4)∥∥f(xy) − x3f(y) − f(x)y3∥∥ ≤ ψ(x, y) (2.5)

for all x, y ∈ A, then there exists a unique cubic derivationD : A→ X such that

‖f(x) −D(x)‖ ≤1

16ψ(x, 0) (2.6)

for all x ∈ A

Proof. Puttingy = 0 in (2.2), we get

‖2f(2x) − 16f(x)‖ ≤ ψ(x, 0) (2.7)

for all x ∈ A. So∥∥∥∥f(x) −

1

8f(2x)

∥∥∥∥ ≤1

16ψ(x, 0)

for all x ∈ A. Hence

∥∥∥∥1

8nf(2nx) −

1

8mf(2mx)

∥∥∥∥ ≤1

16

m−1∑

k=n

1

8kψ(2kx, 0) (2.8)

for all nonnegative integersn,m with n < m. Thus

18n f(2nx)

is a Cauchy sequence in

X. SinceX is complete, there exists a mappingD : A→ X defined by

D(x) := limn→∞

1

8nf(2nx)

for all x ∈ A. Lettingn = 0 andm→ ∞ in (2.6), we get the inequality (2.4).It follows from (2.2) that

∥∥D(2x+ y) +D(2x− y) − 2D(x+ y) − 2D(x− y) − 12D(x)∥∥

= limn→∞

1

8n

∥∥f(2n(2x+ y)) + f(2n(2x− y)) − 2f(2n(x+ y)) − 2f(2n(x− y)) − 12f(2nx)∥∥

≤ limn→∞

1

8nψ(2nx, 2ny) = 0


for all x, y ∈ A. So

D(2x+ y) +D(2x− y) − 2D(x+ y) − 2D(x− y) − 12D(x) = 0

for all x, y ∈ A. HenceD : A→ X is a cubic mapping.LetC : A→ X be another cubic mapping satisfying (2.4). Then

‖D(x) − C(x)‖ =1

8n

∥∥D(2nx) − C(2nx)∥∥

≤1

8n

(‖f(2nx) −D(2nx)‖ + ‖f(2nx) − C(2nx)‖

)

≤1

8n+1ψ(2nx, 0),

whichtends to zero asn→ ∞ for all x ∈ A. So we haveD(x) = C(x) for all x ∈ A. Thisproves the uniqueness ofD.

On the other hand, it follows from (2.3) that

∥∥D(xy) − x3D(y) −D(x)y3∥∥

=

∥∥∥∥1

64n(D(4nxy) − 8nx3D(2ny) − 8nD(2nx)y3)

∥∥∥∥

=

∥∥∥∥ limn→∞

1

64nf(4nxy) − x3 lim

n→∞

1

8nf(2ny) − lim

n→∞

1

8nf(2nx)y3

∥∥∥∥

≤ limn→∞

1

64nψ(2nx, 2ny)

≤ limn→∞

1

8nψ(2nx, 2ny) = 0

for all x, y ∈ A. ThusD(xy) = x3D(y) +D(x)y3

for all x, y ∈ A, as desired.

Corollary 2.2. Letp < 3 andθ be positive real numbers. Iff : A→ X is a mapping suchthat

∥∥f(2x+ y) + f(2x− y) − 2f(x+ y) − 2f(x− y) − 12f(x)∥∥ ≤ θ

(‖x‖p + ‖y‖p

),

(2.9)∥∥f(xy) − x3f(y) − f(x)y3

∥∥ ≤ θ(‖x‖p + ‖y‖p

)(2.10)

for all x, y ∈ A, then there exists a unique cubic derivationD : A→ X such that

‖D(x) − f(x)‖ ≤θ

16 − 2p+1‖x‖p

for all x ∈ A.

Proof. Defineψ(x, y) = θ(‖x‖p + ‖y‖p), and apply Theorem 2.1 to get the desired result.


Theorem 2.3. Supposethat a functionψ : A×A→ [0,∞) satisfies

∞∑

i=1

64iψ( x

2i,y

2i

)<∞ (2.11)

for all x, y ∈ A. If f : A → X is a mapping satisfying(2.2)and (2.3), then there exists aunique cubic derivationD : A→ X such that

‖f(x) −D(x)‖ ≤1

16ψ(x, 0) (2.12)

for all x ∈ A, whereψ(x, y) :=∞∑i=1

8iψ(

x2i ,

y2i

)for all x, y ∈ A.

Proof. It follows from (2.5) that∥∥∥f(x) − 8f

(x2

)∥∥∥ ≤1

2ψ

(x2, 0

)

for all x ∈ A. So

∥∥∥8nf( x

2n

)− 8mf

( x

2m

)∥∥∥ ≤1

16

m∑

k=n+1

8kψ( x

2k, 0

)(2.13)

for all nonnegative integersn,m with n < m. Thus8nf

(x2n

)is a Cauchy sequence in


D(x) := limn→∞

8nf( x

2n

)

for all x ∈ A. Lettingn = 0 andm→ ∞ in (2.11), we get the inequality (2.10).It follows from (2.2) that∥∥D(2x+ y) +D(2x− y) − 2D(x+ y) − 2D(x− y) − 12D(x)

∥∥

= limn→∞

8n

∥∥∥∥f(

2x+y

2n

)+f

(2x−y

2n

)−2f

(x+y

2n

)−2f

(x−y

2n

)−12f

( x

2n

)∥∥∥∥

≤ limn→∞

8nψ( x

2n,y

2n

)≤ lim

n→∞

64nψ( x

2n,y

2n

)= 0


D(2x+ y) +D(2x− y) − 2D(x+ y) − 2D(x− y) − 12D(x) = 0

for all x, y ∈ A. HenceD : A→ X is a cubic mapping.On the other hand, it follows from (2.3) and (2.9) that

∥∥D(xy) − x3D(y) −D(x)y3∥∥

=

∥∥∥∥64n

(D

(xy4n

)−

1

8nx3D

( y

2n

)−

1

8nD

( x

2n

)y3

)∥∥∥∥

=∥∥∥ lim

n→∞

64nf(xy

4n

)− x3 lim

n→∞

8nf( y

2n

)− lim

n→∞

8nf( x

2n

)y3

∥∥∥

≤ limn→∞

64nψ( x

2n,y

2n

)= 0



for all x, y ∈ A.The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4. Let p > 6 and θ be positive real numbers. Iff : A → X is a mappingsatisfying(2.7)and(2.8), then there exists a unique cubic derivationD : A→ X such that

‖D(x) − f(x)‖ ≤θ

2p+1 − 16‖x‖p

for all x ∈ A.


Definition 2.5. Let A, B be algebras. A cubic mappingf : A → B is called acubichomomorphismif f satisfiesf(xy) = f(x)f(y) for all x, y ∈ A.

Example 2.6. Assume thatA is an algebra and thatB is a commutative algebra. Letf : A → B be a homomorphism andF (x) := f(x)3 for all x ∈ A. It is easy to show thatF : A→ B is a cubic homomorphism.

Remark 2.7. By the same methods as in the proofs of the results in this section, one canprove the generalized Hyers–Ulam stability of cubic homomorphisms in Banach algebras.

3. On the Stability of Quartic Derivations on Banach Modules

In this section, we prove the generalized Hyers–Ulam stability of the quartic derivation ona Banach module over a normed algebra.

Theorem 3.1. Suppose that a functionψ : A×A→ [0,∞) satisfies

ψ(x, y) :=∞∑

i=0

1

16iψ(2ix, 2iy) <∞ (3.14)

for all x, y ∈ A. If f : A→ X is a mapping such that

∥∥f(2x+ y) + f(2x− y) − 4f(x+ y) − 4f(x− y) − 24f(x) − 6f(y)∥∥ ≤ ψ(x, y),

(3.15)∥∥f(xy) − x4f(x) − f(x)y4

∥∥ ≤ ψ(x, y) (3.16)

for all x, y ∈ A, then there exists a unique quartic derivationD : A→ X such that

‖f(x) −D(x)‖ ≤1

32ψ(x, 0) (3.17)

for all x ∈ A


Proof. Puttingy = 0 in (3.2), we get

‖2f(2x) − 32f(x)‖ ≤ ψ(x, 0) (3.18)

for all x ∈ A. So∥∥∥∥f(x) −

1

16f(2x)

∥∥∥∥ ≤1

32ψ(x, 0)

for all x ∈ A. Hence

∥∥∥∥1

16nf(2nx) −

1

16mf(2mx)

∥∥∥∥ ≤1

32

m−1∑

k=n

1

16kψ(2kx, 0) (3.19)

for all nonnegative integersn,m with n < m. Thus

116n f(2nx)

is a Cauchy sequence in


D(x) := limn→∞

1

16nf(2nx)


∥∥D(2x+ y) +D(2x− y) − 4D(x+ y) − 4D(x− y) − 24D(x) − 6D(y)∥∥

= limn→∞

1

16n

∥∥f(2n(2x+ y)) + f(2n(2x− y)) − 4f(2n(x+ y))

− 4f(2n(x− y)) − 24f(2nx) − 6f(2ny)∥∥

≤ limn→∞

1

16nψ(2nx, 2ny) = 0


D(2x+ y) +D(2x− y) − 4D(x+ y) − 4D(x− y) − 24D(x) − 6D(y) = 0

for all x, y ∈ A. HenceD : A→ X is a quartic mapping.LetQ : A→ X be another quartic mapping satisfying (3.4). Then

‖D(x) −Q(x)‖ =1

16n‖D(2nx) −Q(2nx)‖

≤1

16n

(‖f(2nx) −D(2nx)‖ + ‖f(2nx) −Q(2nx)‖

)

≤1

16n+1ψ(2nx, 0),

whichtends to zero asn→ ∞ for all x ∈ A. So we haveD(x) = Q(x) for all x ∈ A. Thisproves the uniqueness ofD.


On the other hand, it follows from (3.3) that

∥∥D(xy) − x4D(y) −D(x)y4∥∥

=

∥∥∥∥1

256n(D(4nxy) − 16nx4D(2ny) − 16nD(2nx)y4)

∥∥∥∥

=

∥∥∥∥ limn→∞

1

256nf(4nxy) − x4 lim

n→∞

1

16nf(2ny) − lim

n→∞

1

16nf(2nx)y4

∥∥∥∥

≤ limn→∞

1

256nψ(2nx, 2ny)

≤ limn→∞

1

16nψ(2nx, 2ny) = 0



Corollary 3.2. Letp < 4 andθ be positive real numbers. Iff : A→ X is a mapping suchthat

∥∥f(2x+y)+f(2x−y)−4f(x+y)−4f(x−y)−24f(x)−6f(y)∥∥ ≤ θ

(‖x‖p+‖y‖p

),

(3.20)∥∥f(xy) − x4f(y) − f(x)y4

∥∥ ≤ θ(‖x‖p + ‖y‖p

)(3.21)

for all x, y ∈ A, then there exists a unique quartic derivationD : A→ X such that

‖D(x) − f(x)‖ ≤θ

32 − 2p+1‖x‖p

for all x ∈ A.


Theorem 3.3. Supposethat a functionψ : A×A→ [0,∞) satisfies

∞∑

i=1

256iψ( x

2i,y

2i

)<∞ (3.22)

for all x, y ∈ A. If f : A → X is a mapping satisfying(3.2)and (3.3), then there exists aunique quartic derivationD : A→ X such that

‖f(x) −D(x)‖ ≤1

32ψ(x, 0) (3.23)

for all x ∈ A, whereψ(x, y) :=∞∑i=1

16iψ(

x2i ,

y2i

)for all x, y ∈ A.


Proof. It follows from (3.5) that∥∥∥f(x) − 16f

(x2

)∥∥∥ ≤1

2ψ

(x2, 0

)

for all x ∈ A. So∥∥∥16nf

( x

2n

)− 16mf

( x

2m

)∥∥∥ ≤1

32

m∑

k=n+1

16kψ( x

2k, 0

)(3.24)

for all nonnegative integersn,m with n < m. Thus16nf

(x2n

)is a Cauchy sequence in


D(x) := limn→∞

16nf( x

2n

)


∥∥D(2x+ y) +D(2x− y) − 4D(x+ y) − 4D(x− y) − 24D(x) − 6D(y)∥∥

= limn→∞

16n

∥∥∥∥f(

2x+ y

2n

)+ f

(2x− y

2n

)− 4f

(x+ y

2n

)

− 4f

(x− y

2n

)− 24f

( x

2n

)− 6f

( y

2n

)∥∥∥∥

≤ limn→∞

16nψ( x

2n,y

2n

)≤ lim

n→∞

256nψ( x

2n,y

2n

)= 0


D(2x+ y) +D(2x− y) − 4D(x+ y) − 4D(x− y) − 24D(x) − 6D(y) = 0

for all x, y ∈ A. HenceD : A→ X is a quartic mapping.On the other hand, it follows from (3.3) and (3.9) that

‖D(xy) − x4D(y) −D(x)y4‖

=

∥∥∥∥256n

(D

(xy4n

)−

1

16nx4D

( y

2n

)−

1

16nD

( x

2n

)y4

)∥∥∥∥

=∥∥∥ lim

n→∞

256nf(xy

4n

)− x4 lim

n→∞

16nf( y

2n

)− lim

n→∞

16nf( x

2n

)y4

∥∥∥

≤ limn→∞

256nψ( x

2n,y

2n

)= 0



Corollary 3.4. Let p > 8 and θ be positive real numbers. Iff : A → X is a mappingsatisfying(3.7) and (3.8), then there exists a unique quartic derivationD : A → X suchthat

‖D(x) − f(x)‖ ≤θ

2p+1 − 32‖x‖p

for all x ∈ A.



Definition 3.5. Let A, B bealgebras. A quartic mappingf : A → B is called aquartichomomorphismif f satisfiesf(xy) = f(x)f(y) for all x, y ∈ A.

Example 3.6. Assume thatA is an algebra and thatB is a commutative algebra. Letf : A → B be a homomorphism andF (x) := f(x)4 for all x ∈ A. It is easy to show thatF : A→ B is a quartic homomorphism.

Remark 3.7. By the same methods as in the proofs of the results in this section, one canprove the generalized Hyers–Ulam stability of quartic homomorphisms in Banach algebras.

References


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[4] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximatelyadditive mappings,J. Math. Anal. Appl.184(1994), 431–436.

[5] D. H. Hyers, On the stability of the linear functional equation,Proc. Nat. Acad. Sci.U.S.A.27 (1941), 222–224.

[6] D. H. Hyers, G. Isac and Th. M. Rassias,Stability of Functional Equations in SeveralVariables,Birkhauser, Basel, 1998.

[7] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms,Aequationes Math.44 (1992), 125–153.

[8] K. Jun and H. Kim, The generalized Hyers–Ulam–Rassias stability of a cubic func-tional equation,J. Math. Anal. Appl.274(2002), 867–878.

[9] Pl. Kannappan, Quadratic functional equation and inner product spaces,Results Math.27 (1995), 368–372.

[10] S. H. Lee, S. M. Im and I. S. Hwang, Quartic functional equations,J. Math. Anal.Appl.307(2005), 387–394.

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[12] C. Park, Homomorphisms between LieJC∗-algebras and Cauchy–Rassias stabilityof Lie JC∗-algebra derivations,J. Lie Theory15 (2005), 393–414.


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Chapter 11

TETRAHEDRON ISOMETRY ULAM STABILITYPROBLEM

John Michael Rassias1

National and Capodistrian University of AthensPadagogical Department, Section of Mathematics and Informatics4, Agamemnonos Str, Aghia Paraskevi, Attikis 15342, GREECE

Abstract

In this paper we investigate the tetrahedron isometry Ulam stability problem.

1. Introduction

In 1940 S. M. Ulam gave a talk before the Mathematics Club of the University of Wisconsinin which he discussed a number of unsolved problems. Among these was the followingquestion concerning the stability of homomorphisms.

Ulam stability problem. We are given a group G and a metric group G′ with metric(.,.)ρ . Given 0ε > , does there exist a 0δ > such that if :f G G′→ satisfies( ( ), ( ) ( ))f xy f x f yρ δ< for all ,x y in G, then a homomorphism :h G G′→ exists

with ( ( ), ( ))f x h xρ ε< for all x G∈ ?

By now an affirmative answer has been given in several cases, and some interestingvariations of the problem have also been investigated. We shall call such an :f G G′→ anapproximate homomorphism.

In 1941 D. H. Hyers (on the stability of the linear functional equation, Proc. Nat. Acad.Sci. USA 27(1941), 222-224) considered the case of approximately additive mappings

:f E E′→ where E and E′ are Banach spaces and f satisfies Hyers inequality

1 E-mail address: [email protected]


( ) ( ) ( )f x y f x f y ε+ − − ≤

for all , .x y in E It was shown that the limit

( ) lim 2 (2 )n n

nL x f x−

→∞=

exists for all x E∈ and that :L E E′→ is the unique additive mapping satisfying

( ) ( )f x L x ε− ≤ .

No continuity conditions are required for this result, but if ( )txf is continuous in the realvariable t for each fixed x ,then L is linear ,and if f is continuous at a single point of E then

:L E E′→ is also continuous.A generalization of this result was proved via the following theorems. In the first two

theorems 1-2, we assumed the following weaker condition (or weaker inequality)

( ) [ ]( ) ( ) p qf x y f x f y x yθ+ − + ≤ for all x, y in E,

involving a product of different powers of norms, where 0θ ≥ and real p, q such that1p qρ = + ≠ , and retained the condition of continuity of ( )txf in t for fixed x. Besides

through the last two theorems 1.3-1.4, we investigated that it is possible to replace ε in theabove Hyers inequality by a non-negative real-valued function such that the pertinent seriesconverges and other conditions hold and still obtain stability results. In all the casesinvestigated in this article, the approach to the existence question was to prove asymptotictype formulas of the form ( ) lim 2 (2 )n n

nL x f x−

→∞= , or ( ) lim 2 (2 )n n

nL x f x−

→∞= .

However, in 2002 we (J. Ind. Math. Soc. 69 (2002), 155-160) considered and investigatedquadratic equations involving a product of powers of norms in which an approximatequadratic mapping degenerates to a genuine quadratic mapping. Analogous results could beinvestigated with additive type equations involving a product of powers of norms.

In 1982, J. M. Rassias (“on approximation of approximately linear mappings by linearmappings”, J. Funct. Anal. 46 (1), 5-9) provided a generalization of Hyers’ stability Theoremwhich allows the Cauchy difference to be unbounded, as follows:

Theorem 1.1 ([1], [2], [5]). Let :f E E′→ be a mapping from a normed vector space Einto a Banach space E′ subject to the inequality

( ) ( ) ( ) p pf x y f x f y x yε+ − − ≤

for all ,x y E∈ , where and pε are constants with 0 0 1/ 2and pε > ≤ < . Then thelimit

Tetrahedron Isometry Ulam Stability Problem 133

(2 )( ) lim2

n

nn

f xL x→∞

=

exists for all x E∈ and :L E E′→ is the unique additive mapping which satisfies

22( ) ( )

2 2p

pf x L x xε− ≤

−

for all x E∈ . If 0p < then inequality (1.3) holds for , 0x y ≠ and (1.4) for 0x ≠ . If1/ 2p > then inequality (1.3) holds for all ,x y E∈ and the limit

( ) ⎟⎠⎞

⎜⎝⎛=

∞→ nn

n

xxA2

2lim

exists for all x E∈ and :A E E′→ is the unique additive mapping which satisfies

22( ) ( )

2 2p

pf x A x xε− ≤

−

for all x E∈ . If in addition YXf →: is a mapping such that the transformation ( )txft →is continuous in R∈t for each fixed Xx∈ , then L is R - linear mapping.

The case 1/ 2p = in inequality (1.3) is singular. A counter-example has been given byP. Gavruta (“An answer to a question of John M. Rassias concerning the stability of Cauchyequation”, in: Advances in Equations and Inequalities, in: Hadronic Math. Ser., 1999, 67-71).Our above-mentioned stability is called Ulam - Gavruta - Rassias stability.

Theorem 1.2 ([3]). Let X be a real normed linear space and let Y be a real complete normedlinear space. Assume in addition that YXf →: is an approximately additive mapping forwhich there exist constants 0≥θ and R∈qp, such that 1p qρ = + ≠ and f satisfiesinequality

( ) [ ]( ) ( ) p qf x y f x f y x yθ+ − + ≤

for all Xyx ∈, . Then there exists a unique additive mapping YXL →: satisfying

( ) ( ) 2 2

f x L x x ρ

ρ

θ− ≤

−

for all Xx∈ . If in addition YXf →: is a mapping such that the transformation

( )txft → is continuous in Rt ∈ for each fixed Xx∈ , then L is R - linear mapping.


Theorem 1.3 ([4]). Let X be a real normed linear space and let Y be a real complete normedlinear space. Assume in addition that YXf →: is an approximately additive mapping forwhich there exists a constant 0≥θ such that f satisfies inequality

1 21 1

( ) ( , ,..., )n n

i i ni i

f x f x K x x xθ= =

⎛ ⎞− ≤⎜ ⎟

⎝ ⎠∑ ∑ (*)

for all 1 2( , ,..., ) nnx x x X∈ and 0: ∪+→ RnXK is a non-negative real-valued function

such that

0

( ) ( , ,..., ) ( )j j j jn n

j

R R x n K n x n x n x∞

−

=

= = < ∞∑

is a non-negative function of x, and the condition

1 2lim ( , ,..., ) 0m m m mnm

n K n x n x n x−

→∞=

holds. Then there exists a unique additive mapping YXLn →: satisfying

( ) ( ) ( )n nf x L x R xnθ

− ≤

for all Xx∈ . If in addition YXf →: is a mapping such that the transformation

( )txft → is continuous in R∈t for each fixed Xx∈ , then nL is an R linear mapping.

Replacing 1, 2,...,ix x for i n= = in (*), we obtain1 1

( 1)

0 0

( ) ( ) ( , ,..., ) ( , ,..., )j j j j j j j j

j j

f x n f n x n K n x n x n x n K n x n x n xn

ν νν ν θ θ

− −− − − +

= =

− ≤ =∑ ∑ ,

and( ) lim ( )nL x n f n xν ν

ν

−

→∞= .

Analogous stability results we get if we substitute 1, 2,...,ixx for i nn

= = in (*).

Theorem 1.4 ([4]). Let X be a real normed linear space and let Y be a real complete normedlinear space. Assume in addition that YXf →: is an approximately additive mappingsuch that f satisfies inequality

1 21 1

( ) ( , ,..., )n n

i i ni i

f x f x N x x x= =

⎛ ⎞− ≤⎜ ⎟

⎝ ⎠∑ ∑ (**)


for all 1 2( , ,..., ) nnx x x X∈ and 0: ∪+→ RXN n is a non-negative real-valued function

such that ( )xxxN ,,, … is bounded on the unit ball of X, and

1 2 1 2( , ,..., ) ( ) ( , ,..., )n nN tx tx tx k t N x x x≤

for all 0t ≥ ,where ( ) ∞<tk and

0 0

0

( ) ( )j jn n

j

R R x n k n∞

−

=

= = < ∞∑ .

If in addition YXf →: is a mapping such that the transformation ( )txft → iscontinuous in Rt ∈ for each fixed Xx∈ and YXf →: is bounded on some ball of X,

then there exists a unique R - linear mapping YXLn →: satisfying

( ) ( ) ( , ,..., ),nf x L x MN x x x− ≤

for all Xx∈ , where

( 1)

0( ).m m

mM n k n

∞− +

=

= ∑

Replacing 1, 2,...,ix x for i n= = in (**), we obtain the results of this theorem.

Analogous stability results we get if we substitute 1, 2,...,ixx for i nn

= = in (**).

In 2007, S. Xiang, M. J. Rassias and we [15] investigated the Aleksandrov and triangleperimeter isometry Ulam stability problem.

In this paper we study the tetrahedron edge perimeter isometry Ulam stability problemon bounded domains.

2. Tetrahedron Perimeter Isometry Stability

Let X and Y be real Banach spaces. A mapping :I X Y→ introduced by JohnMichael Rassias, is called a tetrahedron edge perimeter isometry if I satisfies the tetrahedronedge perimeter identity

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )I x I y I y I z I z I x I x I y I z

x y y z z x x y z

− + − + − + + +

= − + − + − + + +(*)

for all , , .x y z X∈


In this section, we establish isometry stability results pertinent to the famous Ulamstability problem and the tetrahedron edge perimeter mapping : ,T X Y→

( ) ( ), , , ,iT x y z T x y z x y y z z x x y z= = − + − + − + + +

with respect to a tetrahedron ABCD of vertices ( ) ( ) ( ) ( )0 , , ,A B x C y D z , and the

corresponding mapping

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ), , , ,fT x y z T f x f y f z f x f y

f y f z f z f x f x f y f z

= = −

+ − + − + + +

as well as the difference operator fD such that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ), , , , , ,f f iD x y z T x y z T x y z f x f y f y f z f z f x

f x f y f z x y y z z x x y z

= − = − + − + −

+ + + −⎡ − + − + − + + + ⎤⎣ ⎦

in the bounded ball ( ): 0 1B x X x r r= ∈ ≤ < ≤ of a real Hilbert space X

associated with an inner product . , . , where the norm . is given by the formula2 ,x x x= .

Theorem 2.1. If a mapping :f X Y→ satisfies the following tetrahedron edge perimeterinequality

( ), , p pfD x y z x y y zϑ ⎡ ⎤≤ − + −⎣ ⎦ , (2.1)

for all ( ), ,x y z B X∈ ⊆ and some 0ϑ ≥ , and 1p > , then there exists a unique linear

tetrahedron edge perimeter :I X Y→ , such that the following inequality

( ) ( )1

122

12

2

2 1

pp

pf x I x xθ

−+

−

⎡ ⎤⎢ ⎥− ≤ ⎢ ⎥

−⎢ ⎥⎣ ⎦

(2.2)

holds for all x B∈ , where


( )12 1 2 0 1pr ifθ ϑ ϑ ϑ−= + ≤ ≤ ≤ (2.3)

and ( ) ( )lim 2 2n n

nI x f x−

→∞= for all x B∈ .

Proof. Let , ,x y z B∈ .Substituting 0x y z= = = in (2.1), one obtains ( )0 0f = . Setting

z y x= = in (2.1), we get ( )f x x= for all x B∈ . Thus replacing

( ), , , ,02xx y z x⎛ ⎞= ⎜ ⎟

⎝ ⎠ in (2.1) and then employing ( )f x x= and the standard triangle

inequality, we obtain

( ) ( )1 1 2 22 2

ppf x f x x xϑ −⎛ ⎞− − ≤⎜ ⎟⎝ ⎠

for all x B∈ , or equivalently inequality

( ) ( )1 1 2 22 2

ppf x f x x xϑ −⎛ ⎞− ≤ +⎜ ⎟⎝ ⎠

, (2.4)

for all x B∈ . Thus

( ) ( ) ( )

( )

2 2221 1 12 2

2 2 2

12 ,2

ppx x f x f x f x f x

f x f x

ϑ−⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ≥ − = +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞− ⎜ ⎟⎝ ⎠

, (2.5)

for all x B∈ . Employing ( )f x x= and (2.5) and the quadratic identity

( ) ( ) ( )2 2

21 1 1 1 12 2 2 ,2 2 2 2 2

f x f x f x f x f x f x⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ) ( )2 2

21 1 12 2 2

f x f x f x f x⎛ ⎞ ⎛ ⎞= − + + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, (2.6)

for all x B∈ , we get the inequality


( ) ( )2 2

21 1 1 1 122 2 2 4 2

f x f x x f x f x⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ) ( ) ( )2

2 1 2221 1 2 2 2 2 2 24 2

p p pp p px x x x xϑ ϑ ϑ+− − −⎛ ⎞≤ − + + = +⎜ ⎟⎝ ⎠

,

or

( ) ( ) ( )2

1 221 1 212 2 2 2 22

p pp pf x f x x xϑ ϑ+− −⎛ ⎞− ≤ +⎜ ⎟⎝ ⎠

( ) ( )2

1112 2 12

ppf x f x r xϑ ϑ +−⎛ ⎞− ≤ +⎜ ⎟⎝ ⎠

,

or the fundamental inequality

( )1 1

1 2 212 2 12

p ppf x f x r x xϑ ϑ θ

+ +−⎛ ⎞− ≤ + =⎜ ⎟

⎝ ⎠, (2.7)

for all x B∈ , where ( )12 1 2 0 1pr ifθ ϑ ϑ ϑ−= + ≤ ≤ ≤ , because

( ) ( ) ( )1

2 1 1 1 12 11 2 1 1 12 2 2 2 4

pp p p p ppp x x x x

−+ + − − +−− ⎛ ⎞= = ⎜ ⎟

⎝ ⎠

( ) 1 11 1 11 112 2

p pp p pr x r x+ +− − −< = .

Therefore, by (or without) induction on n , we obtain the general inequality

( ) ( )1

1 21 1 12 2 2

10 2

1 22 2 21 2

pnpn p pjn n

pj

f x f x x xθ θ

−⎛ ⎞⎜ ⎟−⎛ ⎞− ⎝ ⎠+ +⎜ ⎟− ⎝ ⎠−

=

⎡ ⎤⎡ ⎤ −⎢ ⎥− ≤ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥−⎣ ⎦

∑ , (2.8)

for all x B∈ , on every natural number n , and some 0,ϑ ≥ and 1p > .

From (2.8), it is clear that the sequence ( ) nI x , with ( ) ( )2 2n nnI x f x−= , is a

Cauchy sequence, because X is a complete space and 1.p > Therefore the limit

( ) ( ) ( )lim lim 2 2n nnn n

I x I x f x−

→∞ →∞= = ,

exists and satisfies (*) for all x B∈ , yielding the existence of a tetrahedron edge perimeterisometry : .I X X→

The proof for the linearity and uniqueness of the mapping :I X X→ follows standardtechniques ([1]-[6]).


Corollary 2.2. If a mapping :f X Y→ satisfies the following tetrahedron edge perimeterinequality

( ) ( ), , 2 p qfD x y z x y y zϑ ⎡ ⎤≤ − −⎣ ⎦ , (2.9)

for all ( ), ,x y z B X∈ ⊆ and some 0ϑ ≥ , and 1p qρ = + > , then there exists a unique

linear tetrahedron edge perimeter :I X Y→ , such that the following inequality

( ) ( )1

122

12

2

2 1f x I x x

ρρ

ρθ

−+

−

⎡ ⎤⎢ ⎥− ≤ ⎢ ⎥

−⎢ ⎥⎣ ⎦

, (2.10)

holds for all x B∈ , where ( )12 1 2 0 1r ifρθ ϑ ϑ ϑ−= + ≤ ≤ ≤ , and

( ) ( )lim 2 2n n

nI x f x−

→∞= for all x B∈ .

Proof. Let , ,x y z B∈ .Substituting 0x y z= = = in (2.9), one obtains ( )0 0f = . Setting

z y x= = in (2.9), we get ( )f x x= for all x B∈ . Thus replacing

( ), , , ,02xx y z x⎛ ⎞= ⎜ ⎟



( ) ( )1 1 2 22 2

f x f x x x ρρϑ −⎛ ⎞− − ≤⎜ ⎟⎝ ⎠


( ) ( )1 1 2 22 2

f x f x x x ρρϑ −⎛ ⎞− ≤ +⎜ ⎟⎝ ⎠

for all x B∈ .The rest of the proof is omitted as analogous to that of the above Theorem 2.1.

Corollary 2.3. If a mapping :f X Y→ satisfies the following tetrahedron edge perimeterinequality

( ) ( )2 22, ,3

p p p pfD x y z x y y z x y y zϑ ⎡ ⎤⎛ ⎞≤ − − + − + −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

, (2.11)


for all ( ), ,x y z B X∈ ⊆ and some 0ϑ ≥ , and 1p > , then there exists a unique linear

tetrahedron edge perimeter :I X Y→ , such that the following inequality

( ) ( )1

122

12

2

2 1

pp

pf x I x xθ

−+

−

⎡ ⎤⎢ ⎥− ≤ ⎢ ⎥

−⎢ ⎥⎣ ⎦

, (2.12)

holds for all x B∈ ,where ( )12 1 2 0 1pr ifθ ϑ ϑ ϑ−= + ≤ ≤ ≤ , and

( ) ( )lim 2 2n n

nI x f x−

→∞=

for all x B∈ .

Proof. Let , ,x y z B∈ .Substituting 0x y z= = = in (2.11), one obtains ( )0 0f = .

Setting z y x= = in (2.11), we get ( )f x x= for all x B∈ . Replacing

( ), , , ,02xx y z x⎛ ⎞= ⎜ ⎟



( ) ( )1 1 2 22 2

ppf x f x x xϑ −⎛ ⎞− − ≤⎜ ⎟⎝ ⎠


( ) ( )1 1 2 22 2

ppf x f x x xϑ −⎛ ⎞− ≤ +⎜ ⎟⎝ ⎠

for all x B∈ .

The rest of the proof is omitted as analogous to that of the above Theorem 2.1.

Note 2.4. The “product-sum” of powers of norms

( )2 2p p p px y y z x y y z− − + − + −

in (2.11) was introduced by J. M. Rassias and several specialists have already employed it.

Open Ulam Isometry Stability Problem 2.5.

To investigate the tetrahedron edge perimeter isometry stability on unbounded domains.


References

[1] J. M. Rassias, On Approximation of Approximately Linear Mappings by LinearMappings, J. Funct. Anal. USA 46 (1982), 126-130.

[2] J. M. Rassias, On Approximation of Approximately Linear Mappings by LinearMappings, Bull. Sc. Math.108 (1984), 445-446.

[3] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Th. USA 57 (1989), 268-273.[4] J. M. Rassias, Solution of a stability problem of Ulam, Discuss. Math. 12 (1992), 95-

103.[5] J. M. Rassias, Complete solution of the multi-dimensional problem of Ulam, Discuss.

Math. 14 (1994), 101-107.[6] J. M. Rassias and M. J. Rassias, On some approximately quadratic mappings being

exactly quadratic, J. Ind. Math. Soc. 69 (2002) , 155-160.[7] P. Gavruta, A generalization of the Hyers – Ulam-Rassias stability of approximately

additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.[8] P. Gavruta , An answer to a question of John M. Rassias concerning the stability of

Cauchy equation, in: Advances in Equations and Inequalities, in: Hadronic Math. Ser.,1999, 67-71.

[9] Ch. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,Bull. Sci. Math., 2008, 1-10.

[10] L. Cadariu and V. Radu, The alternative of fixed point and stability results for functionalequations, Euler’s FIDA in: IJAMAS, Vol. 1, 2007.

[11] B. Bouikhalene and E. Elquorachi, Ulam-Gavruta-Rassias stability of the Pexiderfunctional equation, Euler’s FIDA in: IJAMAS, Vol. 1, 2007.

[12] J. M. Rassias, S. Xiang and M. J. Rassias, On the Aleksandrov and triangle isometryUlam stability problems, Euler’s FIDA in: IJAMAS, Vol. 1, 2007

[13] K. Ravi and M. Arunkumar, On the Ulam-Gavruta-Rassias stability of the orthogonallyEuler-Lagrange type functional equation, Euler’s FIDA in: IJAMAS, Vol. 1, 2007.

[14] M. A. Sibaha, B. Bouikhalene and E. Elquorachi, Ulam-Gavruta-Rassias stability for alinear functional equation, Euler’s FIDA in: IJAMAS, Vol. 1, 2007.

[15] J. M. Rassias, S. Xiang and M. J. Rassias, On the Aleksandrov and triangle isometryUlam stability problems, Intern. J. Appl. Math. Stat., IJAMAS, 7 (Fe07) (2007), 133-142[dedicated to Euler’s].

[16] J. M. Rassias, On the stability of a multi-dimensional Cauchy type functional equation,in: Geometry, Analysis and Mechanics (dedicated to Archimedes), 1994, 365-376,World Sci. Publ., River Edge, NJ.

[17] J. M. Rassias, Solution of a Cauchy-Jensen stability Ulam type problem, ArchivumMathematicum (Brno) 37(3) (2001), 161-177.

[18] J. M. Rassias and M. J. Rassias, On the Ulam stability of Jensen and Jensen typemappings on restricted domains, J. Math. Anal. Appl. 281 (2003), 516-524.

[19] J. M. Rassias and M. J. Rassias, Asymptotic behavior of Jensen and Jensen typefunctional equations, PanAmerican Math. J. 15(4) (2005), 21-35.

[20] J. M. Rassias and M. J. Rassias, Asymptotic behavior of alternative Jensen and Jensentype functional equations, Bull. Sci. Mathematiques 129(7) (2005), 545-558.


[21] J. M. Rassias , Alternative contraction principle and Ulam stability problem, Math. Sci.Res. J. 9(7) (2005), 190-199.

[22] J. M. Rassias, On the Cauchy-Ulam stability of the Jensen equation in C*-algebras,Intern. J. Pure & Appl. Math. Stat., 2(1) (2005), 62-70.

[23] J. M. Rassias , Alternative contraction principle and alternative Jensen and Jensen typemappings, Intern. J. Appl. Math. Stat., 2006.

[24] J. M. Rassias, Refined Hyers-Ulam approximation of approximately Jensen typemappings, Bull. Sci. Mathematiques , 131(1) (2007), 89-98.

[25] J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J.Math. 20 (1992),185-190.

[26] J. M. Rassias, On the stability of the non-linear Euler-Lagrange functional equation inreal normed linear spaces, J. Math. Phys. Sci. 28 (1994), 231-235.

[27] J. M. Rassias, On the stability of the general Euler-Lagrange functional equation,Demonstratio Math. 29 (1996), 755-766.

[28] J. M. Rassias, Solution of the Ulam stability problem for Euler-Lagrange quadraticmappings, J. Math. Anal. Appl. 220 (1998), 613-639.

[29] J. M. Rassias, On the stability of the multi-dimensional Euler-Lagrange functionalequation, J. Indian Math. Soc. (N.S.) 66 (1999), 1-9.

[30] J. M. Rassias and M. J. Rassias, On the Ulam stability for Euler-Lagrange type quadraticfunctional equations, Austral. J. Math. Anal. Appl. 2 (2005), 1-10.

[31] J. M. Rassias , On the Ulam stability of mixed type mappings on restricted domains, J.Math. Anal. Appl. 276 (2002), 747-762.

[32] J. M. Rassias , Asymptotic behavior of mixed type functional equations, Austral. J.Math. Anal. Appl. 1(1) (2004), 1-21.



Chapter 12

HYERS–ULAM STABILITY OF CAUCHY TYPE

ADDITIVE FUNCTIONAL EQUATIONS

Matina J. Rassias∗

Department of Statistics, University of GlasgowMathematics Building, Office No. 208, University Gardens

Glasgow G12 8QW, U.K.

Abstract

In 1940 (and 1964) S. M. Ulam proposed the well-known Ulam stability problem.In 1941 D. H. Hyers solved the Hyers–Ulam problem for linear mappings. In thispaper we introduce a Cauchy type additive functional equation and investigate theHyers-Ulam stability of this equation.

2000 Mathematics Subject Classifications:Primary 39B. Secondary 26D.

Key words: Hyers–Ulam stability, Cauchy type additive functional equation.

1. Introduction

In 1940 (and 1964) Stanislaw M. Ulam (5) proposed the following stability problem, well-known asUlam stability problem:

“When is true that by slightly changing the hypotheses of a theorem one can still assertthat the thesis of the theorem remains true or approximately true?”

In particular he stated the stability question:

“Let G1 be a group andG2 a metric group with the metricρ(., .). Given a constantδ > 0, does there exist a constantc > 0 such that if a mappingf : G1 → G2 satisfiesρ(f(xy), f(x)f(y)) < c for all x, y ∈ G1, then a unique homomorphismh : G1 → G2

exists withρ(f(x), h(x)) < δ for all x ∈ G1 ?”

In 1941 D. H. Hyers (1) solved this problem for linear mappings as follows:


144 Matina J. Rassias

Theorem 1.1. If a mappingf : R → R satisfies the approximately additive inequality

∣∣f(x + y) − f(x) − f(y)∣∣ ≤ δ, (1.1)

for some fixedδ > 0 and allx, y ∈ R, whereR is the set of real numbers, then there existsa unique additive mappingA : R → R, satisfying the formula

A(x) = limn→∞

2−nf(2nx), (1.2)

and inequality

|f(x) − A(x)| ≤ δ (1.3)

for some fixedδ > 0 and all x ∈ R. If, moreover,f(tx) is continuous int for each fixedx ∈ R, thenA(tx) = tA(x) for all t, x ∈ R.

A : R → R is a unique linear additive mapping satisfying equation

A(x + y) = A(x) + A(y). (1.4)

In this paper we introduce a Cauchy type additive functional equation and investigatethe Hyers–Ulam stability of this equation.

2. Cauchy Type Additive Functional Equations

Definition 2.1. A mappingf : R → R is called approximately Cauchy type additive, if theapproximately Cauchy additive functional inequality

∣∣f(x + y) + f(x − y) + f(y − x) − f(x) − f(y)∣∣ ≤ ε (2.5)

holds for everyx, y ∈ R with ε ≥ 0.

Theorem 2.2. Assume thatf : R → R is an approximately Cauchy type additive mappingsatisfying(2.5). Define,fn(x) = 2−nf(2nx).

Then, there exists a unique Cauchy type additive mappingA : R → R such that

A(x) = limn→∞

2−nf(2nx) (2.6)

for all x ∈ R andn ∈ N = 1, 2, . . ., which is the set of natural numbers and

|f(x) − A(x)| ≤ 3ε (2.7)

for some fixedε > 0 and all x ∈ R. If, moreover,f(tx) is continuous int for each fixedx ∈ R, thenA(tx) = tA(x) for all t, x ∈ R.

A : R → R is a unique linear Cauchy type additive mapping satisfying equation

A(x + y) + A(x − y) + A(y − x) = A(x) + A(y). (2.8)

Hyers–Ulam Stability of Cauchy Type Additive Functional Equations 145

Proof. Step 1.By substitutingx = y = 0 andx = y in (2.5), respectively, we can observethat

|f(0)| ≤ ε (2.9)

and ∣∣f(x) − 2−1f(2x)∣∣ ≤ 3

2ε. (2.10)

Hence,for n ∈ N − 0∣∣f(x) − 2−nf(2nx)

∣∣ ≤∣∣f(x) − 2−1f(2x)

∣∣ +∣∣2−1f(2x) − 2−2f(22x)

∣∣ + · · ·

+∣∣2−(n−1)f(2n−1x) − 2−nf(2nx)

∣∣

≤3

2

(1 +

1

2+ · · · +

1

2n−1

)ε

= 3(1 − 2−n)ε. (2.11)

Step 2. Following, we need to show that if there is a sequencefn : fn(x) =2−nf(2nx), thenfn converges.

For everym > n > 0, we can obtain∣∣fm(x) − fn(x)

∣∣ =∣∣2−mf(2mx) − 2−nf(2nx)

∣∣

= 2−n∣∣2−(m−n)f(2mx) − f(2nx)

∣∣

≤ 2−n3(1 − 2−(m−n))ε

<3ε

2n→ 0,

for n → ∞. SinceR is completewe can conclude thatfn is convergent. Thus, there is awell-definedA : R → R such thatA(x) = lim

n→∞

2−nf(2nx).

Step 3.Observe that

|f(x) − fn(x)| =∣∣f(x) − 2−nf(2nx)

∣∣ ≤ 3(1 − 2−n)ε,

from which by lettingn → ∞ we obtain

|f(x) − A(x)| ≤ 3ε.

Step 4.By lettingx → 2nx andy → 2ny, from (2.5), we have:∣∣∣f(2n(x + y)) + f(2n(x − y)) + f(2n(y − x)) − f(2nx) − f(2ny)

∣∣∣ ≤ ε.

Next, by multiplying with2−n and by lettingn → ∞, we can conclude that truly existsan A : R → R such that:A(x) = lim

n→∞

2−nf(2nx) satisfies theCauchy-type additivityproperty

A(x + y) + A(x − y) + A(y − x) = A(x) + A(y). (2.12)

Step 5.We need to prove that A isunique.Observe, from (2.12), that

A(0) = 0 and A(2x) = 2A(x).

146 Matina J. Rassias

Therefore, byinductionhypothesiswe can show that

A(2nx) = 2A(2n−1x) = 2nA(x)

or equivalentlyA(x) = 2−nA(2nx).

Assume, now, the existence ofA′ : R → R, such thatA′(x) = 2−nA′(2nx). With the aidof the triangular inequality,

|A(x) − A′(x)| ≤∣∣2−nA(2nx) − 2−nf(2nx)

∣∣ +∣∣2−nf(2nx) − 2−nA′(2nx)

∣∣≤ 2−n3ε + 2−n3ε

→ 0,

asn → ∞. Thus, theuniquenessof A is proved and the stability ofCauchy-type additivemappingA : R → R is established.

Step 6.To complete the proof of Theorem 2.2, we only need to examine whetherA :R → R is a linear Cauchy-type mapping. To be more precise, we need to show that:

(1) A(x + y) + A(x − y) + A(y − x) = A(x) + A(y), and(2) A(rx) = rA(x), ∀r ∈ R.

Recall that we have shown already that(1) holds.Therefore, we only need to show that(2) is valid∀r ∈ R.

For that we will study four cases.Case 1:Let r = k ∈ N = 0, 1, 2, . . ..

For k = 0, from (2), we haveA(0) = 0. This is verified if we substitutex = y = 0 in(2.12).

Assume, thatA((k − 1)x

)= (k − 1)A(x) is true∀k.

Then, we need to prove thatA(kx) = kA(x).Note that forx = x, andy = 0 from (2.12), we can easily obtainA(−x) = (−1)A(x).Let x = x andy = (k − 1)x in (2.12). Then,

A(kx) + A(−(k − 2)x) + A((k − 2)x) = A(x) + A((k − 1)x),

orA(kx) = kA(x), ∀ k ∈ N = 0, 1, 2, . . ..

Case 2:Let r = k ∈ Z.

We only need to observe thatA is odd. Since, we have already proved that(2) is valid∀k ∈ N = 0, 1, 2, ... we can then conclude that

A(kx) = kA(x), ∀ k ∈ Z.

Case 3:Let r = kl∈ Q, for k ∈ Z, l ∈ Z − 0.

Then,A(x) = A(l 1

lx)

= lA(

1lx), for l ∈ Z − 0. Hence,A

(1lx)

= 1lA(x).

Besides,for k ∈ Z, A(

klx)

= A(k 1

lx)

= kA(1lx), from Case2.

Thus,A(

klx)

= klA(x), or A(rx) = rA(x) for r ∈ Q.

Case 4:Let r ∈ R, wherer = qn : rational numbers.

Hyers–Ulam Stability of Cauchy Type Additive Functional Equations 147

SinceR is acompletespace, every sequenceqn converges inR, i.e. limn→∞

qn =q∈R.

Recall thatA(x) = limn→∞

2−nf(2nx) andf(tx) is continuous int for each fixedx in R.

Therefore,A(tx) is continuous int for each fixedx in R. Besides,

limn→∞

A(qnx) = A(

limn→∞

qnx)

= A(qx) (2.13)

andlim

n→∞

A(qnx) = limn→∞

qnA(x) = qA(x). (2.14)

From (2.13) and (2.14)Case 4.is now proved, which completesStep 6.and thus the proofof our Theorem 2.2 .

References

[1] Hyers, D. H. (1941). On the stability of the linear functional equations,Proc. Nat.Acad. Sci., 27, 222–224: The Stability of Homomorphisms and Related Topics,‘Global Analysis – Analysis of Manifolds’,Teubner-Texte zur Mathematik, 57 (1983),140–153.

[2] Rassias, J. M. (1982). On approximation of approximately linear mappings by linearmappings,J. Funct. Anal.46, 126–130.

[3] Rassias, J. M. (1984). On approximation of approximately linear mappings by linearmappings,Bull. Sc. Math.,108, 445–446.

[4] Rassias, J. M. (1989). Solution of a problem of Ulam,J. Approx. Th.,57, 268–273.

[5] Ulam, S. M. (1964). A Collection of Mathematical problems,Interscience Publisher,Inc., No. 8, New York; Problems in Modern Mathematics,Wiley and Sons, New York,1964, Chapter VI.


Chapter 13

SOLUTION AND ULAM STABILITY OF A MIXED TYPECUBIC AND ADDITIVE FUNCTIONAL EQUATION

J.M. Rassias1,a, K. Ravi2,b, M. Arunkumar2,c

and B.V. Senth. Kumar3,d

1Pedagogical Department E. E., Section of Mathematics and Informatics,National and Capodistrian University of Athens, 4, Agamemnonos

Str., Aghia Paraskevi, Athens, Attikis 15342, GREECEhttp://www.primedu.uoa.gr/~jrassias

2Department of Mathematics, Sacred Heart College,Tirupattur-635 601, TamilNadu, India

3Department of Mathematics, C.Abdul Hakeem College of Engg. and Tech.,Melvisharam - 632 509,TamilNadu, India

Abstract

In this paper, the authors investigate the general solution and Ulam stability of mixedtype cubic and additive functional equation of the form

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

3

4 4

f x y z f x y z f x y z f x y z

f x f y f z f x y f x z f y z

+ + + − + + + − + + + −

+ ⎡ + + ⎤ = ⎡ + + + + + ⎤⎣ ⎦ ⎣ ⎦

(*)

introduced by the first author. We also investigate the first author’s stability of the equation(*) controlled by a mixed type product – sum of powers of norms .

2000 Mathematics Subject Classification: 39B52, 39B72, 39B82.

Key words and phrases. Additive function, Cubic function, Hyers-Ulam-Rassias stability,Ulam-Gavruta-Rassias stability, Mixed Type Product-Sum of powers of norms stability.

a E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected].

J.M. Rassias, K. Ravi, M. Arunkumar et al.150

1. Introduction

In 1940, S. M. Ulam [55] raised the following question concerning the stability of grouphomomorphisms: “Under what conditions does there is an additive mapping near anapproximately additive mapping between a group and a metric group ? ”

In 1941, D. H. Hyers [15] answered the stability problem of Ulam under the assumptionthat the groups are Banach spaces. In 1950, Aoki [3] generalized the Hyers theorem foradditive mappings. In 1978, Th. M. Rassias [47] provided a generalized version of thetheorem of Hyers which permitted the Cauchy difference to become bounded. Since then, thestability problems of various functional equations have been extensively investigated by anumber of authors [5], [8]-[10], [12]-[14], [59], [30], [44], [46], [49], [54].

In 1982, J. M. Rassias [39] gave a further generalization of the result of D. H. Hyers andproved theorems using weaker conditions controlled by a product of different powers ofnorms. Very recently, J. M. Rassias introduced the mixed type product sum of powers ofnorms [51]. The investigation of stability of functional equations involving a mixed typeproduct - sum of powers of norms is known as Ulam-Gavruta-Rassias stability.

Theorem 1.1 ([39]). Let :f E E′→ be a mapping from a normed vector space E into a

Banach space E′ subject to the inequality

( ) ( ) ( ) p pf x y f x f y x y+ − − ≤ ε

for all ,x y E∈ , where ε and p are constants with 0>ε and 102

p≤ < . Then the limit

( )2( ) lim

2

n

nn

f xL x

→∞=

exists for all x E∈ and :L E E′→ is the unique additive mapping which satisfies

22( ) ( )

2 2p

pf x L x x− ≤−ε

for all x E∈ .

The above-mentioned stability involving a product of different powers of norms is calledUlam-Gavruta-Rassias stability by M. A. Sibaha et al., [53], Nakmahachalasint [32,33], Raviand Arunkumar [50], Ravi and Senthil Kumar [52]. Besides, J. M. Rassias [42] alsointroduced and investigated the Euler-Lagrange type quadratic mappings, called Euler-Lagrange-Rassias quadratic mappings.


( ) ( ) ( ) ( )2 2f x y f x y f x f y+ + − = + (1.1)

Solution and Ulam Stability… 151

is said to be quadratic functional equation because the quadratic function ( ) 2f x a x= is a

solution of the functional equation (1.1). Quadratic functional equations were used tocharacterize inner product spaces [1], [2], [19]. A square norm on an inner product spacesatisfies the important parallelogram law

( )2 2 2 22 .x y x y x y+ + − = +

It is well known that a function f is a solution of (1.1) if and only if there exists a unique

symmetric bi-additive function B such that ( ) ( ),f x B x x= for all x [1, 26]. The bi-

additive function B is given by

( ) ( ) ( )1, .4

B x y f x y f x y= ⎡ + + − ⎤⎣ ⎦ (1.2)

Functional equation

( ) ( ) ( ) ( ) ( )2 2 2 2 12f x y f x y f x y f x y f x+ + − = + + − + (1.3)

is called cubic functional equation, because the cubic function ( ) 3f x c x= is a solution of

the equation (1.3). The general solution and the generalized Hyers-Ulam-Rassias stability forthe functional equation (1.3) was discussed by K. W. Jun and H. M. Kim [20]. They provedthat a function f between real vector spaces X and Y is a solution of (1.3) if and only if

there exists a unique function :C X X X Y× × → such that ( ) ( ), ,f x C x x x= for all

x X∈ and C is symmetric for each fixed one variable and is additive for fixed twovariables.

The general solution and the generalized Hyers-Ulam-Rassias stability of the cubicfunctional equation

( ) ( ) ( ) ( ) ( )3 3 3 3 48f x y f x y f x y f x y f x+ + − = + + − + (1.4)

on abelain groups was investigated by K. H. Park and Y. S. Jung [36].K. W. Jun and H. M. Kim [22] introduced the following generalized quadratic and

additive type functional equation

( ) ( ) ( )1 1 1

2n n

i i i ji i i j n

f x n f x f x x= = ≤ < ≤

⎛ ⎞+ − = +⎜ ⎟

⎝ ⎠∑ ∑ ∑ (1.5)

in the class of function between real vector spaces. For 3n = , Pl. Kannappan proved that afunction f satisfies the functional equation (1.5) if and only if there exists a symmetric bi-


additive function A and additive function B such that ( ) ( ) ( ),f x B x x A x= + for all

x [26]. The Hyers-Ulam stability for the equation 3n = was proved by S. M. Jung [24].The Hyers-Ulam-Rassias stability for the equation 4n = was also investigated byI. S. Chang et al [7].

The general solution and the generalized Hyers-Ulam stability for the quadratic andadditive type functional equation

( ) ( ) ( ) ( )f x ay af x y f x ay af x y+ + − = − + + (1.6)

for any positive integer a with 1,0,1a ≠ − was discussed by K. W. Jun and H. M. Kim[21].

The general solution and the generalized Hyers-Ulam stability for mixed type of cubicand quadratic functional equation of the form

( ) ( ) ( ) ( ) ( ) ( )6 6 4 3 3 3 2 9 2f x y f x y f y f x y f x y f x+ − − + = + − − + (1.7)

was investigated by I. S. Chang and Y. S. Jung [6].In [34], W. G. Park and J. H. Bae considered the following quartic functional equation

( ) ( ) ( ) ( ) ( ) ( )6 6 4 3 3 3 2 9 2f x y f x y f y f x y f x y f x+ − − + = + − − + (1.8)

and proved that a function f between real vector spaces X and Y is a solution of (1.8) ifand only if there exists a unique symmetric multi–additive function

:Q X X X X Y× × × → such that ( ) ( ), , ,f x Q x x x x= for all x X∈ .

Stability of a functional equation deriving from cubic and quartic functions of the form

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

4 3 3 12

12 2 2 8 192 2 30 2

f x y f x y f x y f x y

f x y f x y f y f x f y f x

⎡ + + − ⎤ = − ⎡ + + − ⎤⎣ ⎦ ⎣ ⎦+ ⎡ + + − ⎤ − − + +⎣ ⎦

(1.9)

and its general solution and the generalized Hyers-Ulam-Rassias stability was discussed byM. Eshaghi Gordji et al., [11].

In this paper, we discuss the general solution and Ulam stability of mixed type cubic andadditive functional equation and investigate the J. M. Rassias stability of Mixed Type product– sum of powers of norms for the equation

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

3

4 4 .



+ + + − + + + − + + + −

+ ⎡ + + ⎤ = ⎡ + + + + + ⎤⎣ ⎦ ⎣ ⎦(1.10)


It is easy to see that the function ( ) 3f x ax bx= + is a solution of the functional equation

(1.10) where ,a b are real constants.

2. General Solution of the Functional Equation (1.10)

Through out this section, we assume that A and B are two real vector spaces.

Lemma 2.1. Let :f A B→ be a function satisfying the functional equation (1.10) then fis an odd function.Proof. Letting ( ), ,x y z be ( )0,0,0 in (1.10), we get

( )0 0f = . (2.1)

Replacing ( ), ,x y z by ( ),0,0x in (1.10), we obtain

( ) ( )f x f x− = − (2.2)

for all x A∈ . Hence f is an odd function.

Lemma 2.2. Let :f A B→ be a function satisfying the functional equation (1.10) then fis a cubic function.

Proof. Replacing ( ), ,x y z by ( ), ,x y y z z x− − − and using (2.1), (2.2) in (1.10), we

obtain

( )( ) ( )( ) ( )( )( ) ( ) ( )

2 2 2

8

f x y f y z f z x

f x y f y z f z x

− + − + −

= ⎡ − + − + − ⎤⎣ ⎦(2.3)

for all , ,x y z A∈ . Setting ( ), ,x y y z z x− − − by ( ), ,u v w in (2.3), we arrive

( ) ( ) ( ) ( ) ( ) ( )2 2 2 8f u f v f w f u f v f w+ + = ⎡ + + ⎤⎣ ⎦ (2.4)

for all , ,u v w A∈ . Replacing ( ), ,u v w by ( ), ,u u u in (2.4), we obtain

( ) ( )2 8f u f u= (2.5)

for all u A∈ . Again replacing ( ), ,x y z by ( ), ,x x x in (1.10), we get


( ) ( ) ( )3 4 2 5f x f x f x= − (2.6)

for all x A∈ . Now with the help of (2.5), we arrive ( ) ( )3 27f x f x= for all x A∈ . In

general for any positive integer n , we obtain ( ) ( )3f nx n f x= for all x A∈ . Therefore

f is a cubic function.

Lemma 2.3. Let :f A B→ be a function satisfying the functional equation (1.10) then fis an additive function.

Proof. Replacing z by x y+ in (1.10), we get

( )( ) ( ) ( ) ( ) ( ) ( ) ( )3 2 2 2 4 4 4 2 4 2f x y f y f x f x f y f x y f x y+ + + + + = + + + (2.7)

for all ,x y A∈ . Again replacing z by x y− − in (1.10), we obtain

( ) ( ) ( ) ( ) ( ) ( )( )2 2 8 8 8 2f x f y f x y f x f y f x y+ + + = + + + (2.8)

for all ,x y A∈ . Multiplying (2.8) by 3 and using (2.7), we get

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 6 5 5 2 2f x f y f x y f x f y f x y f x y+ + + = + + + + + (2.9)

for all ,x y A∈ . Subtracting (2.9) from (2.8), we have

( ) ( ) ( ) ( ) ( )( ) ( )2 2 3 3 2 2f x y f x y f x f y f x y f x y+ + + = + + + − + (2.10)

for all ,x y A∈ . Substituting y by y− in (2.10), we arrive

( ) ( ) ( ) ( ) ( )( ) ( )2 2 3 3 2 2f x y f x y f x f y f x y f x y− + − = − + − − − (2.11)

for all ,x y A∈ . Adding (2.10) and (2.11), we get

( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )

2 2 2 2

6 2 2 2 2

f x y f x y f x y f x y

f x f x y f x y f x y f x y

+ + + + − + −

= + + − + + − − − (2.12)

for all ,x y A∈ . Setting ( ),x y x y+ − by ( ),u v in (2.12), we arrive


( ) ( ) ( ) ( )

( ) ( ) ( ) ( )6 2 2 2 22

f u x f u y f x v f x v

u v f u f v f u f v

+ + + + + + −

+⎛ ⎞= + + − −⎜ ⎟⎝ ⎠

(2.13)

for all , , ,x y u v A∈ . Replacing ( ), , ,x y u v by ( ), , ,z z z z in (2.13), we obtain

( ) ( )2 2f z f z= (2.14)

for all z A∈ . Again replacing ( ), ,x y z by ( ), ,x x x in (1.10), we get

( ) ( ) ( )3 4 2 5f x f x f x= − (2.15)

for all x A∈ . With the help of (2.14), we arrive ( ) ( )3 3f x f x= for all x A∈ . In

general for any positive integer n , we obtain ( ) ( )f nx n f x= for all x A∈ . Therefore

f is an additive function.

Theorem 2.4. A mapping :f A B→ is a function satisfying the functional equation (1.10)for all , ,x y z A∈ , if and only if there exists two mappings :T A A A B× × → and

:R A A A B× × → such that ( ) ( ) ( ), ,f x T x x x R x= + for all x A∈ , where T is

symmetric for each fixed one variable and is additive for fixed two variables and R isadditive.

Proof. Replacing z by z− in (1.10), we get

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

3

4 4



+ − + − + − + − − + + +

+ ⎡ + − ⎤ = ⎡ + + − + − ⎤⎣ ⎦ ⎣ ⎦(2.16)

for all , ,x y z A∈ . Adding (1.10) and (2.16) and dividing by 4, we arrive

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

2 2

2

f x y z f x y z f x f y

f x y f x z f y z f x z f y z

+ + + + − + +

= + + + + + + − + − (2.17)

for all , ,x y z A∈ . This equation (2.17) was already dealt by H.M. Kim [28]. ByTheorem 2.1 [23], the proof is completed.


3. Generalized Ulam Stability of the Functional Equation (1.10)

For convenience, we define,

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

, , 3

4 4

D f x y z f x y z f x y z f x y z f x y z


= + + + − + + + − + + + −

+ ⎡ + + ⎤ − ⎡ + + + + + ⎤⎣ ⎦ ⎣ ⎦for all , ,x y z A∈ .

Theorem 3.1. Let A be a real vector space and B be a Banach space. Let: [0, )A A Aα × × → ∞ be a function such that

( )0

3 ,3 ,327

i i i

ii

x y zα∞

=∑ converges and

( )3 ,3 ,3lim 0

27

n n n

nn

x y zα→∞

= (3.1)

for all , ,x y z A∈ . If :cf A B→ is a cubic function satisfying

( ) ( ), , , ,cD f x y z x y zα≤ (3.2)

for all , ,x y z A∈ , then there exists a unique cubic function :T A B→ satisfying (1.10)and

( ) ( ) ( )0

3 ,3 ,3181 27

k k k

c kk

x x xf x T x

α∞

=

− ≤ ∑ (3.3)

for all x A∈ . Function ( )T x is defined by

( ) ( )3lim

27

nc

nn

f xT x

→∞= (3.4)

for all x A∈ .

Proof. Replacing ( ), ,x y z by ( ), ,x x x in (3.2) and using (2.5), we get

( ) ( ) ( )3 1 , ,27 81

cc

f xf x x x xα− ≤ (3.5)

for all x A∈ . Replacing x by 3x and divided by 27 in (3.5) and adding the resultantinequality with (3.5), we obtain


( ) ( ) ( ) ( )2

2

3 3 ,3 ,31 , ,27 81 27

cc

f x x x xf x x x x

αα⎡ ⎤

− ≤ +⎢ ⎥⎣ ⎦

(3.6)

for all x A∈ . In general for any positive integer n , we have

( ) ( ) ( )1

0

3 3 ,3 ,3127 81 27

n k k knc

cn kk

f x x x xf x

α−

=

− ≤ ∑

( )0

3 ,3 ,3181 27

k k k

kk

x x xα∞

=

≤ ∑ (3.7)

for all x A∈ . We have to show that sequence ( )327

nc

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

converges for all x A∈ .

Replacing x by 3m x and dividing by 27m in (3.7) for any , 0n m > , we obtain

( ) ( ) ( ) ( )

( )1

0

3 3 3 3 31 327 27 27 27 27

3 ,3 ,31 127 81 27

n m m n mc c c m

cm n m m n

k m k m k mn

m kk

f x f x f xf x

x x xα + + +−

=

− ≤ −

≤ ∑

( )

0

3 ,3 ,3181 27

k m k m k m

k mk

x x xα + + +∞

+=

≤ ∑ (3.8)

for all x A∈ . By condition (3.1) the right hand side of (3.8) converges to 0 as n →∞ .

Thus the sequence ( )327

nc

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

is a Cauchy sequence. Due to completeness of the Banach

space B , there exists a mapping :T A B→ such that

( ) ( )3lim ,

27

nc

nn

f xT x

→∞= for all .x A∈

Letting n →∞ in (3.7), we obtain (3.3). To show that T satisfies (1.10), we are setting

( ), ,x y z by ( )3 ,3 ,3n n nx y z in (3.2) and dividing by 27n , we get


( )( ) ( )( ) ( )( ) ( )( )( ) ( ) ( ) ( )( ) ( )( ) ( )( )

( )

1 3 3 3 3 327

4 3 3 3 4 3 3 3

3 ,3 ,3.

27

n n n nc c c cn

n n n n n nc c c c c c

n n n

n



x y zα

+ + + − + + + − + + + −

⎡ ⎤ ⎡ ⎤+ + + − + + + + +⎣ ⎦ ⎣ ⎦

≤

for all , ,x y z A∈ . Taking limit n →∞ and using the definition of ( )T x in the above

inequality, it becomes

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

3

4 4

T x y z T x y z T x y z T x y z

T x T y T z T x y T x z T y z

+ + + − + + + − + + + −

+ ⎡ + + ⎤ = ⎡ + + + + + ⎤⎣ ⎦ ⎣ ⎦

for all , ,x y z A∈ . Therefore T satisfies (1.10). To prove uniqueness of T , suppose thatthere exists another cubic mapping :U A B→ satisfying (3.3) and (3.4). Therefore

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )0

0

127

3 ,3 ,31 227 81 27

3 ,3 ,3281 27

n

k n k n k n

n kk

k n k n k n

k nk

T x U x T x f x f x U x

x x x

x x x

α

α

+ + +∞

=

+ + +∞

+=

− ≤ − + −

≤

≤

∑

∑

for all x A∈ . By condition (3.1), the right hand side goes to 0 as n →∞ and it follows

that ( ) ( )T x U x= for all x A∈ . Hence T is unique. Hence the proof is complete.


127 , ,

3 3 3i

i i ii

x y zα∞

=

⎛ ⎞⎜ ⎟⎝ ⎠

∑ converges and lim 27 , , 03 3 3

nn n nn

x y zα→∞

⎛ ⎞ =⎜ ⎟⎝ ⎠

(3.9)

for all , ,x y z A∈ . If :cf A B→ is a cubic function satisfying (3.2) for all , ,x y z A∈ .

Then there exists a unique cubic function :T A B→ which satisfies (1.10) and


( ) ( )1

1 27 , ,81 3 3 3

kc k k k

k

x x xf x T x α∞

=

⎛ ⎞− ≤ ⎜ ⎟⎝ ⎠

∑ (3.10)

for all x A∈ . The function ( )T x is defined by

( ) lim 273

nc nn

xT x f→∞

⎛ ⎞= ⎜ ⎟⎝ ⎠

(3.11)

for all x A∈ .

Proof. Replacing x by 3x

in (3.5) and proceeding the same way as that of Theorem 3.1 the

proof is complete.

Corollary 3.3. Let A be a real normed space and B be a Banach space. If :cf A B→ isa cubic function satisfying the functional inequality

( ) , , p p pcD f x y z x y zε≤ + + (3.12)

with 3p < (or) 3p > , for some 0ε > and for all , ,x y z A∈ . Then there exists aunique cubic function :T A B→ which satisfies (1.10) and

( ) ( )3,

27 3

33 27

p

p

c p

p

xfor p

f x T xx

for p

ε

ε

⎧<⎪

⎪ −− ≤ ⎨⎪

>⎪ −⎩

(3.13)

for all x A∈ .

Proof. If we choose ( ) , , p p px y z x y zα ε= + + for all , ,x y z A∈ . Then by

Theorem 3.1, we arrive

( ) ( )27 3

p

c p

xf x T x

ε− ≤

−, for all x A∈ and 3p < .

Using Theorem 3.2, we arrive

( ) ( )3 27

p

c p

xf x T x

ε− ≤

−, for all x A∈ and 3p > .


Corollary 3.4. Let A be a real normed space and B be a Banach space. If a cubic function:cf A B→ satisfies the functional inequality

( ), ,cD f x y z ε≤ (3.14)

for some 0ε > and for all , ,x y z A∈ . Then there exists a unique cubic function:T A B→ which satisfies (1.10) and

( ) ( )78cf x T x ε

− ≤ (3.15)

for all x A∈ .

Proof. If we choose ( ), ,x y zα ε= for all , ,x y z A∈ . Then by Theorem 3.1 it follows

that

( ) ( )78cf x T x ε

− ≤ , for all x A∈ .

Corollary 3.5. If :cf A B→ is a cubic function from a normed vector space A into a

Banach space B satisfies

( ), , p p pcD f x y z x y zε≤ (3.16)

for all , ,x y z A∈ , where and pε are constants with 1p < (or) 1p > , then there

exists a unique cubic function :T A B→ which satisfies (1.10) and

( ) ( )

3

3 1

3

3 1

1,81 3

13 81

p

p

c p

p

xfor p

f x T xx

for p

ε

ε

+

+

⎧<⎪

⎪ −− ≤ ⎨⎪

>⎪ −⎩

(3.17)

for all x A∈ .

Proof. If we choose ( ), , p p px y z x y zα ε= for all , ,x y z A∈ . Then by


( ) ( )3

3 181 3

p

c p

xf x T x

ε+− ≤




( ) ( )3

3 13 81

p

c p

xf x T x

ε+− ≤−

, for all x A∈ and 1p > .


( )0

3 ,3 ,33

i i i

ii

x y zα∞

=∑ converges and

( )3 ,3 ,3lim 0

3

n n n

nn

x y zα→∞

= (3.18)

for all , ,x y z A∈ . If :af A B→ is a additive function satisfying

( ) ( ), , , ,aD f x y z x y zα≤ (3.19)

for all , ,x y z A∈ . Then there exists a unique additive function :R A B→ which satisfies(1.10) and

( ) ( ) ( )0

3 ,3 ,319 3

k k k

a kk

x x xf x R x

α∞

=

− ≤ ∑ (3.20)

for all x A∈ . The function ( )R x is defined by

( ) ( )3lim

3

na

nn

f xR x

→∞= (3.21)

for all x A∈ .


( ) ( ) ( )3 1 , ,3 9

aa

f xf x x x xα− ≤ (3.22)

for all x A∈ . Replacing x by 3x and dividing by 3 in (3.22) and adding the resultantinequality with (3.22), we obtain

( ) ( ) ( ) ( )2

2

3 3 ,3 ,31 , ,3 9 3

aa

f x x x xf x x x x

αα⎡ ⎤

− ≤ +⎢ ⎥⎣ ⎦

(3.23)



( ) ( ) ( )1

0

3 3 ,3 ,313 9 3

n k k kna

an kk

f x x x xf x

α−

=

− ≤ ∑

( )0

3 ,3 ,319 3

k k k

kk

x x xα∞

=

≤ ∑ (3.24)

for all x A∈ . We have to show that the sequence ( )33

na

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

converges for all x A∈ .

Replacing x by 3m x and divide by 3m in (3.24) for any , 0n m > , we obtain

( ) ( ) ( ) ( )

( )1

0

3 3 3 3 31 33 3 3 3 3

3 ,3 ,31 13 9 3

n m m n ma a a m

am n m m n

k m k m k mn

m kk

f x f x f xf x

x x xα + + +−

=

− ≤ −

≤ ∑

( )0

3 ,3 ,319 3

k m k m k m

k mk

x x xα + + +∞

+=

≤ ∑ (3.25)

for all x A∈ . By condition (3.18) the right hand side of (3.25) converges to zero as

n →∞ . Thus the sequence ( )33

na

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

is a Cauchy sequence. Due to completeness of

the Banach space B , there exists a mapping :R A B→ such that

( ) ( )3lim ,

3

na

nn

f xR x


Letting n →∞ in (3.24), we establish (3.20). To show that R satisfies (1.10) and isunique, the proof will be similar to that of Theorem 3.1.


1

3 , ,3 3 3

ii i i

i

x y zα∞

=

⎛ ⎞⎜ ⎟⎝ ⎠

∑ converges and lim 3 , , 03 3 3

nn n nn

x y zα→∞

⎛ ⎞ =⎜ ⎟⎝ ⎠

(3.26)


for all , ,x y z A∈ . If :af A B→ is a additive function satisfying (3.19) for all

, ,x y z A∈ . Then there exists a unique additive function :R A B→ which satisfies (1.10)and

( ) ( )1

1 3 , ,9 3 3 3

ka k k k

k

x x xf x R x α∞

=

⎛ ⎞− ≤ ⎜ ⎟⎝ ⎠

∑ (3.27)

for all x A∈ . The function ( )R x is defined by

( ) lim 33

na nn

xR x f→∞

⎛ ⎞= ⎜ ⎟⎝ ⎠

(3.28)

for all x A∈ .



proof is complete.

Corollary 3.8. Let A be a real normed space and B be a Banach space. If :af A B→ isa additive function satisfying the functional inequality

( ) , , p p paD f x y z x y zε≤ + + (3.29)

with 1p < and 1p > , for some 0ε > and for all , ,x y z A∈ . Then there exists a uniqueadditive function :R A B→ which satisfies (1.10) and

( ) ( )1,

3 3

13 3

p

p

a p

p

xfor p

f x R xx

for p

ε

ε

⎧<⎪

⎪ −− ≤ ⎨⎪

>⎪ −⎩

(3.30)

for all x A∈ .

Proof. If we choose ( ) , , p p px y z x y zα ε= + + for all , ,x y z A∈ . Then by


( ) ( )3 3

p

a p

xf x R x

ε− ≤




( ) ( )3 3

p

a p

xf x R x

ε− ≤

−, for all x A∈ and 1p > .

Corollary 3.9. Let A be a real normed space and B be a Banach space. If a additivefunction :af A B→ satisfies the functional inequality

( ), ,aD f x y z ε≤ (3.31)

for some 0ε > and for all , ,x y z A∈ . Then there exists a unique additive function:R A B→ which satisfies (1.10) and

( ) ( )6af x R x ε

− ≤ (3.32)

for all x A∈ .

Proof. If we choose ( ), ,x y zα ε= for all , ,x y z A∈ . Then by Theorem 3.6 it follows

that

( ) ( )6af x R x ε

− ≤ , for all x A∈ .

Corollary 3.10. If :af A B→ is a additive function from a normed vector space A into a

Banach space B satisfies

( ), , p p paD f x y z x y zε≤ (3.33)

for all , ,x y z A∈ where and pε are constants with 13

p < (or) 13

p > , then there

exists a unique additive function :R A B→ which satisfies (1.10) and

( ) ( )

3

3 1

3

3 1

1 ,9 3 3

13 9 3

p

p

a p

p

xfor p

f x R xx

for p

ε

ε

+

+

⎧<⎪

⎪ −− ≤ ⎨⎪

>⎪ −⎩

(3.34)

for all x A∈ .

Proof. If we choose ( ), , p p px y z x y zα ε= for all , ,x y z A∈ . Then by



( ) ( )3

3 19 3

p

a p

xf x R x

ε+− ≤

−, for all x A∈ and

13

p < .


( ) ( )3

3 13 9

p

a p

xf x R x

ε+− ≤−

, for all x A∈ and 13

p > .

Theorem 3.11. Let A be a real vector space and B be a Banach space. Let: [0, )A A Aα × × → ∞ be a function satisfying (3.1), (3.9), (3.18), (3.26) for all

, ,x y z A∈ . If :f A B→ be a function satisfying

( ) ( ), , , ,D f x y z x y zα≤ (3.35)

for all , ,x y z A∈ . Then there exists a unique cubic function :T A B→ and a uniqueadditive function :R A B→ which satisfies (1.10) and

( ) ( ) ( ) ( ) ( )0 0

3 ,3 ,3 3 ,3 ,31 181 27 9 3

k k k k k k

k kk k

x x x x x xf x T x R x

α α∞ ∞

= =

− − ≤ +∑ ∑ (3.36)

and

( ) ( ) ( )1 1

1 127 , , 3 , ,81 3 3 3 9 3 3 3

k kk k k k k k

k k

x x x x x xf x T x R x α α∞ ∞

= =

⎛ ⎞ ⎛ ⎞− − ≤ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑ (3.37)

for all x A∈ .

Proof. Let ( ) ( ) ( )c af x f x f x= + . Then

( ) ( ) ( ) ( ) ( ) ( ) ( )c af x T x R x f x f x T x R x− − = + − −

( ) ( ) ( ) ( )c af x T x f x R x≤ − + − (3.38)

Using Theorems 3.1, 3.6 and (3.38), we arrive

( ) ( ) ( ) ( ) ( )0 0

3 ,3 ,3 3 ,3 ,31 181 27 9 3

k k k k k k

k kk k

x x x x x xf x T x R x

α α∞ ∞

= =

− − ≤ +∑ ∑for all x A∈ .


Again using Theorems 3.2, 3.7 and (3.38), we arrive

( ) ( ) ( )1 1

1 127 , , 3 , ,81 3 3 3 9 3 3 3

k kk k k k k k

k k

x x x x x xf x T x R x α α∞ ∞

= =

⎛ ⎞ ⎛ ⎞− − ≤ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑for all x A∈ .

Corollary 3.12. Let A be a real normed space and B be a Banach space. Let :f A B→be function satisfying the functional inequality

( ) , , p p pD f x y z x y zε≤ + + (3.39)

with 3p < (or) 1p > for some 0ε > and for all , ,x y z A∈ . Then there exists a uniquecubic function :T A B→ and a unique additive function :R A B→ which satisfies(1.10) and

( ) ( ) ( )

1 1 3,27 3 3 3

1 1 13 27 3 3

pp p

pp p

x for pf x T x R x

x for p

ε

ε

⎧ ⎧ ⎫+ <⎨ ⎬⎪ − −⎪ ⎩ ⎭− − ≤ ⎨⎧ ⎫⎪ + >⎨ ⎬⎪ − −⎩ ⎭⎩

(3.40)

for all x A∈ .

Proof. Using Corollaries 3.3, 3.8 and (3.38), we arrive

( ) ( ) ( ) 1 127 3 3 3

pp pf x T x R x xε ⎧ ⎫− − ≤ +⎨ ⎬− −⎩ ⎭

, for all x A∈ and 3p < .

Again using Corollaries 3.3, 3.8 and (3.38), we arrive

( ) ( ) ( ) 1 13 27 3 3

pp pf x T x R x xε ⎧ ⎫− − ≤ +⎨ ⎬− −⎩ ⎭

, for all x A∈ and 1p > .

Corollary 3.13. Let A be a real normed space and B be a Banach space. If a function:f A B→ satisfies the functional inequality

( ), ,D f x y z ε≤ (3.41)

for some 0ε > and for all , ,x y z A∈ . Then there exists a unique cubic function:T A B→ and a unique additive function :R A B→ which satisfies (1.10) and


( ) ( ) ( ) 739

f x T x R x ε− − ≤ (3.42)

for all x A∈ .

Proof. Using Corollaries 3.4 and 3.9 and (3.38), we arrive

( ) ( ) ( ) 739

f x T x R x ε− − ≤ , for all x A∈ .

Corollary 3.14. If :f A B→ be a function from a normed vector space A into a Banachspace B satisfies

( ), , p p pD f x y z x y zε≤ (3.43)

for all , ,x y z A∈ , where and pε are constants, with 1p < (or) 13

p > then there

exists a unique cubic function :T A B→ and a unique additive function :R A B→which satisfies (1.10) and

( ) ( ) ( )

33 1 3 1

33 1 3 1

1 1 1,81 3 9 3

1 1 13 81 3 9 3

pp p

pp p

x for pf x T x R x

x for p

ε

ε

+ +

+ +

⎧ ⎧ ⎫+ <⎨ ⎬⎪ − −⎪ ⎩ ⎭− − ≤ ⎨⎧ ⎫⎪ + >⎨ ⎬⎪ − −⎩ ⎭⎩

(3.44)

for all x A∈ .

Proof. Using Corollaries 3.5, 3.10 and (3.38), we arrive

( ) ( ) ( ) 33 1 3 1

1 181 3 9 3

pp pf x T x R x xε + +

⎧ ⎫− − ≤ +⎨ ⎬− −⎩ ⎭,

for all x A∈ and 1p < .Again using Corollaries 3.5, 3.10 and (3.38), we arrive

( ) ( ) ( ) 33 1 3 1

1 13 81 3 9


⎧ ⎫− − ≤ +⎨ ⎬− −⎩ ⎭

,

for all x A∈ and 13

p > .


4. Mixed Type Product – Sum Stability of FunctionalEquation (1.10)

Through out this section, we assume that A be a normed space and B be a Banach spacerespectively.

Theorem 4.1. If :cf A B→ is a cubic function satisfying

( ) ( ) 3 3 3, , p p p p p pcD f x y z x y z x y zε≤ + + + (4.1)

for all , ,x y z A∈ , where ε and p are constants with 0ε > and 1p < . Then thereexists a unique cubic function :T A B→ defined by (3.4) satisfies (1.10) and

( ) ( ) 33 1

481 3

pc pf x T x xε

+− ≤−

(4.2)

for all x A∈ .

Proof. Setting ( ), ,x y z by ( )0,0,0 in (4.1), we get ( )0 0f = . Replacing ( ), ,x y z by

( ), ,x x x in (4.1) and using (2.5), we get

( ) ( ) 33 427 81

pcc

f xf x xε

− ≤ (4.3)


( ) ( )2 3

32

3 4 3127 81 27

ppc

c

f xf x xε ⎡ ⎤

− ≤ +⎢ ⎥⎣ ⎦

(4.4)


( ) ( )31

3

0

3 4 327 81 27

kn pnpc

cnk

f xf x xε −

=

⎛ ⎞− ≤ ⎜ ⎟

⎝ ⎠∑

33

0

4 381 27

kpp

kxε ∞

=

⎛ ⎞≤ ⎜ ⎟

⎝ ⎠∑ (4.5)



nc

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

is a Cauchy sequence,

replacing x by 3m x and divide by 27m in (4.5) for any , 0n m > , we obtain

( ) ( ) ( ) ( )

31 3

0

3 3 3 3 31 327 27 27 27 27

1 4 3 327 81 27

n m m n mc c c m

cm n m m n

kpn pmm

k

f x f x f xf x

xε −

=

− ≤ −

⎛ ⎞≤ ⎜ ⎟

⎝ ⎠∑31

3

0

4 381 27

k mpnp

kxε

+−

=

⎛ ⎞≤ ⎜ ⎟

⎝ ⎠∑

( )( )3

3 10

4 181 3

p

p k mk

xε ∞

− +=

≤ ∑ (4.6)

for all x A∈ . As 1p < , the right hand side of (4.6) tends to 0 as m →∞ . Thus the

sequence ( )327

nc

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

is a Cauchy sequence. Since B is complete, there exists a mapping

:T A B→ such that

( ) ( )3lim ,

27

nc

nn

f xT x


Letting n →∞ in (4.5), we arrive (4.2). To show T satisfies (1.10) and it is unique theproof is similar to that of Theorem 3.1.

Theorem 4.2. If :cf A B→ is a cubic function satisfying (4.1) for all , ,x y z A∈ , where

ε and p are constants with 0ε > and 1p > . Then there exists a unique cubic function:T A B→ defined by (3.11) satisfies (1.10) and

( ) ( ) 33 1

43 81

pc pf x T x xε

+− ≤−

(4.7)

for all x A∈ .



proof is complete.


Theorem 4.3. If :af A B→ is a additive function satisfying

( ) ( ) 3 3 3, , p p p p p paD f x y z x y z x y zε≤ + + + (4.8)

for all , ,x y z A∈ , where ε and p are constants with 0ε > and 13

p < . Then there

exists a unique additive function :R A B→ defined by (3.21) satisfies (1.10) and

( ) ( ) 33 1

49 3

pa pf x R x xε

+− ≤−

(4.9)

for all x A∈ .


( ) ( ) 33 43 9

paa

f xf x xε

− ≤ (4.10)


( ) ( )2 3

32

3 4 313 9 3

ppa

a

f xf x xε ⎡ ⎤

− ≤ +⎢ ⎥⎣ ⎦

(4.11)


( ) ( )31

3

0

3 4 33 9 3

kn pnpa

ank

f xf x xε −

=

⎛ ⎞− ≤ ⎜ ⎟

⎝ ⎠∑

33

0

4 39 3

kpp

kxε ∞

=

⎛ ⎞≤ ⎜ ⎟

⎝ ⎠∑ (4.12)


na

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

is a Cauchy sequence,

replacing x by 3m x and divide by 3m in (4.12) for any , 0n m > , we obtain


( ) ( ) ( ) ( )

313

0

3 3 3 3 31 33 3 3 3 3

4 39 3

n m m n ma a a m

am n m m n

k mpnp

k

f x f x f xf x

xε+−

=

− ≤ −

⎛ ⎞≤ ⎜ ⎟

⎝ ⎠∑

( )( )3

1 30

4 19 3

p

p k mk

xε ∞

− +=

≤ ∑ (4.13)

for all x A∈ . As 13

p < , the right hand side of (4.13) tends to 0 as m →∞ .

Thus the sequence ( )33

na

n

f x⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

is a Cauchy sequence. Since B is complete, there

exists a mapping :R A B→ such that

( ) ( )3lim ,

3

na

nn

f xR x


Letting n →∞ in (4.12), we arrive (4.9). To show R satisfies (1.10) and it is uniquethe proof is similar to that of Theorem 3.6.

Theorem 4.4. If :af A B→ is a additive function satisfying (4.8) for all , ,x y z A∈ ,

where ε and p are constants with 0ε > and 13

p > . Then there exists a unique additive

function :R A B→ defined by (3.28) satisfies (1.10) and

( ) ( ) 33 1

43 9

pa pf x R x xε

+− ≤−

(4.14)

for all x A∈ .



proof is complete.

Theorem 4.5. If :f A B→ be a function satisfying

( ) ( ) 3 3 3, , p p p p p pD f x y z x y z x y zε≤ + + + (4.15)


for all , ,x y z A∈ , where ε and p are constants with 0ε > and 1p < (or) 13

p > .

Then there exists a unique cubic function :T A B→ and a unique additive function:R A B→ which satisfies (1.10) and

( ) ( ) ( ) 33 1 3 1

1 14 , 181 3 9 3

pp pf x T x R x x for pε + +

⎧ ⎫− − ≤ + <⎨ ⎬− −⎩ ⎭ (4.16)

for all x A∈ . Also

( ) ( ) ( ) 33 1 3 1

1 1 14 ,3 81 3 9 3

pp pf x T x R x x for pε + +

⎧ ⎫− − ≤ + >⎨ ⎬− −⎩ ⎭ (4.17)

for all x A∈ .

Proof. Using Theorems 4.1, 4.3 and (3.38), we arrive

( ) ( ) ( ) 33 1 3 1

1 14 ,81 3 9 3


⎧ ⎫− − ≤ +⎨ ⎬− −⎩ ⎭

for all x A∈ and 1p < .Again using Theorems 4.2, 4.4 and (3.38), we arrive

( ) ( ) ( ) 33 1 3 1

1 14 ,3 81 3 9


⎧ ⎫− − ≤ +⎨ ⎬− −⎩ ⎭

for all x A∈ and 13

p > .

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Chapter 14

STABILITY OF M APPINGS APPROXIMATELY

PRESERVING ORTHOGONALITY

AND RELATED TOPICS

Aleksej Turnsek∗

Faculty of maritime studies and transportUniversity of Ljubljana

Pot pomorscakov 4, 6320 PortorozSlovenia

Abstract

Speaking of thestabilitywe follow the question of S. Ulam [21, p.63]: “when is ittrue that by changing a little the hypotheses of a theorem one can still assert that thethesis of the theorem remains true or approximately true?” In this chapter we studystability of mappings between Hilbert spaces which nearly preserve orthogonality, in-ner product or its absolute value. In the first section we present some results on linearapproximately orthogonality preserving mappings. In the second section we study or-thogonality equation and in the last one we present some stability results on Wignerequation.

2000 Mathematics Subject Classifications:Primary: 39B82; Secondary: 39B72, 46C99,47B99.

Key words: Approximate orthogonality; Orthogonality equation; Wigner equation; Stabil-ity.

1. Mappings Approximately Preserving Orthogonality

Let H andK be real or complex Hilbert spaces with an inner product denoted by〈·, ·〉.As usual, vectorsx andy are said to be orthogonal,x ⊥ y, if 〈x, y〉 = 0. A mappingT : H → K is calledorthogonality preserving, if it preserves orthogonality, that is

x ⊥ y ⇒ Tx ⊥ Ty, x, y ∈ H.


178 Aleksej Turnsek

It is known that a linear mappingT is orthogonality preserving if and only ifT = γU ,whereU is an isometry andγ ≥ 0, see [4]. Let us say that for a givenε ∈ [0, 1) vectorsx, y ∈ H are approximately orthogonal orε-orthogonal, denoted byx ⊥ε y, if

|〈x, y〉| ≤ ε‖x‖ ‖y‖.

Thus one can consider the class ofapproximately orthogonality preservingmappings as allthose satisfying the condition

x ⊥ y ⇒ Tx ⊥ε Ty, x, y ∈ H.

Hence, the natural stability question is whether an approximately orthogonality preservinglinear mappingT : H → K must be close to a linear orthogonality preserving mapping.More precisely: ifT : H → K is a linear approximately orthogonality preserving mapping,is there a linear orthogonality preserving mappingV : H → K andδ(ε) > 0, such that

‖T − V ‖ ≤ δ(ε)min‖T‖, ‖V ‖

and δ(ε) → 0 as ε → 0.

Let us fix some notation. The Banach space of all bounded linear operators fromH toK is denoted byB(H,K) and we writeB(H) for B(H,H). The spectrum of an operatorT

is denoted byσ(T ).In the next lemma we show that linear approximately orthogonality preserving map-

pings are automatically bounded and “almost” multiples of isometries.

Lemma 1.1([4], [20]). LetH andK be Hilbert spaces andT : H → K a nonzero linearapproximately orthogonality preserving mapping,ε ∈ [0, 1). Then

√1 − ε

1 + ε‖T‖ ‖x‖ ≤ ‖Tx‖ ≤ ‖T‖ ‖x‖, x ∈ H.

Proof. We show that for unit vectorsu andv

√1 − ε

1 + ε‖Tv‖ ≤ ‖Tu‖ ≤

√1 + ε

1 − ε‖Tv‖. (1)

If u andv are linearly dependent, then (1) is satisfied. Hence, we may assume thatu andv

are linearly independent unit vectors. Chooseλ ∈ C, |λ| = 1, such that〈u, λv〉 ∈ R. (If His real one can takeλ = 1). Thenu + λv ⊥ u − λv and henceTu + λTv ⊥ε Tu − λTv.Therefore, ∣∣⟨Tu + λTv, Tu − λTv

⟩∣∣ ≤ ε‖Tu + λTv‖ ‖Tu − λTv‖.

This is equivalent to

(‖Tu‖2 − ‖Tv‖2

)2+(2Im〈Tu, λTv〉)2 ≤ ε2

[(‖Tu‖2 + ‖Tv‖2

)2− (2Re〈Tu, λTv〉)2

],

hence ∣∣ ‖Tu‖2 − ‖Tv‖2∣∣ ≤ ε

(‖Tu‖2 + ‖Tv‖2

),

which gives (1). Now it follows from (1) thatT is bounded and then that√

1 − ε

1 + ε‖T‖ ≤ ‖Tu‖ ≤ ‖T‖.

Stability of Mappings. . . 179

Thus foru = x/‖x‖ oneobtains

√1 − ε

1 + ε‖T‖ ‖x‖ ≤ ‖Tx‖ ≤ ‖T‖ ‖x‖, x ∈ H, (2)

and the proof is completed.

Let T ∈ B(H,K) and letT = U |T | be its polar decomposition, where|T | ∈ B(H)is a positive square root ofT ∗T andU ∈ B(H,K) is a partial isometry. Usually the polardecomposition is stated for the operators inB(H), see [18, p.96], however for the operatorsin B(H,K) the proof is the same with obvious modifications.

Lemma 1.2([20], Lemma 2.1). LetH andK be Hilbert spaces,T ∈ B(H, K) andT =U |T | its polar decomposition. If form, M > 0

m‖x‖ ≤ ‖Tx‖ ≤ M‖x‖, x ∈ H, (3)

thenU is an isometry and

‖T − U‖ ≤ max|M − 1|, |m − 1|

.

Proof. Assume first thatH is a complex Hilbert space. From (3) it follows thatT is injectivewhich shows thatU is an isometry. Thus a positive operator|T | satisfies (3) as well, hence|T | is invertible. The inequalities

1

M‖x‖ ≤

∥∥ |T |−1x∥∥ ≤

1

m‖x‖

and (3) imply thatσ(|T |) ⊆ [m, M ]. Thusσ(|T | − 1) ⊆ [m − 1, M − 1] and, sincethe norm of a selfadjoint operator equals its spectral radius, it follows that‖|T | − 1‖ ≤max|M − 1|, |m − 1|. Therefore

‖T − U‖ = ‖U(|T | − 1)‖ =∥∥ |T | − 1

∥∥ ≤ max|M − 1|, |m − 1|

.

If H is a real Hilbert space, letHC be its complexification, see [11, p. 150] for thedetails, and|T |C : HC → HC a positive linear operator defined by

|T |C(x, y) = (|T |x, |T |y).

Then it is easy to check that|T |C also satisfies (3), and by the first part of the proof, itfollows that

‖|T |C − 1‖ ≤ max|M − 1|, |m − 1|

.

The proof is completed using the fact thatU is an isometry and that the latter inequalityimplies‖|T | − 1‖ ≤ max

|M − 1|, |m − 1|

.

Remark 1.1. Let f : H → K be a nonlinear mapping such thatm‖x‖ ≤ ‖f(x)‖ ≤ M‖x‖with m andM both close to1. Suppose that we can approximatef by a linear mappingS,i.e.,‖f(x)−Sx‖ ≤ ε‖x‖ for all x ∈ H and for someε ∈ [0, m). Then we can approximate

180 Aleksej Turnsek

f by a linear isometryU from the polar decomposition ofS. Indeed, from our assumptionsonf andS we obtain

(m − ε)‖x‖ ≤ ‖Sx‖ ≤ (M + ε)‖x‖.

By the previous lemma it follows that‖S −U‖ ≤ max|M + ε− 1|, |m− ε− 1|

. Hence

U is close toS and thenU is also close tof .

Now we can prove stability of linear approximately orthogonality preserving mappings.

Theorem 1.3([20], Theorem 2.3). LetH andK be Hilbert spaces,T : H → K a nonzerolinear approximately orthogonality preserving mapping,ε ∈ [0, 1), andT = U |T | its polardecomposition. ThenU is an isometry and

‖T − ‖T‖U‖ ≤

(1 −

√1 − ε

1 + ε

)‖T‖. (4)

Proof. Combine Lemma 1.1 and Lemma 1.2 withm =√

1−ε1+ε

, M = 1 and T‖T‖

.

Remark 1.2. Theestimate in the previous theorem is sharp. An example of such a mapping

is T =(

a 00 1

): R ⊕ R

n−1 → R ⊕ Rn−1, n ≥ 2, wherea =

√1−ε1+ε

for someε ∈ [0, 1) and

1 in the lower right corner is the identity onRn−1. However, the approximating isometriesare not unique, see [20] for the details.

Remark 1.3. There is also a variant of Theorem 1.3 for mappings between HilbertC∗-modules, see [12, Theorem 4.4]. LetA be aC∗-algebra of compact operators onH andlet V andW be HilbertA-modules. LetT : V → W be anA-linear approximatelyorthogonality preserving mapping with someε ∈ [0, 1). Then there is anA-linear isometryU : V → W such that (4) holds.

Example 1.4.Nonlinear orthogonality preserving or approximately orthogonality preserv-ing mappings need not be approximated by linear mappings at all. Indeed, letf : R

2 → R2

be defined by

f(x1, x2) =

(x1, x2), x1x2 6= 0

(1, 1), x1 6= 0, x2 = 0

(−1, 1), x1 = 0, x2 6= 0

(0, 0), x1 = x2 = 0

Thenf is a nonlinear orthogonality preserving, hence also approximately orthogonalitypreserving mapping. Let us suppose that we can find a linear operatorA such that‖f(x)−Ax‖ ≤ ‖x‖ for all x ∈ R

2. This implies, sincef(±1, 0) = (1, 1), that

‖(1, 1) − A(1, 0)‖ ≤ 1 and ‖(1, 1) + A(1, 0)‖ ≤ 1.

By the parallelogram identity it follows that2 + ‖A(1, 0)‖2 ≤ 1, a contradiction.

Question 1. Is the Theorem 1.3 true also for unitary (pre-Hilbert) spaces? Namely, inLemma 1.2 we used the concept of the polar decomposition for which the spaces need tobe complete.

Question 2. Can we approximate a nonlinear approximately orthogonality preserving map-ping by a nonlinear orthogonality preserving mapping?


1.1. Orthogonality in Normed Spaces

The notion of orthogonality in an arbitrary normed space may be introduced in variousways. One of the possibilities is the following definition introduced by Birkhoff [1], seealso James [13]. LetX be a real or complex normed space; then forx, y ∈ X

x ⊥B y ⇐⇒ ‖x + λy‖ ≥ ‖x‖ for all scalarsλ.

We call the relation⊥B a Birkhoff–James orthogonality. It is easily seen that, for inner-product spaces, this last definition is equivalent to the usual definition of orthogonality.In general normed spaces Birkhoff-James orthogonality is neither symmetric nor additive,but it is always homogeneous. It is clear from the definition that scalar multiples of linearisometries preserve orthogonality. Converse was proved by Koldobsky [15] for real spacesand later by Blanco and Turnsek in general.

Theorem 1.5([2], Theorem 3.1). LetX andY be normed spaces. A linear mapT : X → Yis orthogonality preserving if and only if it is a scalar multiple of a linear isometry.

Our aim is to define an approximate Birkhoff orthogonality generalizing the⊥ε one forinner product spaces; we follow the approach of Chmielinski [5].

LetX be a normed space overK ∈ R, C. The norm inX need not come from an innerproduct. However, see Lumer [16] and Giles [10], there exists a mapping[·, ·] : X×X → K

with the following properties:

(s1) [λx + µy, z] = λ[x, z] + µ[y, z], x, y, z ∈ X , λ, µ ∈ K;

(s2) [x, λy] = λ[x, y], x, y ∈ X , λ ∈ K;

(s3) [x, x] = ‖x‖2, x ∈ X ;

(s4) |[x, y]| ≤ ‖x‖‖y‖, x, y ∈ X .

A mapping satisfying (s1)-(s4) is called asemi-inner product(s.i.p.). Note that there mayexist infinitely many different semi-inner products inX . There is a unique s.i.p. inX if andonly if X is smooth(i.e., there is a unique supporting hyperplane at each point of the unitsphere or, equivalently, the norm isGateaux differentiable). Recall that the norm is Gateauxdifferentiable atx 6= 0 if the limit

limt→0, t∈R

‖x + ty‖ − ‖x‖

t

exists for all y ∈ X . Recall also that in this case this last limit is equal to the real partof fx(y), wherefx is a support functionalat x, i.e., a norm one linear functional suchthat fx(x) = ‖x‖. If we define[y, x] = ‖x‖fx(y), then [·, ·] is a unique s.i.p. Now insmooth spaces, by an analogy with inner product spaces, we definesemi-orthogonalityandapproximate semi-orthogonality:

x ⊥s y ⇐⇒ [y, x] = 0;

x ⊥εs y ⇐⇒ |[y, x]| ≤ ε‖x‖‖y‖,

182 Aleksej Turnsek

for someε ∈ [0, 1).Our next aim is to express the above approximate semi-orthogonality relation without

s.i.p., thus involving only norm, and take it as a definition of approximate Birkhoff orthog-onality in general normed spaceX .

Proposition 1.6([5], Proposition 3.1, 3.2). LetX be a smooth normed space and letε ∈[0, 1). Then|[y, x]| ≤ ε‖x‖‖y‖ if and only if ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖‖λy‖ for allscalarsλ.

Proof. (⇒): Recall that[y, x] = ‖x‖fx(y), wherefx is a support functional atx. Then|fx(y)| ≤ ε‖y‖, and for someθ ∈ [0, 1] and for someϕ ∈ [−π, π] we have

fx(y) = θε‖y‖eiϕ.

For arbitraryλ ∈ K we have

‖x + λy‖ ≥ |fx(x + λy)| =∣∣‖x‖ + λθε‖y‖eiϕ

∣∣

=∣∣‖x‖ + θε‖y‖

(Re(λeiϕ) + iIm(λeiϕ)

)∣∣ ,

hence

‖x + λy‖2 ≥(‖x‖ + θε‖y‖Re(λeiϕ)

)2+(θε‖y‖(Im(λeiϕ)

)2

≥ ‖x‖2 + 2θε‖x‖‖y‖(Re(λeiϕ)

≥ ‖x‖2 − 2θε‖x‖‖y‖|λ|

≥ ‖x‖2 − 2ε‖x‖‖λy‖.

(⇐): Let γ = Arg(fx(y)). Then|fx(y)| = e−iγfx(y) = fx(e−iγy) = Re(fx(e−iγy)).From‖x + λe−iγy‖2 ≥ ‖x‖2 − 2ε‖x‖‖λy‖ it follows that

‖x + λe−iγy‖ − ‖x‖

|λ|≥

−2ε‖x‖‖y‖

‖x + λe−iγy‖ + ‖x‖.

Taking the right and left limits asλ→0 we get|fx(y)|≤ ε‖y‖ and the proof is completed.

Taking into account the last proposition we define the approximate Birkhoff orthogo-nality on any normed space.

Definition 1 ([5]). Let X be a normed space and letε ∈ [0, 1) be given. We say thatx isε-Birkhoff orthogonal toy, x ⊥ε

B y, if ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖‖λy‖ for all scalarsλ.

Question 3. LetX andY be normed spaces and letT : X → Y be a linear mapping whichapproximately preserves Birkhoff orthogonality. Is thenT close to some multiple of anisometry?


2. Stability of the Orthogonality Equation

Sinceorthogonality preserving mappings can be far from being linear or continuous, seeExample 1.4, we will now impose a stronger condition. LetH andK be Hilbert spaces. Amappingf : H → K is calledinner product preservingif it is a solution of theorthogonalityequation:

〈f(x), f(y)〉 = 〈x, y〉 for x, y ∈ H. (O)

It is easy to see thatf satisfies (O) if and only if it is a linear isometry. We say that amappingf : H → K approximately preserves the inner product if it satisfies

∣∣〈f(x), f(y)〉 − 〈x, y〉∣∣ ≤ ϕ(x, y) (AO)

for some appropriate control functionϕ and allx, y ∈ H.

Theorem 2.1([8], Theorem 4.1). If f : H → K satisfies (AO) with a functionϕ : H×H →[0,∞) such thatlimm+n→∞ cm+nϕ(c−mx, c−ny) = 0 for all x, y ∈ H and for some1 6= c > 0, then there exists a unique mappingU : H → K satisfying the orthogonalityequation (O) and such that

‖f(x) − Ux‖ ≤√

ϕ(x, x) for all x ∈ H.

Proof. We give just the idea of the proof which goes back to Hyers. One definesfn(x) =cnf(c−nx) and shows that the sequence is Cauchy, hence convergent. The limitUx satisfiesthe requirements of the theorem.

Let ϕ(x, y) = ε‖x‖p‖y‖p with p ∈ R \ 1. Thenϕ satisfies the conditions of theabove theorem and we get the following result.

Corollary 2.2 ([3], Theorem 2). Let ε > 0 andp ∈ R \ 1 be fixed. Then, for a mappingf : H → K satisfying

|〈f(x), f(y)〉 − 〈x, y〉| ≤ ε‖x‖p‖y‖p, x, y ∈ Hp, (5)

whereHp = H for p ≥ 0 andHp = H \ 0 for p < 0, there exists a unique mappingU : H → K satisfying (O) and such that

‖f(x) − Ux‖ ≤√

ε‖x‖p.

The constant√

ε which appears in the assertion of the previous corollary is the bestpossible. To see it let us consider the example.

Example 2.3 ([8], Example 2.4). Let f : l2 → l2 be a mapping defined byf(x) =(√

ε‖x‖p, x). Then f satisfies(5) and Ux = (0, x) is a solution of (O) such that‖f(x) − Ux‖ =

√ε‖x‖p. That the estimate is indeed sharp follows from the uniqueness.

Remark 2.1. Results of Theorem 2.1 and Corollary 2.2 can be generalised to the setting ofHilbert C∗-modules, see [9].

184 Aleksej Turnsek

2.1. The casep = 1

The casep = 1 seems to be a singular one. So, let us see what we can say about mappingsf : H → K satisfying the following condition:

∣∣〈f(x), f(y)〉 − 〈x, y〉∣∣ ≤ ε‖x‖ ‖y‖, x, y ∈ H, ε ∈ [0, 1). (6)

From (6) it follows that√

1 − ε‖x‖ ≤ ‖f(x)‖ ≤√

1 + ε‖x‖. But we can even assumethatf preserves the norm.

Proposition 2.4([3], Proposition 1). Letf : H → K satisfies(6). Define

g(x) =

f(x)‖x‖

‖f(x)‖if x 6= 0,

0 if x = 0.

Then:

(i) |〈g(x), g(y)〉 − 〈x, y〉| ≤ 2ε‖x‖ ‖y‖,

(ii) ‖g(x)‖ = ‖x‖,

(iii) ‖f(x) − g(x)‖ ≤ (1 −√

1 − ε)‖x‖.

Let H be an n-dimensional Hilbert space andg : H → K a mapping satisfying (6)for someε ∈ [0, 1). Let e1, . . . , en be an orthonormal basis inH and for x ∈ H,

x =n∑

i=1αiei, define a linear operatorS : H → K by Sx =

n∑i=1

αig(ei).

Proposition 2.5 ([20], Proposition 2.5). If g : H → K satisfies(6) and ‖g(x)‖ = ‖x‖,x ∈ H, then

‖g(x) − Sx‖ ≤

√ε(n + 2

√n − 1) ‖x‖.

Proof. Let x =n∑

i=1αiei and denote

αi〈g(x), g(ei)〉 − αi〈x, ei〉 = λi.

Thusαi〈g(x), g(ei)〉 = |αi|

2 + λi,

where|λi| ≤ ε|αi| ‖x‖ because of (6). Note also that|〈g(ei), g(ej)〉| ≤ ε for i 6= j. Then

‖g(x) − Sx‖2 =⟨g(x) − Sx, g(x) − Sx

⟩

= ‖x‖2 − 2Re( n∑

i=1

αi〈g(x), g(ei)〉)

+n∑

i,j=1

αiαj〈g(ei), g(ej)〉

= ‖x‖2 − 2n∑

i=1

|αi|2 − 2Re

( n∑

i=1

λi

)+

n∑

i=1

|αi|2 +

n∑

i6=j,i,j=1

αiαj〈g(ei), g(ej)〉

≤ 2ε‖x‖n∑

i=1

|αi| + ε

(( n∑

i=1

|αi|)2

−n∑

i=1

|αi|2

)

≤ 2ε‖x‖ ‖x‖√

n + ε(‖x‖2n − ‖x‖2

)= ε

(n + 2

√n − 1

)‖x‖2,

andthe result follows.


Now we can prove stability of the orthogonality equation for the finite-dimensionaldomain.

Theorem 2.6. Let H be ann-dimensional Hilbert space andf : H → K a mappingsatisfying(6). Then there exists a linear isometryU : H → K and a continuous functionη : [0,∞) → [0,∞) with η(ε) → 0 asε → 0, given by

η(ε) = min

1 −√

1 − ε +

√8ε(n + 2

√n − 1), 1 +

√1 + ε

,

such that‖f(x) − Ux‖ ≤ η(ε)‖x‖, x ∈ H.

Proof. Since‖f(x)‖ ≤√

1 + ε‖x‖, we get

‖f(x) − Ux‖ ≤ ‖f(x)‖ + ‖Ux‖ ≤(1 +

√1 + ε

)‖x‖

for any isometryU . Assume now thatε < ε0 = 12(n+2

√

n−1)and let δ =

√2ε(n + 2

√n − 1) < 1. Let the functiong : H → K be as in Proposition 2.4. From

Proposition 2.5 one obtains a linear operatorS such that‖g(x) − Sx‖ ≤ δ‖x‖, hence

(1 − δ)‖x‖ ≤ ‖Sx‖ ≤ (1 + δ)‖x‖.

From Lemma 1.2, see also Remark 1.1, it follows that‖S−U‖ ≤ δ, whereU is the isometryfrom the polar decomposition ofS. Thus‖g(x) − Ux‖ ≤ 2δ‖x‖ and

‖f(x) − Ux‖ ≤(1 −

√1 − ε + 2δ

)‖x‖ for ε < ε0.

Sinceδ = 1 for ε = ε0 and since3 −√

1 − ε > 1 +√

1 + ε, the functionη is continuousand the proof is completed.

We can formulate the previous theorem in a more compact form. See [22], [23]forsimilar results on bounded domains.

Theorem 2.7. There is a universal constantC with the following property. LetH be ann-dimensional Hilbert space andf : H → K a mapping satisfying(6). Then there exists alinear isometryU : H → K with

‖f(x) − Ux‖ ≤ C√

ε n ‖x‖

for all x ∈ H.

Remark 2.2. One can show, see [4, Lemma 2], that a mappingf satisfying (6) has to bequasi linearin the following sense:

‖f(x + y) − f(x) − f(y)‖ ≤ 2√

ε (‖x‖ + ‖y‖) (7)

and

‖f(λx) − λf(x)‖ ≤ 2√

ε |λ|‖x‖ (8)

186 Aleksej Turnsek

Remark 2.3. Thesolution of (6) need not be neither additive nor homogeneous. Indeed,consider the mappingf : R

n → Rn+1 given byf(x) = (

√ε‖x‖, x). However, we can

assume with no loss of generality that solutions of (6) are real homogeneous. For this weneed the following observations. Define

Ω(x) = ‖x‖

(f( x

2‖x‖

)− f

( −x

2‖x‖

)),

wheref is a solution of (6). ThenΩ is real homogeneous and satisfies (6). To see thisdenoteλ = 1

2‖x‖ andµ = 12‖y‖ . Then

〈Ω(x),Ω(y)〉 − 〈x, y〉

= ‖x‖‖y‖[(〈f(λx), f(µy)〉 − 〈λx, µy〉

)+(〈λx,−µy〉 − 〈f(λx), f(−µy)〉

)

+(〈−λx, µy〉 − 〈f(−λx), f(µy)〉

)+(〈f(−λx), f(−µy)〉 − 〈−λx,−µy〉

)]

and by using the triangle inequality it follows thatΩ satisfies (6). Furthermore,Ω is closeto f . Indeed,

Ω(x) − f(x) = ‖x‖(f(λx) − f(−λx)

)− ‖x‖

(λf(x) + λf(x)

)

= ‖x‖(f(λx) − λf(x)

)− ‖x‖

(f(−λx) − (−λ)f(x)

).

Using the triangle inequality and (8) it follows that‖Ω(x) − f(x)‖ ≤ 2√

ε ‖x‖.

Let f : Rn → R

n be a map which satisfies the condition (6) for someε ∈ [0, 1).Then by Proposition 2.4 and Remark 2.3 we can assume thatf preserves the norm, that is‖f(x)‖ = ‖x‖, and thatf is homogeneous. Furthermore, if we can approximatef by alinear map, then by Remark 1.1, we can approximatef also with an isometry. By a resultof Kalton, [14, Theorem 2.2], we have the following theorem.

Theorem 2.8. There is a universal constantC with the following property. Letf : Rn →

Rn be a continuous map which satisfies the condition(6). Then there is an isometryU :

Rn → R

n with‖f(x) − Ux‖ ≤ C

√ε (log n + 1)‖x‖

for all x ∈ Rn.

Question 4. Let f : Rn → R

n satisfy (6) for someε ∈ [0, 1). Suppose thatf pre-serves the norm, that is‖f(x)‖ = ‖x‖, and thatf is homogeneous. Letα(f) =infT

sup‖x‖=1

‖f(x)−Tx‖, where the infimum is taken over all linear mappingsT : Rn → R

n.

Let α(n, ε) = supf

α(f), the supremum being taken over allf as above. Then by the Theo-

rem 2.7,α(n, ε) ≤ C√

ε n (α(n, ε) ≤ C√

ε (log n + 1)) if f is continuous). Is it true thatthe “approximation error”α(n, ε) depends on the dimension? Find its lower bound?

Question 5. Is the orthogonality equation with control functionϕ(x, y) = ε‖x‖‖y‖ stablealso in the case of the infinite dimensional domain?


3. Stability of the Wigner Equation

In the Hilbert space formulation of quantum mechanics, which is mainly due to von Neu-mann, several mathematical objects appear whose physical meaning is connected with theprobabilistic aspects of the theory, see [17]. For exampleS(H), the set of all positive trace-class operators onH with trace 1. The elements ofS(H) are called states of the system.The extreme points ofS(H) as a convex set inB(H) are called pure states. It is easy to seethat they are exactly the rank-one projections onH. A rank-one projection can be triviallyidentified with its range or with any unit vector which spans its range. Hence, one can re-gard pure states in three different ways: rank-one projections, one-dimensional subspaces,unit vectors (in this latter case the identification is one-to-one only up to multiplication bya scalar of modulus 1). IfP = x ⊗ x andQ = y ⊗ y are pure states, then thetransitionprobabilitybetween them is defined by tr(PQ) = |〈x, y〉|2, where tr denotes the usual tracefunctional. In this context it is very important Wigner’s theorem which we can formulate indifferent ways, see[17, p. 12]. The classical formulation, see [19], says:

Theorem 3.1. If f : H → K satisfies∣∣〈f(x), f(y)〉

∣∣ = |〈x, y〉| for x, y ∈ H, (W )

thenf is phase-equivalent to a linear or a conjugate-linear isometry.

Recall that functionsf, g : H → K are phase-equivalent if there exists a functionγ : H → S1, whereS1 is the unit circle in the complex plane, such thatg(x) = γ(x)f(x)for all x ∈ H. Recall also that conjugate-linear meansf(λx + µy) = λf(x) + µf(y).

Stabilityof the Wigner equation (W) is explained in the following analogue of the The-orem 2.1.

Theorem 3.2([8], Theorem 2.1). If f : H → K satisfies∣∣|〈f(x), f(y)〉| − |〈x, y〉|

∣∣ ≤ ϕ(x, y) for x, y ∈ H

with a functionϕ : H ×H → [0,∞) such thatlimm+n→∞ cm+nϕ(c−mx, c−ny) = 0 forall x, y ∈ H and for some1 6= c > 0, then there exists a unique (up to a phase-equivalentfunction) mappingU : H → K satisfying the Wigner equation (W) and such that

‖f(x) − Ux‖ ≤√

ϕ(x, x) for all x ∈ H.

This theorem covers also the case of control functionϕ(x, y) = ε‖x‖p‖y‖p for p 6= 1.However the casep = 1 is again a singular one. Only some partial results for the dimensionof the domain equal to 1 or 2 have been obtained.

Theorem 3.3([3], Theorem 3). If dimH = 1 andf : H → K satisfies∣∣|〈f(x), f(y)〉| − |〈x, y〉|

∣∣ ≤ ε‖x‖‖y‖ (9)

with ε ∈ [0, 1), then there exists a mappingU satisfying the Wigner equation (W) and suchthat

‖f(x) − Ux‖ ≤ 2√

ε‖x‖, x ∈ H.

188 Aleksej Turnsek

Theorem 3.4 ([6], Corollary 2). There existsε0 > 0 such that for anyε ∈ (0, ε0) iff : R

2 → R2 satisfies(9), then there exists a mappingU : R

2 → R2 satisfying the Wigner

equation onR2 and such that

‖f(x) − Ux‖ ≤ δ(ε)‖x‖, x ∈ R2

for some functionδ : (0, ε0) → R+ satisfying the conditionlimε→0 δ(ε) = 0.

Question 6. Is the Wigner equation stable also for domains of dimension greater than two?

References

[1] G. Birkhoff, Orthogonality in linear metric spaces,Duke Math. J.1 (1935), 169–172.

[2] A. Blanco, A. Turnsek, On maps that preserve orthogonality in normed spaces,Proc.Roy. Soc. Edinburgh Sect. A136(2006), No. 4, 709–716.

[3] J. Chmielinski, On a singular case in the Hyers-Ulam-Rassias stability of the Wignerequation,J. Math. Anal. Appl.289(2004), No. 2, 571–583.

[4] J. Chmielinski, Linear mappings approximately preserving orthogonality,J. Math.Anal. Appl.304(2005), No. 1, 158–169.

[5] J. Chmielinski, On anε-Birkhoff orthogonality,J. Inequal. Pure Appl. Math.6 (2005),No. 3, Art. 79.

[6] J. Chmielinski, Stability of angle-preserving mappings on the plane,Math. Inequal.Appl.8 (2005), 497–503.

[7] J. Chmielinski, Stability of the orthogonality preserving property in finite-dimensionalinner product spaces,J. Math. Anal. Appl.318(2006), No. 2, 433–443.

[8] J. Chmielinski, Stability of the Wigner equation and related topics,Nonlinear Funct..Anal. Appl.11 (2006), No. 5, 859–879.

[9] J. Chmielinski, M. S. Moslehian, ApproximatelyC∗-inner product preserving map-pings,Bull. Korean Math. Soc.45 (2008), 157–167.

[10] J. R. Giles, Classes of semi inner product spaces,Trans. Amer. Math. Soc.129(1967),436–446.

[11] P. Halmos,Finite-dimensional vector spaces, Reprinting of the 1958 second edition,Undergraduate Texts in Mathematics, Springer-Verlag, New York–Heidelberg, 1974.

[12] D. Ili sevic, A. Turnsek, Approximately orthogonality preserving mappings onC∗-modules,J. Math. Anal. Appl.341(2008), 298–308.

[13] R. C. James, Orthogonality and linear functionals in normed linear spaces,Trans.Amer. Math. Soc.61 (1947), 265–292.


[14] N. J. Kalton, A remark on quasi-isometries,Proc. Amer. Math. Soc.131(2003), 1225–1231.

[15] A. Koldobsky, Operators preserving orthogonality are isometries,Proc. Roy. Soc. Ed-inburgh Sect. A123(1993), 835–837.

[16] G. Lumer, Semi inner product spaces,Trans. Amer. Math. Soc.100(1961), 29–43.

[17] L. Molnar, Selected preserver problems on algebraic structures of linear operatorsand on function spaces, Lecture Notes in Mathematics, 1895, Springer-Verlag, Berlin,2007.

[18] G. K. Pedersen, Analysis now,Graduate Texts in Mathematics,118, Springer-Verlag,New York, 1989.

[19] J. Ratz, On Wigner’s theorem: Remarks, complements, comments, and corollaries,Aequationes Math.52 (1996), 1–9.

[20] A. Turnsek, On mappings approximately preserving orthogonality,J. Math. Anal.Appl.336(2007), 625–631.

[21] S. M. Ulam,Problems in Modern Mathematics, Science Editions, John Wiley & Sons,New York, 1964.

[22] I. Vestfrid,ε-isometries in Euclidean spaces,Nonlinear Anal.63 (2005), No. 8, 1191–1198.

[23] I. Vestfrid, Addendum to “ε-isometries in Euclidean spaces”, [Nonlinear Anal.63(2005), No. 8, 1191–1198],Nonlinear Anal.67 (2007), No. 4, 1306–1307.



Chapter 15

THE FRANKL PROBLEM FOR SECOND ORDER

NONLINEAR EQUATIONS OF M IXED TYPE

WITH NON-SMOOTH DEGENERATE CURVE

Guo Chun WenSchool of Mathematical Sciences, Peking University

Beijing 100871, China

Abstract

In [1]–[6], the authors posed and discussed the Tricomi and Frankl problems ofsome second order equations of mixed type, but they only consider some special mixedequations. In [3], the authors discussed the uniqueness of solutions of Tricomi problemfor some second order mixed equation with nonsmooth degenerate line. The presentpaper deals with the Tricomi and Frankl problems for second order nonlinear mixedequations with non-smooth degenerate curve, we first give the formulation of the Tri-comi problem, and derive some estimates and existence of solutions of the Tricomiproblem for the equations with nonsmooth degenerate line. Finally we discuss theFrankl problem for the above mixed equations with non-smooth degenerate curve.Thus the results obtained in this paper generalize the results obtained by the authors of[2]–[6].

2000 Mathematics Subject Classifications:35M05, 35J70, 35L80.

Key words: Tricomi and Frankl problems, nonlinear equations of mixed type, nonsmoothdegenerate line.

1. Formulation of Tricomi Problem for Mixed Equations

Let D be a simply connected bounded domain in the complex planeC with the boundary∂D = Γ∪L, whereΓ(⊂ x > 0, y > 0) ∈ C2

µ (0 < µ < 1) is a curve with the end pointsz = 1, i, andL = L1 ∪ L2 ∪ L3 ∪ L4, whereL1, L2, L3, L4 are four characteristics withthe slopes−H2(x)/H1(y),H2(x)/H1(y),−H2(x)/H1(y),H2(x)/H1(y) passing through

192 Guo Chun Wen

the pointsz = x+ iy = 0, 1, 0, i respectively as follows

L1 =

−G1(y)=−

y∫

0

H1(t) dt=G2(x)=

x∫

0

H2(t) dt, x∈(0, x1)

,

L2 =

−G1(y)=−

y∫

0

H1(t) dt=

1∫

x

H2(t) dt=G2(1)−G2(x), x ∈ (x1, 1)

,

L3 =

G1(y)=

y∫

0

H1(t) dt=−

x∫

0

H2(t) dt=−G2(x), y∈(0, y2)

,

L4 =

G1(1) −G1(y)=

1∫

y

H1(t) dt=−

x∫

0

H2(t) dt=−G2(x), y∈(y2, 1)

.

(1.1)

Here H1(y) =√|K1(y)|, H2(x) =

√|K2(x)|, K1(0) = 0, K2(0) = 0, K1(y) =

sgny|y|m1h1(y), K2(x) = sgnx|x|m2h2(x) are continuous inD, possess the first orderderivative andyK1(y) > 0 on y 6= 0, xK2(x) > 0 onx 6= 0, m1,m2 (< min(1,m1)) arepositive constants,h1(y), h2(x) inD arecontinuously differentiable positive functions, and(x1, y1), (x2, y2) are the intersection points ofL1, L2 andL3, L4 respectively. There is noharm in assuming that the boundaryΓ of the domainD is a smooth curve, which possessesthe formG2(x) = G2(1) −G1(y) andG1(y) = G1(1) −G2(x) near the pointsz = 1 andi respectively. DenoteD+ = D ∩ x > 0, y > 0,D− = D−

1 ∪D−

2 , D−

1 = D ∩ y < 0,D−

2 = D ∩ x < 0. In this paper we use the notation of the complex number inD+ andthe hyperbolic number inD− (see [10]).

Now we introduce the second order nonlinear equation of mixed type with nonsmoothdegenerate line

Lu=K1(y)uxx+K2(x)uyy+aux+buy+c∗u=−d in D, (1.2)

wherec∗ = c − |u|σ, a, b, c, d are real functions ofz(∈D), u, ux, uy (∈R), σ is a non-negative constant, and suppose that the equation (1.1) satisfiesCondition C:

1) The coefficientsa, b, c, d are measurable inD+ and continuous inD− for anycontinuously differentiable functionu(z) in D∗ = D\Z ′, Z ′ = 0, 1, i, and satisfy

L∞[η,D+] ≤ k0, η = a, b, c, L∞[d,D+] ≤ k1, c ≤ 0 in D+,

C[d,D−]=C[d,D−]+C[dx,D−]≤k1, C[η,D−]≤k0, η=a, b, c,

|a|/H1 = o(1) as y = Imz(z ∈ D−

1 ) → 0, m1 ≥ 2,

|b|/H2 = o(1) as x = Rez(z ∈ D−

2 ) → 0, m2 ≥ 2,

|η|/H1H2, |η|/H21 , |η|/H2

2 = o(1) as z (∈ D−) → 0, η = a, b,

η|x|−m2/2, ηx|x|−m2/2−1, η|y|−m1/2, ηy|y|

−m1/2−1=O(1) as z→ 0, η=c, d,

(1.3)

in whichk0 (≥ max[2√h(y), 1/

√h(y), 1], k1 (≥ max[6k0, 1]) arepositive constants.

Mixed Equations with Nonsmooth Degenerate Line 193

2) For any continuously differentiable functionsu1(z), u2(z) in D∗, F (z, u, uz) =aux + buy + cu+ d satisfies the following condition

F (z, u1, u1z)−F (z, u2, u2z)= a(u1−u2)x+b(u1−u2)x+c(u1−u2) in D,

wherea, b, c satisfythe conditions as those ofa, b, c. Obviously equation (1.1) with theconditionK2(x) = 1, a = b = c = d = 0 is the so-called Chaplygin equation.

If H1(y) = [|y|m1h1]1/2,H2(x) = [|x|m2h2(x)]

1/2 as stated before, then we have

Y = G1(y) =

y∫

0

H1(t) dt, |Y | ≤k0

m1 + 2|y|(m1+2)/2,

X = G2(x) =

x∫

0

H2(t)dt, |X| ≤k0

m2 + 2|x|(m2+2)/2 in D±,

(1.4)

andtheir inverse functionsy = ±|(G1)−1(Y )|, y = ±|(G2)

−1(X)| satisfy the inequalities

|y| = |(G1)−1(Y )|≤

(k0(m1+2)

2

)2/(m1+2)

|Y |2/(m1+2) =J1|Y |2/(m1+2),

|x| = |(G2)−1(X)| ≤

(k0(m2 + 2)

2

)2/(m2+2)

|X|2/(m2+2) = J2|X|2/(m2+2).

(1.5)

The Tricomi problem for equation (1.2) may be formulated as follows:

Problem T . Find a continuous solutionu(z) of (1.1) inD\0, whereux, uy are continu-ous inD∗ = D\1, i, 0, and satisfy the boundary conditions

u(z) = φ(z) on Γ, u(z) = ψ1(x) on L2, u(z) = ψ2(y) on L4, (1.6)

whereφ(1) = ψ1(1), φ(i) = ψ2(i), andφ(z), ψ1(x), ψ2(y) satisfy the conditions

C2α[φ(z),Γ] ≤ k2, C2

α[ψ1(x), L2] ≤ k2, C2α[ψ2(y), L4] ≤ k2, (1.7)

in whichα (0<α<1), k2 are positive constants.

If the boundaryΓ nearz = 1, i possesses the form

G2(x) = G2(1) −G1(y), G1(y) = G1(1) −G2(x)

respectively, we find the derivative for (1.6) according to the parameters = Re z = x onΓ

194 Guo Chun Wen

nearz = 1 and the parameters = Im z = y onΓ nearz = i, and obtain

us =ux+uyyx =ux−H2(x)uy/H1(y)=φ′(x), i.e.

H1(y)ux−H2(x)uy =H1(y)φ′(x) on Γ near z = 1,

us =uxxy+uy =−H1(y)ux/H2(x)+ uy =φ′(y), i.e.

H1(y)ux−H2(x)uy = −H2(x)φ′(y) on Γ near z = i,

us =ux+uyyx =ux+H2(x)uy/H1(y)=ψ′

1(x), i.e.

H1(y)ux +H2(x)uy = H1(y)ψ′

1(x) on L2,

us =uxxy+uy =H1(y)ux/H2(x)+uy =ψ′

2(y), i.e.

H1(y)ux +H2(x)uy = H2(x)ψ′

2(y) on L4,

H2(x)uy(x) = 2R0(x) or H1(y)ux(y) = 0 on L′

0,

H1(y)ux(y) = 2R0(y) or H2(x)uy(x) = 0 on L′′

0,

(1.8)

whereL′

0 = 0 ≤ x ≤ 1, y = 0, L′′

0 = x = 0, 0 ≤ y ≤ 1, L0 = L′

0 ∪ L′′

0, andR0(x), R0(y) are undetermined real functions. It is clear that the complex form of (1.8) isas follows

Re[λ(z)(U + iV )

]= Re

[λ(z)(H1(y)ux − iH2(x)uy)

]/2 = R(z) on Γ ∪ L0,

Re[λ(z)(U + jV )

]= Re

[λ(z)(H1(y)ux − jH2(x)uy)

]/2 = R(z) on L2 ∪ L4,

Im[λ(z)(U+jV )

]z=z1

= Im[λ(z)(H1(y)ux−jH2(x)uy)

]/2|z=z1

=c1,

Im[λ(z)(U+jV )

]z=z2

= Im[λ(z)(H1(y)ux−jH2(x)uy)

]/2|z=z2

=c2,

(1.9)wherej is the hyperbolic unit such thatj2 = 1,

U(z) =H1(y)

2ux, V (z) = −

H2(x)

2vy, d1 = φ(1) = b0,

c1 =1

2√

2

[−H1(y1)ψ

′

1(x1)], c2 =

1

2√

2

[−H2(x2)ψ

′

2(y2)],

and

λ(z)=

(1+i)/√

2,

(1+i)/√

2,

(1+j)/√

2,

(1+j)/√

2,

1 or i,

i or 1,

R(z)=

H1(y)φ′(x)/2

√2 on Γ at z=1,

−H2(x)φ′(y)/2

√2 on Γ at z= i,

H1(y)ψ′

1(x)/2√

2 on L2,

H2(x)ψ′

2(y)/2√

2 on L4,

0 or R0(x) on L′

0

R0(y) or 0 on L′′

0.

(1.10)

whereR0(x), R0(y) are as stated before. Denotingt1 = 1, t2 = i, t3 = 0, we have

eiφ1 =λ(t1 − 0)

λ(t1 + 0)= e0πi−πi/4 = e−πi/4 , γ1 = −

1

4−K1 = −

1

4, K1 = 0,

eiφ2 =λ(t2 − 0)

λ(t2 + 0)= eπi/4−πi/2 = e−πi/4, γ2 = −

1

4−K2 = −

1

4, K2 = 0,

eiφ3 =λ(t3−0)

λ(t3+0)=eπi/2−0πi =eπi/2, γ3 =

π/2

π−K3 =

1

2, K3 =0,

(1.11)


in which we considerRe[W (z)] = H1(y)ux/2 = 0 on L′

0 and Im[iW (z)] =−H2(x)uy/2 = 0 onL′′

0, thus the index ofλ(z) on∂D+ = Γ ∪ L0 is

K = (K1 +K2 +K3)/2 = 0. (1.12)

Obviously the Tricomi problem for Chaplygin equation is a special case of ProblemT forequation (1.2).

Noting thatφ(z)∈C2α(Γ), ψ1(x)∈C

2(L2), ψ2(y) ∈ C2α(L4) (0 < α < 1), we can find

two twice continuously differentiable functionsu±0 (z) in D±

, for instance, which are thesolutions of the Dirichlet problem with the boundary condition onΓ ∪ L2 ∪ L4 in (1.6) forharmonic equations inD±, thus the functionsv(z) = v±(z) = u(z) − u±0 (z) in D is thesolution of the equation in the form

Lv = K1(y)vxx+K2(x)vyy+avx+bvy+c∗v=−d in D (1.13)

satisfying the corresponding boundary conditions

v(z)=0 on Γ∪L2∪L4, i.e. Re[λ(z)W (z)]=R(z) on Γ∪L2∪L4,

v(1) = b0, v(0) = 0,(1.14)

where the coefficients of (1.13) satisfy the conditions similar to ConditionC, W (z) =U + iV = v+

z in D+ andW (z) = U + jV = v−z in D−, hence later on we only discussthe case ofR(z) = 0 on Γ ∪ L2 ∪ L4 andc1 = c2 = d1 = 0 in (1.14) and the case ofindexK = 0, which is called ProblemT , the other case can be similarly discussed. Fromv(z) = v±(z) = u(z) − u±0 (z) in D±, we haveu(z) = v±(z) + u±0 (z) in D±, and

v+(z) = v−(z) − u+0 (z) + u−0 (z) on L0,

uy =v±y +u±0y, v+y =v−y −u+

0y+u−0y =2R1(x), v−

y =2R1(x) on L′

0,

ux =v±x +u±0x, v+x =v−x −u+

0x+u−0x =2R2(y), v−

x =2R2(y) on L′′

0.

2. Representation of Solutions of Tricomi Problem for MixedEquations

In this section, we first write the complex form of equation (1.2). Denote

W(z)=U+iV=1

2

[H1(y)ux−iH2(x)uy

]=uz =

H1(y)H2(x)2[uX−iuY ],

H1(y)H2(x)WZ =H1(y)H2(x)

2[WX +iWY ]=

1

2

[H1(y)Wx+iH2(x)Wy

]=Wz in D+,

(2.1)

196 Guo Chun Wen

we have

H1(y)H2(x)WZ =H1H2[WX +iWY ]/2=H1H2

[(U+iV )X +i(U+iV )Y

]/2

= iH1H2

[(U+V )−i(U−V )

]µ+iν

= iH1H2

[(U+V )+i(U−V )

]µ−iν

=[iH2H1y/H1−a/H1+H1H2x/H2−ib/H2

]W

+[iH1H2x/H2−a/H1−H1H2x/H2 + ib/H2

]W−c∗u−d

/4

=A1(z)W+A2(z)W+A3(z)u+A4(z)=g(Z), i.e.[(U+V )+i(U−V )

]µ−iν

=

2[H2H1y/H1]U + 2[H1H2x/H2]V − i[aux + buy + c∗u+ d]/(4H1H2)

= ig(Z) in D+Z ,

(2.2)

in whichD+Z , D

+τ are the image domains ofD+ with respect to the mappingZ = Z(z) =

X + iY , τ = µ+ iν = τ(z) respectively, and

µ = G2(x) +G1(y) = X + Y, ν = G2(x) −G1(y) = X − Y in D+. (2.3)

Similarly introduce the hyperbolic unitj such thatj2 = 1, we can obtain

W(z)=U+jV =1

2

[H1(y)ux−jH2(x)uy

]=H1(y)H2(x)

2[uX−juY ]=H1(y)H2(x)uZ ,

H1(y)H2(x)WZ =H1(y)H2(x)

2[WX+jWY ]=

1

2

[H1(y)Wx+jH2(x)Wy

]=Wz in D−,

−K1(y)uxx−K2(x)uyy =H1(y)[H1(y)ux−jH2(x)uy

]x

+jH2(x)[H1(y)ux−jH2(x)uy

]y−jH2(x)H1yux+jH1(y)H2xuy

=4H1(y)H2(x)WZ−j[H2H1y/H1]H1ux+j[H1H2x/H2]H2uy =aux+buy+c∗u+d, i.e.

H1(y)H2(x)WZ =H1H2[WX +jWY ]/2=H1H2

(U+V )µe1+(U−V )νe2

=

2j[H2H1y/H1

]U+2j

[H1H2x/H2

]V +aux+buy+c∗u+d

/4

[jH2H1y/H1+a/H1

](W+W )+

[H1H2x/H2−jb/H2

](W−W )+c∗u+d

/4

=[a/H1+H1H2x/H2+H2H1y/H1−b/H2

](U+V )

+[a/H1−H1H2x/H2+H2H1y/H1+b/H2

](U−V )+c∗u+d

e1/4

+[a/H1−H1H2x/H2−H2H1y/H1−b/H2

](U+V )

+[a/H1+H1H2x/H2−H2H1y/H1+b/H2

](U−V )+c∗u+d

e2/4, i.e.

(U+V )µ = A1(U + V )+B1(U − V )+C1u+D1,

(U−V )ν = A2(U + V )+B2(U − V ) +C2u+D2,in D−

τ ,

(2.4)


in whiche1 =(1 + j)/2, e2 =(1 − j)/2,D−

Z , D−

τ are the image sets ofD−

1 with respect tothe mappingZ = Z(z), τ = µ+ jν = τ(z) respectively, and

A1 =1

4H1H2

[a

H1+H1H2x

H2+H2H1y

H1−

b

H2

], C1 =

c∗

4H1H2,

B1 =1

4H1H2

[a

H1−H1H2x

H2+H2H1y

H1+

b

H2

], C2 =

c∗

4H1H2,

A2 =1

4H1H2

[a

H1−H1H2x

H2−H2H1y

H1−

b

H2

], D1 =

d

4H1H2,

B2 =1

4H1H2

[a

H1+H1H2x

H2−H2H1y

H1+

b

H2

], D2 =

d

4H1H2in D−.

(2.5)

For the domainD−

2 , we can also write the coefficients of equation (2.4) inD−

τ , whereτ = µ+ jν = G1(y) +G2(x) + j[G1(y) −G2(x)]. It is clear that a special case of (2.2),(2.4) is the complex equation

WZ = 0 in D+Z ∪D−

Z . (2.6)

The boundary value problem for equations (2.2), (2.4) with the boundary condition (1.14)and the relation: the first formula in (2.7) below will be called ProblemA. Here we mentionthat if we denoteµ = x+G1(y), ν = x−G1(y) inD−

1 , andµ = G2(x)+y, ν = G2(x)−yin D−

2 , then the last system in (2.4) is true still.

Now we state and verify the representation of solutions of ProblemT for equation (1.2).

Theorem 2.1.Under ConditionC, any solutionu(z) of ProblemT for equation(1.2) in Dcanbe expressed as follows

u(z)=u(x)−2

y∫

0

V (z)

H2(x)dy=2Re

z∫

1

[Rew

H1(y)+

(i

−j

)Imw

H2(x)

]dz + b0 in

(D+

D−

),

w(z) = Φ(Z) + Ψ(Z) = Φ(Z) + Ψ(Z), T (Z) = −1

π

∫ ∫

Dt

f(t)

t− Zdσt,

Ψ(Z) = T (Z) + T (Z), Ψ(Z)=T (Z) − T (Z) in D+Z ,

w(z) = φ(z) + ψ(z) = ξ(z)e1 + η(z)e2 in D−,

η(z)=−

ν∫

0

g2(y)

4H1(y)H2(x)dν=θ(z)+

y∫

0

g2(z) dy=

∫

S2

g2(y) dy+

y∫

0

g2(z) dy

=

|y|∫

y0

g2(z) dy, z ∈ s2,

(2.7)

198 Guo Chun Wen

ξ(z) = ζ(z) +

y∫

0

g1(z) dy, z ∈ s1,

gl(z)=Al(U+V )+Bl(U−V )+2ClU+Dlu+El, l = 1, 2,

ξ(z)=ζ(z)+

x∫

0

g1(z) dx, z ∈ s1, η(z)=θ(z)+

x∫

0

g2(z) dx, z ∈ s2,

gl(z)=Al(U+V )+Bl(U−V )+2ClV +Dlu+El, l = 1, 2,

(2.7)

in which Z = X + iY = G2(x) + iG1(y), f(Z) = g(Z)/H1H2, U = H1ux/2,V = −H2uy/2, ζ(z)e1 +θ(z)e2 is a solution of(2.6) in D−

Z , s1, s2 are two families ofcharacteristics inD−:

s1 :dx

dy=H1(y)

H2(x), s2 :

dx

dy= −

H1(y)

H2(x)(2.8)

passingthrough the pointz = x+ jy ∈ D−, S1, S2 are the characteristic curves from thepoints onL1, L2 to two points onL′

0 respectively,

θ(z)=

∫

S2

g2(z) dy, η(z) = −

ν∫

0

[g2(z)/4H1(y)H2(x)

]dν

is the integral along characteristic curves1 from a pointz0 = x0 + jy0 on L2 to thepoint z = x + jy ∈ D−

Z , θ(x) = −ζ(x) on L′

0, andζ(z) = −θ(G2(x)−G1(y)) on thecharacteristic curves ofs1, s2 passing through the pointz = x respectively, and

w(z) = U(z) + jV (z) =1

2H1ux −

j

2H2uy,

ξ(z)=Reψ(z)+Imψ(z), η(z)=Reψ(z)−Imψ(z),

A1 =1

4

[h1y

h1+H1h2x

H2h2−

2b

H22

], B1 =

1

4

[h1y

h1−H1h2x

H2h2+

2b

H22

],

A2 =1

4

[h1y

h1+H1h2x

H2h2+

2b

H22

], B2 =

1

4

[h1y

h1−H1h2x

H2h2−

2b

H22

],

A1 =1

4

[H2h1y

H1h1+h2x

h2+

2a

H21

], B1 =

1

4

[H2h1y

H1h1−h2x

h2+

2a

H21

],

A2 =1

4

[−H2h1y

H1h1−h2x

h2+

2a

H21

], B2 =

1

4

[−H2h1y

H1h1+h2x

h2+

2a

H21

],

C1 =a

2H1H2+m1

4y, C2 =−

a

2H1H2+m1

4y, D1 =−D2 =

c∗

2H2,

E1 = −E2 =d

2H2in D−

1 , C1 = −b

2H1H2+m2

4x,

C2 =−b

2H1H2−m2

4x, D1 =D2 =

c∗

2H1, E1 = E2 =

d

2H1in D−

2 ,


in whichH1(y) =

[|y|m1h1(y)

]1/2, H2(x) =

[|x|m2h1(x)

]1/2,

hereinh1(y), h2(x) are positive continuously differentiable functions.

Proof. Here and later on we only discuss the integrals inD−

1 , the case inD−

2 can be

similarly discussed. From (2.4) it is easy to see that equation (1.2) inD−

1 canbe reducedto the system of integral equations: (2.7). Moreover we can extend the equation (2.4) ontothe the symmetrical domainDZ of D−

1Z with respect to the real axisImZ = 0, namelyintroduce the functionW (Z) as follows:

W (Z) =

W [z(Z)],

−W [z(Z)],u(z) =

u(Z) in D−

1Z ,

−u(Z) in DZ ,

and then the equation (2.4) is extended as

Wz = A1W + A2W + A3u+ A4 = g(Z) in D−

1Z ∪ DZ ,

where

Al(Z)=

Al(Z),

Al(Z),l=1, 2, 3, A4(Z)=

A4(Z),

−A4(Z),

gl(Z)=

gl(z) in D−

1Z ,

−gl(Z) in DZ ,l=1, 2,

(2.9)

hereA1(Z) =A2(Z), A2(Z) =A1(Z), A3(Z) = A3(Z), and we mention that in generalu(z) onL′

0 may not be continuous. It is easy to see that the system of integral equations(2.9) can be written in the form

η(z)=θ(z)+

y∫

0

g2(z) dy=

y∫

y0

g2(z) dy,

ξ(z) = ζ(z) +

y∫

0

g1(z)dy=

y∫

y0

g1(z) dy, z = x+ jy = x+ j|y| in D−

1 ,

(2.10)

wherex0+jy0 is the intersection point ofL2 and the characteristic curves1 passing throughz = x + jy ∈ D−

1 , the functionζ(z) is determined byθ(z), i.e. the functionζ(z) can bedefined byζ(z) = −θ(z) = −θ(G2(x) − G1(y)), for the extended integral, which can beappropriately defined inD−

1Z , for convenience later on the above formg2(z) is written still,and the numbersy − y0, t− y0 will be written by y, t respectively.

3. Existence of Solutions of Tricomi Problemfor Mixed Equations

For proving the existence of solutions of Tricomi problem for mixed equations with nons-mooth degenerate line inD, we first give the estimates of the solutions of ProblemT for

200 Guo Chun Wen

(1.2) inDZ = D+Z . It is clear that ProblemT is equivalent to ProblemA for the complex

equation

WZ =1

H1H2

[A1W +A2W +A3u+A4

]in DZ ,

A1 =iH2H1y

4H1+H1H2x

4H2−

a

4H1−

ib

4H2, A3 =

−c∗

4,

A2 =iH2H1y

4H1−H1H2x

4H2−

a

4H1+

ib

4H2, A4 =

−d

4,

(3.1)

with the boundary condition

Re[λ(z)W (z)

]= R(z) on Γ ∪ L2 ∪ L4, u(1) = d1, u(0) = 0, (3.2)

and the relation

u(z)=u(x)−2

y∫

0

V (z)

H2(x)dy=2Re

z∫

0

[ReW

H1(y)+i

ImW

H2(x)

]dz in D+. (3.3)

As stated in Section 1, we can assumeR(x) = 0 onΓ∪L2∪L4 in (3.2),d1 = 0, u(0) = 0,because the indexK = 0 of λ(z) on∂DZ . In the following we first prove that there existsa solution of ProblemA+ for (3.1), (3.3) with the boundary condition (3.2) onΓ and

Re[−iW (x)] = −1

2H2(x)R1(x) on L′

0, Re[W (iy)] =1

2H1(y)R2(y) on L′′

0,

and the boundary value problem for (3.1), (3.3) with the boundary condition (3.2) onL2∪L4

and

Re[−jW (x)]=H2(x)

2R1(x)=R(z) on L′

0, Re[W (jy)]=1

2H1(y)R2(y)=R(z) on L′′

0

will be called ProblemA−, whereR1(x), R2(y), R1(x), R2(y) are as stated in (1.14).From the method and result in [8]–[11], we know that ProblemA+ for equation (3.1), (3.3)in D+ has a solutionW (z). Hence in the following we only prove the unique solvability ofProblemA− for (3.1), (3.3) inD−, which is the Darboux type problem (see [2]).

Theorem 3.1. If equation (1.1) satisfies Condition C, then there exists a solution[w(z), u(z)] of ProblemA− for (3.1)–(3.3).

Proof. We can only discuss inD−

1 , because the case inD−

2 can be similarly discussed.

By using the method in [10], we may only discuss the problem inD∗ = D−

1 ∩ (0 ≤)a0 = δ0 ≤ x ≤ b0 = 1 − δ0 (< 1), −δ ≤ y ≤ 0, ands1, s2 are the characteristics offamilies in Theorem 2.1 emanating from any two points(a0, 0), (b0, 0) (0 ≤ a0 < b0 < 1),whereδ, δ0 are sufficiently small positive numbers. In this case, we can omit the functionK2(x), and may only consider the functionK(y) = K1(y) = −|y|mh(y) = −|y|m1h1(y),wherem = m1, h(y) = h1(y) is a continuously differentiable positive function inD−

1 .

It is clear that for two characteristicss1, s2 passing through a pointz = x + jy ∈ D−

1


andx1, x2 arethe intersection points with the axisy = 0 respectively, for any two pointsz1 = x1 + jy ∈ s1, z2 = x2 + jy ∈ s2, we have

|x1−x2|≤|x1−x2|

=2

∣∣∣∣∣∣

y∫

0

√−K(t) dt

∣∣∣∣∣∣≤

2k0

m+2|y|1+m/2≤M |y|m/2+1 for −δ≤y≤0, (3.4)

whereM is a positive constant as stated in (3.6) below, andd is the diameter ofD−

1 . FromConditionC, we can assume that the coefficients of (2.7) possess continuously differen-tiable with respect tox ∈ L′

0 and satisfy the conditions

|Al|, |Alx|, |Bl|, |Blx|, |Dl|, |Dlx| ≤ k0 ≤ k1/6,

|El|, |Elx|≤k1/2, 2√h, 1/

√h, |hy/h|≤k0≤k1/6 in D, l = 1, 2,

(3.5)

and we shall use the constants

M = 4 max[M1,M2,M3], M1 = max

[8(k1d)

2,M3

k1

],

M2 =(2 +m)k0d

2

δ2+m

[4k1+

4ε0+m

δ

], M3 =2k2

1

[d+

1

2H(y′1)

],

γ = max

[2k1dδ

β +4ε(y)+m

2β′

]<1, −δ ≤ y ≤ 0,

(3.6)

andMl (l = 1, 2, 3) are positive constants,d is the diameter ofD, β′ = (1+m/2)(1−3β),ε0 = max

D−ε(z), 1/2H(y′1) ≤ k0[(m + 2)a0/k0]

−m/(2+m), δ, β are sufficiently smallpositive constants, andy′1 is an appropriately negative number. We choosev0 = 0, ξ0 = 0,η0 = 0 and substitute them into the corresponding positions ofv, ξ, η in the right-handsides of (2.7), and by the successive iteration, we find the sequences of functionsvk,ξk, ηk, which satisfy the relations

vk+1(z) = vk+1(x)−2

y∫

0

Vk(z)dy=vk+1(x)+

y∫

0

(ηk − ξk) dy,

ξk+1(z) = ζk+1(z) +

y∫

0

g1k(z) dy =

y∫

y0

glk dy,

ηk+1(z) = θk+1(z) +

y∫

0

g2k(z)dy =

y∫

y0

g2k(z) dy,

glk(z) = Alξk+Blηk+Cl(ξk+ηk)+Dlvk+El, l=1, 2, k=0, 1, 2, . . . ,

(3.7)

settingglk+1(z) = glk+1(z) − glk(z) (l = 1, 2) and

y= y−y1, t= t−y1, vk+1(z)=vk+1(z)−vk(z), ξk+1(z)=ξk+1(z)−ξk(z),

ηk+1(z)=ηk+1(z)−ηk(z), ζk+1(z)=ζk+1(z)−ζk(z), θk+1(z)=θk+1(z)−θk(z),

202 Guo Chun Wen

wherev(x) = u(x) − u0(x) onL′

0 as stated before, andz1 = x1 + jy1 is the intersectionpoint of the characteristic curves1 and the boundaryL2. Moreover we can prove thatvk,ξk ηk ζk, θk in D∗ satisfy the estimates

|vk(z) − vk(x)|, |ξk(z)−ζk(z)|, |ηk(z)−θk(z)|≤M′γk−1|y|1−β ,

|ξk(z)|, |ηk(z)| ≤M(M2|y|)k−1/(k − 1)!≤M ′γk−1,

∣∣ξk(z1)−ξk(z2)−ζk(z1)−ζk(z2)∣∣≤M(M2|y|)

k−1|x1−x2|1−β/(k−1)!

≤M ′γk−1[|x1−x2|

1−β+|x1−x2|β |t|β

′], |ηk(z1)−ηk(z2)−θk(z1)−θk(z2)|,

|vk(z1)−vk(z2)|, |ξk(z1)−ξk(z2)|, |ηk(z1)−ηk(z2)|≤M(M2|t|)k−1

×|x1−x2|1−β/(k−1)! ≤M ′γk−1

[|x1−x2|

1−β+|x1−x2|β|t|β

′],

∣∣ξk(z) + ηk(z)−ζk(z) − θk(z)∣∣≤M ′γk−1|x1−x2|

β|y|β′

,

|ξk(z)+ηk(z)| ≤M(M2|y|)k−1|x1−x2|

β |y|β′

/(k−1)!

≤M ′γk−1|x1−x2|1−β , 0≤|y|≤δ,

(3.8)

wherez = x+ jy, z = x+ jt is the intersection point ofs1, s2 passing through the pointsz1, z2, β′ = (1 + m/2)(1 − 3β/2), β is a sufficiently small positive constant, such that(2 +m)β < 1, andM ′ is a sufficiently large positive constant.

On the basis of the above estimate (3.8), the convergence of two sequences of functionsM(M2|y|)

k−1/(k − 1)!, M ′γk−1|y|β′

and the comparison test, we can derive thatvn, ξn, ηn in D∗ uniformly converge tov∗, ξ∗, η∗ satisfying the system of integralequations

v∗(z) = v∗(x)−2

y∫

0

V∗ dy=u∗(x)+

y∫

0

(η∗−ξ∗) dy,

ξ∗(z) = ζ∗(z)+

y∫

0

[A1ξ∗+B1η∗+C1(ξ∗+η∗)+D1v∗+E1

]dy, z∈s1,

η∗(z) = θ∗(z)+

y∫

0

[A2ξ∗+B2η∗+C2(ξ∗+η∗)+D2v∗+E2

]dy, z∈s2,

and the function[W∗(z), v∗(z)] = [(ξ∗ +η∗ + jξ∗− jη∗)/2, v∗(z)] is a solution of ProblemA− for equation (3.1). Moreover the functionu(z) = v∗(z)+u0(z) is a solution of ProblemT for (1.2) inD−. The proof is finished. Besides by the similar method, we can prove theuniqueness of solution of ProblemA− in D1 in (3.1)–(3.3) withc = c∗.

From the above discussion, we obtain the following theorem.

Theorem 3.2.Let equation(1.2) with c = c∗ satisfy Condition C. Then the above Tricomiproblem(ProblemT ) for (1.2) with c = c∗ has a unique solution.


4. The Frankl Problem for Mixed Equations

Now we consider some general domains with non-characteristic boundary and prove thesolvability of Frankl problem for equation (1.2).

1) LetD be a simply connected bounded domainD in the complex planeC with theboundary∂D = Γ ∪ L, whereΓ, L are as stated before. Now, we consider the domainD′

with the boundaryΓ∪L′

1∪L′

2∪L′

3∪L′

4, as stated in Section 1, the curveΓ can be replacedby another smooth curveΓ′, because it can be realized through a conformal mapping. Theparameter equations of the curvesL′

1, L′

2, L′

3, L′

4 are as follows:

L′

1 =γ1(s)+y=0, 0≤s≤s′1

, L′

2 =x−G(y)=1, l1≤x≤1

,

L′

3 =γ2(s)+x=0, 0≤s≤s′2

, L′

4 =y−G(x)=1, l2≤y≤1

,

(4.1)

whereY =G1(y) =y∫0

√|K1(y)| dy in D1, X =G2(x) =

x∫0

√|K2(x)| dx in D2, γk(s) on

Sk = 0 ≤ s ≤ s′k (l1 =G1[−γ1(s′

1)], l2 =G2[−γ2(s′

2)] arecontinuously differentiable,γk(0) = 0, γk(s) > 0 on 0 < s ≤ s′k (k = 1, 2), G′

1(y) = H1(y), G′

2(x) = H2(x),the slope of the curvey = −γ1(s) at the intersection pointz∗1 of L′

1 and the characteristiccurve ofs1 : dy/dx = 1/H1(y) in x + jy−plane is not equal to that of the characteristiccurve at the point, and the slope of the curvey = −γ2(s) at the intersection pointz∗2 ofL′

3 and the characteristic curve ofs1 : dy/dx = 1/H1(y) in x + jy−plane is not equal tothat of the characteristic curve at the point,z′1 = l1 − jγ1(s

′

1), z′

2 = −γ1(s′

1) + jl2 are theintersection point ofL′

1, L′

2 andL′

3, L′

4 respectively. Actually we can permit that the curveL′

1 with any characteristic curve ofs1 : dy/dx = 1/H1(y) has at most one intersectionpoint, similarly we can discuss the curveL′

3. From the above conditions, we can determinethex−coordinate ofL′

1 andy−coordinate ofL′

3. Here we mention that in [2], under thenon-characteristic curvey = −γ(x) satisfying0 < γ′(x) ≤ 1 on L0 = 0 ≤ x ≤ 1,A.V. Bitsadze discussed the mixed equationsgnyuxx + uyy = 0 by the method of integralequations, even though the reasoning occupied 26 pages (pp. 379–406, [2]), the Franklproblem had not been completely solved.

We consider the Frankl problem (ProblemF ′) for equation (1.2) inD′ with the bound-ary conditions

u(z)=φ(z) on Γ, u(z)=ψ(z) on L′, i.e. Re[λ(z)w(z)]=R(z), z ∈ Γ ∪ L′,

Im[λ(z)w(z)]∣∣z=z′

1

= c1, Im[λ(z)w(z)]∣∣z=z′

2

= c′2, u(t1) = d1.

(4.2)Herein denoteL′ = L′

2∪L′

4, t1 = 1, w(z) = uz, λ(z) = a(x) + ib(x) on Γ, λ(z) =a(x) + jb(x) onL′, andλ(z), r(z), cl (l = 1, 2), d1 satisfy the conditions

C1α[λ(z),Γ] ≤ k0, C

1α[r(z),Γ] ≤ k2,

C1α[λ(z), L′]≤k0, C

1α[r(z), L′]≤k2, |cl|, |d1|≤k2, l=1, 2,

maxz∈L′

2

1

|a(x) + b(x)|, max

z∈L′

4

1

|a(y) + b(y)|≤ k0,

(4.3)

whereλ(z), r(z) are as stated in (1.6)–(1.7), andα (0 < α < 1), k0, k2 are positiveconstants.

204 Guo Chun Wen

SettingY = G1(y) =y∫0

√|K1(t)| dt, X = G2(x) =

x∫0

√|K2(t)| dt. By the con-

ditions in (4.1), the inverse functionx = σ1(ν) = (µ + ν)/2 of ν = x − G1(y) canbe found, i.e. µ = 2σ1(ν) − ν, 0 ≤ ν ≤ 1 and the curveL′

1 can be expressed byµ = 2σ1(ν) − ν = 2σ1(x + γ1(s)) − x − γ1(s) on S1, andy = σ2(ν) = (µ − ν)/2of µ = G2(x) + y can be found, i.e.µ = 2σ2(ν) + ν, 0 ≤ ν ≤ 1 and the curveL′

3

can be expressed byµ = 2σ2(ν) + ν = 2σ2(γ2(s) + y) + γ2(s) + y on S2. We make atransformation

µ =[µ− 2σ1(ν) + ν

]/[1 − 2σ1(ν) + ν

], ν = ν, 2σ1(ν) − ν ≤ µ ≤ 1,

µ =[µ− 2σ2(ν) − ν

]/[1 − 2σ2(ν) − ν

], ν = ν, 2σ2(ν) + ν ≤ µ ≤ 1,

(4.4)

whereµ, ν are real variables, their inverse transformations are

µ =[1 − 2σ1(ν) + ν

]µ+ 2σ1(ν) − ν, ν = ν, 0 ≤ µ, ν ≤ 1,

µ =[1 − 2σ2(ν) − ν

]µ+ 2σ2(ν) + ν, ν = ν, 0 ≤ µ, ν ≤ 1.

(4.5)

It is not difficult to see that the transformation in (4.4) maps the setD′ ontoD. Denote by

Z = x+ jY = x+ jG1(y) = f(x+ jY ) = f(Z),

Z = x+ jY = x+ jG(y) = f−1(Z)(4.6)

the above transformation and its inverse transformation respectively, wherex = [µ+ ν]/2,Y = [µ− ν]/2, and by

z = x+ jy = z(Z) = z[f(Z(z))] = f(z),

z = x+ jy = f−1(z),(4.7)

the corresponding transformation and its inverse transformation respectively. In this case,the last system of equations in (2.4) can be rewritten as

ξµ = A1ξ +B1η + C1(ξ + η) +Du+ E,

ην = A2ξ +B2η + C2(ξ + η) +Du+ E,z ∈ D′−. (4.8)

Suppose that (1.2) inD′ satisfies ConditionC, through the transformation (4.4), we obtainξµ = [1 − 2σ1(ν) + ν]ξµ, ην = ην in D′

1 = D′ ∩ y < 0, andξµ = [1 − 2σ2(ν) − ν]ξµην = ην in D′

2 = D′ ∩ x < 0, whereξ = U + V, η = U − V , and then

ξµ =[1 − 2σ1(ν) + ν

] [A1ξ +B1η + C1(ξ + η) +Du+ E

],

ην = A2ξ +B2η + C2(ξ + η) +Du+ E in D1 =D∩y<0,

ξµ =[1 − 2σ2(ν) − ν

] [A1ξ +B1η + C1(ξ + η) +Du+ E

],

ην = A2ξ +B2η + C2(ξ + η) +Du+ E in D2 =D∩x<0,

(4.9)

and through the transformation (4.6), the boundary condition (4.2) is reduced to

Re[λ(f−1(z))W (f−1(z))

]= R(f−1(z)), z ∈ Γ ∪ L2 ∪ L4,

Im[λ(f−1(zk))W (f−1(zk)

]= ck, k=1, 2, u(1)=d1,

(4.10)


in which zk = f(z′k), k = 1, 2. Therefore the boundary value problem (4.8), (4.2) (ProblemA′) is transformed into the boundary value problem (4.9), (4.10), i.e. the correspondingProblemA in D. On the basis of Theorem 3.2, we see that the boundary value problem(4.9),(4.10) has a solutionw(z), and

u(z)=2R

z∫

0

[ReW

H1(y)+

(i

−j

)ImW

H2(x)

]dz + d1 in

(D+

D−

)

is just a solution of ProblemF ′ for (1.2) inD′ with the boundary condition (4.2).

Theorem 4.1. If equation(1.2) in D′ satisfies ConditionC in the domainD′ with theboundaryΓ∪L′

1∪L′

2∪L′

3∪L′

4, whereL′

1,L′

2,L′

3,L′

4 are as stated in(4.1), then ProblemF ′ for (1.2) with the boundary conditions(4.2) has a solutionu(z).

2) Next let the domainD′′ be a simply connected domain with the boundaryΓ ∪ L′′

1 ∪L′′

2 ∪ L′′

3 ∪ L′′

4, whereΓ is as stated before, which can be replaced by another smooth curveΓ′′, and similarly to the case 1, the parameter equations of the curvesL′′

1, L′′

2, L′′

3, L′′

4 are asfollows:

L′′

1 =γ1(s)+y=0, 0≤s≤s′1

, L′′

2 =γ2(s)+y=0, 0≤s≤s′2

,

L′′

3 =γ2(s)+x=0, 0≤s≤s′3

, L′

4 =γ4(s)+x=0, 0≤s≤s′4

,

(4.11)

in which γk(0) = 0, γk(s) on Sk = 0 ≤ s ≤ s′k (k = 1, 2, 3, 4) are continuouslydifferentiable,z′′1 = l1 − jγ1(s

′

1), z′′

2 = −γ2(s′

2) + jl2 are the intersection points ofL′′

1,L′′

2 andL′′

3, L′′

4 respectively, the slope of curveL′′

2 at the intersection pointz∗1 of L′′

2 and thecharacteristic curve ofs1 : dy/dx = −1/H1(y) is not equal to that of the characteristiccurve at the point, and the slope of curveL′′

4 at the intersection pointz∗2 of L′′

4 and thecharacteristic curve ofs2 : dx/dy = −1/H2(x) is not equal to that of the characteristiccurve at the point. The curvesL′′

1, L′′

3 satisfy some conditions as stated below.The so-called Frankl problem (ProblemF ′′) for equation (1.2) in the domainD′′ is to

find a solution of (1.2) inD′′ satisfying the boundary conditions

u(z)=φ(z) on Γ, u(z)=ψ(z) on L′′, i.e. Re[λ(z)w(z)

]=R(z), z ∈ Γ ∪ L′′,

Im[λ(z)w(z)]∣∣z=z′′

1

= c1, Im[λ(z)w(z)]∣∣z=z′′

2

= c2, u(t1) = d1.

(4.12)Herein denotew(z) = uz, L′′ = L′′

2 ∪L′′

4, λ(z) = a(x)+ ib(x) onΓ, λ(z) = a(x)+ jb(x)onL′′, andλ(z), r(z), cl (l = 1, 2), d1 satisfy the conditions

C1α[λ(z),Γ] ≤ k0, C

1α[r(z),Γ] ≤ k2,

C1α[λ(z), L′′]≤k0, C

1α[r(z), L′′]≤k2, |cl|, |d1|≤k2, l=1, 2,

maxz∈L′′

2

1

|a(x) + b(x)|, max

z∈L′′

4

1

|a(y) + b(y)|≤ k0,

(4.13)

whereλ(z), r(z) are as stated in (1.6)–(1.7), andα (0 < α < 1), k0, k2 are positiveconstants.

By the conditions in (4.11), the inverse functionx = τ1(µ) = (µ + ν)/2 of µ =x+G1(y) can be found, i.e.ν = 2τ1(µ)−µ, 0 ≤ µ ≤ 1, the inverse functiony = τ2(µ) =

206 Guo Chun Wen

(µ − ν)/2 of µ = G2(x) + y can be found, i.e.ν = −2τ2(µ) + µ, 0 ≤ µ ≤ 1, and thecurveL′′

2, L′′

4 can be expressed by

ν = 2τ1(µ) − µ = 2τ1(x− γ2(s)) − x+ γ2(s),

ν = −2τ2(µ) + µ = −2τ2(x− γ4(s)) + x− γ4(s).(4.14)

We make a transformation

µ = µ, ν =ν − 2τ1(µ) + µ

2τ1(µ) − µ+ 1, 0 ≤ ν ≤ 2τ1(µ) − µ,

µ = µ, ν =ν + 2τ2(µ) − µ

−τ2(µ) + µ+1, 0 ≤ ν ≤ −2τ2(µ) −µ.

(4.15)

It is clear that their inverse transformations are

µ = µ, ν = (ν − 1)(2τ1(µ) − µ) + 2τ1(µ) − µ, 0 ≤ µ, ν ≤ 1,

µ = µ, ν = (ν − 1)(−2τ2(µ) + µ) − 2τ2(µ)+µ, 0 ≤ µ, ν ≤ 1,(4.16)

Denote by

Z = x+ jY = x+ jG(y) = g(x+ jY ) = g(Z),

Z = x+ jY = x+ jG(y) = g−1(Z),

the above transformation and its inverse transformation respectively, wherex = [µ+ ν]/2,Y = [µ− ν]/2, and by

z = x+ jy = z(Z) = z[g(Z(z))] = f(z),

z = x+ jy = g−1(z),(4.17)

the corresponding transformation and its inverse transformation respectively. In this case,the last system of equations in (2.4) can be rewritten as

ξµ = A1ξ +B1η + C1(ξ + η) +Du+ E,

ην = A2ξ +B2η + C2(ξ + η) +Du+ E,z ∈ D′′−. (4.18)

Through the transformation (4.15), we obtain

(u+ v)µ =(u+v)µ, (u−v)ν =[2τ1(µ) − µ](u−v)ν in D′′

1 = D′′ ∩ y < 0,

(u+ v)µ =(u+v)µ, (u−v)ν =[−2τ2(µ) + µ](u−v)ν in D′′

2 = D′′ ∩ x < 0.(4.19)

System (4.18) inD′′− is reduced to

ξµ = A1ξ +B1η + C1(ξ + η) +Du+ E,

ην =[2τ1(µ)−µ

] [A2ξ+B2η+C2(ξ+η)+Du+E

]in D′

1,

ξµ = A1ξ +B1η + C1(ξ + η) +Du+ E,

ην =[− 2τ2(µ)+µ

] [A2ξ+B2η+C2(ξ+η)+Du+E

]in D′

2.

(4.20)


Moreover, through the transformation (4.17), the boundary condition (4.12)on L′′

2, L′′

4 isreduced to

Re[λ(g−1(z))W (g−1(z))

]= R[g−1(z)], z ∈ Γ ∪ L′

2∪L′

4,

Im[λ(g−1(z′k))W (g−1(z′k))

]= ck, k = 1, 2, u(1) = d1,

(4.21)

in which z′k = g(z′′k), k = 1, 2. Therefore the boundary value problem (4.8), (4.12) inD′′

is transformed into the boundary value problem (4.20), (4.21), where we require that theboundariesL′

k = g(L′′

k) (k = 1, 3) satisfy the similar conditions in (4.1). According to themethod in the proof of Theorem 4.1, we can see that the boundary value problem (4.20),(4.21) has a solutionu(z), and then the correspondingu = u(z) is a solution of ProblemF ′′ of equation (1.2).

Theorem 4.2. If the mixed equation(1.2) satisfies ConditionC in the domainD′′ withthe boundaryΓ ∪ L′′

1 ∪ L′′

2 ∪ L′′

3 ∪ L′′

4, whereL′′

1, L′′

2, L′′

3, L′′

4 are as stated in(4.11), thenProblemF ′′ for (1.2) in D′′ has a solutionu(z).

References

[1] L. Bers,Mathematical aspects of subsonic and transonic gas dynamics, Wiley, NewYork, 1958.

[2] A. V. Bitsadze,Some classes of partial differential equations, Gordon and Breach,New York, 1988.

[3] J. M. Rassias, Uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem,Complex Variables47 (2002), 707–718.

[4] M. S. Salakhitdinov and B. Islomov, The Tricomi problem for the general linear equa-tion of mixed type with a nonsmooth line of degeneracy,Soviet Math. Dokl.34(1987),133–136.

[5] M. M. Smirnov, Equations of mixed type,Amer. Math. Soc.,Providence RI, 1978.

[6] H. S. Sun, Tricomi problem for nonlinear equation of mixed type,Sci. in China(SeriesA) 35 (1992), 14–20.

[7] I. N. Vekua,Generalized analytic functions, Pergamon, Oxford, 1962.

[8] G. C. Wen, Conformal mappings and boundary value problems,Translations of Math-ematics Monographs106, Amer. Math. Soc., Providence, RI, 1992.

[9] G. C. Wen and H. Begehr,Boundary value problems for elliptic equations and systems,Longman Scientific and Technical, Harlow, 1990.

[10] G. C. Wen,Linear and quasilinear complex equations of hyperbolic and mixed Type,Taylor & Francis, London, 2002.

[11] G. C. Wen, Solvability of the Tricomi problem for second order equations of mixedtype with degenerate curve on the sides of an angle,Math. Nachr.281 (2008), 1047–1062.

INDEX

A

Abelian, 120 alternative, 70, 115, 116, 141, 142 analog, 25 antagonistic, viii, 15, 16 application, 92 applied mathematics, viii Archimedes, 117, 141 assumptions, 15, 17, 39, 43, 76, 180 asymptotic, 86, 132 asymptotically, 173

B

Banach spaces, 1, 13, 14, 52, 53, 57, 67, 70, 86, 92, 94, 107, 109, 111, 113, 115, 117, 119, 128, 129, 131, 135, 150, 172, 174

boundary conditions, 97, 193, 195, 203, 205 boundary value problem, 74, 81, 95, 105, 197, 200,

205, 207 bounded linear operators, 178 Brno, 46, 141

C

C*-algebra, 67, 68, 69, 93, 129, 142, 180 Cauchy problem, 97 components, 32 conjugation, 96 continuity, 44, 58, 86, 132 contractions, 115 control, 2, 16, 20, 108, 120, 183, 186, 187 convergence, 27, 60, 62, 202 convex, 41, 116, 187 coordination, 97 cosine, 67 CRC, 35 Cybernetics, 35, 36

D

decomposition, 179, 180, 185 degenerate, ix, 191, 192, 199, 207

derivatives, 76 differential equations, viii, 38 differentiation, 80

E

energy, vii, 35 equality, 40, 41, 44, 55, 101, 114, 120 equilibrium, 16 Euclidean space, 189 evolution, 17 expansions, 75

F

factorial, 75 Feynman, vii filtration, 19, 24 financial support, 66 Fourier, 105

G

games, viii, 15, 16, 20, 33, 35 gas, 105, 207 gene, 105 generalization, 2, 14, 52, 67, 68, 86, 93, 108, 116, 119,

128, 132, 141, 150, 173, 174 generalizations, 105 Ger, 37, 38, 39, 40, 41, 42, 44, 46 graduate students, viii groups, 150, 151

H

Hilbert space, 52, 70, 136, 177, 178, 179, 180, 183, 184, 185, 187

homomorphism, 56, 57, 59, 61, 62, 63, 85, 110, 124, 128, 131, 143

homomorphisms, viii, 1, 14, 55, 57, 58, 59, 68, 69, 85, 107, 109, 119, 124, 128, 131, 141, 150

hybrid, 34, 35 hyperbolic, 75, 83, 105, 192, 194, 196, 207

Index

210

I

identification, 187 identity, 75, 76, 135, 137, 180 independence, 22 indices, 24, 25, 33 induction, 4, 5, 6, 7, 8, 9, 10, 11, 40, 59, 62, 89, 138,

146 infinite, 186 injection, 88 interval, 17, 36, 37, 38, 41, 44, 46, 58, 96 inversion, 100 iteration, 201

K

Korean, 69, 129, 172, 173, 174, 188

L

linear function, 13, 14, 52, 68, 93, 96, 116, 128, 131, 141, 147, 173, 175, 181, 188

M

maritime, 177 meanings, 2 measures, 18, 31, 33 memory, 107, 119 metric, 47, 57, 85, 86, 108, 113, 115, 131, 143, 150,

173, 188 metric spaces, 173, 188 modeling, 35 models, 16 modules, viii, 58, 67, 69, 107, 110, 129, 180, 188 modulus, 187 Monte Carlo method, vii multiples, 178, 181 multiplication, 187 multivariate, 16

N

natural, 56, 120, 138, 144, 178 nonlinear, ix, 48, 68, 116, 142, 179, 180, 191, 192,

207 normal curve, 96 normed linear space, 2, 39, 42, 43, 45, 48, 86, 108,

111, 133, 134, 142, 188 norms, 1, 57, 58, 86, 87, 132, 140, 149, 150, 152 nuclear weapons, vii

O

observations, 18, 19, 33, 186 operator, 26, 30, 101, 104, 136, 178, 179, 180, 184,

185 organizations, viii orthogonality, ix, 177, 178, 180, 181, 182, 183, 185,

186, 188, 189

P

parabolic, 75, 105 parameter, 19, 30, 193, 194, 203, 205 partial differential equations, 74, 75, 95, 207 permit, 203 perturbation, 57, 68 physical sciences, viii physicists, viii physics, vii, viii Poisson, 15, 18, 22, 31, 32, 69 powers, 57, 58, 86, 87, 132, 140, 149, 150, 152 probability, 18, 87, 187 probability theory, 87

Q

quantum, 35, 187 quantum mechanics, 187 quasilinear, 207

R

radius, 179 random, vii, 15, 16, 17, 18, 20, 24, 25, 31, 32, 33 random numbers, vii random walk, 15, 16, 20, 32 range, 187 real numbers, 37, 63, 86, 107, 114, 115, 122, 124, 126,

127, 144 reasoning, 40, 43, 44, 113, 203 recall, 87, 108 referees, 34 reflexivity, 41 regular, 75, 101, 104 relevance, vii research, viii, 15, 87 residues, 105 revolutionary, viii

S

scalar, 181, 187 shock, 105 simulation, 35 singular, viii, 86, 95, 100, 133, 184, 187, 188 singularities, 81, 97 solutions, viii, 2, 38, 73, 74, 75, 76, 78, 79, 81, 83, 105,

186, 191, 195, 197, 199, 207 spectrum, 178 statistics, 87 stochastic, 15, 16, 34, 35, 87 stochastic processes, 16 strikes, 18 students, viii, 66 subsonic, 207 systems, 207

Index

211

T

theory, vii, 2, 14, 15, 35, 46, 74, 105, 108, 187 thermonuclear, vii thresholds, 16, 20, 24 topological, 41 topology, vii transformation, 13, 28, 52, 67, 86, 92, 108, 111, 113,

115, 128, 133, 134, 135, 172, 204, 206, 207 transformations, 26, 32, 92, 204, 206 transition, 187

V

validity, 38 values, 15, 18, 20, 29, 31, 33, 39 variable, 17, 30, 86, 115, 132, 151, 155 variables, 34, 68, 73, 74, 75, 93, 96, 151, 155, 172,

173, 174, 204 vector, 2, 4, 6, 9, 11, 12, 13, 39, 45, 56, 57, 132, 150,

151, 152, 153, 156, 158, 160, 161, 162, 164, 165, 167, 187, 188

FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY NOTIONS (F.U.N.)

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