SEMIGROUPS OF OPERATORS, APPROXIMA TION AND SA ...

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APPROXIMA TION AND SA TURA TION

IN BAN A CH SPA CES

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TR diss 1598

CA. Timmermans

STELLINGEN

behorende bij het proefschrift

SEMIGROUPS OF OPERATORS, APPROXIMATION AND SATURATION

IN BANACB SPACES

door

CA. Timmermans

De openbare verdediging van hel proefschrift en de stellingen vindt plaats op

donderdag 17 december 1987 om 16.00 uw-

in de aula van de Technische Universiteit Delft,

Mekelveg te Delft

U bent hierbij van harte welkom. No afloop van de promotie is er een receptie.

Theorem 1 ■

I Let X = C[0.1]. equipped with the supretnum norm. Suppose *>'■ [0.1 ]x [R —> P. be continuous. Assume there exists a continuous function •en on (0.1) such that I 0

0 < *0(x) < -p(x.f) for x € (0.1). f € P..

Let A: D(A) C X —• X be defined by

D(A) = {u € C[0.1] O C2(0.1): lim f(x.u(x)) u"(x) = 0. j = 0.1)

(Au)(x) = *(x.u(x)) u"(x) for u € D(A) and x € (0.1).

Then the operator A is densely defined and m-dissipative in X.

References: 1. J.A. Goldstein. C-Y.Lin: Singular nonlinear parabolic boundary value

problems in one space dimension. Journ. Diff. Equations 68. pp. 429-443. (19S7).

2. J.A. Goldstein. C-Y.lin: Highly degenerate parabolic boundary value problems, October 19S7, preprint.

3. C-Y.Lin: Degenerate nonlinear parabolic boundary value problems. Thesis, Tulane Univ.. july 19S7.

Theorem 2 The sufficient condition

lim inf sup | (n (B f-f) - g) (x) | = 0. n-«° x€[0.1] n

where B is the n-th Bernstein operator and f.g € C[0,1] for the assertion f" € C(0.1) and x(l-x) f(x)/2 = g(x). 0 < x < 1. in [Mi], Th 3.2, is also a necessary condition, provided that g(0) = g(l) = 0 .

Reference: [Mi]: Micchelli, C.A. The saturation class and iterates of the Bernstein polynomials. Journ. of Approx. Th. S. pp 1-1S. (1973).

Theorem 3 Let the sequence (K ), n = 1,2,... of Kantorovic operators be defined by

V cCo.iD — c[o.i] n (<k-H)/(n+l)

(K f)(x) = (n+l) 2 p (x) f(t)dt. n = 1.2.... n k=0 n , K Jk/(n+l)

where Pn k(x) = [ £ J x (l-x)n" .

Let the space C; [0.1] (nt i 1) be equipped with the norm II«II defined by

llfll := 2 II D f II . f € d"[0.1]. m k=0

k where D f denotes the k-th derivative and II»II the supremum norm. Then for f € CÔ.l]

i) II KRf - f llm = o(n_1) . n —»» if and only if f is a polynomial of degree < 1.

i i ) II KJ - f llm = 0 ( n - 1 ) . n - ♦ »

i f and only i f u:= Dmf s a t i s f i e s u € C ^ O . l ) . u- € AC. (0 .1) n LÔ. l ) ,

#u" € L " ° ( 0 . 1 ) . where *(x) = x ( l - x )

Theorem 4 With the definitions and notations of chapter 1 of the thesis, the following holds:

f a € L (x0>r_). then r„ is not natural and moreover

) r2 is regular <=* W € LJ(x0>r2) A (aW)'1 € l\xQ.T2)

i) r2 is exit <=» W € L 1 ^ . ^ ) A (aW)"1 « L ^ X Q , ^ )

ii) r2 is extrance c* W C L ^ X Q . ^ ) A (oW)"1 € L ^ X Q . ^ ) and

-V> \ fa C L (x_. r2). then r_ is not regular and moreover ) r2 is natural =* W € L 1 ^ . ^ ) V (oW)"1 € L ^ x ^ ) i) W € L (xQ.r2) * r2 is exit or natural

ii) (aw) € L (x0.r2) * r2 is entrance or natural

Theorem 5

Let J = (r r_). - a> < r < x. < r, < • ; j is the two points compact ification of J: a.0 € C(J). a > 0 on J.

W(x) = exp | - J^ Pa-1dt j . x € (lyi^).

Assume p. .p„ € ( - = • . j r ) . X = C(J) equipped with the supremum norm, let A: D(A) C X —. A be defined by

D(A) = {u € C(J) n C ^ J ) ! lim u(x)sin p$ ♦ (W"V)(x)cos p. = 0. i = 1.2}.

Au = ou" ♦ 0u' for all u € D(A)

and suppose that the boundary points r. are regular in the sense of Feller, i.e.

W € l\j). (aW)"1 € LJ(J).

Let u. and u- be solutions of

u - Au = 0,

satisfying

-1 .1-1 lim Uj(x) = 0. lim (W V ) ( x ) = (-1) .1 = 1 < -» r. x -» r,

Finally, let G be the open subset of (- | . | ) x (- | . | ) in the xy-plane bounded by the curves x = - =• , y = =■ .

„-1 (u2(rj) tan x + (W u2)(rj)) tan y + tan x = 0. x i 0. y < 0. and „-1 (Uj(r2) tan y ♦ (W uj)(r2)) tan x - tan y = 0, x > 0 . y i 0-

Then A is densely defined and m-dissipative in X if and only if (p.,p„) € C.

Theorem 6 Lei Let Let X. A. G. u.. u_. W, p and p„, be defined as in the preceding theorem.

H = {(x.y) € (-|.|)x ( . | , | ) | - | < x < - arctan (W1 u ^ M r , ) .

arc tan (({u2(rj)tan x - ( w ' ^ M r j )) _ 1 - ( W ^ u j H ^ ) ) / u,(r2)) ^ y < TL

Then we have G C H. and H\C * <}>. Moreover, if (p., p„) € H\G. then i) A is densely defined.

11) I-A: D(A) —»X is a bijection. (I-A)"1 is positive, but (1-A)"1 is not a contraction on X and thus

ill) A is not m-dissipative.

Stelling 7

In R (n \ 2) is B een bol met straal a en R een overal even dikke bolschil met uitwendige straal b en inwendige straal c. n> punten worden op willekeurige wijze op het inwendige van B gestrooid. Indien p € W. zodat

/% n n* p < "'t0 - c ) (a + b ) n

dan is de bolschil altijd zodrjiig neer te leggen dat de bolschil p + 1 punten bevat.

Stelling 8 De regels van het spel Mastermind (Invicta P.Ltd) in de oorspronkelijke vorm met 6 kleuren en 4 plaatsen worden bekend verondersteld. Een strategie voor Mastermind is een algoritme om door middel van het stellen van vragen en het verkrijgen van de betreffende antwoorden volgens de regels van het spel een Mastermind code te breken. Zij A de verzameling der strategieën om een willekeurige Mastermind code te breken, en zij B de verwachting van het aantal vragen bij een strategie T € A. Dan geldt

JS ^ f S S h 4 - 3 8 8

Litt. R.W. Irving, Towards an optimum Mastermind Strategy. J. Recr. Math. Vol 11(2). pp. S1-S7 (19S7).

Ste l l ing 9 Bij een tennistoernooi voor gemengde tu t t i - f rut t i dubbels met 16 manlijke en 16 vrouwelijke deelnemers kan een wedstrijdrooster voor acht - en niet meer dan acht - volledige ronden van acht wedstrijden worden opgesteld op een zodanige wijze dat elke deelnemer elke andere deelnemer in de wedstrijden hoogstens éénmaal ontmoet a l s dubbelpartner of a l s tegenspeler. Een vol ledig wedstrijdrooster i s af te lezen uit het onderstaande schema, waarbij de vrouwelijke deelnemers genummerd zijn van 1 t/m 16 en de mannelijke van 17 t/m 32.

(14e Hompestomper tennistoernooi, okt. '67. Heerenveen)

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Opmerking: In dit achem hebben "kleine letters" dezelfde bcickrni* a l s hoofdletter». De "kleine let ler"- infori«t ie kan afpclrid worden uit de hoofdletter- informtie .

A speelt tegen dubbel B ♦ C D speelt tegen dubbel E * F C speelt tegen dubbel H * I J speelt tegen dubbel K ♦ L M speelt tegen dubbel N ♦ O P speelt tegen dubbel 0 ♦ R S speelt t c g c dubbel T < V V speelt tegen dubbel ï * Z

Men nene in elke r i j voor de diverse letters de correspondcicndc kolomnummers.

Voorbeeld 6e ronde'dubbel I ♦ 22 - dubbel 4 •• 23 dubbel 2 * 21 - dubbel 3 ♦ 24 dubbel 5 •» 18 - dubbel 6 •• 19 enz. .

« e d s i r i j d Ie ronde: 2e ronde.' 3e ronde: 4e ronde: ?* ronde: fic ronde: 7e ronde: 6c ronde:

r o o s t e r dubbel dubbel dubbel dubbel dubbel dubbel dubbel dubbel

X X X X X X X X

Stelline 10

De aantrekkelijkheid van het voetbal als kijkspel kan verhoogd worden door de afmetingen van het doel te vergroten.

Stelling 11

Zoals de leerlingen in het voortgezet onderwijs een schoolrapport ontvangen opgesteld door hun leraren, dienen omgekeerd de leraren een rapport te ontvangen opgesteld door de klassen waaraan ze lesgeven.

Stelling 12

Leraren bij wie het ontvangen van slechte rapporten - als bedoeld in de vorige stelling - zich voordoet als een chronisch verschijnsel, dienen verplicht te worden een bijscholingscursus te volgen.

Stelling 13

In de discussie over de mogelijke oorzaken van de zure regen wordt ten onrechte meestal niet genoemd: de invloed van electromagnetische straling opgewekt door radio- en radarzenders. Wetenschappelijk onderzoek naar deze invloed dient ter hand genomen te worden.

Stelling 14

De demokratisearring fan bestjoer fan de Fryske wetterskippen is earst foldwaande motivearre as ek de demokratisearring fan it behear fan it Fryske oerflaktewetter syn beslach krijt en yn hannen lein wurdt fan in wetter- en suveringskip. wylst de provinsje him beheine moat ta tafersjoch.

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SEMIGROUPS OF OPERATORS, APPROXIMATION AND SATURATION IN BAN ACH SPACES

CA. Timmermans

Delft University Press

TR diss 1598

2

SEMIGROUPS OF OPERATORS, APPROXIMATION AND SATURATION IN BAN ACH SPACES

Aan Hanny Ellen Henkjaap

Aan mijn moeder

3

SEMIGROUPS OF OPERATORS, APPROXIMATION AND SATURATION IN BANACH SPACES

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Hogeschool Delft,

op gezag van de rector magnificus, Prof.dr. J.M. Dirken,

in het openbaar te verdedigen ten overstaan van een commissie door

het College van Dekanen daartoe aangewezen op donderdag 17 december 1987

te 16.00 uur

door

CORNELIS ALBERTUS TIMMERMANS, geboren te Dordrecht, wiskundig ingenieur

Delft University Press

4

Dit proefschrift is goedgekeurd door de promotor Prof.dr. Ph.P.J.E. CLÉMENT

Samenstelling van de commissie:

Prof.dr. Ph.P.J.E. Clément (promotor) Prof.dr. O. Diekmann Prof.dr. A.W. Grootendorst Prof.dr.ir. R. Martini

Prof.dr. CL. Scheffer Prof.dr.ir. F. Schurer Dr.ir. C.J. van Duijn

5

CONTENTS

INTRODUCTION

CHAPTER 1 - SOLUTIONS OF GENERAL SECOND ORDER DIFFERENTIAL EQUATIONS IN VARIOUS SPACES 13

1.1. Solutions of u - aD u - 0Du = f in C(J) and L°°(J). Classification of boundary points 13 1.1.1. Preliminaries 13 1.1.2. Solutions of the homogeneous equation 14 1.1.3. Classification of the boundary points 30 1.1.4. Solutions of u - QD 2 U - 0Du = f in C(J) 40 1.1.5. Solutions of z - aD2z - /3Dz = k in L°°(J) 48

1.2. Solutions of v - D(aDv -/9v) = g in L ](J) 50 1.2.1. Solutions of v - D(aDv - 0v) = 0 50 1.2.2. Solutions of v - D(D(av) - 0v) = g in L^J) 56

1.3. Solutions of w - (D(aDw) - jJDw) = h in NBV(J) 60 1.3.1. Solutions of w - (D(aDw) - 0Dw) = 0 62 1.3.2. Solutions in NBV(J) of w - (D(aDw) - j5Dw) = h 66

CHAPTER 2 - SEMIGROUPS GENERATED BY DIFFERENTIAL OPERATORS SATISFYING VENTCEL'S BOUNDARY CONDITIONS AND THEIR DUALS 67

2.1. Semigroups in Banach spaces 67 2.2. Two propositions j)n adjoint operators 72 2.3. Semigroups in C(J) 77 2.4. Dual semigroups in 3R x NBV(J) x ]R 88 2.5. Restricted dual semigroups in 3R x L^J) x }R 102 2.6. "Bidual" semigroups in K x L°°(J) x ]R 106

6

CONTENTS (continued)

2.7. On CQ-semigroups in a space of bounded continuous functions in the case of entrance boundary points 116

2.8. Dual semigroups in NBV(J), NAC(J) and L ](J) 126 2.9. "Bidual" semigroups in L°°(J) 130

CHAPTER 3 - SATURATION PROBLEMS FOR BERNSTEIN OPERATORS IN Cm[0,l] 135

3.1. Introduction 135 3.2. A unified approach to pointwise and uniform

saturation for Bernstein polynomials 136 3.3. Saturation and Favard classes 153 3.4. Application: Uniform saturation class for Bernstein

operators on C[0,1 ] 159 3.5. Uniform saturation classes for Bernstein operators in

Cm[0, l] norms (m > 1) 163

REFERENCES 186

SAMENVATTING 189

ACKNOWLEDGEMENT 191

7

INTRODUCTION

This thesis deals with aspects of the theory of semigroups of operators in Banach spaces as well as aspects of the theory of approximation by means of linear operators. In fact we investigate operators in Banach spaces which generate semigroups of operators and we apply the obtained results to saturation problems in approximation theory. Saturation is an interesting phenomenon in approximation theory. This concept was introduced by Favard in 1947, [Fa].

Definition (cf. [BB], p. 87) Let (L ) be a sequence of linear operators in a Banach space X strongly convergent to the identity operator in X. Then the sequence is (uniformly) saturated if there exists a sequence of positive numbers (<f> ), which tends to infinity if n tends to infinity, and a class s(L ) closed in X such that the following holds

(i) lim 6 ||f - L f|| = 0 n—K» u n

if and only if f e s(L ),

(ii) there exists at least one f e X\s(L ) for which

(1) 'I W - f) l l = 0 ( l ) , n - o o .

The class of functions f, satisfying (1) is called the saturation class of the approximation process (L ), and 0(#~ ) is the saturation order.

Thus the saturation problem concerns the determination of the optimal order O(0 ) of approximation and the (non-trivial) class of elements which can be approximated with this optimal order. Saturation problems in approximation theory are investigated by many authors by different methods. In particular we mention the theorems of Lorentz and Schumaker ([LS], Th. 4.3) and Becker and Nessel ([BN], Satz 3.6). Many approximation processes (L ) in a Banach space X strongly convergent to

the identity operator in X, satisfy the so-called Voronowskaya property

(2) lim <j> (L f - f) = Af, f e D(A) C X, n—►<» nx n

where A : D(A) c X —► X is the infinitesimal generator of a strongly continuous semigroup T . in X. An important class of approximation processes for which (2) holds is the class of (^„-semigroups T . on X. In that case L := T(l/n) and <j> = n. It is known that a semigroup, considered as an approximation process, is saturated ([BB], p. 88). The saturation class of T . is sometimes called the Favard class of T . . It is known that if the condition (2) is satisfied, the saturation class of the approximation process'(L ) is the same as the Favard class of T . , denoted by Fav (T ) (see e.g. [BN], Satz 3.6 and ([B], 3.1). A nice example of an approximation process which is saturated is the sequence (B ) of Bernstein operators, defined by

(3) '

B : C[0,1] -+ C[0,1] (equipped with the sup-norm),

(B f)(x) = V f(k/n) p (x), II k - Q 11,K

with p (x) = (P )x (1 - x) n" . n,k

This sequence has the Voronowskaya property

(4) lim ||n(B f - f) - Af|| = 0, f G D(A), n—►<» n

where A is defined by

(5)

D(A)= {f G C [ 0 , l ] | f e C 2 ( 0 , l ) , lim x(l - x)f"(x)= lim x(l - x)f"(x) = o\, x—0 x—1 J

. Af(x) = x(l - x)P(x)/2.

In fact, Voronowskaya proved that

9

lim n(B f(x) - f(x)) = x(l - x)f'(x)/2 n—*oo n

for functions f which are twice differentiable in x, [Vo]. In 2.3 it will be proved that A, defined by (5), is densely defined and m-dissipative, while in 3.3 it will be proved that

(6) D(A) = {f | lim n(B f - f) exists}. *• n—»oo n i

Berens and Lorentz proved (4), [BL]. The description of D(A) in [BN] and [Fb], namely D(A) = {f G C[0,1] | <£f' G C[0,1]}, where <f>(\) - x(l - x)/2, has to be replaced by D(A) as given in (5).

Applying Theorem 3.3.1 to the sequence of Bernstein operators we obtain that the uniform saturation class of the Bernstein operators is Fav (T . ) . The remaining problem is to describe Fav (T ). Several methods are known. Butzer (cf. [BB], p. 92) characterized the Favard class for certain restriction of a dual semigroup T® as the domain of the infinitesimal generator of the non-restricted dual semigroup T . Berens [Be] gave a characterization by means of the concept relative completion of a Banach subspace in a Banach space. For practical purpose the following characterization is useful (cf. [CH], 3.36):

(7) Fav (T ) = D(A®*) n X,

where A© is the adjoint of A®, and A® is certain restriction of the adjoint of A. We compute D(A® ) in section 3.4. It is known that for a function f G C [0,1] the m-th derivative of B f converges to the m-th derivative of f in the uniform norm topology. In section 3.5 we compute the uniform saturation class of the Bernstein operators in C [0,1], equipped with its usual norm, by using the same method as in section 3.4. Partial results concerning the associated Voronowskaya formula for Bernstein operators in C [0,1] can be found in [Fb].

10

As mentioned above, for the uniform saturation class of an approximation process satisfying the Voronowskaya property it is sufficient to have a characterization of D(A®*), where A is a densely defined m-dissipative operator. In section 3.4 as well as in section 3.5 the considered operator A is a second order differential operator. It appears in section 3.4 that

' D ( A ) = { f e C [ 0 , l ] | f SC 2(0,1) , lim x(l - x ) f (x )= lim x(l - x)f"(x) = o ) , x—0 x—1 >

{ (Af)(x)=x(l - x)f'(x)/2.

In chapter 2 we investigate the general case

(8)

' D(A)= {f e C [ r i , r 2 ] | f e C 2 ( r r r 2 ) ,

lim (aD2f + /3Df)(x) = 0, i = 1,2), x - q J

Af = QD f + £Df.

We give necessary and sufficient conditions on a and fi for A to be the infinitesimal generator of a semigroup in C[r . , r - ] , equipped with the supremum norm. Moreover, for m-dissipative operators of this type we will give an explicit characterization of D(A® ).

It is interesting to note that A defined by (8) is an infinitesimal generator if and only if both boundary points are not entrance boundary points in the terminology of Feller.

We shall use the Feller classification of boundary points in regular, exit, entrance and natural boundary points. In section 3.4, where a(x) = x(l - x)/2 and f) = 0, both boundary points are exit boundary points. In section 3.5 however, it appears that the considered operator A is such that both boundary points are entrance boundary points.

In Chapter 1 we give a self-contained explanation of Feller's theory and the classification of boundary points in a more modern setting. The results obtained

11

here are slightly more general, since we do not assume any differentiability on a, as is done in Feller's paper [Fe, 1]. The classification rests on properties of two 'minimal' positive solutions of the homogeneous equation

u - Au = 0.

We also study solutions of second order differential equations in other spaces. More explicitly, we investigate solutions of the other following problems

' u - Q D U - £Du = 0

u e C(J) n C^J), Du e ACJoc(J).

' v - D(D(av) - 0v) = 0

1 1 v e C(J) n L (J), av G C (J), D(av) - 0v e AC(J).

w - D(aDw) - 0Dw = 0

1 w € C (J) n NBV(J), aDw e AC loc(J), D(aDw) - 0Dw G NBV(J).

Here NBV(J) is the space of functions of bounded variation which are normalized by

(i) f(xf)) = 0, x_ G J is a fixed point, (ii) f(x) = (f(x+) + f(x-))/2 for all x G J.

By means of two special positive solutions of the homogeneous problem we construct a Green operator to solve the corresponding inhomogeneous problem. A number of results in Chapter 1 are conveniently arranged in state diagrams in section 1.4.

13

CHAPTER 1 - SOLUTIONS OF GENERAL SECOND ORDER DIFFERENTIAL EQUATIONS IN VARIOUS SPACES

2 OO 1.1. Solutions of u - aD u - ffDu = f in C(J) and L (J).

Classification of boundary points

1.1.1. Preliminaries

Let J be a non-empty open interval of 1R (not necessarily bounded), J the two points compactification of J and dJ := J\J. C(J) denotes the set of real-valued continuous functions on J, C (J) (k = 1,2, ) is the set of functions in C(J) which are k-times continuously differentiable, X := C(J) is the Banach space of real-valued continuous functions on J equipped with the supremum norm. r . (resp. r«) denotes the left (resp. right) boundary point of J, and xft denotes an arbitrary fixed point in J. Thus -oo < r . < x„ < r_ < oo, and for each f G C(J) the limits lim f(x) and lim f(x) exist. Since limits at the

x—»rj+ x—>r2~ boundary points are always one-sided we shortly denote these limits by

lim f(x), i = 1,2. C (J) denotes the set of functions in C(J) with compact x—+rj c

support inside J and CQ(J) the set of functions f in C(J) with lim f(x) = 0. 1 x—»3J

L (J) is the space of equivalent classes of Lebesgue measurable functions on J for which |f|dx < oo. This space, equipped with the usual L -norm,

J J

l lf l l j :- J Ifldx,

is a Banach space. L ( r i ,x n ) and L (*0,r_) have a similar meaning. For sake of convenience we sometimes denote L (x n , r . ) for L (r. ,xA Let a and fi be real-valued continuous functions on J with a(x) > 0 for all x 6 J. a and /J may be unbounded on J, and a(x) may tend to zero if x tends to one of the boundary points. We consider the differential expression

(1.1.1) Au = aD 2u + /3Du

14 2 2

for u e C (J). Here Du and D u denote the first, respectively the second

derivative of u. In order to investigate the solutions of the ordinary differ

ential equation (1.1.2) u - A A u = f

we introduce the real-valued functions W, Q and R on J as follows: .x

(1.1.3) W ( x ) : = e x p { - f (/3a 1)(s)ds}, x e J, 1 J x o J

Y

(1.1.4) Q(x):=(aW)_1(x) f W(s)ds, x e J, J x 0

Y

(1.1.5) R(x) :=W(x)f (aW)_I(s)ds, x e J. J x o

Note that W(x ) = 1, W(x) > 0 for x e J; Q(x) > 0 and R(x) > 0 for x > x • Q(x) < 0 and R(x) < 0 for x < xQ.

1.1.2. Solutions of the homogeneous equation

We start with an investigation of the homogeneous equation

(1.1.6) u - (aD2u +/3Du) = 0

with u e C (J).

Proposition 1.1.1 Let a and p be real-valued continuous functions on J with a(x) > 0 for all x e J. Then there exists a unique positive increasing solution u. of (1.1.6) and a unique decreasing solution u- of (1.1.6) satisfying u.(x„) = 1 (i = 1,2), such that u. (resp. u?) is minimal on (r ,x~) (resp. (xn,rj, i.e. if u. (resp. u-) is any positive increasing (resp. decreasing) solution of (1.1.6) satisfying u.(x„) = 1, then u.(x) < uJx), x e (r x „ / (resp. uJx) < uJx), xefX(),r2)).

15

For the proof see Lemma 4 of section 2.3. D

The importance of Proposition 1.1.1 is that u. and u» are uniquely determined independent monotone positive solutions of (1.1.6). Thus each solution of (1.1.6) is a linear combination of u. and u».

Remark. Since for all positive values of A the functions Aa and \p satisfy the same conditions as a and 6, Proposition 1.1.1 remains valid if (1.1.6) will be changed into

(1.1.6a) u - A(aD2u + /?Du) = 0, A > 0.

As a consequence all assertions in this section remain valid if (1.1.6) will be changed into (1.1.6a). Then of course u. and u . depend on A. Finally it is remarked that the function W defined by (1.1.3) is independent of A. For many purposes it is convenient to rewrite (1.1.6) in an other form:

(1.1.7) u - Q W ( W " 1 U 1 ) ' = 0.

or

(1.1.8) (aW)_1u - (W" 1 ! ! ' ) ' = 0.

The next relation follows from (1.1.8) and is very useful:

(1.1.9) u ,(x) = W(x).{u'(x0)+ | ((aW)"1u)(s)ds}.

Note that W(xQ) = 1.

The next lemma is also a direct consequence of (1.1.8).

Lemma 1.1.2

For each positive decreasing solution u of (1.1.6) W u ' is a negative in

creasing function with Urn (W u')(x)<0. x—*.r?

16

In order to make a useful classification of the boundary points r and r- we will investigate the behaviour of the "minimal" solutions u. and u_. We define

M. := lim u.(x) = sup u.(x) 1 x-»r2 " * XGJ " ]

M, := lim u-(x) = sup uJx) 1 x-»rj " z x £ j ~z

m. := lim u.(x) = inf u.(x), i= 1,2. 1 x-»rj _ 1 xeJ - 1

The following lemmata will be proved for the boundary point r«. It is clear that the proofs for the boundary point r. are similar.

Lemma 1.1.3 M < oo ^=>. R e L (x0,r ) ,

M < oo ^=» R e L (r xQ).

Proof Necessity. From (1.1.9) we obtain for all x e J

x t (1.1.10) Uj(x) = J W ( t ) { | ((aWf ^jXsJds + Uj 'U^Jdt

+ U l (x 0 ) .

Since u. is positive increasing we have for x > x_

X X t

0 < U j ( x ) [ R(t)dt< [ W(t) [ ((aW)"1u1)(s)ds<u (x) < M J x o J xo J xo

Thus, if M. < oo, then R e L (xn,r«).

Sufficiency. Since u . is increasing we have from (1.1.9) for x e (xn , r-)

Y

0 < Uj'(x)< Uj'(x0)W(x) + Uj(x)W(x) [ (aW)"l(s)ds J x 0

or

17

(*) O < u|(x) < C.W(x) + R(x).Uj(x),

with C = UJ'UQ) > 0. Following [Fe,l], p. 483 we consider the differential equation

(**) y ' = CW + Ry, y e c ' f J ) ,

y(xQ) = 1

where C > 0, R G L (x»,r-) and thus also W G L (x 0 , rA It is standard that this equation has a unique solution y with

.x y(x) = v(x) exp

with

{ƒ R(.)d.} xo

x t f W(t).exp/- f R(s)ds]-v(x) = C | W(t).exp

C0 d t+ 1.

Since

and

v '(x) = CW(x).expl - f R(s)ds]- > 0 J x Q

Y

y «(X) = {v '(x) + v(x)R(x)} exp-T f R(t)dt]- > 0

for x G (x f t ,r-), it follows that v and y are increasing for x G (x0,r„). Moreover, since R G L (*0,r_) and W G L (x0,r~) we see that v and y are bounded. Thus

N := lim y(x) = sup y(x) < oo. x ^ r 2 xG(x0,r2)

We define the function z on [xn,r_) by

z := Uj - y.

18

Then z ' ( x ) < 0 , x e [xQ ,r2) ,

I z ( x 0 ) = - 1 ( V " y ( x0 ) = 0'

Thus z(x) < 0 for x e [x0 , r2) , or

Uj(x) < y(x)

and thus also

M. < N < oo.

Since u_ is positive decreasing, u- is also bounded on (x_,r-), so we have

Lemma 1.1.4 Each solution of (1.1.6) on J is bounded on (xn,r.) •<==► R e L (x~,r.), i = 1,2. a

As a consequence of Lemma 1.1.3.we have

Lemma 1.1.5 There exists a positive monotone solution u of (1.1.6) with lim u(x) = oo

j x^r; R $ L (xn,r.), i = 1,2 . Or

D

Lemma 1.1.6 For all a,b e TR the boundary value problem (1.1.6) with u(xn) = a,

1 lim u(x) = b, has a unique solution <=*■ R = L (xn,r.), i = 1,2 .

nT'

19

The next lemma concerns the functions R and Q.

Lemma 1.1.7 If R G L (xQ,r ) and Q G L (xQ,r ) then (i) for each solution u of (1.1.6) lim (W u% )(x) exists,

x-*r2 he "minimal" decreasir

have (ii) for the "minimal" decreasing solution u. (see Proposition 1.1.1) we

(1.1.11) lim uJx) = 0, lim (W'1 u\)(x) = L < 0, x-*r2 '^ x-*r2 '^

(Hi) there exists a positive decreasing solution u , of (1.1.6) satisfying

(1.1.12) lim uJx) = 1, lim (W'^lXx) = 0. x-*r2

J x~~+r2

Proof Let R G L ^ X Q , ^ ) and Q G L 1 ^ , ^ ) . ad (i). Let u be a solution of (1.1.6). From Lemma 1.1.4 we know that u is bounded and since Q G L (xQ,r2) we also have (Q\V)~ G L (x0 , r2) . From (1.1.8) it follows that (W- u 1 ) ' is integrable, hence lim (W~ u')(x) exists.

x - r 2 ad (ii). That lim u_(x) = 0 is a consequence of Lemma 1.1.6. Since

-1 1 X ^ r 2 (aW) G L (xQ , r2) w e h a v e f r o m (1.1.9)

(1.1.13) (W_1u2 ')(x) = u2 '(x0) + ƒ 2 ((aW)_1u

I 2)(s)ds

2((aW) - 1u.(s)ds.

The function W u ' is negative increasing so lim (W- uJ)(x) < 0, and since ~* x—»r2 "l

J 2((aW)'1u. lim I ((aW) u0)(s)ds = 0 it follows that r 2

9 X—tri J Y - ^

20

(1.1.14) L := lim (W_1u')(x) X—*T2

= u 2 ' ( x 0 ) + | 2 ((aW_1u2)(s)ds<0. ""0

Suppose L = 0. Then we have from (1.1.13) and (1.1.14) for all x e (xQ ,r2)

0 < - ((u2)_1u^Xx) = (u2)"1(x).W(x) j " 2 (aW)"1(s).u2(s)ds

< W(x) f 2 (aW)_1(s)ds. J x

Integration of this inequality gives

0 < - log u2(x) < f W(x) f 2 (aW)_1(s)ds < oo. J xQ J x

This implies that lim u„(x) is positive, which is a contradiction. Thus L < 0. X - T 2

l

ad (Hi). Let u denote the unique increasing (bounded) solution of (1.1.6) satisfying u(x.) = 0, u ' ( x

n ) = 1- Then it follows from (1.1.9) that W u ' is a positive bounded increasing solution and thus

k:= lim (W"1u')(x)> 0. x - r 2

We define the function ü by ü := u« - Lk" u. It follows that

m := lim ü(x) = -Lk . lim u(x) > 0, x—*X2 x—»r2 ~

lim (W"1ü')(x)= lim (W_1u')(x) c-»rT x-+ro '*■ x->r2

- L k " 1

x ^ r 2

Lk l. lim (W"1u ,)(x) = 0,

and

(W"1G l) l(x) = (W" 1u') ' (x) - Lk _ 1 (W" 1 u , ) ' (x )> 0

21

for x G (x„,r-). So W G' is a negative increasing function and thus ü is decreasing. We see that the function u , := m ü is a positive decreasing solution of (1.1.6) satisfying (1.1.12). D

We continue with two lemmata concerning the function Q.

Lemma 1.1.8 There exists a positive decreasing solution u of (1.1.6) with Urn u(x) > 0

1 x~*r2 Q G Ll(xQ,r2).

Proof Necessity. Let u be a positive decreasing solution of (1.1.6) with lim u(x) =

x - + r 2 L > 0. Since u *(x) < 0 for all x G J it follows from (1.1.9) that for x £ (x0 , r . )

Y

0 < f ((aW)_1u)(s)ds< -u ' (x 0 ) .

Moreover x , „x

0 < L f (aW)_1(s)ds< f ((aW)_1u)(s)ds " J x Q J x 0

for all x G (x0 ,r_), so

(1.1.15) (aW)"1 eLl(x0,r2).

From Lemma 1.1.2 we know that lim (W~ u ')(x) exists, and that . j x - r 2

k := lim (W u')(x) < 0. It follows from (1.1.8) and (1.1.11) that for all x-+r2

x G (x 0 , r 2)

(1.1.16) - (W _ 1 u ' ) (x)= -k + f ^V((aW)_1u)(s)ds. J x

22

So

or

-(W V x x ^ L . f 2 (aW)_1(s)ds> 0 " J x

-u '(x) >L.W(x) f 2 (aW) 1(s)ds>0. 0

Since u ' e L (x0 , r-) we have

u(xQ) - L > L v0

and with Fubini's theorem we get

.s

f 2 W(t) f 2 (aW)_1(s)dsdt

r s u(x ) > L . [ 2 (aW)"'(s) [ W(t)dtds

J XQ J XQ

J xr = L . I 2 Q(s)ds,

x0

so Q E L (xQ,r2).

Sufficiency. Assume Q G L (x«,r») and let u be a positive decreasing solution of (1.1.6) with lim u(x) = L. Then L > 0. As above lim (W_ 1u'Xx) =

X—*T2 ~ X-*T2

k < 0, and also (1.1.16) holds. Then we obtain for all x e (x„,r_)

(1.1.17) -u ' (x)=-kW(x) + W(x) f 2 (aW)_1(s)u(s)ds

thus

-u'(x) < -kW(x) + u(x)W(x) f 2 (aW) 1(s)ds. J x

If k = 0, then we get

0 < -u'(x)/u(x) < W(x) f 2 (aW)_1(s)ds

23

and integration over (x_,r«) and Fubini's theorem shows that

0 < -lim log (u(x)/u(xj) < f 2 W(t) [ 2 (aW)_1(s)dsdt x—r2

u J XQ J t r s

= f 2 (aW)"'(s) f W(t)dtds J x0 J xQ

= [ 2 Q(s)ds. J x 0

Thus, if k = 0, then lim u(x) = L > 0. x ^ r 2

If k < 0 we obtain from (1.1.17) for all positive decreasing solutions of (1.1.6)

-u ' (x) > -k.W(x)> 0

for all x G (x_,r«), and since u ' e L (x„,r„) we see

(1.1.18) W E L V X Q , ^ ) .

Since Q e L (x 0 , r . ) it is clear that

(1.1.19) (aW)"1 GLl(xQ,T2),

and from (1.1.5) it follows that R e L (x0,r_). In this case the assertion will follow from Lemma 1.1.7(iii).

D

The following lemma is a corollary of Lemma 1.1.8.

Lemma 1.1.9 For each positive decreasing solution u of (1.1.6) lim u(x) = 0 holds <=>■

1 x-^ri Q 0 LJ(x0,r2). D

24

Together with Lemma 1.1.7(iii) the next lemma shows the importance of the condition R £ L (xn ,r„), while Q G L (x- . r . ) , for the value of lim (W u')(x) if u, is a positive decreasing solution of (1.1.6).

Lemma 1.1.10 If R £ L (x„,r ) and Q G L (xn,r.) then each positive decreasing solution

of (1.1.6) satisfies

lim u(x)>0, lim (W'^' )(x) = 0. x->r2

J x-*r2 6

Proof Assume Q G L (x_,r.A From Lemma 1.1.8 we know there exists a positive decreasing solution u_. of (1.1.6) with L~ := lim u„(x) > 0. Let u, := L„ u„.

x—^2 Then u, is a positive decreasing solution of (1.1.6) with lim u.(x) = 1. As in

•* x—►r? ■* _1 ^

the proof of Lemma 1.1.8 we have k = lim (W u')(x) < 0, and also x—»r2 ■*

(1.1.16) holds. From (1.1.16) we obtain for all x G (xn,r„)

(1.1.20) u^(x) = k.W(x) - W(x) f 2 (aW)_1(s).u3(s)ds.

Assume moreover R £ L (x„,r_). Since Q G L (xn,r~) we then have

(aW)"1 G Ll(x0,r2) and W £ L 1 ^ , ^ ) . From (1.1.16) we obtain

u^(x)<k .W(x)<0 , xG( X ( ) , r 2 ) .

Since u^ G L (xQ,r2) it follows that k = 0. So from (1.1.20) we get

( W _ 1 u p ( x ) = - J 2 (aW)"1(s).u3(s)ds

and thus lim (W_ 1u')(x) = 0. x—>ro 3

D

25

The next lemma gives the relation between the function W and strictly monotone solutions of (1.1.6).

Lemma 1.1.11 Let u j be a positive increasing and u ? be a positive decreasing solution of (1.1.6), then

(1.1.21) u)u2~ U2U1 = K0W Witk K0 = (~U\U2 ~ U2U1^X0^ > °'

Proof Since u. en u- satisfy (1.1.6) we have

(au"j +)8uJ)u2= (ou"2 + / 8 u p u r

So

or

U1U2 - U2U1 A U 5 U 2 " U 2 U 1 ~a

( " j " 2 - u 2 u l > ' _ 0 u!u - - u ' u . a '

Integration over (x»,x) leads to (1.1.21). D

Lemma 1.1.12 If u is a positive increasing solution of (1.1.6), then

Urn (W~1ux)(x) < oo «=► Q(EL}(xn,r ) . x-*r2 ° z

Proof

Necessity. By (1.1.8) W~ u ' is a positive increasing function. Assume

M:= lim (W_1u')(x) < oo. x ^ r 2

26

Then there exists an x G (xn' r2^ s u c h t h a t f o r a11 x e ^ x l ' r 2^

( W " 1 u ' ) ( x ) > ^ .

Thus for x > x. we have by integration

u(x)>u(x 0 ) + ^ f W(t)dt, X0

or

and thus

((aW)_1u)(x) > u(x0).(aW)_1(x) + ^ .Q(x),

(W_ 1u')(x) - (W"1u ,)(x_)= f ( W ' ^ ' J ' W d t 0 J x Q

= f ((aW)_1u)(t)dt (by (1.1.8)) J x 0

X X

>u(x0).J (aW)_1(t)dt + ^ J Q(t)dt.

By taking the limit for x —► r . we obtain

M - u ' (x n ) > u(xA f 2 (aW)_1(t)dt + ^ f 2 Q(t)dt. u u J x 0 z J x 0

Thus Q G L (xQ ,r-).

Sufficiency. If Q e L (x~,r») there exists a positive decreasing function u . with lim u.(x) = 1. Since W~ u ' is a positive increasing function there

x-+r2 * exists an N, N G (0,oo], such that

N := lim (Wû 'Xx) . x ^ r 2

So

27

lim (W V x x J . u (x) = N. x - r 2

3

From Lemma 1.1.11 we have for all x e (xn,r»)

(W"1u')(x).u3(x)<K0<cx>.

Thus N < K» < oo. This completes the proof of the lemma. D

Lemma 1.1.13 1, Let u.M-f and Kn be as in Lemma 1.1.11. If R E L (x-,r ) and

(aWf1 £ L](x0,r2),then

(1.1.22) lim (W'KlXx) = -K / lim u(x)<0. x—*r2 ^ u x—*r2 l

Proof Let R e Ll(x0,r2) and (aW)"1 £ L 1 ^ , ^ ) . Then clearly Q jÊ L1(x ( ),r2). Moreover u. is a bounded positive increasing function, thus M := lim u.(x) 1 x—>r2

exists and is positive. The function W u i is negative, and since (W u i ) ' = -1 -1 -1

(QW) U . > 0, W u i is increasing. Thus lim (W ul)(x) exists and is 1 l x-+r2

non-positive, say

(1.1.23) lim (W - 1u ' ) (x) = L < 0. x—»r2

On the other hand, since Q £ L (x„,r_) it follows from Lemma 1.1.12 that

lim (W_1u!)(x) = oo. X—*T2

Furthermore, as Q £ L (x»,r») we have from Lemma 1.1.9 we see

28

lim u9(x) = 0. x—»r2

For the product of W~ u! and u- we have by Lemma 1.1.11 and (1.1.23)

(1.1.24) lim (W"1u!u-)(x) = K n + LM, X—*T2

thus lim (W u!u-)(x) exists and since u! > 0 this limit is non-negative, x—»r2

say

(1.1.25) lim (W _ 1 u!u , ) (x)> 0. X—»T2 '

Let e > 0. By (1.1.23) there exists a number x^ e (x„,r-) such that for all x,t e (x»,r.) with x- < t < x < r»

(1.1.26) 0 < (Wu p(x) - .(Wu p( t ) < c/(2M).

Let T e (x-,r_). Then for all x e (T,r-) we have

0 < (W_ 1u;u2)(x) = u2(x) {J* (W_1uJ)«(s)ds + u | ( x 0 ) ] .

= u2(x) { f (aW)"1(s)u1(s)ds + u j ( x 0 ) } 0

< u2(x) {u j ( r 2 - ) | (oW)" \%)ds + u J(X())]>

T = u2(x) { u j ( r 2 - ) [ (aW)_1(s)ds + u j ( x 0 ) }

X0

+ u2(x)Uj(r2-) ƒ (aW)_1(s)ds

T < u2(x) { u , ( r 2 - ) J" (aW)" ^sjds + u " (XQ)}

Y

+ U j ( r 2 - ) J (aW)_1(s)u2(s)ds

29 .T

= u2(x) { u j ( r 2 - ) | (aW)"1(s)ds + u ' ( x 0 ) } O x

Uj(r2-) ^ ( W ^ u ^ d s

= u2(x) { u ! ( r 2 - ) J (aW)_1(s)ds + u\(xQ)}

+ Uj(r2-) { ( W ^ u ^ x ) - (W _ 1 up(T)}

T (1.1.27) < u2(x) {u j ( r 2 - ) | (aW)_1(s)ds + u ^ ) } + e/2.

Since lim u-(x) = O there exists a number S e (T,r_), such that for all x-+ro 2 z

x e ( S , r 2 )

T (1.1.28) 0 < u2(x) < {u j ( r 2 - ) j* (QW)"1(s)ds + u'1(x()))-"1£/2.

From (1.1.27) and (1.1.28) we obtain for all x e (S,rJ

0 < (W _ 1 uJu 2 ) (x )<e .

Thus

(1.1.29) lim (W_ 1u!u-)(x) = K +LM = 0. x—»r2 l l u

Combining (1.1.23), (1.1.24) and (1.1.29) we obtain

lim (W_ 1u')(x) = L = - K M " 1 < 0. x—r2

l u

This completes the proof of the lemma. D

Remark. This lemma improves the result of Feller ([Fe,l], Th. 11) which stated L < 0.

30

1.1.3. Classification of the boundary points

After the investigations on the solutions of the homogeneous equation (1.1.6) it is possible to give a very useful classification of the boundary points r and r . of the interval J. We give the classification for the right boundary point r_. The classification for the left-boundary point is similar.

The classification given here is Feller's classification [Fe,l]. We mention that Feller assumed that a is positive and continuously differentiate on J, whereas we only assume that a is positive and continuous on J. Thus here we drop the condition on the differentiability of a.

Firstly we give the classification of Feller [Fe,l].

Definition 1 Let a and P be as in section 1.1.1. The boundary point r . is called Regular if W e \}{xQ,x2), (aW)"1 G L\XQ,T2),

Exit if (aW)'1 É L ^ X Q , ^ ) , R G L ^ X Q , ^ ) ,

Entrance if W £ L (xQ ,r2), Q G L (xQ ,r2),

Natural in all other cases.

It is also possible to state the criteria only using the functions Q and R. An equivalent definition is

Definition 2 Let a and p be as in section 1.1.1. The boundary point r_ is Regular if Q G L 1 ^ , ^ ) , R G L1(XQ,T2),

Exit if Q £ L ^ X Q , ^ ) , R G L\X0,T2),

Entrance if Q G L (xQ ,r2), R £ L (x 0 ) r 2 ) ,

Natural if Q £ L V Q , ^ ) , R £ L 1 ^ , ^ ) .

From now on we will use Definition 2. From this definition and the lemmata

31

of section 1.1.1, it follows that the criteria for the boundary points can be given by means of monotone solutions of the homogeneous equation (1.1.6). We will summarize and prove - as far as not proved before - these results in the following Lemmata 1.1.18 - 1.1.21.

State diagram for the boundary point r?

state-1

r . Regular

r . Exit

r . Entrance

r» Natural (i) (ii) (iii)

W G L ^ X Q , ^ )

yes

yes

no

yes no no

( a W r ^ L ^ X Q , ^ )

yes

no

yes

no yes no

R S L ^ X Q , ^ )

yes

yes

no

no no no

Q€L1(x ( ) , r2)

yes

no

yes

no no no

Remark. The equivalence of the Definitions 1 and 2 follows from the implications:

(i) Q G L ^ X Q , ^ ) =► (aW)"1 G h\x0,T2)

(ii) RGL1(X0,T2)=> W G L ^ X Q , ^ )

(iii) W 6 L \ x 0 , r 2 ) A (aW)"1 e L 1 ^ , ^ ) => Q,R G L 1 ^ . ^ ) .

Simple examples

1. J = (-1,1), XQ = 0, a(x) = 1, 0(x) = 0. The points -1 and 1 are regular boundary points. The functions u and u2 with u (x) = sinh(l - x), u.(x)

= sinh(l + x) form a fundamental set of solutions of u - u" = 0, 2

u G C (-1,1), satisfying

32

lim u . (x)= lim u_(x) = 0. x-+l l x—»-l z

2. J = (-7T/2,IT/2), xQ = 0, Q(X) = 1, £(x) = 2 tan x. The points TT/2 and -ir/2 are ex/7 boundary points. The functions u. and u- defined by

u.(x) = cos x + (TT/2 + x) sin x

u^(x) = cos x - (7T/2 - x) sin x

form a fundamental set of solutions of

u - u" - 2 tan( . )u ' = 0, u e C2(-n/2,ir/2),

satisfying

lim u. (x)= lim u,,(x) = 0. X^-TT/2 l X-+7T/2 2

3. J = (-ir/2,jr/2), xQ = 0, a(x) = 1, /?(x) = -2 tan x. The points -JT/2, TT/2 are entrance boundary points. The functions u . and u . defined by

u,(x) = (TT/2 - x)/cos x

U 4 (X ) = (7T/2 + X)/C0S X


u - u" + 2 tan( . )u ' = 0,

satisfying

lim u .(x) = lim u,(x) = 1. X^-TT/2 4 X -7r /2 3

33

4. J = (-00,00), x„ = O, Q(X) = 1, y9(x) = 0. The "points" -00 and 00 are natural boundary points. The functions u. and u» defined by

/ \ x Uj(x) = e

u2(x) = e


u - u" = 0, u £ C(-oo,oo)

satisfying

lim u.(x) = lim u»(x) = 0. x—»-oo * x—*oo ^

Lemma 1.1.14 If r. is a regular boundary point, then there exists a positive decreasing solution u~ of (1.1.6) satisfying

lim u(x) = 0, lim (W'1 u\)(x) = -1 x—r2

z x^r2 z

and a positive decreasing solution « , of (1.1.6) satisfying

lim uJx) = l, lim (W~1 u\)(x) = 0. x-*r2

J x->r2 i

Each solution of (1.1.6) is bounded on (xn,r?), but a solution u of (1.1.6) is positive and decreasing if and only if there are non-negative numbers p?

and p-., not both zero, such that u = p?u7 + p,u,. There also exists a positive increasing solution u. of (1.1.6) for which lim W u$x) < 00.

1 x-+r2 J

Proof The existence of u_, resp. u- follows from Proposition 1.1.1, and the Lemmata 1.1.7 and 1.1.8. Assume u = P2

U-) + Pû? *s positive decreasing with p» > 0,

34

p > O, not both zero. Then (W_1u'Xx) = p 2 (W _ 1 up(x) + p3(W_1u^)(x) < 0 for x G (xft,r_), so u is decreasing. Conversely, if u is a decreasing solution of (1.1.6) then there are p - , p , such that u = P 2

U T + P i u i - **y taking the limits for x -* r . we see

lim u(x) = p . . lim . u,(x) = p . > 0 x—>T2 x—*r2

and

lim (W" lu')(x) = p lim (W_1u»)'(x) = p > 0. x-+r2 x-+r2

If u is decreasing then p . , p , are not both zero. Since each (positive increasing) solution u of (1.1.6) is a linear combination of u ? and u- , lim (W u)(x) is finite.

z ■* x—»r? D

Lemma 1.1.15 If r is an exit boundary point, then there exist a positive decreasing solution u- of (1.1.6) satisfying

lim uJx)'0, lim (W'1 u$(x) = -1, x-+r2 l x-*r2 z

and a bounded positive increasing solution u. with lim (W u \)(x) = oo. 1 x^r2

l

Proof The existence of u_ follows from Proposition 1.1.1 and Lemma 1.1.13. Let u. be a positive increasing solution of (1.1.6). By Lemma 1.1.3 u. is bounded.

_1 l

From Lemma 1.1.12 it follows that lim (W u')(x) = oo. x—»ro

D

Lemma 1.1.16

If r? is an entrance boundary point, then there exists a positive decreasing solution u, of (1.1.6) satisfying

35

lim u(x)=l, lim (W~1 u\)( x) = O, x-+r2

5 x-^r2 J

and an unbounded positive increasing solution u. of (1.1.6) with

lim (W~1u))(x)<oo. x^r2

1

Proof The existence of u , follows from the Lemmata 1.1.8 and 1.1.10. The existence of a positive increasing solution u. of (1.1.6), such that lim u.(x) = oo fol-

x ^ r 2 ] _j

lows from Lemma 1.1.5. From Lemma 1.1.12 it follows that lim (W u!)(x) x—*X2

< oo.

D

Lemma 1.1.17 If r7 is a natural boundary point, then there exists a positive decreasing solution Uy of (1.1.6) satisfying

lim u(x) = 0, lim (W~}u\)(x) = 0, x—>r2

z x-*r2 l

and an unbounded positive increasing solution u. of (1.1.6) with

lim (W~1u\)(x) = oo. x—r2 i

Proof The existence of a positive decreasing solution u . of (1.1.6) with lim u.(x)= 0

follows from Proposition 1.1.1 and Lemma 1.1.9. Let lim (W u')(x) = L, x—►ro z r2

as;

(1.1.6) follows from Proposition 1.1.1 and Lemma 1.1.4. Assume L < 0, then then L < 0. The existence of an unbounded positive increasing solution u. of

lim (-W"1u»)(x).u.(x) = oo, x-+rT z 1

36

in contradiction with the boundedness of W u i u . , which follows from (1.1.21). Thus L = 0.-That lim (W~ u !)(x) = oo follows from Lemma 1.1.12.

x—^2 D

State diagram for monotone solutions of u - aZ) u - f)Du = 0, u e C (J)

r- Regular

r_ Exit

r- Entrance

r_ Natural

I

yes

yes

no

no

II

yes

no

yes

no

III

yes

yes

no

yes

IV

no

yes

no

yes

I - There exists a positive increasing solution u. with lim u.(x) < oo. 1 x-»r2 l

II - There exists a positive decreasing solution u , with lim u(x) = 1. ■* x-+r2

HI - There exists a positive decreasing solution u_ with lim u(x) = 0. ^ x—>r2

IV - For each positive decreasing solution u lim u(x) = 0 holds. x - r 2

Remark. The type of the boundary point r- is completely determined by the columns I and II. The columns III and IV give extra information.

37

-1 2 2 State diagram for lim (W u ' )fx), where u - aD u - 0Du = 0, u € C (J) x^r2

r„ Regular

r . Exit

r» Entrance

r„ Natural

I

yes

no

yes

no

II

yes

yes

no

no

III

yes

no

yes

yes

IV

no

no

yes

yes

I

II

III

IV

For each positive increasing solution u lim (W u ')(x) is finite. x - r 2

There exists a positive decreasing solution u-with lim (W u l ) (x )<0 . ^ x—>r*> ^

There exists a positive decreasing solution u , with lim (W ul)(x) = 0. ■* X—>f) J

For each positive decreasing solution u lim (W u ')(x) = 0 holds. x ^ r 2

Remark. The type of the boundary point r- is completely determined by the columns I and II. The columns III and IV give extra information.

Now we will show that the classification of the boundary points is intrinsic in the following sense. If <j> is a twice differentiable diffeomorphism from an open interval J not necessarily bounded) with the two points compactification J , onto J, such that the boundary points jr. and £- of J are carried into the boundary points r respectively r_, then the boundary points £. and r. (i = 1,2) are of the same type. We will state this in the next lemma.

38

Lemma 1.1.18 Let the continuous bisection <f>: J —► J satisfy the following conditions: <t> e C2(J), '<j> *(t) > 0 for all t e J, <f>(tQ) = xQ, <j>d.) = r. (i - 1,2) and let <f> be normalized by <j> %(tn) = 1. Let a and 0 as in Proposition 1.1.1, u := u o <f>, let the differential expression Au be given by (1.1.1) and assume that <t> carries Au into the differential expression

(1.1.30) Au := aD2u + fSDu.

Let the function W on J be defined by (1.1.3) and let W on J be defined by

(1.1.31) W(t) = exp{-\ (ior1)(s)ds].

Then the boundary points £ . and r. are of the same type, (i = 1,2).

Note that a and £ are defined by (1.1.30).

Proof

Let x = <j>(t). Then u'(t) = u '(\).<f> '(t), u"(t) - u"(x).(0 '(t))2 + u '(x)^"(t). Substitution of u'(x) and u"(x) in (1.1.1) and a comparison with (1.1.30) shows that

a(t) = (aoMt). ' (* '( t))~2

g(t) = (/J o <j>)(t).(<t> '(t))"1 - (a o Mt).f(t).(tf '(t))"3 .

Now

f (£a_1)(s)ds = f tfa^Xfls^'feJds J t 0 J t 0

" It, * (sW(s)) - 1ds v0

• *(t) f YK ' - 1 J x Q

39

So by (1.1.31) we have

W(t) = (W o <t>)(t)-<t> *(0

and it follows that

Moreover,

hence

W e L 1 ^ ) <=* W e L\X0,T2).

(«W)"1(t) = ((aW)"1 o 0(t).* "(t),

(aW)"1 G h\t0,T2) ^ (aW)"1 G Ll(xQ,r2).

Let the functions Qand R. be defined similar to (1.1.4) and (1.1.5):

9(t):=(aW)_1(t) f W(s)ds, °

R(t):= W(t) f (aW)_1(s)ds.

0

Then it is easily verified that

Q(t) = (Q o 0(t).tf '(t),

R(t) « (R o <f>)(t).<f> '(t).

Hence

Q G L ! ( t 0 , r 2 ) * ^ Q e L ^ X Q , ^ ) ,

R. G L ! ( t 0 , r 2 ) ^ * R 6 Ll(x0,r2).

With Definition 2 it is easily seen that £ - and r- are boundary points of the

same type. This proves the lemma. D

40

1.1.4. Solutions of u - aD2u - QDu = f in C(J)

A well-known technique to obtain solutions of an inhomogeneous equation is the construction of solutions by means of Green functions, cf. [Y,l]. In this section we will define a special Green function V: JxJ -+1R, called a regular Green function in Feller's terminology, in order to obtain a special solution of the inhomogeneous equation

(1.1.32) u - (aD2u + 0Du) = f, f e C(J), u e C(J) n C2(J),

denoted by uf. Then the general solution of (1.1.32) in C(J) is given by

u = u f + c .u . + cû2

where u. and u» are two independent solutions of the homogeneous equation (1.1.6) and where c.,c_ GlR. We will see that if r. (resp. r-) is an entrance or a natural boundary point c_ (resp. c.) is equal to zero. If r . (resp. r_) is a regular or an exit boundary point an extra condition will be necessary for uniqueness of the solution of (1.1.32).

In particular we are interested in the case we have so called Ventcel's boundary conditions on D(A):

(1.1.33) lim (QD 2 U + £Du)(x) = 0, i = 1,2. x-> r i

It will appear that in the case of a natural boundary point this condition is automatically satisfied.

Let u. and u . be the unique special solutions of (1.1.6) as given in Proposition 1.1.1. Thus u.(x_) = 1, u. is an increasing solution which is minimal on

(r ,x„) and u„ is a decreasing solution which is minimal on (x„ , rA From the Lemmata 1.1.7, 1.1.9 and (1.1.10) we know that lim u.(x) = 0 if

x-*r: _ 1

41

and only if R e L (x„,r.) or Q £ L (x-,r.), i.e. in the case that r. is a

regular, an exit or a natural boundary point, and thus lim u.(x) > 0 if

and only if R £ L (x„,r.) and Q e L (x„,r.), i.e. r. is an entrance boundary

point.

We define the Green function T: JxJ —>3R of the equation (1.1.6) as follows:

(1.1.34) T(x,s) = ( K 0 Q W ) " 1 ( S ) U 1 ( S ) U 2 ( X ) f o r r j < s < x < r2

= ( K . Q W ) " (s)u-(s)u . (X) for r. < x < s < r-.

Here K~ is the positive constant defined (as in Lemma 1.1.11) by KQ = u j(xQ).u2(x0) - U^XQXU^XQ) = U J U Q ) - u2'(x0). For each f e C(J) we

define the function u f by

(uf)(x) = | ^ (1.1.35) (u f)(x)= | 2 r(x,s)f(s)ds. 1

Then the mapping f - » u f induces an operator K on C(J). We have

Proposition 1.1.19 Let for each f G C(J ) Kf be defined by Kf = uf Then (i) K is a positive, linear contraction operator from C(J) into C(J), (ii) Kf is a particular solution of (1.1.32).

Proof (i) Let f G C(J), then f is bounded, and

|u,(x)| < ||f|| f 2 r(x,s)ds J r l

= llfll {u2(x) J ((K0aW)"1u1)(s)ds

Uj(x) J 2 (K 0 aW) ' 1 u 2 ) (s )ds}

1

+

42

^ j ' l l f H { u 2 ( x ) J (W"1up'(s)ds

Uj(x) ƒ 2 ( W _ 1 u p ' ( s ) d s } .

1

+

Since W u ! is positive increasing and W uJ is negative increasing both integrals exist. It follows that for each x € ( r . , r . ) uf(x) is well-defined. We have for all x e (r. ,r-)

uf(x) - u2(x) I ((KQaW)"J u j f)(s)ds

+ Uj(x) J 2((K ( )aW)"1u2f)(s)ds.

-Xhen-u^-is-twice-continuously-dif-ferentiable-with

Y

u£(x) = u^x) J ((K0aW)_1Ulf)(s)ds

H{W fx 2((K0aW)"1u2f)(s)ds

and

Y

u"{(x) = u2(x) I ((K0aW)" l u j f)(s)ds

u"(x) J 2((KnaW)"1u0f)(s)ds ' 0 " " ' -2 1

( (uJu 2 -uû 1 XK 0 aW)" 1 f ) (x) .

From Lemma 1.1.11 we have

(u ju 2 - uû1)(x) = K0.W(x).

By substitution it is easily verified that for all x e (r ,r0)

43

uf(x) - au'j.(x) - pu '(x) = f(x).

It is clear that K is linear and since T is a positive function the operator K is positive.

Let f be a non-negative function in C(J), and let ((a ,b )) be a sequence of intervals such that

r . < < a . < a < < a. < b . < < b < b . < < r„. 1 n+1 n 1 1 n n+1 2

Moreover, let (f ) be a sequence of continuous functions such that

fn(x) = f(x) x e [an ,bn],

0 < f (x) < f(x) x e (a , , a ) u ( b , b . ) , nv ' v ' v n+1 n n n+1"

f (x) = 0 x € (r. ,a , ) u (b , , r . ) . nv ' v 1 n+1' v n+1' 2'

Then for each x e (r ,r_) the sequence (f (x)) is non-decreasing and lim f (x) = f(x).

n—»oo " Let n e l . Since K is positive uf is a non-negative solution of (1.1.32) with

' n f = f . On the subinterval (r. ,a ,) we have n . 1' n+1'

u f ( x ) - ( au" )(x)-(/Ju!. )(x) = 0, r n ln ln

and for x e (r. ,a .) we have by the definition of u r v 1 n+1 ' f

u f (x) = Uj(x) . | 2 ((KoaW)_1u2fnXs)ds > 0 n+1

and similarly for x e (b .,r_)

u f (x) = u2(x) ■ ƒ n + 1 ((K0aW)"1u2fn)(s)ds > 0.

44

Thus u f is positive increasing on (r. ,a .) and positive decreasing on

(b , ,r_). Hence u,. has a positive maximum at a point x e (a , ,b .). v n+1 2' fn m n+1 n+1 Then u ' (x ) = 0 and u" (x ) < 0. So fn

v m ; fnv m'

u , (x ) = f (x ) + (au" )(x ) fnv m' nv nv v fn' nv

< f (x ) < ||f||. - n \ m / ii ii

Thus certainly

(1.1.36) l |u f | |< | | f | | 'n

for all n e l . Because of~the posiïiVity of-KT~the sequence (uf (x))_is~increasing for each

n x G J, and since

[ 2 r(x,s).(f . J r j n

(s) - f(s)) ds

converges to zero, for each x G (r.,r_) the sequence (u r (x)) converges from l Z t n

below to uf(x). Thus also

(1.1.37) ||uf || < ||Uf||.

Applying Lebesgue's Monotone Convergence Theorem on the integral representation (1.1.35) for u f and using (1.1.36) we obtain

' n

(1.1.38) ||Kf|| = ||uJ| = sup uf(x) = sup lim u f (x) < sup sup \xf (x) 1 XGJ l XGJ n-+oo l n XGJ nQN ^n

< sup sup ||uf || = sup ||uf || < sup ||f|| = ||f|| XGJ n&N In nelT rn nGJN

for all non-negative functions in C(J). Because of the positivity of K, (1.1.38) holds for all functions in C(J).

45

The boundedness of u f follows from (1.1.38), but this does not guarantee that u f e C(J). So we have to prove that lim uf(x) exists for i = 1,2. We will 1 x—>r; ' prove the existence of the limit for the boundary point r-. Let f G C(J). Then by (1.1.35)

(1.1.39) uf(x) = af(x) Uj(x) + bf(x) u2(x)

with

a f(x)= f 2 ( ( K a W ) " ' u f)(s)ds

and

b f ( x ) = J ((K0aW)_ 1 Ulf)(s)ds.

We have to investigate the various cases. We formulate the results in two lemmata.

Lemma 1.1.20 If r. is a regular or an exit boundary point, then

(1.1.40) lim (Kf)(x) = lim u/x) = 0. x-+r2 -x-r2 I

Proof If r„ is a regular or an exit boundary point, then lim u.(x) = 0, lim u.(x)

z x-*i"2 ~l x—>*2 is finite and lim af(x) = 0. Moreover, if r» is regular, then also lim bf(x)

x-+r2 * l x-»r2 f

is finite. In view of (1.1.39) we then have lim uf(x) = 0. x-+r2 *

If r_ is an exit boundary point, then clearly lim af(x) u.(x) = 0, so it re-z x-»r2 '

mains to show that

(1.1.41) lim bf(x) u-(x) = 0. x-+r-j r ~l

46

we have |bf(x)|.u2(x) < K Q 1 ||f||.u2(x) ƒ ((aW)"1u1)(s)ds

= KQ 1 | | f | | . u 2 (x ) | (WûjVWds

= KQ1 | |fl | .u2(x){(W"1u1 ' )(x)- lim (W _ 1u;Xt)}

sK^l l f l l .u^xMWûjXx) .

From (1.1.29) and the proof of Lemma 1.1.13 we have

lim u (x).(W"1u')(x) = 0. x-+r2

Thus also (1.1.41) holds. The proof for r . is similar.

Lemma 1.1.21 If r 7 is a natural boundary point, then

(1.1.42) lim (Kf)(x) = lim u/x) = lim f(x). x—*r2 x-*r2 ' x-^rj

Proof Let r» be a natural boundary point and let e > 0. Since f e C(J) there exists,a point b e (xQ,r-) such that for all x e (b,r«)

|f(x) - lim f(x)| < e. x - r 2

Then we have for x G (b,r_)

(1.1.43) uf(x) = f(x) f 2 r(x,s)ds + g(x) J r l

with g(x) = | 2 r(x,s).(f(s) - f(x))ds. Thus *

47

2'

By the definition of V (1.1.34) we have

(1.1.44) |g(x)| < e f 2 r(x,s)ds, x Ê ( b , r , ) . J r l

2 (W"1uj)'(s)ds (by 1.1.8)

r Y

f 2 r(x,s)ds = u2(x) ƒ (K0aW)"1(s)u1(s)ds

+ Uj(x) J 2(K0aW)_1(s)u2(s)ds

= U 2 ( X ) . K Q 1 . | ( W ' ^ V ^ d s

+ u 1 (x ) .K- 01 . ^ 2

= u ^ . K ^ . ^ W ^ U j ' X x ) - ( W ^ ' X r , * ) }

+ u^xJ -KJ^ . /o - (W_ 1u^)(x)\ (by Lemma 1.1.17)

= 1 - KQ1.u2(x).(W"1u1l)(r1+) (by Lemma 1.1.11).

Since lim (W u.')(x) is finite (compare Lemma 1.1.2) and lim u_(x) = 0 x-+ri ] x—^2 l

if r , is a natural boundary point, it follows that

-P »r-> J r .

(1.1.45) lim I 2 T(x,s)ds= 1 x-+r2 J r j

if r_ is a natural boundary point. Finally it follows from (1.1.43) - (1.1.45)

that

lim u.(x) - f(x) f 2 T(x,s)ds = 0 i-*2 f J r l

and with (1.1.45) we obtain (1.1.42). D

48

Returning to the proof of Proposition 1.1.19, in all occurring cases Kf e C(J). This completes the proof of the Proposition.

□

As a corollary of Lemma 1.1.21 we have

Corollary 1.1.22. If r? is a natural boundary point, then for each solution u in

C(J) of u - Au = f, f e C(J), we have

Urn (Au)(x) = 0. x—r2

D

Corollary 1.1.23

i rr? < 1. sup T(x,s)ds

xeJ \Jr

Proof If eJs) = 1 for all s e J, then

(1.1.46) | | 2 r(x,s)ds| = | J 2 r(x,s).e0(s)ds

= l(Ke0)(x)| < ||Ke0H < ||e0H = 1

which proves the corollary. D

1.1.5. Solutions of z - aD z - 0Dz = k in L (J)

oo Since (C(J), ||.||) is a closed subspace of (L (J), ||.|| ) the results of section 1.1 are valid in the space L (J). Here ||.|| denotes the norm defined by

|z|| = ess sup |z(x) 0 0 XGJ

Let us consider the equation

49

(1.1.47) z - aD2z - /?Dz = k

in L (J). For each k £ L (J) we define the function z. by

(x) - f '2

J r j

(1.1.48) zR(x)= | 2 r(x,s)k(s)ds,

where T is the regular Green function defined by (1.1.34). Then the mapping

k - t z , induces a linear operator K on L (J). As an extension of Proposition

1.1.19 we have

Proposition 1.1.24 Let for each k G L (J) Kk be defined by Kk = z , . Then (i) K is a positive, linear contraction operator from L (J) into L (J). (ii) Kk is a particular solution of (1.1.47).

Proof The proof is essentially the same as the proof of Proposition 1.1.19, so we can refer to that proof.

oo Since k e L (J) and T is continuous, Tk is locally integrable. Moreover, for each x e J we have by (1.1.46)

(1.1.49) |zk(x)| < HkH^. | r2 r(x,s)ds < HkH^ < oo. ' l •J;

z.(x) is well-defined for each x e ( r . , r - ) . It follows from (1.1.48) that z, G C (J), z ' G AC. (J) and that z. is a particular solution of (1.1.47). That K is a contraction follows from (1.1.49). D

50

1.2. Solutions of v - D(aDv - ftv) = g in L (J)

In this section we will investigate solutions of the equation

(1.2.1) v - D(aDv - M = g

in L (J), where J is an open interval ( r . , r . ) (not necessarily bounded) and a and P are real-valued continuous functions on J with a(x) > 0 for all x e J just as in section 1.1. The differential expression

(1.2.2) Bv = D ( a D v - £ v )

which appears in (1.2.1) is the formal adjoint of the differential expression Au given in (1.1.1). So we may expect that there is a relation between the solutions of the homogeneous equation (1.1.6) in C(J) and

(1.2.3) v - D(aDv - £Dv) = 0

in L. (J). Here L. (J) denotes the set of all Lebesgue measurable functions f 00

fb °° for which |f(x)|dx < <x> for each compact subset [a,b] of J.

J a It will be shown that indeed there is a close relation between the solutions of (1.1.6) and (1.2.3). In the various lemmata and propositions we will exploit this relation. We will start with an investigation of the solutions of the homogeneous equation (1.2.3).

1.2.1. Solutions of v - D(aDv - 0v) = 0

Let V be the set of all solutions of the homogeneous equation (1.2.3)

v - D(D(av) - /Sv) = 0,

51

satisfying

(1.2.4)

(a)v 6 L ^ J ) ,

(b)ov e AC loc(J),

(c)(av) ' - fiv e AC loc(J).

Note that v e V also satisfies v e C(J), av e c ' ( J ) , (av)1 - /9v e C (J), and

aWv e C2(J).

Let U be the set of all solutions of the homogeneous equation (1.1.6)

u - (aD2u + /9Du) = 0

with u e C (J). Then we have

Lemma 1.2.1 The mapping $.■ U -* V, defined by

(1.2.5) <t>(u) = (aW)~1u

is a linear bijection.

Proof 1 $(U) c V. Assume u 6 U and set v := (aW) u. Clearly v satisfies (a).

Since av = W u w (b) holds. Moreover,

Since av = W _ 1u with u e C2(J) and W"1 e C (J) we have av e C (J), thus

(av) ' - pw = W - 1 (u ' + /3a 'u) - j5.(oW)"'u

= W" 1 u ,)

i i

which implies that (av) ' - j J v e C (J), thus (c) holds.

52

It is standard that U and V are two dimensional linear spaces. (Here we use the assumption a > 0). Finally, $ is linear and injective (since (aW) > 0), hence surjective. a

Lemma 1.2.2 Let v E V be positive and such that aWv is non-increasing, then

Urn (W }(aWv) ')(x) exists x-*r2

and is non-positive and v £ L (xn,r~).

Proof Assume v e V such that aWv is non-increasing. Since (1.2.3) can be rewritten

as

(1.2.6) v - ( W ' ^ a W v ) ' ) ' = 0

and since QWV is a twice continuously differentiable function it follows that

Y

(1.2.7) W" l(aWv) '(x) = f v(t)dt + (aWv) ' ( xA Jx 0 0

Thus W (QWV) ' is a non-positive non-decreasing continuous function, so lim W (QWV)'(X) exists and is non-positive. Further, from (1.2.5) we

x-»r2 1 -1

then obtain v e L (xn ,r ). Similarly, if v e V, then lim (W (aWv)')(x) exists and is non-negative, and v e L ( r . ,x n )

x—>r

In the Lemmata 1.1.14 - 1.1.17 we gave for the d i f fe ren t types of bounda ry points special solutions u . , u« and u , of the homogeneous equat ion (1.1.6). F rom the preceding Lemma 1.2.1 i t follows that the functions v . = $ ( u . )

1 1 1

(i = 1,2,3) are special solutions of (1.2.3) in L. (J). In the next Lemmata 1.2.3 - 1.2.6 we will summarize some properties of these solutions.

53

Lemma 1.2.3 Let r . be a regular boundary point, let u- and u , be as in Lemma 1.1.14, and let V: = $(u.), i = 2,3. Then v and v, are positive continuous solutions of (1.2.3),

v. G L (xn,r7), aWv7 is decreasing, Urn (aWvy)(x) = 0, 2 U i i x—*r2

av7GC}(J) and Urn ((av)' - pv )(x) = -1; 2 x->r2 z

v, e L (xn,rj, aWv, is decreasing, Urn (aWv,)(x) = 1, 5 u 2 3 x—>r2

i

av G Cl(J) and Urn ((av ) • - /3vJ(x) = 0. i x-yrj J J

For each solution v of (1.2.3) we have v e C(J), v G L (x„,r 2), aWv is bounded on (x„,r .), av G C (J) and Urn ((av) ' - fiv)(x) exists. There also

u ^ x—*r2 exists a positive solution v. for which aWv. is increasing. For each positive solution v of (1.2.3) we have Urn ((av)' - 0v)(x) < 0 if

x->r2 and only if

V = P2V2 + P3V3

with p2> 0, p.. > 0, not both zero.

Lemma 1.2.4 Let r? be an exit, boundary point, let u. and u? be as in Lemma 1.1.15 and let v.*9(u.), i = 1,2. Then v . and v2 are positive continuous solutions of (1.2.3),

v fc L (x~,r ) , aWv. is increasing and bounded,

av EC (J) and Urn ((av ) • - /?v .)(x) = oo; 1 x-*r2 1 1

v. G L (x',r), aWv j is decreasing, Urn (aWv 2)(x) = 0,

54

av.eC^J) and Urn ((av.) • - 0v )(x) = -1. 1 x->r2 J 1

Only solutions of (1.2.3) which are scalar multiples of v? belong to

Ll(xQ,r2). D

Lemma 1.2.5 Let r- be an entrance boundary point, let u. and u , be as in Lemma 1.1.16, and let v. := $(u.), i = 1,3. Then v. and v, are positive continuous solutions of (1.2.3),

v. G L (x„,r .), aWv. is increasing, Urn (aWv.) = oo, x *r2

av G C (J) and Urn ((av) ' - 0v)(x) < oo; 1 x-+r2

v, e £ (x0,r2), aWv ? is decreasing, Urn (aWvJfx) = 1, x *r2

av G C!(J) and Urn ((av) ' - 0v)(x) = 0. J x-*r2

For each solution v of (1.2.3) we have

v e L (xn,r ) and Urn ((av) ' - 0v)(x) u z x-+r2

exists. a

Lemma 1.2.6

Let r2 be a natural boundary point, let u. and u- be as in Lemma 1.1.17', and let v. = *(u.), i = 1,2.

i i Then v. and v, are positive continuous solutions of (1.2.3),

v * fi L (XQJ.), aWv J is increasing, Urn (aWv J(x) = oo, x~*r2

av G C (J) and Urn ((av ) ' - PvJ(x) = oo; 1 x-*r2 1 1

55

v7 6 L (xn,r 7), aWv- is decreasing, Urn (aWv7)(x) = O,

av G C](J) and Urn ((av ) ' - fiv )(x) = 0. z x—*r2

Only solutions of (1.2.3) which are scalar multiples of v . belong to LJ(x0,r2). a

The proofs of these lemmata rest on the proofs of Lemmata 1.1.14 - 1.1.17. The only thing we have to prove yet is v £ L (xf t,r-) in the case of an exit or a natural boundary point. This is a consequence of

Lemma 1.2.7 Let u be a positive increasing solution of (1.1.6) and let v = $(u), then

v G L (xQ,r2) <s=s* Q G L (xQ,r2).

Proof By (1.2.5) we have

X X

f |v(t)|dt= f ItfaWûXOIdt J x o J x o

Y

= f (W_ 1u') ' ( t )dt (by (1.1.8)) J x„ v0

( W " 1 u ' ) ( x ) - u ' ( x o ) .

Then the assertion follows from Lemma 1.1.12. D

We define the functions v and v- by

(1.2.8) y. = (aW)_1u., i - 1,2,

56

where u and u_ are the special solutions of (1.1.6) as given in Proposition 1.1.1. It follows that if r . and r» are not entrance boundary points, then

(1.2.9) lim (aWv.)(x) = 0, i = 1,2. x-»r; ' i

The general solution of (1.2.6) is given by

(1.2.10) v = c l ^ l + c 2 - 2 ' C1'C2 e ' R '

From the Lemmata 1.2.3-1.2.6 the following lemma is easily obtained

Lemma 1.2.8 Let the general solution of (1.2.6) be given by (1.2.12). If r■ is an exit or a natural boundary point, then v G L (xn,r J implies c . = 0. If r. is a regular or an entrance boundary point, then for uniqueness of the solution of (1.2.6) an extra condition on v is needed. a

1.2.2. Solutions of v - D(D(av) - 0v) = g in L](J)

As in section 1.1 we will construct solutions of the inhomogeneous equation

(1.2.11) v - D(D(ov) - M = g, g e L ^ J ) , v e L ^ J )

by means of special Green functions. It will appear that it is possible to use the

Green function T, defined in (1.1.34), to obtain solutions of (1.2.11). Let u. and u_ be the unique "minimal" solutions of (1.1.6) as given in Proposition 1.1.1 and let the Green function T be defined by (1.1.34). For each g e L (J) we define the function v by

(1.2.12) v g ( X ) : = I 2 r (s ,x)g(s)ds-

57

The mapping g - » v induces an operator L on L (J). As a counterpart of Proposition 1.1.19 we have

Proposition 1.2.9 Let for each g G L (J) Lg be defined by Lg = v . Then

8 ] j (i) L is a positive linear contraction operator from L (J) into L (J). (ii) Lg is a particular solution of (1.2.11).

Proof Let g e L (J) and x e J. Then

(1.2.13) vg(x) = ( K ^ W r ^ U j U ) j 2 u2(s)g(s)ds +

Y

+ (KQaW)~l (x).u2(x) j " u j (s)g(s)ds,

where K» is defined by (1.1.21). Since

j 2 u2(s)g(s)ds < u2(x)J 2 |g(s)|ds < u2(x) . Hgllj

and

J r l (s)g(s)ds < U j ( x ) | 1s(s)|ds < Uj(x) . Usllj

1

the integrals in (1.2.13) are convergent. Thus the function v := x -♦ v (x), x 6 J is well-defined and the function aWv is locally absolutely continuous. For the derivative of aWv we have

g

v )'(X) = K ; V V ^ f'2 (1.2.14) (aWvg)'(x) = K- JU l ' ( x ) f 2 ? (s)g(s)ds 2

x K ^ u ^ x ) ] " u1(s)g(s)ds

Thus (aWv ) ' is also locally absolutely continuous.

58

We obtain

(W"1(aWvg) ' ) ' (x)= K Q V W ' ^ P ' W ƒ 2 u2(s)g(s)ds

+ K Q 1 ( W ' 1 U ^ X X ) J u1(s)g(s)ds

+ K Q 1 W " 1 ( X ) ( - U 1 ' U 2 + UÛ1)(X) g(x) a.e.

= K ; 1 ( a W " 1 u 1 ) ( x ) | 2u2(s)g(s)ds ^0

KQ1(aW"1u2)(x) [ X u1(s)g(s)ds - g(x) a.e.

= v (x) - g(x) a.e.

It follows that

(1.2.15) v - (W" ' (aWv ) ' ) ' = g,

thus v is a particular solution of (1.2.11) in L. (J).

Note that aWv e c ' ( J ) . g

That L is linear is clear, and that L is positive follows from the fact that T is positive, so for all g e L (J) |v | < v. and in contraction, it is sufficient to prove the inequality positive, so for all g e L (J) |v | < v. and in order to prove that L is a

IIVgHj <||g||j

for functions g in L (J) which are positive a.e. Since C (J), the set of functions 1 c

in C(J) with compact support in J, is dense in L (J) it is even sufficient to restrict ourselves to positive functions in C (J). Let g G C (J) be positive. Then

c — c there is a compact interval [a,b] C J such that g(x) = 0 for x e J\[a,b]. Because of the positivity of L the function v is non-negative and satisfies the equation

59

v(x) - ((av)' - pv) '(x) = 0 for x e J\[a,b].

As in the proof of Proposition 1.1.19 there are positive constants c. and c„

such that

v (x) = CjV^x), x G ( r r a )

= c2y2(x), x G (b,r2),

where the functions v and v . are defined by (1.2.8). Regarding Lemma 1.1.2 we see that lim (W"^(aWv )')(x) exists (i = 1,2)

x - n 8 and moreover

(1.2.16) lim (W ^QWV ) ' ) ( X ) > 0,

x—»r j 8 lim (W"'(aWv )')(x) < 0.

L x - r 2 8

By integration of (1.2.15) with g := g we obtain

I 2 v (s)ds= f 2 g(s)ds+ lim (W_ 1(aWvJ'Xx) J r . § J r, " x - r 2

lim (W_1(aWv ) ' ) (X) . x—»ri 8

In view of (1.2.16) we have

l|vg||j = j vg(s)ds < J g(s)ds = ||g|i

This implies that L is a contraction in L (J), and this completes the proof of Proposition 1.2.9.

D

60

1.3. Solutions of w - (D(aDw) - ftDw) = h in NBV(J)

Let BV(J) denote the set of all functions h which are of bounded variation over J, i.e. for which the total variation over J, Var-(h) is finite (cf. [Tay], Ch. 9). It is well-known that functions of bounded variation are continuous except at most on a countable set. Moreover, for each x e J the limits h(x+) and h(x-) exist and thus at points of discontinuity the jumps are finite. It is useful to identify two functions h. and h- if for each x 6 J

(1.3.1) hj(x+) - hj(x-) = h2(x+) - h2(x-).

This means that h and h . have the same points of continuity, respectively points of discontinuity. Let J, be the subset of J of points of continuity of h.

n Then it follows that two functions h. and h_, which differ from a constant and for which the jumps h.(x+) - h.(x-) for x G JV»h- (i = 1,2) are equal, will be identified. Clearly VarT(h.) = VarJh . ) and Var (h ) = 0 if and only if h. is constant. In each class of functions which are identified with the function f G BV(J), there is exactly one function h for which

(1.3.2) (i) h(x) = (h(x+) + h(x-))/2 for all x G J,

(ii) h(x0) = 0.

Such a function is called normalized.

Let NBV(J) denote the set of all functions h of bounded variation over J

which are normalized by (1.3.2). NBV(J) is equipped with the total variation

norm IUIyar ( J ) , defined by

' | h | lVar(J) : = V a r J ( h > '

also denoted T |dh|, see e.g. [Tay], Ch. 9.

61

An important closed subspace of NBV(J) is the space NAC(J) of all functions in NBV(J) which are absolutely continuous on J. If h e NAC(J), then it is well-known that h is differentiable almost everywhere, and that

Var(J)= Ijldh ' = Jj lh 'Wtfx-llhV

and conversely, if k e L (J), then the function h, defined by h(x) := i Jxo

belongs to NAC(J). Thus NAC(J) and L (J) are isometric isomorphic spaces.

k(s)ds

In this section solutions w>G NBV(J) of the equation

(1.3.3) w - (D(aDw) - /3Dw) = h, h G NBV(J)

satisfying a) w e C (J) b) aw • G AC loc(J)

will be investigated. Since it will be not necessary that D(QDW) - /?Dw is normalized, (1.3.3) is an equation in BV(J).

We see that h and D(aDw) - £Dw are continuous in the same points of J, and in the points of discontinuity we have

(W_1(aWw')')(x+) - (W"1(aWw') ,)(x-) = h(x+)- h(x-).

Thus the pointwise interpretation of equation (1.3.3) is

(1.3.4) w(x) - (W"1(aWw'))'(x) = h(x) + c

where c = (QWW ') ')(x ) = ((QWW ') '(x0+) + (aWw •) \x -))/2.

The differential expression

Cw = D(aDw) - £Dw

62

is closely related to the differential expression Bv in (1.2.2), so we may expect that there is a close relation between the solutions of (1.1.6), (1.2.3) and

w - (D(aDw) - 0Dw) = 0

in BV. (J). Here (N)BV. (J), respectively (N)AC 1 Q C ( J ) denotes the set of functions h with h G (N)BV([a,b]), respectively (N)AC([a,b]) for each compact subset [a,b] of J. In the next subsection this relation will be investigated.

1.3.1. Solutions of w - (D(aDw) - /3Dw) = 0

Let W be the set of all solutions in NBV. (J) of the homogeneous equation

(1.3.5) w(x) - ( W " 1 ( Q W W ' ) ' ) ( X ) = 0,

satisfying a) w e AC. (J), b) aw • G AC loc(J).

1 1 2 Note that w also satisfies w G C (J), o w ' e C ( J ) and aWw' e C (J). Let V be defined as in section 1.2.1. Then we have

Lemma 1.3.1

The mapping *.- W -* V, defined by V(w) = w' is a bijection.

Proof

(i) V(W) c V. Let w G W and v = w ' . Then v satisfies (1.2.3) and the conditions (a), (b) and (c) of (1.2.4).

(ii) * is infective. If w ' = 0 then w = 0 since w is normalized. (iii) * is surfective. If v G V, then it is easily verified that w defined by

x w(x) = f v(t)dt

v0

belongs to W. D

As a corollary we have

63

Corollary 1.3.2 The mapping Ü := $ <P: W —* U is a bijection. Moreover,

nw - aWw', w eW.

The mapping $ and the set U are defined in section 1.2.1). D

Lemma 1.3.3 Let r? be a regular boundary point, let Uy and w, be as in Lemma 1.1.14 and let w. = n" 7 u. - Urn (n'}u.)(x), i = 2,3.

' ' x-*r2 '

Then n> and vv> are continuously differentiable solutions of (1.3.5) in

BVloJJ>-- "

w2 < 0, w' > 0, w' G L (xn,r J, aWw' is decreasing, Urn (aWw \)(x) = 0,

x-^r2 2

aw* £ C](J) and Urn ((aw ' J « - f3w\)(x) = -1; 1 x-*r2 z z

w, < 0, w' > 0, w' G L (xn,r ) , aWw' is decreasing, Urn (aWw\)(x) = 1,

x—r2 *

aw• G CJ(J) and Urn ((aw\)' - pw\)(x) = 0. J x—*r2 •> •>

Each solution w of (1.3.5) belongs to AC(xn,r J while

w EL C (J), w*G L (xn,r.), aWw' is bounded on (xn,r2)

aw% G C (J) and Urn ((aw ' ) • - 0w \)(x) exists. x->r2 3 5

64

There also exists a solution w., for which w. > 0, w' > 0 and aWw' is increasing. For each increasing solution w of (1.3.5) we have Urn ((aw *) ' - fiw *)(x) < 0

x->r2 if and only if

w = p2w2 + p3w3

with p ?> 0, p , > 0 not both zero.

Lemma 1.3.4 Let r? be an exit boundary point, let u. and u? be as in Lemma 1.1.15 and let w. := n"7M. - Urn (ü'^.Xx), i - 1,2. 1 ' x-*ri l

Then w. and w- are continuously differentiable solutions of (1.3.5) in

BVloc(J> :

w. > 0, w' > 0, w' £ L (xn,r.), aWw' is bounded,

aw\ eC (J) and Urn ((aw ' J • - 0w \)(x) = oo; 1 x->r2

1 l

w2 <0,w2>0,w'2GL (xQ,r ) , Urn aWwUx) = 0, x—*r2

aw' E CJ(J) and Urn ((aw \) • - pw \)(x) = -1. 1 x-*r2 . z z

Only solutions of (1.3.5) which are scalar multiples of vv? belong to BV(xQ,r2).

Lemma 1.3.5 Let r2 be an entrance boundary point, let u. and « , be as in Lemma 1.1.16

and let w := if u - Urn ((l~!u )(x), w .= fT7u - Urn (n" 7 u J f x j . 1 v x-*r i l i i x—>r2 i

Then w. and w, are continuously differentiable solutions of (1.3.5) in BVloc^

65

w. > O, w\ > O, w\' G L (xn,rj, Urn (aWw\)(x) = oo, l l 1 UZ x—*r2

aw' e C1 (J) and Urn ((aw \) • - /5w'\)(x) < oo; 1 x - r 2 l l

, < 0, w • > 0, w' e L](xn,r ) , Urn (aWw \)(x) = 1 5 j 5 UZ x—►/•? w 3

,1 aw ' e C' (J) and Urn ((aw \) • - pw \)(x) = 0. J x—^2 J •?

For each solution w of (1.3.5) we have

w G BV(xn,r7) and Urn ((aw*)* - fiw')(x) exists. u l x-^>r2

Lemma 1.3.6

Let r be a natural boundary point, let u. and u., be as in Lemma 1.1.17, and

let w. := n~]u. - Urn (Cl~}u.)(x), i = 1,2. x—>r

Then w. and w , are continuously differentiable solutions of (1.3.5) in

BVloc^:

w. > 0, w\ > 0, w\ £ L (xn,rj, Urn (aWw\)(x) = oo, l l l u z x—*r2 1

aw\ G C (J) and Urn ((aw \) • - Pw \)(x) = oo; 1 x - r 2 J 1

H> < 0, w* > 0, w ' G L1(xn,r ) , Urn (aWw\)(x) = 0,

aw' G C!(J) and Urn ((aw ' J ' - pw \)(x) = 0. z x-*r2 z z

Only solutions of (1.3.5) which are scalar multiples of vv? belong to BV(x0,r2). D

The proofs of the Lemmata 1.3.3-1.3.6 rest on the proofs of the Lemmata

1.1.14-1.1.17, 1.1.3-1.2.7 and 1.3.2 will therefore be deleted here.

66

1.3.2. Solutions in NBV(J) of w - (DfaDw) - 0Dw) = h

In the sections 1.1 and 1.2 we constructed solutions of inhomogeneous equations in C(J) and L (J) by means of special Green functions. In this section we will also use the Green function T, defined in (1.1.34). To obtain a particular solution w, in NBV(J) of the equation

(1.3.7) w - (D(aDw) - 0Dw) = h, h e NBV(J)

the Green function r , defined in (1.1.34) can also be used. Note that (1.3.7) has the pointwise interpretation (1.3.4). For each h e NBV(J) we define the function w, by

«=o? wh(x) = | | 2 r(s,t)dh(s)dt.

The mapping h —► w, induces an operator M on NBV(J). As a counterpart of the Propositions 1.1.19 and 1.2.9 it can be proved (cf. [Fe,l], Th. 13.3)

Proposition 1.3.7 Let for each h E NBV(J) Mh be defined by Mh = w,. Then (i) M is a linear contraction operator from NBV(J) into NBV(J), (ii) Mh is a particular solution of (1.3.7), (Hi) if h is non-decreasing, then Mh is non-decreasing. D

67

CHAPTER 2 - SEMIGROUPS GENERATED BY DIFFERENTIAL OPERATORS SATISFYING VENTCEL'S BOUNDARY CONDITIONS AND THEIR DUALS

2.1. Semigroups in Banach spaces

Let X be a real Banach space with norm ||.|| and let L(X) be the Banach algebra of all bounded linear operators from X into itself. Let T be an operator-valued function, mapping the non-negative real axis into the Banach algebra L(X) and satisfying the following two conditions:

(2.1.1) (i) T(s + t) = T(s)T(t), s > 0 , t > 0 ;

(") T(0) = I,

where I denotes the identity operator. Then the family (T(t) | t > 0}, shortly denoted by T, is called a semigroup of bounded linear operators in L(X) or a semigroup on X. The semigroup T is said to be of class C„ if it satisfies in addition the condition

(2.1.2) (Hi) lim ||T(t)f - f||Y = 0 tj.0 x

for each f e X.

A semigroup of class C„ in L(X) is strongly continuous at each point of the positive real axis. Therefore, a semigroup of class C~ is also called a strongly continuous semigroup. For each strongly continuous semigroup in L(X) there exist positive constants M and w, not depending on t or f e X, such that for each f e X, and each t > 0

68

(2.1.3) l|T(t)f|lx < MeWt||fHx.

A strongly continuous semigroup in L(X), satisfying (2.1.3) with w = 0 and

M = 1 is called a contraction semigroup in L(X).

The set of all strongly continuous contraction semigroups in L(X) will be

denoted by CSG(X).

Let T e CSG(X). The infinitesimal generator of the semigroup T is an opera

tor A in X with domain D(A) c X defined by

(2.1.4)

D(A) = ( f e X | lim t_1(T(t)f - f) exists) ^ tj.0 J

Af = lim t" J(T(t)f - f) for all f e D(A). t |0

The first part of the Generation Theorem of Hille-Yosida ([BB], (1.3.2);

[CH], Ch. 2) tells that A satisfies the following properties:

(2.1.5)

(i) for each f G D(A)

l|f|lx < llf - AAf|lx for every A > 0, (ii) R(I - A A) = X for every A > 0, (iii) D(A) is dense in X.

Here R(I - AA) denotes the range of I - AA. A linear operator A: D(A) c X -♦ X, satisfying (2.1.5)(i) is called dissipative. A dissipative operator A satisfying (2.1.5)(H) is called m-dissipative. We denote the set of all densely defined m-dissipative operators in X by D(X).

Remark. A linear operator A is called (m-)accretive if -A is (m-)dissipative. It follows from (2.1.5)(i) and (ii) that A is a closed operator. Therefore the domain of A, D(A), equipped with the graph norm l l l l n / A V i.e.

69

(2.1.6) l |u|lD ( A ) = | |u | |x + | |Au||x , u e D ( A ) ,

is a Banach subspace of X.

The second part of the Generation Theorem of Hille-Yosida states that the mapping G: CSG(X) — D(X) which associates with T e CSG(X) its infinitesimal generator A := G(T), is a bijection.

Let A e D(X), then it follows from (2.1.6)(i) and (ii) that for each A > 0, I - AA is a bijection from D(A) onto X, and that R,(A) := (I - AA) is a bounded operator from X onto D(A) C X satisfying

I |RA(A)| |<1, (A>0).

The operator R.(A) is sometimes called the resolvent of A for A > 0.

Example

Let BUC(IR) be the Banach space of all bounded uniformly continuous real-valued functions on TR, equipped with the supremum-norm. For each f £ BUC(IR) and each x GTR, the semigroup T of Gauss-Weierstrass is defined

by

(T(t)f)(x) = (27rt)"1/2 f exp / - j t _ 1(x - s)2Vf(s)ds if t > 0,

(T(0)f)(x) = f(x).

It appears ([Y], Ch. IX, 5; ex. 2) that T is a strongly continuous semigroup and that the domain of its infinitesimal generator is the subspace of functions f in BUCCR) for which the second derivative f" exists and belongs to BUC(IR), and that Af = - | f", f e D(A).

Next we will recall some facts on dual semigroups on the dual space X . For these results we refer to [BB], Ch. 1.4 and [CH], Ch. 3.4.

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Let X be the space of all bounded linear functionals u on X. X is a Banach

space with the norm

|u*|| = sup |<u*,u>|. | |u | lx<l,u€X

Here <.,.> denotes the pairing X x X -+3R. Let L be a linear operator with domain D(L) dense in X and range R(L) in X. Then the J defined by Then the adjoint (or dual) operator L of L with domain D(L ) C X is

' D(L*) = / u* G X* | (3v* e X*)(Vu G D(L)) (<v*,u> = <u*,Lu>)}

(2.1.7) and

L*u* = v* for all u* G D(L*).

Let T be a C„-semigroup of linear contraction operators on X. Then the dual semigroup T* := {T*(t) | t > 0} of T, where T*(t) is the adjoint of T(t) in the dual space X , is also a semigroup of linear contraction operators. T is not necessarily a semigroup of class C„. If X is a non-reflexive Banach space we on-ly know that T is weak -continuous. Such a semigroup is called of class Cn in X . The dual operator A of the infinitesimal generator A of the semigroup T is equal to the weak infinitesimal generator of the dual semigroup T of T, i.e. u* G X* belongs to D(A*) if and only if for all u G X, lim < t" l (T*(t)u* - u*),u>

tj.0 _. exists, and u* G D(A*). A*u* is defined as the weak* limit of t~ (T*(t)u* - u*). We use the notation X» for the closed subspace of X on which T is strongly continuous. The subspace X~ is invariant under T (t), t > 0 and the restriction of T to this subspace is denoted by T». Then T_ is again a strongly continuous semigroup on X„. Its infinitesimal generator is denoted by Afl. We have the following

71

Proposition 2.1.1 (i) D(A*Q) C D(A*) C X*Q, . (ii) X' is equal to the strong closure of

D(A*) and D(A*Q), (Hi) A* is the part of A* in X* i.e. it is the largest restriction of A with

both domain and range in X~, (iv) if X is reflexive, then X' = X and A() = A .

In this chapter we will use the notation X® for a specific representation of the space X-, and T® for the corresponding representation of T . in X®. (Thus X-and X® will be isometrically isomorphic.) Let T e CSG(X), then T® 6 CSG(X®). We can imbed X into X®*, the dual space of X®, by means of the natural imbedding i defined by

(i(u))(u®) := <u®,u>

for all u e X and all u® e X®. It is known that i is an isometry not necessarily surjective, and thus i(X)

is closed in In what follows, we identify X with i(X). For all u e X, u® £ X® and t > 0 we have

<u®,T(t)u> = <T(t)*u®,u> = <T(t)®u®,u> = <u®,T(t)®*,u>

where T(t)® denotes the adjoint of T(t)®. Thus T(t) C T®*(t) := T(t)®* for all t > 0. We denote by X®0 the closed subspace of X®* on which the semigroup T® is strongly continuous. The restriction of this semigroup to X®® is denoted by T®®. Its infinitesimal generator is denoted by A®®. From Proposition 2.1.1 we immediately obtain

72

Proposition 2.1.2 (i) Jif00 is equal to the strong closure of the domain D(A® ) of A® in X®*, (ii) / 4 0 Q is the part of AQ* in X 0 0 .

Note that X C X©0 , T C T 0 0 and A c A 0 0 .

For application in approximation theory we define the Favard class of a

strongly continuous contraction semigroup on X.

Definition: Let T e CSG, then the Favard class Fav(T) of T is defined as:

Fav(T) := / u e X: SUD ||t_1(T(t)u - u ) | | x < 00V

Two characterizations of the Favard class of T are given by

Proposition 2.1.3 ffCHJ, Ch. 3.4) Let T e CSG(X), then (i) Fav(T) = X n D(AQ*), (ii) Fav(T) = \u e X] there is a constant M > 0 and a sequence (u ) in

D(A) such that (a) lim \\u - u\\Y = 0

(b) \\Au^\x <M, n = 7,2,...}.

2.2. Two propositions on adjoint operators

In this section we establish two propositions on adjoint operators. If L: D(L) C X —► X is a linear operator with domain dense in X, then the adjoint L* D(L*)cX* -f X* is defined by (2.1.7). If <.,.> denotes the pairing X x X -*1R, then in many cases the characterization of the elements of D(L ) gives difficulties. This characterization must be obtained in one way or another from the definition of L , especially from the equation

73

<v ,u> = , (see (2.1.7)).

Since in our case the pairing is defined by means of integrals, the difficulties arise from the non-integral terms after integration by parts. These difficulties can be highly reduced if one starts with functions in a suitable dense subset of X and X . Afterwards the obtained results are extended to spaces X and X by means of continuous extension of the considered operators. The next two propositions will serve to that purpose.

Proposition 2.2.1 Let A: D(A) C X —» X be a linear m-dissipative operator with domain D(A) dense in X, and let B: D(B) c X —» X be a closed linear operator. Let there exist a dense linear subspace M of X and a dense linear subspace N of X* such that

(i) N c R(I - B) (ii) <v,Au> = <Bv,u>

for all u G D(A) such that u - Au G M and all v e D(B) such that v - Bv G N.

Then B = A*.

Proof I - B is infective. Let v e D(B) such that v - Bv = 0. Since A is dissipative, I - A is injective. For f e M set u f = (I - A)" f, then for all f G M we have

<v,f> = <v, u f - Au~> = <v,uf> - <v,Auf> = <v,u~> - <Bv,uf> = <v - Bv, uf> = 0.

Since M = X it follows that v = 0. Thus I - B is injective.

74

I - B: D(B) -* X* is surjective and \\(I - B)~ \\ < 1. Let g e X*, then we have to prove that there exists a v e D(B) such that v - Bv = g. Since N is dense in X , there exists a sequence (g ) in N such that

Hm ||g - g || = 0. n—»oo u *

Since N c R(I - B) there exists a sequence (v ) in D(B) such that v - Bv = g . For all f e M we have

<vn,f> = <vn, u f - Auf>

■ <Vn " B v n ' V = <gn, ( I - A)_1f>.

Then for all f G M = X we also have

<v ,f> = <g , (I - A) !f>. n ön v '

So for all n £ l

llvJL = sup l<v_,f>| ||f||<l,feX n

sup |<g (I - A)" f>| l |f | |<i,fex

< sup | | ( I - Af^lLHg II l|f| |<l,fex

< sup ||f||.||gJL | |f | |<l,fex n

(because of the m-dissipat ivi ty of A), thus

l | v n N * - l l 8 n ! l * ' n G : N-

In the same way we find

75

l | vn " VmN* * l l8n " 8 n A , m.n eW.

The sequence (g ) is a Cauchy sequence. From the last inequality it follows that (v ) c D(B) is also a Cauchy sequence in X . Let v := lim v in X . Since Bv = v - g , lim Bv exists and is equal to n-»oo n n n n „^QO n v - g. Since B is closed we have v G D(B) and Bv = v - g. Thus v - Bv = g. This proves the surjectivity of I - B. Moreover, for all g e X we have

||(I - B)_1g|| =||v|| = lim ||v

< lim | | g j | = ||g|| . n—+00 u * *

Thus ||(I - B ) _ , | | < 1.

B = A*. For f e M and g G N set u f = (I - A)_ 1f and v = (I - B)_1g. For all

f E M and all g e N we have

< ( I - B ) _ 1 g , f> = <vg,f> = <vg, u f - Auf> = <V ,\lr> ~ <V , A U r >

g' f g f = <v ,uc> - <Bv ,uc>

g' f g f = <v - Bv , u r>

g g f

= <g, ( I - A)_1f>.

Since (I - A) and (I - B) are contractions we obtain by continuous exten

sion for all f G X and all g e X

<(I- B) - 1g, f> = <g, ( I - A ) _ 1 f > .

76

By using ([K], Theorem 5.30) we obtain from the last relation for all f e X

and all g 6 X*

< ( I - B ) _ 1 g , f> = <((I- A) -1)*g, f>

= <((I- A)*)_1g, f>.

So for all g S X we have

( I -B) _ 1 g = ((I- A ) V g . So

( I - B ) " 1 = ( I - A*) - 1

or I - B = I - A*.

Thus B = A . This completes the proof of the proposition. D

Remark. If B is not closed, but B is closable and the conditions (i) and (ii) are satisfied, then the conclusion of the theorem reads B = A , where B denotes the closure of B.

If in the preceding proposition N = X, then we have from condition (i) R(I - B) = X, and condition (ii) holds for all u G D(A) such that u - Au G M and all v G D(B). So one has more information about B from the conditions (i) and (ii) than in the case that N is a proper subspace of X. It appears that the condition on the closedness of B can be dropped. In fact we have

Proposition 2.2.2 Let A: D(A) C X —» X be a linear m-dissipative operator with domain D(A) dense in X, and let C: D(C) c X —► X be a linear operator, such that

(i) R(I - C) = X\

77

Let there exists a dense linear subspace M of X, such that

(ii) <v,Au> = <Cv,u> for all u G D(A) with u - Au G M and all v G D(B).

Then C = A*.

Proof As in the proof of the preceding proposition it follows from the m-dissipativity of A and condition (ii) that I - C is injective. The surjectivity of I - C is given by condition (i). Thus, if g G X , there exists a unique v G D(C) such that v - Cv = g. Then we have (as in the proof of the preceding proposition) for all f G X and all g G X*

<v f> = <g, (I - A)" f>

or

< ( I - C ) " 1 g , f> = <g, ( I - A)_1f>

= <(I- A*)_1g, f>.

So (I - C)~ ] = (I - A*)" l and it follows that C = A*. D

2.3. Semigroups in C(J)

As in chapter 1, J := (r, ,r_) denotes an open interval o fH (not necessarily bounded), J the two points compactification of J, dJ := J\J and X := C(J), equipped with the supremum-norm. Let x„ denote an arbitrary but fixed point in J. Thus -oo < r. < x» < r» < oo.

Let a and P be real-valued continuous functions on J with a(x) > 0 for all x G J. Let D(A) be given by

78

D(A) := / u e C(J)|u e C2(J), lim (aD2u + £Du)(x) . o\ { x-*d3 J x-*dJ

and let

(2.3.1) Au := aD2u + £Du for all u e D(A).

Then (2.3.1) defines an operator

A: D(A) c C(J) -v C(J).

In 1975 Martini [Ma,2] gave sufficient conditions on a and /? for A to be the generator of a C--contraction semigroup on C[J], for a bounded interval J. In this section we give sufficient and necessary conditions for A to be the generator of a C„-contraction semigroup on C(J), where J is not necessarily bounded.

MATHEMATICS Proceedings A 89 (4), December 15, 1986

On Co-semigroups generated by differential operators satisfying Ventcel's boundary conditions

by Ph. Clément and C A . Timmermans

Department of Mathematics, University of Technology, Delft, the Netherlands

Communicated by Prof. A.C. Zaanen at the meeting of April 28, 1986

1. INTRODUCTION

In [4], [5], [6], Martini and Boer investigated contraction semigroups (7V)),>o of class C0 on the Banach space C[a, b] (equipped with the supremum norm) generated by an operator A of the form

(1.1) Au = aD2u + PDu,

where a and /? are continuous real-valued functions on [a,b], with a(x)>0 for a<x<b. The domain of A, denoted by D(A), is given by

(1.2) D(A): = {ueC[a,b]\ueC2(a,b), lim Au(x)= lim Au(x) = 0}. x-*a x-<b

The boundary conditions lim ._a Au(x) = \imx-.b Au(x) = 0 are usually called Ventcel's boundary conditions [9], [10] and arise in a natural way in approximation theory [6].

In [5, Th. 1 p. 17-18] sufficient conditions on a and /? are given for A to be the generator of a Co-contraction semigroup on C[a,b], namely it is assumed that a and /? satisfy: a) a,peC2(a,b)nC[a,b], b) a(x)>0 for xe(,a,b), a(a) = P(b) = 0, c) a"1 is not integrable over neighbourhoods of a and b, d) a/?_i is bounded on (a, b).

The goal of this note is twofold: firstly we consider not necessarily bounded open intervals J and secondly we give necessary and sufficient conditions for

79

A to be the generator of a C0-contraction semigroup on C(J), where J denotes the two points compactification of J. In section 4 it is shown that our results drastically extend those of [5], even when a vanishes at a and b. The proofs are partly based on arguments of Feller [3] concerning semigroups generated by second order differential operators. Since we are concerned with special boundary conditions and since the lecture of [3] is not completely straightforward, we shall give a self-contained proof here. In Proposition 1 of section 2, we observe that under very general assumptions on a and /?, A defined by (1.1) and (1.2) is a densely defined closed dissipative [7] operator on C(J) with positive resolvent. In Theorem 2 of section 2 we give necessary and sufficient conditions for A to satisfy

(1.3) R{I-A) = C(J),

where R(I-A) denotes the range of I-A. It is known [7, p. 14] that the condition (1.3) for closed densely defined dissipative operators A in a Banach space (X, || I) is necessary and sufficient for A to be the generator of a Co-contraction semigroup on X.

2. MAIN RESULTS

Let J denote a non-empty open interval of IR (not necessarily bounded), 7 the two points compactification of J and dJ: =J\J. C(J) is the Banach space of real-valued continuous functions on J equipped with the supremum norm, denoted by | • |.

PROPOSITION 1. Let a and /? be real-valued continuous functions on J with a>0 on J. Let

D(A): = {ueC(J)\ueC2(J), lim (aD2u + 0Du)(x) = O} x~dJ

and let Au = aD2u + (]Du

for u e D(A). Then A : D(A)-*C{3), and the following conditions are satisfied: (i) D(A) is dense in C(J), (ii) A is closed,

(iii) if we have u - kAu>Q for some A > 0 and some u eD(A), then either u is strictly positive on J or u is identically zero,

(iv) A is dissipative.

In order to state the conditions on a and P in Theorem 2, we define the function W as follows:

W(x):=exp f- j.-£ (t)dtl. I *o a )

Moreover, L (resp. R) denotes the left (resp. right) boundary point of J and x0 denotes a point in J.

80

THEOREM 2. Let A be defined as in proposition 1. Then the range of I-A is C(J), i.e. A is the generator of a C0-semigroup on C(J), if and only if a and p satisfy

(HL) WeL\L, x0) or j W(x) J a"' W~ \t)dt dx=oo or both L L

and R R

(HR) WeL\x0,R)or J W(x) ] a~lW-\t)dt dx=oo or both. x0 x

REMARK. The reader familiar with Feller's terminology will recognize that (HL) (resp. (HR)) is satisfied if and only if L (resp. R) is not an entrance boundary point. Note that in the original paper of Feller a misprint occurs in the definition of entrance boundary point [3, p. 516], but it is clear from the context that it is meant that R is an entrance boundary point if

R R

W(SL\x0,R) and j W(x) J a-lW~l(t)dt dx<oo, X0 X

or, equivalently W$Ll(x0,R) and QeLl(x0,R), where

Q(x): =a~llV-\x) j W(t)dt. x0

Observe that QeL\xQ,R) implies a"1 W~x eL\x0,R).

3. PROOFS

PROOF OF PROPOSITION 1

(i) D(A) is dense in C(J). Let us denote by C*(7) (resp. Ci(7)) the subset of functions of C{J) (resp. of C(J)C\C\j)) which are constant outside of a bounded closed subinterval of J. Clearly C*(7) is dense in C(J) and a simple regularization procedure shows that C\(J) is dense in C(J). Finally we note that Cl(J)cD(A).

(ii) A is closed. Let (un)CD(A), u and v in C(J) be such that lim,,^ | U „ - H | = 0 and lim,,-^ || 4u„ —1>|| = 0. We have to show that usD{A) and Au = v. Let us denote A un by v„. For every a, b e [R such that [a, b]CJ there is a constant c>0 such that a(x)>c for all xe [a,b]. Therefore the restriction on [a, b] of aD2u + pDu is a regular Sturm-Liouville operator [2]. Since u„(a) (resp. u„(b)) converges to u(a) (resp. u(b)), it follows from the classical theory [2] that u'„ and «Ó' a r e Cauchy sequences in C[a,b], and therefore that ueC2[a,b], and au" + Pu' = u on [a,b]. Since a and b are arbitrary, ueC2(J) and au" + Pu' = v on J. Since t»eC(J), \\mxâj un(x) = 0 and u„ converges uniformly to v, we see that lim,<._ay («" + Pu')(x) exists and is equal to zero. Thus ueD(A) and Au = v.

(iii) IfX>0 and ueD(A) then u-XAu>0 implies w>0. It follows from the boundary condition that limx_ay u(x)>0. If there is any eJ such that u(y)<0,

81

then w possesses a negative minimum at some point zsJ. But then u'(z) = 0 and u"(z)>0, thus u(z) - Aau"(z) - Xfiu'(z) <0, a contradiction. Note that from the strong maximum principle [8, p. 6, Th. 3] it follows that either u(x)>0 for every xeJ or u is identically zero.

(iv) A is dissipative. It follows from (iii) that for X>0, I-XA :D(A)~* -*R(I-XA) is injective. Indeed, if ueD(A) and u-XAu = 0, then «>0 and - u > 0 , hence w = 0. Let us define Jx = (I-XA)'1. Then 7 is a positive operator (w>0=>JA«>0)) and JX 1 = 1, where 1 is the constant function equal to 1. (Note that \eD(A)). Then it follows that ||yA«|< |« | for every ueR(I-XA). This completes the proof of Proposition 1.

The proof of Theorem 2 is based on the following lemma.

LEMMA 3. Let A be defined as in Proposition 1. Then I-A is surjective in C(J) if and only if there exist two functions uL and uR, such that (i) uL and uR are in C2(J) and satisfy

(3.1) u-aD2u-0Du = O

on (L,R), (ii) uL is positive, increasing on (L,R) and lim^._i uL(x) = 0,

(iii) uR is positive, decreasing on (L,R) and lim^^^ uR(x) = 0.

PROOF OF LEMMA 3 Necessity. We assume R(I-A) = C(J). Let a,be IR be such that L<a<b<R.

Let ƒ e C(J) be positive on (a, b) and zero outside of (a, b). Let w be the unique function in D(A) satisfying a - aD2ü - fiDü =ƒ. From Proposition 1, (iii), it follows that ü is positive on J. Moreover, since \\mx^dJ f(x) = Q, we have limx_ay u(x) = 0. ü satisfies a - aD2Q - 0Du - 0 outside of (a,b). For x>b, u'(x)i=0. Otherwise there is an x>b such that ü'{x) = 0 and u"(x)>0, hence w has a local minimum of a. Since w>0 and limA._R Q(x) = 0, there must be a local maximum at some y>x, hence u'(y) = 0 and Q"(y) = (a-!ü)(y)>0, a contradiction. Thus « is positive and decreasing on (b,R) with lim^-.^ «(x) = 0.

Next define uR to be the unique solution of (3.1) on (L,R) satisfying uR(b) = a(b) and u'R(b) = a'(b). Note that u'R(b)<0 and u'R does not vanish on (L, Z?), because otherwise «R would have a local minimum >, which (by a similar argument as above) is impossible. Hence u'R<0 on (L,R). It follows that uR satisfies the conditions of the lemma. Similarly we prove the existence of an increasing function uL satisfying the conditions of the lemma.

Sufficiency. Since A is closed and dissipative, R(I-A) is closed. Since C*(7) is dense in C(J), it is sufficient to prove that R(I-A)DC+(J). Let ƒ be in C*(7). Thus there are a, b,c,deU such that L<a<b<R,f(x) = c for x< a and f(x) = d for x>b. Note that u'L(a)>0 and u'R(b)<0. For y.JelR we define:

uy = c\ + yuL on (L,a], with l(x)= 1,

«,5 = rfl+<5Mfl on [Ö,-/?).

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Then uyeC2(L,a]r\C[L,a], uy satisfies uy-auy-Pu'y=f on (L,a] and limx_L

(au" + Pu')(x) = 0. Similarly for uó on [b,R). Let us denote by Uj the unique solution in C2[a,b] of u- au" -fiu'=f satisfying Uf(a) = uf(b) = 0. For/i, veIR, we set

on [a,b]. If we can find y,ö,n,ve(R, such that u defined by uy on (L,a), u^v

on [a,Z?] and u6 on [&,/?) is CX(L, R), then it follows from the differential equation that « e C2(J), u belongs to D(A) and u- Au = ƒ. The continuity of u and u' is insured at a and 6 if and only if the following system possesses a solution:

A B C D

E F G H

-A -B

0 0

0 0

-G -H

V

Y [s J

c c -K

d -M

where/I = Mi.(a)>0, B = u'L(a)>0, C = uL(b)>0, D = u'L(b)>0, E=UR(a)>0, F=u'R(a)<0, G = uR(b)>0, H=u'R(b)<0, K=u'f(a) and M=u'f{b), and where Uj (a) = Uj (b) = 0. The determinant of the system is equal to (AF-BE)(GD-CH)<0. Thus JU, v, y and S are uniquely determined. This completes the proof of Lemma 3.

In the next lemmata we shall find necessary and sufficient conditions for the existence of uL and uR as in Lemma 3. Since the conditions for uL are similar to those for uR we shall restrict ourselves to the case of uR. Since uR is positive on (L, R) we can assume without loss of generality that uR(x0) = 1 for some x0e(L,R). From now on x0 denotes an arbitrary fixed element of (L,R).

LEMMA 4. Let a and /? be as in Proposition 1. Then there exists a unique minimal positive decreasing solution a of (3.1) satisfying Q(x0) = 1, i.e., if u is any positive decreasing solution of (3.1) satisfying u(x0)= 1, then Q<u*

PROOF )F. ForcoeIR, let us denote by uw the unique solution of (3.1) satisfying H(XO)=1, U'(X0) = CD. Set

B: ={we[R|there is a £e(x0,R) such that uw(£)<0}.

For <x> = 0 the solution uw satisfies u^(x0) = 0, u"(x0) = (u(x0)/a(x0))>0. Thus « is increasing in a right neighbourhood of x0, and since no solution has a positive maximum it follows that uw is strictly increasing for x>x0. Thus 0$B. Moreover, B is not empty. Indeed, let £e(x0,R). Since a > 0 on [x0,(^], there is a unique « satisfying (3.1) and ü(x0) = \, u(£)- - 1 . B is open. For coeB, let £e(x0,fl) be such that «a)(<J)<0. Then there is an e > 0 such that "»/(£)<0 f° r | /7-cu|<£- Moreover, if u>xeB and if ( y 2 ^ ^ i , then U>2BB. Finally, if a>eB, then u'w<0. First note that M^(JC0)<0. Let JP denote the first zero of u'w, if it exists. If uui(x)>0, then by (3.1) uw has a minimum at Jf and since uw cannot have a positive maximum and uw vanishes somewhere, this is

* ) on [ x ,R) o 83

not possible. If uw(x) = 0, then uw has to be identically zero, which is not the case. If uu(x)<0, then x would be by (3.1) a negative maximum of uw which is also impossible. Thus u'w<0 on (x0,R). Set cö = sup B and ü = ua. Then a is the desired function. Indeed, «>0 on (x0,R) since co$B, B being open. Furthermore, ü is non-increasing since a is the supremum of decreasing functions. ü>0 follows from the maximum principle [8, Th. 3 p. 6]. Finally if u>0 is a decreasing solution of (3.1) on [x0,r] satisfying M(JC0)= 1 and «<S, then w: =a-u satisfies (3.1) on [x0,R), w>0 and either w = 0 or w'(x0)>0. If w'(x0)>0, then u'(x0)«ü and u'(x0)eB, a contradiction. Thus w = S, and w is minimal. This completes the proof of Lemma 4.

REMARK. In [3], Lemma 4 is proved under the additional assumption ore eCl(J). We shall define

y = lim ü(x)= inf ü(x). x—R xe(xo,R)

Then it is clear that condition (iii) of Lemma 3 is satisfied if and only if y = 0. It will be easier to find necessary and sufficient conditions for y to be positive. This is done in the next lemmata.

LEMMA 5. Ify>0, then W<SLl(x0,R).

PROOF. We shall denote by a the solution of Lemma 4. First we define u to be the unique solution of u-au"-fiu' = 0 on [x0,R) satisfying w(x0) = 0 and u'(x0) = 1. Note that v and v' are positive on (x0,R). Let

M= sup v(x).

Then, if y>0, M=oo. Otherwise, M: =ü-yM~]v would be decreasing, satisfying lim_f_J? u(x) = 0, and «<«, contradicting the minimality of Q. Next observe that W=o'ü-ü'v. Indeed (v'ü-ü'v)(x0)= 1 and an easy computation shows that

(v'a-a'vy _ p v'ü-ü'v a'

(note that u'a-a'v>0). Hence

(v'ü-ü'u)(x) = exp j - j (^-\t)dtl = W(x).

Since -ü'v>0 we have v'ü< W, and since u'>0, u(x0) = 0 and y<«, we get

yu{x)< \ W(t)dt.

Since y>0 and M=<x, we obtain W$Ll(x0,R).

LEMMA 6. If a'lW'xeL](x0,R), then

.84

(i) 7i: = ü'{x0) + J* üa~' W~ ldt<0, (ii) yZ-nW< -ü'<üZ-r\V, where ü denotes the solution defined in Lemma 3 and

Z(x): = W{x) J a~lW-\t)dt.

PROOF. From the differential equation satisfied by Q we get

(3.2) ü\x) = W(x){ü'(x0) + J f l a " ' r {dt}.

Since u'{x)<0, W(x)>0, n = l\mx-.R u\x)W~\x)<0). Next rewrite (3.2) as

ü\x) = nW(x)-{ \ üa-lW-1dt}W(x). X

Since y<u, we get

yZ-nW^-Q'.

On the other hand, since « is positive decreasing and Z is positive, we have

- Q \x) < - n W(x) + a(x)Z(x).

LEMMA 7. Ify>0,thenZeL\x0,R).

PROOF. From (3.2) we get, for every xe(x0,R),

y \ a'^-'dts j ö a - ' w V ^ - ö ' W .

Since y>0, it follows that a~]Wl eLl{x0,R). By using Lemma 6, we get yZ< -a'. But y>0 and -fi 'eL'fXo.^) imply ZeL\x0,R).

LEMMA 8. IfZeLl(x0,R) and W<SL\x0,R) then y>0.

PROOF. If ZeLl(x0,R), it is easily seen that a~liV~l eLl(x0,R). From Lemma 6 it follows that -nW< -a', with 7r<0. Since Q'eL\xQ,R) and W$Lx(x0,R), it follows that 7r = 0. By using Lemma 6 again, we obtain -u'<üZ, and since w>0,

0 < - — <ZeL ' (x 0 , / ? ) . a

It follows that

|log ü(i?)| = |log y - l o g w(*0)| = j <oo.

Hence y>0 .

;85

COROLLARY. It follows from Lemmata 5, 7 and 8 that y = 0 is satisfied if and only if WeL\x0, R) or Z$Lx(x0,R). This completes the proof of Theorem 2.

4. COMPARISON WITH MARTINI'S RESULTS

In this last section we show how the conditions a)-d) given in the introduction can be improved. Let a,PeC[a,b] be such that a > 0 on (a,b). We look for sufficient conditions which imply that b is not an entrance boundary point. If P = P + - / ? " , where p + (resp. /?") denotes the positive (resp. negative) part of P, it is easily seen that WeLl((a +b)/2,b) if a""1/?" eLl((a + b)/2,b) and thus b is not an entrance'bouhdary point. In particular, if a(b)>0, or a(b) = 0 and a~,eLl((a + b)/2,b), then a~xP~ eLl((a + b)/2,b). These cases are not contained in Th. 1 of [5].

Moreover, if a(b) = 0, a~^L\{a + b)/2,b) and P(b)>0, then a'xP' e eL\(a + b)/2,b). This case is also not included in Th. 1 of [5].

Next we show that, if a " ' $Ll((a +b)/2,b) and er*/8~ eL°°((a + b)/2,b), then b is no entrance boundary point. Observe that condition HR of Theorem 2 is satisfied for a and P if and only if it is satisfied for la and Xp with A>0. Thus, without loss of generality we can assume a~' $L'((a + ö)/2,b) and a~^P' < 1 on ((a + 6)/2,6). By using Proposition 1 and Lemmata 3 and 4 it is sufficient to show that y = 0. Let w denote the function defined in Lemma 4. We have

ü = aü" + Pü' on {a, b),

and since w'<0 on (a,b),

a<au" + P'(-a') on (a,b).

Since a > 0 and a "•/?"< 1 on ((a + 6/2), b), we obtain

_i _2 - „ - - t / ,x (a + b , . a u<uu+a ' « ( - « ) on (—7- ,6 ).

Hence, using Young's inequality,

f a u dt< -a (a + fc)/2

ö + 6\ _( a + b (o + 6)/2

+ | J a"'fi2rff + ! J (Q'fdt for xe (o + W/2 (a + fc)/2 V 2

a + b ,b

Thus

j a~iü2dt<2 (o + W/2

If y>0 , we obtain

y2 J a~ldt<2 (D + W / 2

fl + 6

,( a + b

'a + b\ . f a + b , • Ml — ] for x e ( ,b

'a + b\ (a + b , , «I -^r~ . xe[ ——-,6 ,

contradicting the fact that a ' $ L '((A + £)/2, è). Thus y = 0

86

We have shown that the conditions a~l0~ e L\(a + b)/2,b) and a~-fi~ e e Lx((a + b)/2, b) are sufficient for b not to be an entrance boundary point. If a = 0, b= 1, a(x)- 1 - x and /}{b) < 0, both conditions are not satisfied. An easy computation shows that if fi(b)> - 1, then b is not an entrance boundary point. Thus, if a~x $Ll((a+ b)/2,b), 0(b) may even be negative.

ACKNOWLEDGEMENT

We would like to thank C. Scheffer for bringing reference [3] to our attention, and A.C. Zaanen for his critical reading and his valuable suggestions.

REFERENCES

1. Arendt, W., P.R. Chernoff and T. Kato - A generalization of dissipativity and positive semigroups, J. Operator Theory 8, 167-180 (1982).

2. Coddington, E.A. and N. Levinson - Theory of Ordinary Differential Equations, Mc Graw Hill, N.Y., 1955.

3. Feller, W. - The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. 55, 468-519 (1952).

4. Martini, R. and W.L. Boer - On the construction of semi-groups of operators, Indag. Math., 36, 392-405 (1974).

5. Martini, R. - Differential operators degenerating at the boundary as infinitesimal generators of semi-groups. Thesis, TH Delft (1975).

6. Martini, R. - A relation between semi-groups and sequences of approximation operators, Indag. Math., 35, 456-465 (1973).

7. Pazy, A. - Semi-groups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sc. Vo. 44, Springer-Verlag (1983).

8. Protter, M. and H. Weinberger - Maximum Principles in Partial Differential Equations, Prentice-Hall, Englewood Cliffs, New-Jersey, (1967).

9. Taira, K. - Semi-groups and Boundary Value Problems JI, Proc. Japan Acad. 58, Ser. A, 277-280 (1982).

[10] Ventcel', A.D. - On boundary conditions for multidimensional diffusion processes, Th. of Prob. Appl., 4, 164-177 (1959).

87

88

2.4. Dual semigroups in 3R x NBV(J) x 3R

It is known that the dual space C(J) of C(J) is isometrically isomorphic to the space NBV(J) the space of all normalized real functions w of bounded variation on J. The normalization chosen here is — just as in section 1.3 — given by

(2.4.1) (i) W( X Q ) = 0

(ii) w(x) = (w(x+) + w(x-)}/2

for each x e J, but other choices for (i) and (ii) are possible, (cf. [Tay], Ch. 9). NBV(J) is a closed subspace of BV(J), the space of all real functions of bounded variation on J. The pairing C(J) x C(J) -+ 3R is given by

',f> = f. J J

<w,f> = _ fdw.

NBV(J) is a Banach space under the total-variation norm, i.e.

||w|| - := Var(w) = | d|w|. Var(J) (J}

Considered as representations for elements from the dual space C(J) , w and w2 , satisfying (2.4.1)(ii) are equal in (N)BV(J) if and only if for all f e C(J)

<w ,f> = <w»,f>

or

h _ fd(W ] - w2) = 0,

thus if and only if w - w- = c, where c is a real constant. If w e NBV(J) and w. e NBV(J) then c = 0, but in general c e R

89

Because of the boundary conditions on D(A) it is handier to isolate possible jumps in the boundary points r . and r_. So we identify the space NBV(J) with the space X = TR x NBV(J) xJR with the same normalization (2.4.1). The corresponding pairing X x X -» 1 is

(2.4.2) <(w. ,w,w-),u>* := (w(r. +) - w, ).u(r +) 1 ' ' " i ' - ^ i + (w2 - w(r2-)).u(r2-) + J ( r i , r

udw, b r 2)

and for the norm of X we have

||(w1,w,w2)||!(i = |w(rj+) - Wj| + |w2 - w(r2-) | + 1 d|w|.

* * * * Let A : D(A ) c X —► X denote the adjoint of A, defined in section 2.3, then we have the following representation theorem for A :

Theorem 2.4.1 Let X = C(J), let a and 8 be continuous real-valued functions defined on J satisfying

(2.4.3) WGL^Xpr.) or Q if: L1 (xQ,r.) (i - 1,2)

where W and Q are defined by (1.1.3) and (1.1.4) and let A: D(A) C X -► X be defined as in Proposition 1 of section 2.3. Then the adjoint operator A of A with domain D(A ) is given by:

(2.4.4) .

D(A*) = {(w]tw,w2) e X*\we NAC(J), aw1 e ACl (J),

Caw';1 - Pw' - ((aw*)' - pw*)(xQ) G NBV(J), Urn (oWw *)(x) = 0 if r. is a regular boundary

x—>rj l

point (i = 1,2)}.

A*(wrw,w ) = (0, (aw') ' - Bw\ 0) for all (w w.wj e D( A*).

90

Note that from (2.4.3) it follows that r. is either a regular, or an exit or a

natural boundary point.

Proof We start with the definition of an operator B with domain D(B) c X * = l x NBV(J) x TR and range in X as follows:

(2.4.5)

' D(B) = {(Wj,w,w2) e X* | w G NAC(J), aw • E A C ^ J ) , (aw • ) ' - /3w • - ((aw • ) ' - /?w ' X x J G NBV(J), and lim (aWw ')(x) = 0 if r. is a regular boundary

x - q » point, ( i= 1,2)}.

B(w r w,w 2 ) = (0, (aw 1 ) 1 - j8w', 0) for all (Wj.w.Wj) G D(B).

Thus D(B) is equal to the right-hand side of the first equality of (2.4.4). We will show that the conditions of Proposition 2.2.1 are satisfied. Then it follows that B = A and Theorem 2.4.1 will be proved.

We divide the proof in 5 steps.

Step 1 A is a linear m-dissipative operator with domain dense in X. The linearity of A is clear. In [CT, 1], Proposition 1 it is proved that D(A) is dense in C(J) and that A is dissipative. Further, since r. and r- are not entrance boundary points it follows from [CT,1], Theorem 2 that R(I - A) = C(J). So I - A is surjective, and then it follows from [CH], Proposition 3.7 that A is m-dissipative.

Step 2

B is a closed linear operator. The linearity of B is clear. Let ((w w ,w -))

be a sequence in D(B), converging in X to (w w,w.) and assume that the

sequence ( B ( wn l ' w

n > wn 2 ) ) converges in X* to (y r y ,y 2 ) .

91

The assertion is proved if we have shown that (w w,w_) G D(B) and •B(w rw,w2) = ( y r y , y 2 ) . Since lim B(w w w ) = lim (0,(aw •) • - fiw • 0) = (y. ,y,y.) in X*

n—>oo u l u ll/- n—xx> it follows that y. = y- = 0 and

(2.4.6) lim Var ((aw ' ) • - £w • - y) = 0. nôo (j) n n

Note that (aw ' ) ' - /?w ' = W ' ^ a W w ' ) 1 and y(xA) = 0. v n' ^ n v n' 0 Let us denote c n := ((aw * ) ' - /?w ' )(x_). nO n n / v 0 Let x G J. Regard ing the pointwise interpretation of an equation in (N)BV(J), we then have

| ( ( o w ^ ) ' - /Jw^Xx) - c n Q - y(x)|

= | ( W " 1 ( a W w ^ ) ' K x ) - c n 0 - y ( x ) |

= |(W_ 1(aWw^)' - y)(x) - (W - 1(aWwJ) ' - y)(x0)|

(2.4.7) < Var ( W ' ^ a W w ' ) ' - y), (J) "

and moreover

|(aWw^)'(x)-W(x).(y(x) + cn 0) |

= W(x)|(W"1(aWw^),)(x) - cn() - y(x)|

(2.4.8) < W(x).Var((W" l (aWw •) • - y). (J) "

From (2.4.7) we see that the sequence W (aWw ' ) ' - c „ converges pointwise

to y on J. Thus for each x G J we have

lim W" \ a W w •) ' (x) - c . = y(x) n—>oo n n ^

and

(2.4.9) lim ( ( a W w ' ) ' ( x ) - W(x).c n ) = (Wy)(x). n—+oo n nU

92

Further, since lim (w w ,w _) = (w w,w_) in X we have w. = lim w . (i = 1,2). Since w (xn) = w(xn) = 0 and 1 n—oo ni n ° °

lim Var (w - w) = 0 n -oo (j) n

the sequence of functions (w ) converges pointwise to w on J.

Let a,b e TR, such that r < a < x~ c and W(x) > c for all x e [a,b]. Moreover, if e > 0, then we obtain from (2.4.6) and (2.4.7)

sup |(aWw 'J '(x) - W(x).(y(x) + cnQ)| xe[a,b]

 'nO

holds uniformly on [a,b]. Then by integration we obtain

or

or

lim (aWw')'(s) - c AW(s))ds = (Wy)(s)ds n-»oo J x_ n n u J xn

A X A X

lim (aWw'Xx) - ( aWw'XxJ - c _ W(s)ds = (Wy)(s)ds, n-»oo " n U nu J x J x

n ï o o { w n ( x ) - ( a W ) " 1 ( x ) - ( Q W w n ) ( x O ) " C n O ( a W ) 1 ( x ) | W ( s > d s } 0

= (aW)"'(x) f (Wy)(s)ds J x Q

93

uniformly in x on [a,b]. Again by integration we get, since w (x_) = 0 for all n G UN,

lim (w n (x ) - (aWw^)(x0) f (aW)-1(s)ds

X t X t f ( aW) _ 1 ( t ) f W(s)dsdt)-= f ( aW) _ 1 ( t ) f (Wy)(s)dsdt. "Cn0

Thus for all x G [a,b]

w(x) - l im { ( a W w ' X x J f (aW)_1(s)ds n—«x> *- n u J x~

X t X t f (aW)_1(t) f W(s)dsdt)-= f (aW)_1(t) f (Wy)(s)dsdt. Cn0

w0 0 0

So w G C [a,b], and by differentiation and substitution of x»

(2.4.10) ( a W w ' X x J - lim (aWw • )(x ) " n—«x> n u

and

(2.4.11) (aWw'Xx) - (aWw' ) (x n )= l im c n . f W(s)ds u n-+oo n U J x»

+ f (Wy)(s)ds. J x o

By differentiation and dividing by W we get

(W"'(aWw ') ')(x) = lim c n + y(x), x G [a,b]. n-+oo n u

Thus

(2.4.12) c 0 = (W"1(aWw') ')(x ( ))= lim cn()

and

94

'0 (2.4.13) (W '(aWw ') ')(x) - c n = y(x), x G [a,b].

Since a and b are arbitrary, and y G NBV(J), we have aWw • G AC. (J), w G CJ(J) n NBV(J) and W ' ^ a W w 1 ) ' - cQ G NBV(J). Hence (Wj,w,w2) G

D(B) and B(Wj,w,w2) = (0,y,0).

Finally we have to verify that lim (aWw')(x) = 0 if r. is a regular x ^ r j 1

boundary point. Assume r . is a regular boundary point. Since a and b are arbitrarily chosen we have from (2.4.11) and (2.4.12) for each x G J

(2.4.14) (aWw »)(x) = (aWw ')(x ) + f W(s).(y(s) + cQ)ds. J x 0

Since (w ,w ,w _) is a sequence in D(B) we have

(2.4.15) lim (aWw')(x) = 0. x-^r2

n

Since r» is a regular boundary point we have W G L (x„,r2) and by the boundedness of y we have W(y + c„) e L (*0,r-) and then by (2.4.15)

(aWw^)(x0)+ f 2 W(s).(y(s) + cn())ds = X0

= - f 2 ( (aWwp 1 - W(y + cnQ))(s)ds + lim (aWw^)(x) J X„ X—*T2

= - [ 2 ((aWw^)' - W(y + cn0))(s)ds. J x 0

Then from (2.4.8) we obtain

|(aWw^Xx0) + f 2 (Ws).(y(s) + cn0)ds| x 0

J x

2 | ( ( aWw^) ' - W(y + cn0))(s)|ds

95

< {var (W _ 1 (aWw') ' - y)} . f L(J) n J J x

J :

2 W(s)ds

< e . I 2 W(s)ds J x o

for sufficiently large values of n. Thus

(2.4.16) lim (aWw')(x )+ f 2 W(s).(y(s) + c ft)ds = 0. n—»oo n u J x . n "

Then finally from (2.4.14), (2.4.12), (2.4.10) and (2.4.16) we have

lim (QWW ')(X) = (QWW'XXQ) + f 2 W(s).(y(s) + cQ)ds

*" J: x - r 2

= lim (aWwn)(x )+ I 2 W(s)(y(s) + c n 0)ds X0

= 0.

If r. is a regular boundary point, the proof is similar.

Step 3 Next we define the linear subspace NBV^J) in NBV(J) by

NBV*(J) := {w e NBV(J) | there exist numbers a,b e IR, such that r < a < x- 0. Then we have to prove that there exists an element (w. ,w,w.) G IT, such that

96 ||(Wj,w,w2) - (Wj,w,w2|| t < e.

Take w. = w. (i = 1,2) then we have only to prove the existence of a function

w e NBV*(J) such that

(2.4.17) Var (w - w) < e. (J)

Let (a ) be a sequence converging to r . and (b ) be a sequence converging to

r . . We define the functions w , n e M in NBV+(J) by

wn(x) = w(x) , X £ (an ,bn),

wn(x) = w(an) , X e ( r r a n ] ,

wn(x) = w(bn) , X e [ b n , r 2 ) .

Then we have

Var (w - w ) = Var (w - w ) + Var (w - w ) (J) n ((ri,an]) n ([bn,r2)) n

= Var (w - w(a )) + Var (w - w(b )) ((ri,an]) " ([bn,r2)) n

(2.4.18) = Var (w) + Var (w). ((ri,an]) ([bn,r2))

Since w is of bounded variation

lim Var (w) = lim (w) = 0. n _ t 0 ° (ri.an) ([bn,r2))

Thus, if we take w = w with n sufficiently large then it follows from

(2.4.18) that (2.4.17) holds.

97

Step 4 N c R(I - B). Let (z z,z.) G TR x NBV*(J) x R S o there are real numbers a, b, c and d such that r . < a < x_ b. Note that z(x.) = 0. Let us consider the equation

(I - B)(w rw,w 2 ) = (Zj,z,z2)

for (w w,w2) G D(B). By writing out we obtain

1 ( W j , w - W (aWw1) ' , w2) = (Zj,z,z2)

so

and w. — z , , w_ — z .

(2.4.19) w(x) - (W~ ,(aWw') ,)(x) + cQ = z(x), cQ = (aWw')'(xQ).

Since r . and r_ are not entrance boundary points there exist two functions w and w . satisfying the following conditions (see Lemmata 1.3.4-1.3.7)

1 2 w. e C (J), aWw.' G C (J), w. is a solution in BV. (J) of the homogeneous equation

,-1 w - W (aWw') ' = 0

w is positive increasing and w. is negative increasing and lim (aWw.')(x) = 0, - 1 ~z x—»rj _ 1

i = 1,2. Note that

W ](a) > 0, wâ) > 0, Wj(b) > 0, w^b) > 0,

w2(a) < 0, w2'(a) > 0, w2(b) < 0, w^b) > 0.

We return to the equation (2.4.19).

98

Since z(x) = c for x 6 (r ,a], the restriction of the solution of (2.4.19) to (r ,a] has to be of the form

(2.4.20) w = (c - cQ)e0 + ivr{ + /iWj,

where e„ is the constant function with e„(x) = 1, and 7,5 are constants. Since (w.,w,w_) has to belong to D(B) we have the following conditions on

W | ' ,a] %l,s

(2.4.21) (i) w ' G L ^ r j . X Q )

(ii) lim (aWw')(x) = 0 if r . is a regular boundary point. x ^ H !

In the Lemmata 1.3.3, 1.3.4 and 1.3.6 we gave some properties which will be used here. If r is an exit or a natural boundary point, then w ' e L ( r . ,x„) and w ' £ L (r.,x-)thus n = 0 in (2.4.20). If r . is a regular boundary point we have from (2.4.20) (2.4.21 (ii)) and Lemma 1.3.3

lim (aWw ')(x) = n . lim aWw'(x) = 0, x—»ri x—»r] ~z

and since lim aWw'(x) > 0 we have n = 0. x—>r\ ~l

Thus if r . is either a regular, or an exit or a natural boundary point, then H = 0 and

l â r^ - 'o^v^ iWr ieM-The restriction of the solution wl . . can be investigated in a similar way. This motivates the next definitions. For p,q e I w e define the functions

w p := (c - c0)e0 + pwj on ( r r a ] ,

Wq : = ( d " C0 )e0 + qv^2 o n [ b ' r 2 ) -

99

Then w e C (r. ,a] and w satisfies (2.4.19) on (r. ,a]. Similarly for w .

Let us denote by w the particular solution of (2.4.19) as given in Proposition 1.3.7.

For r,s e 3R we define the function w : [a,b] —► 3R by r,s

wr S W = ™ZW + r ^ j (x ) + sw2(x), x e [a,b].

It is the aim to determine p, q, r and s such that the function w, defined by w on (r. ,a), by w on [a,b] and by w on (b,r.) is continuously different ia te . Then it follows from the differential equation (2.4.19) that aWw' G ACj (J), a w ' e AC. (J), (z w,z.) e D(B) and w satisfies (2.4.19). w is continuously differentiable on J if and only if

lim w (x) = w (a) pv ' r,sv ' x—»a-

hm w *(x) = lim w ' (x) p ' r,sv ' x—»a- x—»a+

hm w (x) = w (b) . q r,sv ' x—»b+

lim w ' ( x ) = lim w ' (x).

By writing out these equations we obtain the following system:

c - c„ + pA = rA + sE + K pB = rB + sF + L qG = rC + sG + M qH = rD + sH + N

d - cQ + qG = rC + sG + M

where

100

A = Wj(a),

B = w«(a),

C = Wj(b),

D = w ^ b ) ,

E = w2(a),

F = wj(a),

G = w2(b),

H = w2Hb),

K = wz(a),

L = wz'(a),

M = wz(b),

N = wz'(b),

or ' A E

B F

C G

D H

-A

-B

0

0

0 '

0

-G

-H

r

s

P

q

' c - cQ - K '

- L

d - cQ - M

- N

The determinant of the system is equal to (AF - BE)(GD - CH) < 0. So p, q, r and s are uniquely determined, and the proof of step 4 is completed.

Step 5 Let M := C„,(J) be the subset of functions of C(J) which are constant outside of a bounded closed subinterval of J. It is clear that CM) is dense in C(J). In this step we will prove: For all u e D(A), such that u - Au G M and all (w .,w,wj 6 D(B) such that (z .,z-z(x„),z2) e iV where (z .,z,z.) := (w,,w,w-) - B(w .,w,w?), we have

(2.4.22) <(w .,w,w2),Au>% = <B(w .,w,w ■)),u>%-

Assume u e D(A) and (w w,w„) e D(B) satisfy the above-mentioned conditions.

Let (Zj,z,z2):= (I - B)(w1,w)w2).Then(Zj,z-z(xQ),z2)G N =HRx NBV^J)xIR, z. = w z« = w 2 , z(x) = w(x) - (W~ (aWw')')(x). Thus there are constants a,b e 3R, such that

101

r < a < x- < b < r •

z(x) = z(a) and w(x) = z(a) - c„ + p.w for all x e (r . ,a] ,

cQ = (aWw V(x 0 ) , P j G3R;

z(x) = z(b) and w(x) = z(b) - c» + P?w_ for all x e [b,r»), pÊ IR.

Then we have from Lemmata 1.3.4, 1.3.5 and 1.3.7

lim (QWW ')(x) - p. lim (aWw.')(x) = 0. x—»rj i x—>rj 1

From the proof of Lemma 3 of section 2.2 it follows that (W u ')(x) = 7(W ul)(x) near the boundary point r«. Since W uJ is a negative in-

-1 -1 creasing function lim (W u ' )(x) exists, and similarly lim (W u ')(x)

x—>r2 x-»ri exists as a finite number. Then

<(I - B)(w1,w,w2),u>+

= <(Zj,z,z2),u>+

= (z(rj+) - z1)u(r1+) + (z2 - z(r2-)).u(r2-) + udz

= (z( r ]+) - ZjMrjH-) + (z2 - z(r2-)).u(r2-) + J udw

- f ud(W"1(aWw')')

= (z(rj+) - ZjMrj+J + (z2 - z(r2-)).u(r2-) + I udw

- ( ( W ' V w w V . u X ^ - ) + ((W"1(aWw ,) ,.u)(r1+)

+ (W u ')d(aWw ') (by integration by parts)

= (w(rj+) - WjXuCrjH-) + (w2 - w(r2~).u(r2-)

+ f udw + lim ( (W'^ 'MaWw'JXx) J J x—^2~

102

lim ((Wû'MaWw'JXx) - f aW(W ' u ' ) ' d w —>rj+ J

(since z. = w., z = w - W (aWw ' ) ' and integration by parts)

(w(rj+) - w1).u(r]+) + (w2 - w(r2~)).u(r2-) + (u - Au)dw

(since lim (W u')(x) is finite and

lim (aWw')(x) = 0, (i = 1,2) x-+r;

= <(w1,w,w2),(I -A)u>„,.

Then (2.4.22) follows immediately. At this stage all conditions of Theorem 2.2.1 are satisfied and it follows that B = A . This completes the proof of Theorem 2.4.1.

2.5. Restricted dual semigroups in 3R x L (J) x TR

From section 2.1 we know that A is the weak infinitesimal generator of the dual semigroup T of T.

For the closed subspace X„ of X on which T is strongly continuous we have by Proposition 2.2.1

X* = D(A*).

More explicitly we have

Lemma 2.5.1 X*Q = OR x NAC(J) x 1R.

103

Proof Since NAC(J) is a closed subspace of NBV(J), IR x NAC(J) x JR is a closed subspace of IR x NBV(J) x TR. Let (w w,w ) G 3R x NAC(J) x IR, and let (. > 0, then we have to prove that there exists an element (w w,w ) in D(A ) such that

(2.5.1) l l (w rw,w2) - (w r w,w 2 ) | | + < e.

Since C (J) is dense in L (J), there exists a z. G C (J) such that

J | w ' - Z j | d x < e/2.

Here C (J) denotes the set of continuous functions with compact support inside J.

Set S := supp z. and m := min (aW)(x), then m > 0. Now aWz G C (J), so 2 XGS I C

there exists a z . e C (J) with supp z_ <r S such that

|aWz - z Jdx < m.e/2.

Now we define the function w on J by

w ( x ) = [ (aW)_1(t).z2(t)dt. 0

Then w is constant outside the support of z-, w G NAC(J) n C (J), oWw' G C2(J), and ( w r ^ a W w » ) ' G CJ(J) c BV(J). If w = w w = w we see (w.,w,w_) G D(A ). Moreover,

||(wj,w,w2) - (Wj,w,w2)||#

= Var(w - w) (J)

104

= |w ' - w '|dx

< |w ' - z.|dx + |z. - w ' |dx

< |w • - z.|dx + m |aWz. - zJdx

< e. G

Now we are able to give the infinitesimal generator A„ of the C„-contraction semigroup T~ on X . , being the largest restriction of T which is strongly continuous.

Lemma 2.5.2 X* = JR x NAC(J) x ]R D(A*0) = {(Wj,w,w2) e X*0\we NAC(J), a>v' G AC, (J),

(aw')' - Pw' G AC(J), Urn aWw %(x) = 0 if r. is a regular boundary point (i = 1,2)}.

x—>rj '

A*0(wrw,w2) = (0, (aw*)' - pw', 0) for all (wyw,w2) G D(A*Q). D

Set X® := H x L (J) x ]R, equipped with the norm ||.|| defined by

ll(v1,v,v2)| |0 = |v1 | + |v2 | + J |v|dx

for all (v v,v„). With this norm X® is a Banach space. (The sign O is called "sun".)

We define the mapping

(2.5.2) f Ï: X*-+X®

I(Wj,w,w2) = (w(rj+) - Wj, w ' , w2 - w(r2-)).

105

Since

l |I(w1 ,w,w2)| |0 = |w(r1+) - Wj| + |w2 - w(r2-)| + J iw i |d x

= ll(w rw,w2 | |# ,

and since I(w w,w«) = 0 implies (w w,w.) = (0,0,0), I is an isometric isomorphism. So as Banach spaces, X„ and X® may be identified. Moreover,

(2.5.3) I" (Vj,v,v2) = (y(rj+) - Vj, y, y(r2~) + v2) where

y(x)= f v(t)dt. J x o

In X® we define the operator A®: D(A°) c X® — X® by

(2.5.4) D(A®) = f(D(A*))

A® = I o A* o I 1


' D(A®) = {(V],v,v2) G X® | av G AClQc(J), (av) ' - /?v G AC(J),

(2.5.5) .

lim (aWv)(x) = 0 if r. is a regular x-vr; boundary point),

A®(Vl ,v,v2) = (((av)'- /3v)(r1+),((<*v)»- M »,-((av) •- /?v)(r2-))

for all (v p v,v 2 ) G D(A®).

The pairing <.,.>~: X® x X -> 1R, induced by I, is given by

7-1. <(Vj,v,v2),f>0 = +

= Vj^rj+J + v2f(r2-) + j vfdx.

106

By the definition of A® it follows that A® is the infinitesimal generator of a C„-contraction semigroup T® := (T®(t); t > 0 } i n E x L (J) x JR.

2.6. "Bidual" semigroups in 3R x L°°(J) x 3R

The dual space X®* := ( 1 x L (J) x TR)* of 1R x L (J) x 1 is isometrically

isomorphic is given by isomorphic to the space ]R x L°°(J) x 1R. The pairing <.,.>fl*: X® x X® —► IR

<(z ] ,z,z2),(v1 ,v,v2)>0+ = ZJVJ + z2v2 + I zvdx.

]R x L°°(J) x TR is a Banach space under the norm

||(z1,z,z2)| |0+ = maxflZjUz^, ess sup |z(x)|}. xeJ

For the adjoint operator A®*: D(A®*) c X®* -» X®* of A® we will prove the following representation theorem:

Theorem 2.6.1 Let A®: D(AQ) c X® — X® be defined by (2.5.5). Then the adjoint A®* of A® is given by

(2.6.1)

D(A®*) = {(zvz,z2) e X®* | z e C^J), z • 6 AChc(J), n r / i t r " ■* I t • *- / v \ f r \ aW(W z')' e L°°(J), and Urn z(x) exists and is equal to z.

if r. is a regular or an exit boundary point, i = 1,2},

AQ*(zyz,z2) = (0,aW(W 1 z ') \0), for all (zyz,z2) e D(A®*).

107

Note that from the definition of A, and thus from the definition of A® it follows that r. (i = 1,2) is a regular, an exit or a natural boundary point.

Proof The main line of the proof of this theorem is the same as in the proof of Theorem 2.4.1. We start with an operator C: D(C) c X®* -► X®* and we will prove in a number of steps that there is satisfied the conditions of Proposition 2.2.2. Then it follows that C = A®* and Theorem 2.6.1 is proved. So we start with the definition of the operator C with domain D(C) C X°* = 3R x L°°(J) x JR and range in X® as follows:

(2.6.2)

D(C) := {(z r Z ,z2) e X®* | z G CJ(J), z ' e A C ^ J ) ,

aW(W 1 z ' ) ' GL°°(J), and lim z(x) exists and is equal to z. if r. x—>q 1 l

is a regular or an exit boundary point,

i = 1,2}.

C(z r z ,z 2 ) = (0, aW(W z 1 ) ' , 0), for all (Zj,z,z2) e D(C).

Thus D(C) is equal to the right-hand side of the first equality of (2.6.1). Note that if z • e AC. (J), then

aW(W" I z ' ) 1 = az" + fiz\

In 3 steps we will verify the conditions of Proposition 2.2.2.

Step 1 A® is a linear m-dissipative operator with domain dense in X®. The linearity is clear. Since A® is the infinitesimal generator of the C„-contraction semigroup T® it follows from the first part of the Generation Theorem of Hille-Yosida (section 2.1) that A® is m-dissipative with domain dense in X®.

108

Step 2 C is a linear operator and R(I - C) = X®*. Let (kj,k,k2) e X ° * = 3R x L°°(J) x 3R. We consider the equation

(2.6.3) ( I - C)(z rz,z2) = ( k r k , k 2 )

for (z. ,z,z.) G D(C). By writing out we obtain

( z 1 , z - a W ( W " 1 z ' ) ' , z 2 ) = (k1 ,k,k2)

Z l = k l ' Z2 = k 2 '

z - aW(W _ 1 z ' ) ' = k.

Let the function z, be defined by (1.1.48), then we know from Proposition 1 1.1.24 (and its proof) that z, e C (J), z ' e AC. (J) and that z, is a

particular solution of (2.6.4). Thus the general solution of (2.6.4) is given by

Z = Z k + C l ^ l + C2 y 2 ' C l ' C 2 e n R '

where u and u- are the special solutions of (1.1.6) as given in Proposition 1.1.1.

If r (resp. r„) is a natural boundary point, then lim u«(x) = oo (resp. 1 l x—>x\ l

lim u.(x) = oo), thus C- = 0 (resp. c, = 0). X—*T2 l l l

If r is a regular or an exit boundary point, then

lim z(x) = lim z.(x) + c . . lim u.(x) + c«. lim u„(x) x-+rj x—>rj K l x—*rj _ 1 l x—»rj ~z

= c- • lim u.(x) 1 x—»r] " z

so

and

(2.6.4)

and we have

109

c 2 = z r(u 2(r 1+))" =k1(u2(r1+)) .

If r» is a regular or an exit boundary point, then we have

C l = Z 2 - ( u l ( r 2 - ) ) " 1 = k 2 ( ^ l ( r 2 - ) ) " 1 -

It follows that if r. and r» are not entrance boundary points the equation (2.6.3) has a unique solution in D(C), which completes the proof of step 2.

Step 3 E x C (J) x JR is a dense linear subspace of ÜR x L (J) x JR and for all (y v,v J such that (I - AQ(v.,v,v ) E IR x C (J) x 3R, and all (z ,z,z J e D(C)

<(zrz,z2). A®(vrv,v2)>Q% = <C(zrz,z2, (vrv,v2)>Q,.

That IR x C (J) x IR is dense in 3R x L (J) x JR follows from the density of I c

C (J) in L (J). We denote by V the subset of tripels (v ,v,v.) in D(A0) such C J Z

that (I - A®)(v v,v.) G IR x C (J) x ]R Assume (z.,z,z2) G D(C) and ( v r v , v 2 ) G V.

Then we have by integration by parts

<C(z1,z,z2),(v1,v,v2)>0„ t - <(z1,z,z2),A°(v1,v,v2)> =

= <(0,aW(W~lz')',0),{vrv,v2)>Qt

-<(z ] ,z,z2),((W"1(aWv) ,)(r1+),(W"1(aWv)') ' ,-(W"1(aWv)')(r2-))>0< t

110

= f aWCW^z'J ' .vdx- f z.(W_1(aWv)')'dx

- z1.(W"1(oWv)')(r1+) + z ^ W ' ^ a W v ) ' ) ^ - )

= [(aWvXxMW^z'Xx)] J2 - f W_1(aWv)'dz

- [ z C x ^ W ' ^ a W v J ' X x ) ] ^ + f W'âWvJ 'dz

- z].(W"1(aWv)'Xr1+) + z ^ W ' ^ a W v ) ' X ^ - )

= lim (aWvXx).(W_1z'Xx) - lim (aWvXx).(W~!z')(x) x->r2 x—rj

- lim (z(x) - z.).(W"1(aWv)'(x)) X - T 2

Z

(2.6.5) + lim (z(x) - Z . ) . (W" 1 (QWV) ' (X) , x - r i

l

provided that these limits exist. In the next lemmata we will prove that these limits exist and are equal to zero.

Lemma 2.6.2 (a) If r. is a natural boundary point, and (v .,v,v?) E V, then

lim (W~1(OLW\) %)(X) = 0, i = 1,2. x—*rj

(b) If r. is a regular or an exit boundary point, and (v .,v,vj G D(A®), then

lim (W~1(aWv)%)(x) exists. x—>r

Proof

It is sufficient to prove the lemma for the boundary point r„. (a) Assume r- is a natural boundary point, and let (v. ,v,v ) e V. Set g := v - (W~ (aWv) ' ) ' , then g e C (J). So there exists a b G J, such that

for all x G (b,r.) , g(x) = 0.

I l l

On (b,r_) v satisfies the homogeneous equation v - (W (aWv)') ' = 0. Let u. -1 and u . be as in Proposition 1.1.1. From Lemma 1.2.6 we have v := (aW) u.

£ L (x~,r_), and it follows from the fact that v e L (x0 , r-) and Lemma 1.2.2 that for x G (b,r-)

v(x) = cQv2(x) := c0((aW)"1u2)(x), cQ G TR.

Then (aWv)(x) = c,.u2(x), x G (b,r«), so

lim (W_1(aWv)')(x) = c n lim (W -1uI)(x) x—^2 u x—>T2

= 0

by Lemma 1.1.17. (b) Next we assume that r^ is a regular or an exit boundary point, and let (v v,v_) G D(A®). Then the assertion follows from the fact that W (aWv)' belongs to AC(J). D

Corollary of Lemma 2.6.2 (a) then (a) If r. 'is a natural boundary point, (v .,v,v-) G V and (z .,z,z .) G D(C),

(2.6.6) lim (W âWv) %)(x).(z(x) - z.) = 0, (i = 1,2). x—>rj l

(b) If r. is a regular or an exit boundary point, (v .,v,v?) G D(A®) and (zrz,z2) G D(C), then (2.6.6) holds.

Lemma 2.6.3 (a) If r. is an exit or a natural boundary point, (v.,v,v.) G V, and (z z,z ) G D(C), then

(2.6.7) lim (aWv)(x).(W~1z,)(x) = 0, (i = 1,2). x-*rj

112

(b) If r. is a regular boundary point, (v ,.v,v?y) G D(A®) and (z ,,z,z ) e

D(C), then (2.6.7) holds.

Proof (a) It suffices to prove the lemma for the boundary point r-. Assume r- is an exit or a natural boundary point, let (v.,v,v_) G V and let (z.,z,z„) e

D(C). From the Lemmata 1.2.2, 1.2.4 and 1.2.6 we know that (aW)~ u g 1 - 1 1 L (xft,r„) and (aW) u- e L (x,.,r»). Then, as in the proof of Lemma 2.6.2(a)

we have in a sufficiently small left neighbourhood (b,r-) c (x0 ,r„) of r .

V(X) = C 0 ( ( Q W ) " 1 U 2 ) ( X ) , c Q e K .

Thus

lim (a\Vv)(x) = lim c_u»(x) = 0. x-+r2 x—*T2

Further, since aW(W z ' ) ' G L°°(J), there exists a number K > 0, such that on J

KW^z'Xx)! < K(aW) -1(x) (a.e.)

Then we have for x e (x n , r . )

|(W"1z')(x)| = | z , (x 0 )+ f (QW)"1(t)(aW(W"1z')'Xt)dt|

< |z '(x0) | + J" (aW)"1(t).|(QW(W"1z'),)(t)|dt 0

.x |z'(x0)| + K. [ (aW)_1(t)dt.

x0

So for x G (b,r_) we have

113

(2.6.8) |(aWv)(x).(W" l z 'Xx)! <

|.|z ,(x0)|.u2(x) + |c0|.K.u2(x) | (aW)" 2(t)dt. <|c0

If x tends to r_, the first term in the right-hand side of (2.6.8) tends to zero, and it remains to prove that the second term in the right-hand side of (2.6.8) tends to zero. Since u is decreasing we have for x„ < y < x

u2(x) ƒ (aW)_1(t)dt < u2(x).My + | ((aW)"1u2Xt)dt

fy -1 (with M = ((aW) )(t)dt)

y J x 0

f'2 -1 (26.9) <u2(x).My + J ((aW) '^XOdt.

Let e > 0. Because of Lemmata 1.2.4 and 1.2.6 we have v . = (aW) u« e L (x 0 , r . ) , so there exists a number y„ such that for all y with y_ < y < r

J v

r 2 -1 (2.6.10) 0 < | ((aW) u-)(t)dt < e/2.

y ~z

Let y. G (y 0 , r . ) . By Lemmata 1.1.15 and 1.1.17 we have lim u.(x) = 0, so x~*r2

there exists a number x. with y. < x. < r ? such that for all x G (x. ,r_)

(2.6.11) 0 < u.(x).M <e/2. - 2 yj

Then from (2.6.9)-(2.6.11) it follows that for all x G ( y . , ^ ) we have

x r . u2(x) j (aW)_1(t)dt < u2(x).My + J ((aW)_1u2Xt)dt

< e/2 + e/2 = e.

This completes the proof of part (a).

114

(b) If r_ is regular, and (z z,z-) e D(C), then

(W"1z')(x) = z , (x 0 )+ [ (aW)~ W a W f W ^ z ' V X O d t , J x o

which is bounded since (aW)"1 e L ^ x ^ r . ) and aW(W _ 1 z ' ) ' e L°°(J). Moreover, for all (v.,v,v_) e D(A®), by assumption lim (aWv)(x) = 0. Then it follows that (2.6.7) holds. 1

D

We return to (2.6.5). It follows from the lemmata just proved that the occur

ring limits are equal to zero. Hence

<C(z1,z,z2),(v],v,v2)>0<1 - <(z1,z,z2),A®(v],v ;y2)> = 0,

proving the assertion in Step 3. In the Steps 1-3 we proved that the conditions of Proposition 2.2.2 are satisfied. Then from this proposition it follows that C = A® . This completes the proof of Theorem 2.6.1. □

As in section 2.5 we know that A® is the weak infinitesimal generator of the dual semigroup T® of T®. For the closed subspace X®® of X® , on which T® is strongly continuous we have

(2.6.12) X®® = D(A®).

Explicitly we have

Lemma 2.6.4 If r j and r. are regular or exit boundary points, then

(2.6.13) *®Q = {(z z,zJelRxC(J)xlR\ lim z(x) = z., i = 1,2\. K l * I x-*rj ' J

115

Proof

Assume r. and r„ are regular or exit boundary points. Let Z be the set in the oo right-hand side of (2.6.13). Then Z is a closed subspace of E x L (J) x 1R.

Let (z z,z_) G Z. i z 2

Then there exists a function z G C+(J) such that hm z. = z., ï = 1,2. Set ~ V if — 1 1

2 * ^ r j y :- z - z, then y G C„(J). Let e > 0. Since C (J) is dense in CJJ ) in the sense

2 of the supremum norm, there exists a n y e C (J) such that ||y - y|| < e. Now it is easily verified that (z. , y + z, z») G D(A® ) and

||(z1,z,z2) - ( z r y + z, z2) | |0 + < e.

Hence D(A®*) is dense in Z and thus Z = X®®. D

Let r ,r» be regular or exit boundary points. We define the mapping

(2.6.14)

r I : X 0 0 - + X = C(J)

I ( z r z , z 2 ) = z.

Then

and

(Zj ,z,z2 | |0 O = IKZj ,z,z2)||OJ|t

= max { IZjUz^JIzH^}

= z

since z = 0 G C(J) implies (z. ,z,z_) = (0,0,0) I is an isometric isomorphism. So the spaces X®® and X may be identified. If this is the case, the space X is 0-reflexive with respect to the operator A. Note that then also A®® = A.

116

2.7. On CL-semigroups in a space of bounded continuous functions in the case of entrance boundary points

CA. Timmermans Delft University of technology

Reports of the faculty of Mathematics and Informatics, no. 87-24 (1987) Submitted for publication

1. INTRODUCTION

In [1], Clément and the author investigated contraction semigroups (T(t)), t > 0, of class C„ on the Banach space C(I), where I is the two points compactification of an open interval ( r . , r - ) , -oo < r < r . < oo, (equipped with the supremum norm) generated by an operator A„ of the form

(1.1) AQu = QD 2 U + £Du.

Here a and /? are continuous real valued functions on I, with a(x) > 0 for

x e I. The domain of A-, denoted by D(An), is given by

(1.2) D(A ) := ( u S C(I) I u e C2(I), lim A u(x) = 0, lim Anu(x) = o } . u *• x—»rj u x—»r2 u ^

The boundary conditions in (1.1) are usually called Ventcel's boundary conditions, [7].

In order to state the conditions on a and /9 in the next theorems, we define the functions W and Q as follows

W(x):=exp{- f (/?.<*" J)(t)dt}, L J x o J

Y

Q(x):=(aW)_1(x) f W(t)dt, J x .

117 v

R(x):= W(x) f (aW)_1(t)dt. J x o

Here x~ denotes an arbitrary fixed point in I. In [1] we proved the following theorem, which gives necessary and sufficient conditions for An to be the generator of a C„-contraction semigroup on C(I).

Theorem 1 Let a and p be real-valued continuous functions on I, and let A„ be given by (1.1) and (1.2), then An is the generator of a C^-semigroup on C(I) if and only if a and fi satisfy

(Hj) W G l}(rrx0) or Q$ L^r^) or both,

(H2) WEL^Xpr^ or Q $ L1\xQ,r2) or both.

The reader familiar with the terminology of Feller [3], will recognize that (H.) (i = 1,2) is satisfied if and only if r. is not an entrance boundary point, thus if r. is a regular, an exit or a natural boundary point. Let

(1.3) D(A) := {u G C(I) | u € C2(I), QD 2U + £Du G C(ï)},

and let A: D(A) — C(I) be defined by

(1.4) Au = aD2u + 0Du.

In this paper we will give necessary and sufficient conditions for A to be the generator of a C„-semigroup on C(I).

118

2. MAIN THEOREMS

Let I denote a non-empty open interval of IR (not necessarily bounded), I the two points compactification of I, and 31 := I\I. C(I) is the Banach space of real-valued continuous functions on I equipped with the supremum norm, denoted by ||.||. The main theorems read:

Theorem 2

Let a and p be real-valued continuous functions on I with a > 0 on I. Let

(2.1) D(A) := (u G C(I) | u E C2(I), aD2u + 0Du G C(~I)},

and let 2 Au = a.D U + fiDu

for u G D(A). Then A: D(A) —* C(I), and the following conditions are satisfied: (i) D(A) is dense in C(I) (ii) A is closed (Hi) I - A is surjective in C(I).

Theorem 3

Let A be defined as in Theorem 2, then A is the generator of a C„-semigroup on C(I) if and only if the following condition

(K) f ° R(x)dx = oo and [ 2 R(x)dx = <x> J r y )x0

is satisfied.

We also have:

119

Theorem 4 Let a and fi be real-valued continuous functions on I with a > 0 on I. Let

D(A}) := \u e C(I) | u G C2(I), aD2u + fiDu e C(I),

Urn ((aD2u) + f3Du)(x) = o\, X—»ry J

and let

A .u = aD u + (3Du

for u e D(A .). Then A .: D(A .) —* Cfl) is the generator of a C^-semigroup on C(I) if and only if the following conditions are satisfied: (i) W&L^TJ.XQ) or Q$ L*(rrxQ) orboth,

(ii) \ 2 R(x)dx = <x>. JxQ

In the next section we will prove Theorems 2 and 3. The proof of Theorem 4 is similar.

3. PROOFS

The proof of Theorem 2 is partly based on the following lemmata.

Lemma 5 Let a and fi be as in Theorem 2, and let X > 0. Moreover, let A . (resp. A?) be the set of positive increasing (resp. decreasing) solutions on I of

(3.1) u - X(aD2u + pDu) - 0

satisfying ufx„) = 1. Then there exists a unique u. G A. (resp. a unique Uy e A2) which is minimal on (r xnJ (resp. [x„,r )), i.e. if u e A . (resp. u e A.) then u. , „ _ , / . M < oo (resp. M < <x>) if and only if R £ L (xQ,r ) (resp. R e L (r^XQ)).

Proof We only prove the lemma for M..

Necessity. Since u. satisfies the equation

(3.2) (W" 1 ! ! ' ) ' =(aW)_1Uj

we have for all x G (r. ,r»)

x t (3.3) Uj(x) = | W(t) { ƒ (aW)_1(s).u(s)dx + u j(x0)}dt + U^XQ) .

Since u. is positive increasing we then have for x > xfi

X X t

Uj(x0).f R(t)dt< f W(t) f (QW)"1(s).u(s)ds<u1(x)<M ]. J x 0 J x0 X0

So, if M. < oo, then R G L (x - , rA

Sufficiency. Since u. is increasing we have for x G (x„,r»)

0 < Uj'(x) < Uj'(x0).W(x) + Uj(x).W(x) J x (aW)_1(s)ds

= Ul'(x0).W(x) + Ul(x).R(x).

121

Here R e L (x,.,r«), and thus also W e L (x..,r.A It is standard that the differential equation

y • - Ry = C.W, y e C 1 ^ , ^ ) , C e l

y(xQ)= 1

has a unique bounded solution on [x~,r_). Since y1 > 0 and y(*0) = 1 w e have y is increasing. Moreover we have

0 < u.(x) < y(x) < lim y(x) < oo. - l x—r2

Thus, if R £ L (x n , r - ) , then M. < oo. D

Let V be the Wronskian u!u» - u l u . , then clearly V > 0. We define the Green function G: Ixl -► JR for (3.1) by

G(x,s) = U1(X).(QV)" (S).U2(S), X < S,

= u 2 W-( a V ) " (s).Uj(s), x > s,

and for each g e C(I) the function u : I —► 1R by o

(3.4) ug(x) = J 2 G(x,s)g(s)ds.

Then it follows by verification that u G D(A), and that u satisfies the g g

equation

(3.5) u - AAu = g.

(See also [3], Th. 13.1).

122

Lemma 7 The mapping / - A/4: D(A) —► R(I - A) is injective if and only if condition (K) holds. (I is the identity operator).

Proof If condition (K) is satisfied it follows from Lemma 6 that M. =M_ = oo. Then the only bounded solution of (3.1) is u = 0. Thus I - AA is injective. Conversely, if I - AA is injective, then u is the unique solution of (3.5).

o However, if M. < oo, i = 1 or 2, then for each constant C also u + Cu. should

i g i

be an element of D(A) by the fact that AAu. = u. e C(I). This is a contradiction, thus M. = oo, and, with Lemma 6, condition (K) holds. D

Proof of Theorem 2 (i) D(A) is dense in C(I). In [1, prop. 1] it is proved that DQ(A) (see (1.2))

is dense in C(I). Since Dn(A) <r D(A) for D(A) the same holds. (ii) A is closed. Let (u ) c D(A), u and v in C(I) be such that

lim ||u - u|| = 0 and lim ||Au - v\\ = 0. We have to show that n—»oo n n—»oo n

u G D(A) and Au = v. Let us denote Au by i/ . For every a,b e IR

such that [a,b] C I there is a constant c > 0 such that Q(X) > c for all 2

x e [a,b]. Therefore the restriction on [a,b] of aD u + /JDu is a regular Sturm-Liouville operator [2]. Since u (a) (resp. u (b)) converges to u(a) (resp. u(b)), it follows from the classical theory [2] that u • and u" are

2 Cauchy sequences in C[a,b], and therefore that u e C [a,b], and

2 au" + £u ' = v on [a,b]. Since a and b are arbitrary, u E C (I) and au" + /3u' = v on I. Since v e C(I), we see that lim (au" + /3u ')(x)

x->dl exists. Thus u € D(A) and Au = v.

(iii) I - A is surjective in C(I). This is a direct consequence of (3.4)-(3.5) with A= 1.

D

123

For the proof of Theorem 3 we need the following lemmata.

Lemma 8 If X > 0 and u G D(A), then u - XAu > 0 implies u > 0 if and only if condition (K) is satisfied.

Proof If R £ L (r ,x_) and R £ L (x-,r_), then I - AA is injective and since G > 0 it follows from (3.4) that g := u - AAu > 0 implies u = u > 0.

1 1 ^ On the other hand, if R e L ( r . ,x . )o r R e L (x„,r_)then M. < oo or M_ < oo. Say M. < oo. Let ü e D(A) be such that u > 0 and u - Au > 0. Let

-I -C > (AM.) lim u(x) and let u = u - Cu Then u - AAu > 0, however not

1 x—*r*> " 1 - -u > 0. D

Lemma 9 A is dissipative if and only if condition (K) holds.

Proof From Lemma 7 we know that condition (K) holds if and only if I - AA is injective. Assume I - A A is injective. We define J. = (I - A A) . Then J. is a positive linear operator (u > 0 =>■ J. u > 0, Lemma 8) on R(I - AA), and J. e„ = e„ e D(A), where e„ is the constant function e„(x) = 1. Then it follows that IUAgll < llgll for all g 6 R(I - AA), or

(3.6) ||u|| < ||u - Au|| for all u e D(A),

thus A is dissipative. Conversely, assume A is dissipative, then it follows from (3.6) that u - AAu = 0 implies u = 0, thus I - AA is injective. D

124

Proof of Theorem 3 It is known [4, Th. 3.1] that a necessary and sufficient condition for a closed operator A, with dense domain and for which R(I - A) = C(I) to generate a

strongly continuous semigroup of contraction operators, is that A is dissipative. Then Theorem 3 is a direct consequence of Theorem 2(i),(ii),(iii) and Lemma

9. D

Examples 2 2

1. Let A be defined as in Theorem 2 with I = TR, x» = 0, a(x) = (1 + x ) and )9(x) = 0. Then W(x) = 1, (aW)_1(x) = (1 + x 2 ) " 2 and R(x) = -^ arctan x + 1 x 2 1+x2 '

Thus condition (K) is satisfied and A is the infinitesimal generator of a C„-semigroup on C([-oo,oo)].

2. Let A be defined as in Theorem 2 with I = ^- -z-, -z) , x„ = 0, a(x) = 1, £(x) = -2 tan x. Then W(x) = (cos x) , (QW)~ (X) = cos2x, R(x) = -= tan x + z— . Thus condition (K) is satisfied and A is the infinitesimal

2cos2x generator of a CL-semigroup on C ( [- -r, -z] J . Remark. This example can be obtained from example 1 by means of the

C TT TT ~\

diffeomorphism <j>: (-00,00) —» [- — — J with <f>(x) = arctan x.

Remark. Let a and /3 be such that condition (K) is satisfied. Then if moreover condition (H.), i = 0 or 2 is satisfied, then one can prove that lim Au(x) = 0.

1 x ^ r i In this case the boundary point r. is called natural (see Feller [3]). In the case that both boundary points are natural Theorems 1 and 3 are both applicable.

Example I = 1R, xQ = 0, a(x) = 1, 0(x) = 0. Then W(x) = 1, (aW)~l(x) = 1, Q(x) = R(x) = x. D(AQ) = {u £ C([-oo,oo]) I u e C2(-oo,oo), u" e C([-oo,oo])}, D(A.) = {u e C([-oo,oo]) I u G C2(-oo,oo) lim u"(x) = 0}. 1 x—»ioo Then D(A„) = D(A.), and A« = A. is the infinitesimal generator of a CQ-semigroup in C([-oo,oo]), the so-called Gauss-Weierstrass semigroup.

125

ACKNOWLEDGEMENT The author would like to thank Ph. Clément for his valuable comments.

REFERENCES

1. Clément, Ph. and CA. Timmermans, On C~-semigroups generated by differential operators satisfying Ventcel's boundary conditions.

2. Coddington, E.A. and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, N.Y. (1955).

3. Feller, W., The parabolic differential equations and the associated semigroups of transformations, Ann. of Math. 55, 468-519 (1952).

4. Lumer, G. and R.S. Phillips, Dissipative operators in a Banach space, Pacific. J. Math. 11, 679-698 (1961).

5. Martini, R., A relation between semigroups and sequences of approximation operators, Indag. Math. 35, 456-465 (1973).

6. Protter, M. and H. Weinberger, Maximum principles in partial differential equations, Prentice-Hall, Englewood Cliffs, N.J. (1967).

7. Ventcel's, A.D., On boundary conditions for multidimensional diffusion processes, Th. of Prob. Appl. 4, 164-177 (1959).

126

2.8. Dual semigroups in NBV(J), NAC(J) and L V J )

In this section we will derive a characterization of the adjoint operator A : NBV(J) — NBV(J) of the operator A: C(J) — C(J) defined by 2.6, (1.3)-(1.4) in an analogous way as in section 2.4. We will prove the following theorem.

Theorem 2.8.1 Let X = C(J), let a and 0 are continuous real functions defined on J, satisfying

(aW)'1 G L^x^r.) and R £ L^x^r.), (i = 1,2),

where W and Q are defined by (1.1.3) and (1.1.4), and let A: D(A) c X -» X be defined as in Theorem 2 of section 2.6. Then the adjoint operator A of A with domain D(A) C X = NBV(J) is given by

(2.8.1)

D(A*) = {w £ X* \ w e NAC(J), aw' G AC. (J), (aw %) • - Pw • - ((aw')' - 0w %)(xn) G NBV(J)

0J

Urn (aWw*)(x) = 0, i = 1,2},

A w = (aw') ' - /3w', for w G D(A ) .

Note that r . and r 7 are entrance boundary points.

127

Proof We define an operator B with domain D(B) c X and range in X as follows

(2.8.2)

D(B) = {w £ X | w G NAC(J), aw G A C j ^ J ) , (aw • ) ' - /3w • - ((aw ') • - jSw ')(x0) G NBV(J), lim (aWw')(x) = 0, i = 1,2), x-»r:

Bw = (aw ' ) ' - Bw', for all w G D(B).

Thus D(B) is equal to the right-hand side of the first equality of (2.8.1). We follow the line of the proof of Theorem 2.4.1.

Step 1 A is a linear m-dissipative operator with domain dense in X. This follows from Theorem 2 and Lemma 9 in section 2.7.

Step 2 B is a closed linear operator. Let (w ) be a sequence in D(B), converging in - * *

X to w, and assume the sequence (Bw ) converges in X to y. Then we have to prove w G D(B) and Bw = y. The proof is similar to that of Step 2 in Theorem 2.4.1.

Step 3 NBV*(J) is dense in X* and NBV*(J) C R(I - B).

Step 4 For all u G D(A) such that u - Au G C*(J) and all w G D(B) such that w - Bw + Bw(xn) G NBV*(J) we have

<w,Au>* = <Bw,u>.

128

Let u G D(A) and w G D(B) such there is satisfied the assumptions. Let z := w - Bw.

Then there are constants a,b G E , such that r . < a < x,. y3

for all x G [b,r2), p 3 G 3R.

Here w and w are the special functions given in Lemma 1.3.6. From the boundary conditions on D(B) we have

lim (aWw ')(x) = p lim (aWw )(x) = 0. x—H2 ■* x—*r2 i

From Lemma 1.3.7 we have lim (aWw')(x) = 1, so p . = 0. Analogously x—>r2 ~ ^ i

p. = 0. Thus, on (r. ,a] we have

w(x) = z(a) - (Bw)(x0),

on [b,r.) we have

w(x) = z(b) - (Bw)(x0).

As in the proof of Theorem 2.4.1 we have lim (W u ')(x) exists as a finite x—»rj

number. Then we have

<Bw,u> + =f udCW'âWw') ' )

= u(x).W"] (aWw ') '(x) ] Tr2

+ - f (W"l u ')d(aWw ')

r 1 - J

= uW.W'Vww'VW]^ 2 . , . - ( W ' u ' K x M a W w ' X x ) ] ^

129

!-°* + I aW(W"1u1) ' dw

(2.8.4) = <w,Au>^

provided that

lim u(x).(W 1 ( Q W W ' ) ' ) ( X ) = 0 , i = 1,2, x->r; (2.8.5)

(2.8.6) lim (W iu ,)(x).(aWw')(x) = 0, i = 1,2. x-»r;

r 1 „ ii

(2.8.5) follows from the boundedness of u and (2.8.3). (2.8.6) follows from the boundedness of W u ' and the boundary conditions on D(B). So we have

<Bw,u>* = <w,Au>*.

Now all conditions of Proposition 2.2.1 are satisfied and it follows that B = A . This completes the proof of Theorem 2.8.1. D

Analogous to Lemma 2.5.2 and (2.5.5) it is easy to prove that

(2.8.7)

XQ = D(A ) = NAC(J)

D(Aj) = {w G X* | aw • G AC 1 Q C ( J ) , ( a w ' ) ' - pw • e AC(J), lim (aWw ')(x) = 0}.

A* w = (aw ' ) ' - 0w ' , for all w G D(A*),

and

130

(2.8.8)

f X® = L1(J)

D(A®) = {v € X® | av e AC loc(J), (av) • - /3v e AC(J), lim (aWv)(x) = 0}. x-+rj

A®v = ((av) • - 0v) ' , for air v e D(A®).

Note that A® is an infinitesimal generator of a Q,-contraction semigroup T® in LJ(J).

2.9. "Bidual" semigroups in L (J)

It is known that X®* = (hl(J)f = L°°(J). Analogous to Theorem 2.6.1 we will prove the following theorem on the adjoint operator A® .

Theorem 2.9.1 Let AQ: D(AQ) c X® -» X® be defined by (2.8.8). Then the adjoint A®* of A® is given by

(2.9.1)

D(A®*) = {zex®*\ze C!(J), z• e ACjJJ),

aW(W~1zi)% eL°°(J)}

A®*z = aW(W~]z V • = az" + pz '.

Note that from the definition of A, and thus from the definition of A® it follows that r. and r- are entrance boundary points.

Proof The proof is similar to the proof of Theorem 2.6.1. We define the operator C with domain D(C) c X® and range in X® as follows:

131

(2.9.2)

f D(C) = {z e X®* | z e c ' d ) , z ' e AC loc(J),

aW(W"1z1) ' G L°°(J)}

Cz = aW(W"1z»)'.

Thus D(C) is equal to the right-hand side of the first equality in (2.9.1). Since A® is an infinitesimal generator of a (^.-contraction group in X® this operator is linear and m-dissipative. C is a linear operator and R(I - C) = X® . This is easily seen as follows. Let k e L and let

(2.9.3) z - aW(W 1 z ' ) ' - k.

Let the function z, be defined by (1.1.48). Then it follows from the proposition that z. G D(C). Thus the general solution of (2.9.3) is given by

z = zR + CJUJ + c3u3 , C J , C 3 G K

where u. and u , are given in Lemma 1.1.16. Since r . and r_ are entrance boundary points, the functions u. and u . are unbounded on J, thusc. = c . = 0, and (2.9.3) has a unique solution in D(C). Finally, we will show that for all v such that v - A®v G C (J), and all z G D(C):

< Z ' A ° V >0 * = < C z-v >0*-

Therefore assume z e D(C) and v - Av G C (J). Then we have

<Cz,v>0+ - <z,A®v>Q t =

= f aW(W_ 1z') ' .vdx - f z.(W ' ( a W v ' J ' d x

132

= [(aWv)(x).(W lz')(x)~]T2 - f W âWvJ'dz 1 J J

= [z(x).(W_1(aWv)»(x)]'2~ + f W'VaWvVdz

(2.9.3) = lim (oWv)(x).(W"1z'Xx)- lim (aWv)(x).(W"'z ')(x) x-+r2 x—»rj

lim z(x).(W_1(aWv) ')(x) + 1 x—»r2 x—>rj

- lim z(x).(W âWvJ 'Xx)* lim Z(X).(W"1(QWV)')(X)

provided that these limits exist. In the next lemmata we will prove that these limits exist and are equal to zero.

Lemma 2.9.2 If r. is an entrance boundary point and v - /4®v G C (J), then

lim (W~1(aWv)%)(x) = 0, i = 1,2. x-*rj

Proof

If v - (W" (aWv) ' ) ' = g G C (J), then there exists a b G J, such that for all x G (b,r_): g(x) = 0. Let u . and u . be as in Proposition 1.1.1. From Lemma 1.2.5 we have

v 1 : = ( a W ) - 1 u 1 ^ h\xQ,r2).

Then from Lemma 1.2.2 it follows that

v(x) = cQv3(x) := c0(aW)_ 1u2(x), cQ G 1R, for x G (b,r2),

and from Lemma 1.1.16 with u . := u- that

lim (W"1(aWv)')(x) = cn lim (W - 1u')(x) = 0. x-+r2 u x-»T2 ~z

133

The proof for r . is similar. D

Lemma 2.9.3 If r. is an entrance boundary point, v - A®v G C (J) and z G D(C), then

(2.9.4) lim (aWvXx).(W'1 z')(x) = 0, i = 1,2.

Proof Assume v - A®v G C (J) and z G D(C). As in the proof of Lemma 2.9.2 there exists a number b G (x n , r . ) such that

v(x) = c0((QW)"1u3)(x), c Q G K .

Thus

(2.9.5) lim (aWv)(x) = cn . lim u,(x) = c x—>X2 x—»T2

Set g := aW(W" 1 z ' ) ' , then g G L°°(J)

(W"1z ,)(x) = z , (x 0 )+ [ ((aW)_1g)(t)dt. J x „ 0

Since (a\V)" G L (xn ,r ) and g is bounded, it follows that lim (W_ z')(x) u z x—»r?

exists and

(2.9.6) lim (W"1z ,)(x) = z'(x f ,)+ f ((aW)_1g)(t)dt. x - r 2 0 J xQ

Moreover x t

z(x) = ƒ {z '(x0) + I ((aW)"lg)(s)ds} W(t)dt + z(xQ).

Since W £ L (*0 ,r2) and z is bounded it is necessary that

134

f ' -1 (2.9.7) lim z ' ( x . ) + ((oW) g)(s)ds = 0

t^r2 ° J xQ

Finally, (2.9.4) follows from (2.9.5)-(2.9.7). The proof for r. is similar. D

Returning to (2.9.3) it follows from the Lemmata 2.9.2 and 2.9.3 that

<Cz,v>Q* = <Cz,v>0!|t

for all v such that v - A®v G C (J) and all z G D(C). Then the conditions of C O*

Proposition 2.2.2 are satisfied and it follows that C = A . Theorem 2.9.1 is proved now. D

0* * As in section 2.6 A is a weak infinitesimal generator of the dual semigroup T°* of T®. For the closed subspace X®® of X®*, on which T®* is strongly continuous we have

X®® = D(A®*).

135

CHAPTER 3 - SATURATION PROBLEMS FOR BERNSTEIN OPERATORS IN C^CU]

3.1. Introduction

In this chapter we shall consider saturation classes for Bernstein operators in various spaces. In section 3.2 we first consider saturation problems both uniform and pointwise in C[0,1] for Bernstein operators. We will do this in a unified way in the sense that the pointwise saturation theorem of Lorentz [Lo] and the uniform saturation theorems ([BN], [LS]) are obtained as special cases of two general theorems. The pointwise saturation class is well-known. The uniform saturation class is investigated, in different ways by different authors. Concerning the investigations of Becker and Nessel, [BN] and Felbecker [Fe], there seems to be a mistake in the definition in the domain of

the limit operator lim n(B - I) and in the paper of Lorentz and Schumaker n—KX> n

[LS] there seems to be a mistake in Remark 2 on Theorem 4.3 and the application with Bernstein polynomials. In section 3.2 these claims will be justified, and an other approach to saturation problems will be given. In section 3.3 we state and prove a variant of a result of Becker and Nessel about uniform saturation. This theorem associates the saturation class of an approximation process with the Favard class of the semigroup generated by the operator occurring in the so-called Voronowskaya formula. The proof given here is direct and does not use the Theorem of Trotter-Kato and the corresponding semigroup. In section 3.4 we use the theorem of section 3.3 in order to obtain the uniform saturation class of the Bernstein operators for the supremum norm. Finally, in section 3.5 we give a characterization of the uniform saturation class for the Bernstein operators with respect to the usual C [0,1]- norm (and thus for all equivalent norms). By a suitable transformation we can reduce this problem to problem of uniform saturation of positive contractions in C[0,1]. Again we use the general theorem of section 3.3 and for the characterization of the Favard class we invoke the results of Chapter 2.

136

3.2. A unified approach to pointwise and uniform saturation for Bernstein polynomials

by

Ph. Clément and CA. Timmermans Delft University of Technology

Report 86-45

1. INTRODUCTION AND MAIN RESULTS

In [9] Lorentz proved the following saturation theorem for the Bernstein operators Bn: C[0,1] -► C[0,1], n = 1,2,..., defined by

(Bnf)(x) = j ^ ( " J xk(l - x ) n " k f [±] , x e [0,1],

with f G C[0,1].

Theorem 1.1 (pointwise saturation) If f is a continuous function on [0,1], and M > 0, then

(1-1) \(Bnf)(x) - f(x)\ < (M/n)a(x)

for all x G C[0,1], n = 1,2,3,..., is equivalent to

(1.2) ƒ ' G C!(0,1) and

\f'(x) - f'(y)\ < M\x - >i for 0 < x,y < 1.

(1.3) Here a(x) = ~ x( 1 - x).

137

On the other hand Becker and Nessel [1] proved

Theorem 1.2 (uniform saturation)

If f is a continuous function on [0,1 J, then

(1.4) \(Bnf)(x) - f(x)\ < M/n

for some M > 0 and all x e [0,1 J, n = 1,2,3 is equivalent to

(1.5) a(x)\(D2hf)(x)\<M>

for some M' > 0 and for all x,h such that x+h,x-h E (0,1), h > 0. Here

(1.6) (D2hf)(x) = h~2(f(x + h) - 2f(x) + f(x - h)),

and a(x) as in Theorem 1.1.

The goal of this paper is to provide a unified approach to these two saturation problems. In order to do that we first need to compare conditions (1.2) and (1.5). This is done with the next lemma.

Lemma 1.3 Let "if be a non-negative continuous concave function on [0,1 J and let f be a continuous function on [0,1 J. a. If f satisfies

(1.7)

r a(x) \D2hf(x)\ < *(x)

for all x,h such that x-h,x+h e (0,1), h > 0

then the following condition holds

138

(1.8) fee1 (0,1)

X

\f'(x) - f*(y)\< f y(t)>OL~1(t)dt,forallx,ymthO<y<x<l. J v

b. If (1.8) holds, then we have

(1.9) a(x) \D2hf(x)\ < 29(x)

for all x,h such that x-h,x+h G (0,1), h > 0.

We give a proof of this lemma in the Appendix. In Theorem 1.4 we will show that pointwise and uniform saturation are special cases of a general saturation problem.

Theorem 1.4 Let ty be a non-negative continuous concave function on [0,1]. If f is a continuous function on [0,1 ] satisfying

(1.10) \(Bnf)(x)-f(x)\<j;V(x)

for x G [0,1 J, n > 1, then f satisfies (1.8).

Remark. If * = a we recover one part of Theorem 1.1 and if * = M we get by using Lemma 1.3 one part of Theorem 1.2 with M ' = 2M. For the converse we have

Theorem 1.5 Let * (x) = ap(x), x G [0,1], 0<p<l.lffe C[0,1] n CÔ.l) and satis

fies

X

(1.11) \f'(x) - f'(y)\ < \ a1(t)^(t)dt y

for 0 < y < x < 1, then

139

(1.12) \(Bnf)(x) - f(x)\ < 21-p^p(x)/n

for O < x < 1 and n > 1.

Remarks. If p = 1 we recover the second part of Theorem 1.1. If (1.5) is satisfied with M ' = 1 then it follows from Lemma 1.3 that (1.11) holds with * = * = 1. Then from Theorem 1.5, with p = 0, we obtain (1.4) with M = 2. Hence we also recover part two of Theorem 1.2.

1 n One may ask whether the constant 2 in Theorem 1.5 could be 1 in order to have a true converse of Theorem 1.4 with * = * .

P This is not the case as it is shown in section 5. Concerning the proofs, we partly follow the analysis of [1] for Theorem 1.5. As in Micchelli [12], we introduce the semigroup generated by the iterates of Bernstein operators. In section 5 we mention a mistake occurring in the literature (cf. [1], [6]) concerning the definition of the domain of the infinitesimal generator of this semigroup.

2. PROOF OF THEOREM 1.5

As observed in [2], we may restrict ourselves to functions in C[0,1], which vanish at the endpoints, since the Bernstein operators preserve affine functions. Moreover, since (B f)(0) = f(0) and (B f)(l) = f(l) for all f G C[0,1] it is sufficient to prove (1.11) for x e (0,1). Let p e [0,1] and

let f e C [ 0 , l ] n C !(0,1) satisfy (1.11) and f(0) = f(l) = 0.

Then

|f '(x) - f'(y)| < [ aP_1(s)ds, for 0 < y < x < 1. J y

It follows that f' is locally Lipschitz continuous on (0,1), therefore f"(x) exists a.e. on (0,1) and |f'(x)| < a p - 1 (x ) a.e. on (0,1). Let the kernel k e C([0,l]x[0,l]) be defined by

140

k(x,t) = . t ( l - x), 0 < t < x < 1,

x( l - t), 0 < x < t < 1.

If f is as above and 0 < x < 1, we have

k(x,t) |P(t)| < 2

almost everywhere in t on (0,1), and thus k(x,»)f"(«) is integrable on (0,1).


(2.1)

and

(2.2)

f(x) = - f k(x,t).f"i Jo

(t)dt

(Bnf)(x) = - J (Bnk(.,t))(x>P(t)dt, for 0 < x < 1,

and n = 1,2,3,.

Hence we obtain

(2.3) f(x) - (Bnf)(x) = J (k(x,t) - (Bnk(.,t))(x)) aP _ 1(t)dt ,

for 0 < x < 1.

Note that k(x,t) - (B k(«,t))(x) is non-negative since k(»,t) is concave for all t, [4]. From (2.1) and (2.2) with f = a and the fact that a - B a = —, we have

n n '

(2-4) J {k(x, t)-Bnk(. , t )}(x)dt = ^ - , x e ( 0 , l ) .

Since for each x e (0,1)

na J(x) f {k(x,t)- Bnk(.,t)(x)}-dt = 1

it follows from Jensen's inequality for integrals with 1-p e (0,1) that

-1 na

O

141 1

(x) j {k(x,t) - Bnk(.,t)(x)}(a-1(t)1_Pdt

< {na _ 1 (x) | (k(x,t) - Bnk(.,tXx)}a" ^ O d t } 1 " "

(2.5) = ( n a - ^ x W ^ ^ K X x ) - K(x)}1 _ P ,

where K e C[0,1] is defined by

K(x) = 2x log x + 2(1 - x) log (1 - x), 0 < x < 1, K(0) = K(l) = 0,

by observing that K" = a

Thus, from (2.3) and (2.5) we get

(2.6) |f(x) - (Bnf)(x)| < [ ^ - ) P.{(BnK)(x) - K(x))1 _ P .

In order to estimate the right-hand side of (2.6) we need the following.

Lemma 2.1 For all x e f 0,1] and all n G U

(2.7) n((BnK)(x) - K(x)) < 2.

Proof Since (B K)(x) - K(x) = (B K)(l - x) - K(l - x) it is sufficient to restrict ourselves to x e [0,1/2]. For x = 0 the left-hand side of (2.7) vanishes, so (2.7) is trivially satisfied. For x e (0,1/2] we define the function

by

F x : [ 0 , l ] - K

2 Fx(t) = ^ " ^ - {{K(t) - K(x) - (t - x)-K '(x)}.

142

Then

Further,

thus

Fx(0) = J^J - \(-K(x) + xK '(x)} = j ^ + log (1 - x) > 0

F x ( l ) = I ^ + l o g x > 0

F (x) = 0. xv '

■^-«-Wi-*». F '(x) = 0, Hm F '(t) = oo and lim F '(t) = -oo.

x t _o+ x x->l- x

2 1 F"(t) = xv ' x(l - x) t(l - t) '

so F"(x) > 0, and thus F has a minimum in x. Observe that F" has two zeroes x x x x. and x_, with 0 < x. < x < x« < 1. It follows that F is concave on [0,x.] , strictly convex on (x . ,x . ) and concave on [x„,l]. Thus F possesses only one local minimum on (0,1), which is taken in x and has the value zero. It follows that for each x e (0,1/2], F is a non-negative function on [0,1], and since B is a positive.operator, that B F is a non-negative function. Especially,

(BnFx)(x) = £ - | (BnK(x) - K(x)) > 0.

This proves the lemma. D

Returning to the proof of Theorem 1.5, from (2.6) and (2.7), (1.12) immediately follows. D

143

3. THE SEMIGROUP GENERATED BY THE ITERATES OF BERNSTEIN

OPERATORS

2 It is wel l -known that for u G C[0,1] n C (0,1)

lim n(I - B )u(x) = -a(x)u"(x), x e (0,1). n—»oo n

This is the so-called Voronowskaya formula, see ([8], p. 22). Moreover it is known ([2], p. 703) that for f,g G C[0,1] the following holds

lim ||n(B f - f) - g|| = 0, n—KX) il

2 where || || denotes the supremum norm, if and only if f e C (0,1), X(1 2 X) r'(x) = 8to, x G «M>. and

lim g(x) = lim g(x) = 0. x-»0 x-*l

(For the sake of completeness we give a proof of the if-part in Step 3 of the proof of Theorem 3.1). Therefore it is natural to define the operator

A: D(A)CC[0 ,1 ]^C[0 ,1 ] , by

D(A):= -fu G C[0,1]| u e C 2 ( 0 , l ) a n d lim a(x)u"(x) ^ x—0

= lim a(x)u"(x) = o } x—>1 J

Au(x) = -a(x)u"(x), x G (0,1), for u G D(A).

It is known ([11],[3]) that -A generates a positive contraction C„-semigroup on C[0,1] (equipped with the supremum norm), which we shall denote by T(t), t > 0. Then we have

144

Theorem 3.1 Let f be in C[0,1], then

Urn max \\B[nt]f - T(t)f\\ = 0 n^°° tefOJJ n

for every T > 0, where {T(t), t > 0} denotes the semigroup generated by -A and [nt] denotes the largest integer not larger than nt.

Proof We divide the proof in 4 steps. Step 7. Let f e C2[0,1]. Then

|n(I - Bn)f(x)| < a(x)||P||, x G (0,1)

by Theorem 1.1. Thus

|n(I - Bn)f(x) - Af(x)| < 2a(x)||ni, x e (0,1).

2 1 So for f e C [0,1] and for each e > 0 there exists a 5 with 0 < 8 < y , such

that for x e [0,5] u [1-5, 1] and for all n > 1,

|n(I - Bn)f(x) - Af(x)| < e.

Moreover, it follows from ([9], Th. 2) that for n sufficiently large

|n(I - B )f(x) - Af(x)| < €, for x [5, 1-6].

Then we have for all f e C2[0,1], lim ||n(I - B )f - Af|| = 0. n—too n

2 Step 2. Next we show that C [0,1] is dense in D(A), with respect to the graph norm of A:

Hu||A := ||u|| + ||Au||, for u e D(A).

145 2

Given u e D(A), if suffices to construct a sequence (u ) c C [0,1], such that

(3.1) lim ||u - u|| = 0 and lim ||Au - Au|| = 0. m—>oo m m—too m

Following [5], we define the functions u (m = 3,...) on [0,1] by

u (x) = nr '

f u ( 1 ^ ) + u . ( l ) . C x - l ) + l » - ( i i ) . ( x . i i ) 2 . x e [ 0 l 1

UW' x e & J ' m) u ( l - l ) + U ' ( l - i - ] . ( x - l + - ^ ) +

*- m-* v mJ mJ

+ | u " ( l - ^ ) . ( x - l + - ^ ) 2 , X G [ l - - L , l ] .

An easy computation shows that (u ) satisifes (3.1).

Step 3. Next we prove that

lim ||n(I - B )f - Af|| = 0 n—*oo

for all f e D(A). It follows from Theorem 1.5 that ||n(I - B )f|| < 2 ||Af||

every f e D(A) and thus

||n(I - Bn)f|| < 2 | | f | lA

for every f e D(A).

We see that the operators, n(I - B ) are uniformly bounded on (D(A), | |« | |A , n - A

while lim n(I - B )f = Af for f belonging to the dense subset C [0,1] of n—«x> n

(D(A), | | . | IA) .

It follows that

146

lim ||A f - Af|| = O n—>oo n

for all f e D(A).

Step 4. Now we define the family of contractions {V(t); t > 0} in C[0,1] by

means of

V(0) = I, V(t) = B [ 1 / t ] , f o r t > 0 .

Then Step 3 yields

lim i|t_1 (I - V(t)f - Af|| = 0 U0

for all f e D(A). Then from Chernoff's product formula (see [7], Th. 8.4) we have for each f e C [ 0 , l ]

lim ( v ( ^ ) n f = T(t)f n—«x> n

uniformly for t in compact subsets of ]R , where (T(t); t > 0} is the semigroup by -A. It follows that for each f e C[0,1]

lim ||B™ f - T(t)f|| = 0. m—»oo l m / i l

Let T > 0, 0 < t < T and set [■"] = n, then

[nt] < m < [nt + t] < [nt] + [t] < [nt] + [T].

Since lim Bkf = f, k = 1,2,...,[T], it follows that lim | |B [ n t ]f - T(t)f|| = 0 n—»oo n n—*oo n

for all f e C[0,1]. This proves the theorem.

147

4. PROOF OF THEOREM 1.4

As in Micchelli [12] the proof of Theorem 1.4 will be done in two steps. First we show that if f and \P are as in Theorem 1.4 then

(4.1) |f - T(t)f| (x) < ttf(x), x e (0,1) and t > 0.

Then we show that (4.1) implies (1.3). Thus we obtain a generalization of the results of [12], where * = M > 0. For k e l we have

k k~l I B f - f = V B (B f - f), w i t h f e C [ 0 , l ] .

to n n t

Since B are positive operators, we obtain by (1.10)

|B„f-f |< E | B ( B f - O I < ^ E B;*. £=0 £=0

Since * is positive and concave,

([4]), hence

k-1 k - 7 B 0.

For s > 0, we define

fs:-7j„S™fdl-

148

It is well-known that f e D(A). Moreover

(4.3) Af - -f(I - T(s))f, and lim ||f - f|| = 0. s s sjO s

From (4.2) and (4.3), we have

|Afs(x)| < *(x), x G ( 0 , l ) .

Thus

(4.4) a(x).|P(x)| < *(x) on (0,1), and |f'(x) - f'(y)| < f a_ 1*(t)dt, S S S J y

holds for 0 < y < x < 1. Since f converges uniformly to f, it follows from (4.4) that (f') is bounded

S ï*

and uniformly equicontinuous on [e, 1 - e], for each e e (0,1/2).

Thus by Arzela-Ascoli's theorem, we obtain

|f '(x) - f'(y)l < [ a - 1tt(t)dt, for 0 < y < x < 1. J y

This completes the proof of Theorem 1.4.

5. REMARKS

1. An easy calculation shows that

(5.1) (BjK) ( y ) - K ( y ) = log 4 = 1,386294

It follows that the smallest value of the constant M, such that

(5.2) 0 < n(B K(x) - K(x)) < M

holds for each n and each x e [0,1], is strictly larger than 1. From Lemma 2.1 and (5.1) we see that for this smallest value Mft

149

log 4 < M < 2.

This result improves the result of Berens and Lorentz ([2], Lemma 3) who proved (5.2) with M = 7. Although we suspect that M„ = log 4 we did not succeed in giving a proof of this fact until now.

1 r\

2. We moreover see from (5.1) that the constant 2 in Theorem 1.5 cannot be replaced by 1 as it is claimed in [10], p. 422. Thus Theorem 1.4 has not a true converse.

3. It is worthwhile to note that it is claimed in [1], p. 39 and [6] that if f 6 C[0,1] and x(l - x)f"(x) is bounded on [0,1], then lim ||A f - Af||

n—*oo n °° = 0. However since n(B f(0) - f(0)) = n(B f(l) - f(l)) = 0, this cannot be true if lim x(l - x)f'(x) # 0 or lim x(l - x)f"(x) * 0. The function

x-»0 x->l f = K (from Lemma 2.1) yields a simple counterexample.

6. APPENDIX

Here we will give a proof of Lemma 1.3.

Part (a). Suppose f e C[0,1] satisfies (1.7). Then

(6.1) |(D2f)(x)| < ^ f 4 , h>0 1 n /v " a(x) '

for all x,h such that x-h,x+h e (0,1). As in [1] we define

_h _h 2 rl fh(x) := - y f h f h f(x + s + t)dsdt (h > 0),

then f^(x) = (D^f)(x). Thus, with y < x,

(6 X X X

2) |fj|(x) - f (y)| < J |f^(t)|dt = ƒ |(Dj;f)(t)|dt < ƒ *(t)a_1(t)dt.

150

Let [a,b] c (0,1), a < y < x < b. Since *a~ is bounded on [a,b], the set

F = {f' | h > 0, 0 < a - h, b + h < 1} is uniformly equicontinuous on [a,b]. Moreover, f = converges uniformly to f on [a,b] if h tends to zero. From the theorem of Arzela-Ascoli it follows that there is a subsequence in F

converging uniformly on [a,b] to a g G C[a,b], and since f, (x) - f,(a) =

f'(t)dt, we obtain from the limit of this expression that f G C (a,b), and

fj(x) converges to f '(x) for each x G (a,b). Taking the limit in (6.2) we obtain (1.8) for x,y G (a,b). Since [a,b] is arbitrarily chosen inside (0,1), (1.8) holds for x,y G (0,1) with 0 < y < x < 1.

Part b. Suppose f G CÔ.l) and (1.8) holds. Then for each [a,b] c (0,1), f G C [a,b] and f' G AC[a,b]. Hence f • is differentiable a.e. on [a,b] and

|f"(x)| <*(x>a _ 1(x) a.e. on [a,b].

Since [a,b] is arbitrary

or

|f'(x)l <*(x)a_ 1(x) a.e. on (0,1)

a(x)-|f"(x)| < *(x).

Now, with x-h,x+h G (0,1), h > 0

a(x) |(D^f)(x)| =

= " ^ - |f(x + h) - 2f(x) + f(x - h)|

h (h - u){f"(x + u) + f"(x - u)} du

151

x(l - x) " h -JÜLL f (h - u) I * ( x + U)

h2 J 0( h U ; l(x + u)(l - x - u)

^(x - u) 1 . + 7 \Tt— \ r d u-

(x - u)(l - x + u)J With the elementary inequalities, (see [ 1 ]), for O O, x + h < 1,

x h - u

and

we obtain

x + u 1 - x - u

1 - x h - u 1 - x + u x - u

•(1 - x ) < h ,

• x < h,

*(x + u) + *(x - u) < 2*(x)

a(x) |(DjJf)(x)| < -£- f {tf(x + u) + *(x - u)} du

.h i i < ¥ Io 2*(x)du

o = 2*(x)

This completes the proof of the lemma.

152

[1] M. Becker and R.J. Nessel, Iteration von Operatoren und Saturation in lokal konvexen Raümen, Forschungsbericht des Landes Nordhein-West-falen, Nr. 2470, Westdeutscher Verlag Oplagen, 1975, pp. 27-49.

[2] H. Berens and G.G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J., 21 (1972), pp. 693-708.

[3] Ph. Clément and CA. Timmermans, On Cn-semigroups generated by differential operators satisfying VentceFs boundary conditions, Indag. Math., 89 (1986), pp. 379-387.

[4] P.J. Davis, Interpolation and approximation, Blaisdell Publ. Company (1963).

[5] C.J. van Duijn, personal communication. [6] G. Felbecker, Linearkombinationen von iterierten Bernsteinoperatoren,

Manuscripta Mathematica, 29 (1979), pp. 229-248.

[7] J.A. Goldstein, Semigroups of linear operators and applications, Oxford University Press (1985).

[8] G.G. Lorentz, Bernstein polynomials, Univ. of Toronto Press (1953). [9] G.G. Lorentz, Inequalities and saturation classes of Bernstein poly

nomials. In: On Approximation Theory, pp. 200-207, Proc. Conf. Oberwolfach, 1963, Birkhaüser Verlag, Basel (1964).

[10] G.G. Lorentz and L.L. Schumaker, Saturation of positive operators,

Journ. of Approx. Theory, 5 (1972), pp. 413-424. [11] R. Martini, A relation between semigroups and sequences of approxi

mation operators, Indag. Math., 35 (1973), pp. 456-465. [12] C. Micchelli, The saturation class and iterated of the Bernstein poly

nomials, J. Approx. Theory, 8 (1973), pp. 1-18.

153

3.3. Saturation and Favard classes

Let X be a real Banach space and let (L ) be a sequence of contractions in X,

strongly convergent to the identity operator I in X. Moreover, let us assume

there exists a positive function <j>: U —♦ UN with lim <£(n) = oo and a densely n-+oo

defined m-dissipative operator A such that

lim |Wn)(L f - f) - Af|| = 0 n—>oo n

for all f e D(A) C X.

Let us define for f e X

AQf := 0(n)(Lnf - f), n = l , 2 , . . . .

It is easy to prove (see [CH], Lemma 2.9) that for each n £ {1,2,...} A is a

bounded m-dissipative operator.

In the next theorem it will be shown that if

lim 0(n)(L f - f) exists, n—voo n

then f G D(A) and that

sup fln) HflnXL f - f)|| < oo n>l

if and only if f £ Fav(T^), where T ^ is the semigroup with infinitesimal

generator A. It follows that Fav(T^) is the saturation class of the approxima

tion process (L ) (see the Introduction). Indeed, if

tf(n)(Lnf - f ) = o(l), n ^ o o ,

then we have f £ D(A) and Af = 0. Moreover,

154

|L n f - f | | = O ( ^ n ) ' ] )

if and only if f e Fav(TA). Thus, if A # 0, then N(A) c Fav(TA), and N(A) ^ Fav(TA), where N(A) denotes the null space of A. Theorem 3.3.1 is a variant of a theorem of Becker and Nessel ([BN], Satz 3.6). They proved their theorem by means of the Theorem of Trotter ([Tr]). Here we shall give a direct proof without using the iterates of L and without

using the semigroup generated by A. We shall use Theorem 3.3.1 in the last sections in order to find the saturation class of the Bernstein operators in the usual C [0,l]-norm (m = 0,1,2,...).

There we shall use the fact that

(3.3.1) Fav(TA) - D(A0*) n X

(see Proposition 2.1.3) and the results of Chapter 2 concerning D(A® ).

Theorem 3.3.1 Let X be a Banach space. Let A: X —► X be linear and m-dissipative with dense domain D(A) and let for n = 1,2,... A : X —► X be a linear, bounded

n and dissipative operator with D(A ) = X. Then we have: if

(3.3.2) Urn A x = Ax, for all x e D(A), n—Hx n

then (i) \x G X \ Urn A x existsf = D(A) ^ n—►oo n J

(ii) \x&X\ sup \\A x\\ < oo} = *■ nëN J

{x G X | sup \\A,x\\ < oo} = Fav(TA), *■ h>0 }

where Ah := h'Ê* - I), E^ ; - (I - hA)"1, h > 0.

155

Proof

that (3.3.2) holds. Define the operator B: D(B) c X -+ X (i) Assume A and (A ) satisfy the conditions of the theorem and assume

(3.3.3) ' D(B) : = | x e X | lim A x exists}-

Bx := lim A x , x 6 D(B). n-K» n

By the dissipativity of A we have for all x e D(B)

|x|| < ||x - AA x||, for all A > 0.

By taking the limit for n —» oo it follows that

||x|| < ||x - ABx||, for all A > 0

and all x e D(B), thus B is dissipative.

From (3.3.3) it follows that

A C B.

Since A is m-dissipative, hence maximal dissipative (see [CH], Prop. 3.8) we have A = B.

This completes the proof of (i).

(ii) We recall (see [CH], Prop. 3.18) that

(3.3.4) Fav(TA) = {x G X | sup \\A x|| < oo}. L h>0 J

Assume x e X such that there exists a positive number M. for which

(3.3.5) l | A n x | | < M , n = l , 2 , .

156

Firstly we will prove that for each n A is m-dissipative. Let n e ]N. n

Since A is dissipative we have only to prove that I - hA is surjective for all h > 0. It is sufficient to prove that R(I - h_A) = X for some h„ > 0 (see e.g. [CH], Prop. 3.7). Let us take h„ > 0 such that

h 0 H A n l | < l .

We consider the equation

(3.3.6) x - hAA x = f, f e X. v O n

We define the operator T: X —» X by

Tx := h~A x + f. 0 n

Then for all x,y e X we have

||Tx - Ty|| < h0 | |Anx - Any||

< h0||An||.||x - y||.

By the Banach Contraction Theorem it follows that there exists a fixed point x such that Tx„ = x_. So x~ is a solution of (3.3.6). This proves the surjectivity of I - hAA . 0 n

Secondly, we will prove that x G Fav (T^), or with (3.3.4)

sup \\A, x|| < oo. h>0

Set A , := h ' W n - I), where E, n := (I - hA ) l , h > 0 and ||E. n | | < 1 n,h v h ' h v n' ' " h " ~ by the dissipativity of A . Then

157

So

-1 -1 A n A , = (-h + h E, )x n,h h -1 A n -1 A n

= (-h I E hn . ( I - h A n ) + h J E h V

A n = E, A x . h n

"Vhx|l = l | EhnVi' <| |Eh

n | | . | |Anx| |

< l|Anx||

< M j

A n ^ by assumption (3.3.5). It follows from (3.3.2) that E, x converges to E, x, therefore that A , x converges to A.x. Thus \\A,x\\ < M. and then from (3.3.4) we see x e Fav (T^).

For the converse suppose x e Fav(T^). Then by (3.3.4) there exists a positive number M- such that for all h > 0

P h x | | < M2 .

A Since E, x = x + hA, x we have h - - h-

(3.3.7) | |E A x - x|| < M2h.

Let X ^ be the Banach space D(A) equipped with the graph norm 11-11™ A Y

Since (A ), n e 3N, is a seq all x G X A and all n e 3N Since (A ), n e U , is a sequence of bounded linear operators in X we have for

HAnx|| < ||Anl|.||x|| < HAnl|.||x||D(A).

Thus for each n 6 3N, A is a bounded linear operator from X A into X. Since for each x e X A lim A x exists it follows that for each x e X A n-+oo n

158

sup ||A x|| < oo. nëlN n

Then by the uniform boundedness principle there exists a constant M. such

that for all x e X A

HAnx|| < M3 l |x | |D ( A )

(3.3.8) = M3(||x|| + ||Ax||).

For x e Fav(T^) we obtain

||Anx|| < HAnE^x|| + ||Anx - AnEJ^x||

< M3 (||E*x|| + | |AE^X| |) + ||An||.|| x - E ^ x || (by (3.3.8))

< M3 (||x|| + |Mhx||) + ||An|| M2h (by (3.3.7)),

<M 3 ( | | x | | + M2) + ||An||.M2h.

Since the left-hand side is independent of h we have

||Anx|| < M3 (||x|| + M2),

thus also

sup ||A. x H < ob. neU n

This completes the proof of the theorem.

159

3.4. Application: Uniform saturation class for Bernstein operators on CfO, 11

Let (B ) be the sequence of Bernstein operators in C[0,1]. As well-known for n k

each n e ]N B is a positive operator and B e« = efl. Here e, (x) = x for all x e [0,1], k = 0,1,... .In section 2.3 we proved that

lim n(B f - f) = Af, for f € D(A), n—HX> n

where A is defined by

' D(A)= {f e C [ 0 , l ] | f e C2(0,1),

lim x(l - x)f"(x) = lim x(l - x)f"(x) = o \ , x->0 x—1 J

(Af)(x) = x(l - x)P(x)/2, for f e D(A).

Let for n G 1 A : C[0,1] -» C[0,1] be defined by

Anf = n(Bnf - f), f e C [ 0 , l ] ,

where (B ) is the sequence of Bernstein operators on C[0,1]. Since the Bernstein operators are contractions in C[0,1], it follows that A is m-dissipative in C[0,1], (see the discussion preceding Theorem 3.3.1).

The operator A is a special case of the operators investigated in section 2.3, [CT,1], with a(x) = <f>(x) = x(l - x)/2, ^(x) = 0, J = (0,1). With the notations of Chapter 1 we find after calculations with x_ = 1/2

W(x)= 1, Q(x)=(2x - l)/(x(l - x ) ) , R(x) = 2 log x - 2 log (1 - x).

So

Q £ L ^ O . l ^ ) , R e (0,1/2) Q £ \}(\/2,\\ R e (1/2,1)

160

and it follows that 0 and 1 are exit boundary points. Then it follows from section 2, Theorems 1 and 2 that A is m-dissipative with domain D(A) dense in C[0,1], and thus A is the generator of a (^-contraction semigroup on C[0,1]. Therefore the conditions of Theorem 3.3.1 are satisfied. It follows that

(i) D ( A ) = { f e C [ 0 , l ] | lim n(B f - f) exists}. ^ n—«x> n J

(ii) The saturation class of the sequence of Bernstein operators (B ) is Fav(T^), i.e. the Favard class of the semigroup on C[0,1] with infinitesimal generator A.

As an application of the theory developed in sections 2.4-2.6 we will determine the saturation class of the sequence of Bernstein operators by means of Proposition 2.1.3, which says

Fav(TA) = X n D(A0*).

From Theorem 2.4.1 we obtain

' X * = l x NBV(0,1) x TR, xQ = 1/2, flx) = x(l - x)/2,

D(A*) = { ( w r w , w 2 ) G X* | w G NAC(J), 0w' G A C ^ J ) ,

( ^w ' ) ' - ( ^ w ' ) ' (1/2) G NBV(0,1)|-,

A*(Wj,w,w2) = (0, Ww')», 0), for all (Wj,w,w2) G D(A*).

By Lemmata 2.5.1 and 2.5.2 we have

161

f X*Q = M x NAC(O.l) x ÜR, xQ = 1/2,

D(A*) = { ( w r w , w 2 ) e X*Q | w G NAC(O.l), ^w • G AC1(JC(0,1),

W w ' ) ' G AC(0,1)\

[ A 0 ( w r w , w 2 ) = (0, (^w 1 ) ' , 0), f o r a l l (w r w,w 2 )GD(A*) .

By means of the mapping I : X j - X 0 = ] R x L°°(0,1) x JR defined in (2.5.2)

we obtain by (2.5.5)

f X 0 = n R x L ] (0 , l )xIR,

D(A®) = {(Vj,v,v2) G X® | ^v G AC lQc(0,l),

(0v)' GAC(0,1)}

L A®(V l ,v,v2) = ((^v)'(0+), (*v)", (tfv) '(I-)).

Finally, by Theorem 2.6,1 we obtain

oo. f X®* = 3 R x L ^ ( 0 , l ) x l R ,

D(A®*) = { ( z r z , z 2 ) G X®* | z G CÔ.l),

z»GAC l o c (0 , l ) , «^Z»GLW(0,1),

lim z(x) and lim z(x) exist, x-»0 x-+l lim z(x) = z. and lim z(x) = z- V x->0 l x—1 2J

[ A® (z1,Z,z2) = (0,^z",0).

With the identifieation £ defined a second representation for A® , namely

162

f X©* = L°°[0,1]

D(A0*) = {z e L°°[0,l] | z e c\o,A) n C[0,l], oo r z ' 6 A C l o c ( 0 , l U z " 6 L [0,1]}

A®*z = </>z", for all z G D(A©*).

Thus with (3.3.1) and Theorem 3.3.1 we have for the saturation class of the

Bernstein operators:

Theorem 3.4.1 (flvf, Th. A) For f e CfOJJ, (i) and (ii) are equivalent, where (V \\f-Bnf\\00 = 0(n-1), « - c o ,

(ii) f e C-(0,1), ƒ ' 6 AChc(0,l), <t>f" e L°°(0J).

Remark: If f belongs to the saturation class of the sequence of Bernstein operators, then f • does not necessarily belong to AC(0,1). The function K, defined by

K(x) = 2x log x + 2(1 - x) log (1 - x),

K(0) = K(1) = 0

introduced in section 2.1 belongs to the saturation class of the Bernstein operators. How< K ' £ AC(0,1). operators. However K '(x) = 2(log x - log(l - x)). Thus K ' e AC. (0,1) and

163

-,mr 3.5. Uniform saturation classes for Bernstein operators in C [0,1] norms

In this section we will introduce a class of positive operators approximating the identity, constructed from the Bernstein operators. We will investigate a number of its properties and determine its saturation class with the help of the theory of the Chapters 1 and 2 and Theorem 3.3.1. This class will be used for the determination of the uniform saturation class of the Bernstein operators in Cm[0, l] .

Let m be a positive integer, then we define the differential operators D (k = l,2,...,m) by

D1 = D: C ^ O . l l - ^ Q O . l ] , Df = f',

and D k : C k [ 0 , l ] - C [ 0 , l ] , Dkf = D(Dk~1f), k = 2,...,m.

Moreover we define the operators D~ (k = l,2,...,m) by

and

D ' ^ Q O J l - C ^ O . l ] ,

(D_1f)(x)= [ f(t)dt, Jo

D " k : C [ 0 , l ] ^ C k [ 0 , l ]

D"kf = D _ 1 (D" k + 1 f ) , k = 2,...,m.

Note that for f 6 C[0,1] we have

D m D" m f = f,

164

and that for f G Cm[0,l]

D D f - f is a polynomial of degree < m.

Next we define the sequence (F ), n = 1,2,..., by

(3.5.1) F m , „ = C [ 0 , l ] - C [ 0 , l ]

F f = DmB D"mf, m,n n '

where B is the n-th Bernstein operator. For investigations of this sequence the following lemma is useful:

Lemma 3.5.1 f f Da I, Th. 6.3.3) Let m and n be positive integers with n > m. If there are constants p and P such that

P < (Dmf)(x) <P, 0<x< 1, then

_ m (n - m)! ,nm„ ,:, . n n P < n J j - i - (D B f)(x) < P, 0 < x < 1.

n! n ' - • D

In the next proposition we list some properties of the sequence (F ).

Proposition 3.5.2

Let the sequence (F ) (with fixed m > 1) be defined on the Banach space C[0,1 ], equipped with the supremum norm, by (3.5.1), then (i) for each n, F is a positive contraction on C[0,1J, (ii) for each f&C[0,l]

Urn F f = f, „-oo m,n'

(Ui) for each n: n(F - I) is m-dissipative. 1 m,n

165

Proof (i) Assume f > 0 and let u := D" f, then u > 0 and D u > 0. Applying Lemma 3.5.1 with p = 0 we obtain DmB u > 0 and then

n ~

F f = DmB D"mf = DmB u > 0. m,n n n

Thus F is a positive operator. m,n _ _1 Moreover F e„ = D B D~ e„ = D B u with u = (m!) e . Then m,n 0 n 0 n m m v ' m

D u = en. If we apply Lemma 3.5.1 with P = 1 we obtain m m 0 :

, m ( n - m ) ! ( D m B u ) ( x ) ^ ,

Since

we get

- m n! , n .—, r. < 1 (n - m)

| F e J | = | |D m B u || < 1. m,n 0" " n m" ~

Thus for all f G C[0,1],

l | F m,n f | 1 * " f | | - | | F m , n e 0 " < ^

So F is a contraction. m,n (ii) Let f G C[0,1] and let u = D _ m f . Then u G C m [ 0 , l ] and it follows from [Lo], par. 1.8, that

or

lim D m B u = D m u n—*oo n

lim DmB D _ m f = D m D" m f = f n—KX> n

in the uniform norm topology.

166

(iii) This is a direct consequence of the discussion preceding Theorem 3.3.1.

D

In what follows we denote n(F - I) by C . Moreover, we define the m,n m,n

operators G and C in C[0,1] by:

D(Gm) = { f e C[0,1] | f e C2(0,1), 4>(" + m^'f • e C[0,1]}, where ^(x) = x(l - x)/2,

G f :=^f" + m^'f • - / " " V V ) ' , for all f e D(Cm).

D(C ) = D(G ), v m nv C m f := G m f " f-D f' where ^) := m ( m " ] ) / 2 ' for a11 f e D ( c

m ) For G the following lemma holds: m

Lemma 3.5.3 G is densely defined and m-dissipative.

Proof 2

Since C [0,1] C D(G ) is dense in C[0,1] it follows that G is densely defined. Next we investigate the boundary points 0 and 1 of the interval [0,1]. With the notations of Chapter 1, and x„ = 1/2 we have

Q(X) = ^(X) = X(1 - x)/2,

£(x) = m(l - 2x)/2,

W(x) = 2 - 2 n V m ( l - x f m ,

(aW)-1(x) = 2 1 + 2 m x m - 1 ( l - x ) m - 1 ,

, - 1 . ' x Q(x) = (aW) '(x) f __ W(t)dt,

J 1/ 1/2

R(x) = W(x) f (aW) !(t)dt. J 1/2

For x e (1/2,1) we have for m = 2,3,...

V

0 < Q(x) = 2x (1 - x) t (1 - t) dt J 1/2

Y «m+1 m - 1 , , ,m-l f ,. ^ - m , . < 2 x (1 - x) (1 - t) dt

J 1/2

= 2m + 1 (m - i r ' x ^ O - x)m-l{(l - x) 1 -" 1 - 2 m - !

< 2 m + 1 { l - 2 m _ 1 ( l - x ) m _ 1 }

For x

< 2 m + 1 .

e (1/2,1) and m = 1 we

O<QW-2J;/2

J 1/2

< -4 log (

have

r !(i -

(i - o"

1 - X ) .

t ) - 1

■'dt

Thus for m = 1,2,... Q e L (1/2,1). In a similar way we can prove that

Q e L ^ O . l / Z ) . Next we investigate the integrability of R on (1/2,1) for m = 1,2,... . For x e (1/2,1) we have

168 x _,, . _ -ni/, x-m f .m-1 . . ^ m - l j * R(x) = 2x (1 - x) t (1 - t) dt

J 1/2 /2 Y

»2-m,. ,-m f ., ,m-l , > 2 (1 - x) (1 - t) dt J 1/2

->2-ni - 1 , . \-m, ,. .m ~-m. = 2 m (1 - x) {-(1 - x) + 2 )

= 2 m {-1 + 2 (1 - x) }.

Thus R 0 L (1/2,1). In a similar way it can be proved that R g L (0,1/2). Then it follows that 0 and 1 are entrance boundary points. Therefore, the theory of section 2.7 is applicable. Lemma 9 of that section shows that G is dissipative. The surjectivity of I - G m follows from Theorem 2 of that section. Then also I - AG (A > 0) is surjective, ([CH], Th. 3.7). This completes the proof of Lemma 3.5.3. D

For C an analogous lemma holds. m

Lemma 3.5.4 C is densely defined and m-dissipative.

Proof Since D(C ) = D(G ) C is densely defined. Assume A > 0. From the dissi-m m m 3

pativity of G it follows that

(1 + A ( m ) rtfll < ||f|| < ||f - A(l + A ( m ) )_1Gmf||.

Hence

llfll < llf + A ( m ) f - AGmf||

or

llfll < llf - AC f||.

169

This proves the dissipativity of C . For the surjectivity of I - AC let us consider

f - A C m f = g, g e C [ 0 , l ] , A > 0 ,

or (1 + A ( m ) )f - AGmf = g,

or

f-Ad + A ^ r ^ j - d + A («b-V

From the surjectivity of I - pG for each n > 0 it follows that there exists a solution f. This proves the surjectivity of f - AC for each A > 0. D

The next proposition is of crucial importance for the application of Theorem 3.3.1.

Proposition 3.5.5 Let D(A) - just as in section 2.3 - be given by

D(A):={feC[0,l]\feC2(0,l), Urn x( 1 - x)f(x) = Urn x( 1 - x)f'(x) = 0}, x-»0 x— 1

and let m> 1. Then

D(Cm) = C1fO,lJnD(A).

Proof CÔ.l] n D(A) c D(C ) is clear, so we have only to prove D(Cm) c c ' [ 0 , l ] n D(A).

(i) D(C ) c C]fO,lJ. Let f s C [ 0 , l ] n C2(0, l)and g e C[0,l]such that

<t>f" + m0 l f ' = g,

where

170

Then

or

thus

or

0(x) = x(l - x ) / 2 .

,1 -m, ,m~,. ,

(4>mfy = g4>m-1 6C[o , i ] ,

( A ' ) ( x ) = f g^m_1dt + K, xG(0 , l ) , K e E JO

(*) f'(x) = ^_m(x) f g*m - 1dt + K0~m(x). Jo

By I'Hospital's Rule we have

Y

lim <j>~m(x) g0m" dt \-*0 J 0

= lim ( g ^ X x V C m . - r ' W ^ ' W ) x—0

= lim g(x)/(m(l - 2x)) x—0

= g(0)/m.

Hence the function ip on [0,1) defined by V>(0) = g(0)/m, X

Jo

belongs to C[0,1). From (*) it follows that

1/2 , c 1/2 r t . f(x) = f [\] - j ^ m ( t ) j o (g^m- l Xs)dsdt

f 1/2 K tf"m(t)dt.

J x

171

Since <j> £ L (0,1/2) it follows from the boundedness of f near zero that K = 0. Thus

f'(x)=^"m(x) f g^m_1dt Jo

implying f' e C[0,1). In a similar way we can prove f' e C(0,1]. Thus

f ' e C[0,l]or f e c ' [ 0 , l ] ,

(ii) D(C ) c D(A). Let f and g be as in part (i). Since f' e C[0,1] and g e C[0,1], it follows that <j>r = g - m<f>H • e C[0,1]. So lim (<£f")(x) and

x—>0 lim (<f>f")(\) exist. Assume x—>1

L := lim (tff")(x) > 0. x-»0

Then there exists a number 6 > 0, such that for all x e (0,5)

x( l - x)f"(x)/2 > L/2

and thus for x e (0,5)

f"(x)> L(x(l - x ) )" 1 > L x _ 1 > 0.

Then it follows that

f1/2 -1 f ( l /2)> f'(x) + L t dt

= f '(x) - L log 2x.

Since f' is bounded and -log 2x is positive and unbounded for x e (0,1/2), L has to be zero, a contradiction. Similarly if L < 0 we also obtain a contradiction. Therefore L = 0. Similarly lim (4>P)(x) = 0 and thus D(C ) C D(A). D

172

In order to apply Theorem 3.3.1 it is sufficient to prove that

lim C f = C f n-»oo m>n m

for all f G D(C ). With D"mf = u we have v m

C f = DmA D " m f = D m A u m,n n n

where

A u = n(B u - u). n v n '

Concerning the convergence of D A u, Felbecker [Fe] proved that for

u G C m + 2 [ 0 , l ]

lim DmA u = Dm(<iu") n—+00 n

in the uniform norm topology. For our purpose we need a sharper result.

First we will prove the following.

Lemma 3.5.6

For all u e (f1*1 [0,1]

lim n(D Bu-B D u) = mê%D u - I ,1 D u


Proof TV» J- 1

Let u e C [0,1] and n > m. Following the lines of Lorentz, [Lo], par. 1.4 we first compute (D B u)(x), 0 < x < 1, where

n (3.5.2) (B u)(x) = E u(k/n).pn . (x),

n k = Q n,K

173

, , fnï k,, .n-k Pn,k(x) = U x ( 1 - X ) •

Let Au(x) denote the first order difference u(x + 1/n) - u(x) of the function u at the point x. In general, the difference (A uXx) of £-th (£ > 1) order corresponding to the increment 1/n at x is defined by

(A1u)(x) = Au(x), (A£u)(x)= A(A£_1u(x)).

Then, as well-known,

£ (3.5.3) (A£u)(x)= E ( - l / M ( f ) u ( x + i/n).

i=0

By differentiation of 3.5.2 we obtain

n (DB u)(x) = E u(k/n)(Dpn )(x) n k = Q n,K

= E u(k/n) . (" ){kx k " 1 ( l - x ) n _ k - (n - k)xk(l - x ) " ^ " 1 } k=0 K

n-1 n-1 = E u((k+ l)/n).n.p ( x ) - E u(k/n).n.p (x)

k=0 n _ 1 k=0 n _ I , K

n-1 = n. E (Au)(k/n).pn (x).

k=0 n _ 1 , K

By repeated differentiation we obtain in this way

n-m E

k=0

From Taylor's Theorem we have

m+1

n-m (3.5.4) (DmB u)(x) = , " ! ,, E (Amu)(k/n).p (x).

v n /v ' (n - m)! i,_o n m ' K

u((k + i)/n) = _E i C ^ ^ u X k / n ) , - ^ - ^ (i)m+1h(k/n,^k/n)

174

where h is continuous on

V : = { ( x , t ) e [ 0 , l ] x [ - l , l ] | 0 < x + t < 1}

with £. . G [0,i/n] and h(k/n,0) = 0. Applying this formula we obtain for I,K/ n

(Amu)(k/n):

m (Amu)(k/n)= E ( - D ^ r j u C k + iVn)

i=0 *

(3.5.5)

m E i=0

m+l = E (- i r 'Cp E j (t)Vu)(k/n)

:_n » j = 0 J- n

v , nm-ifm") 1 r i ï m + 1 , , . , t .. + £0

{-l) îJ(SrTT)rW h<k/n i,k/n>

= mE+1 {nVu)(k/n).E ( - ^ ( " V j=0 J ' i=0

1 -m-1 + ~, rrr n (m+ 1)! ^ ( - 1 ) l . J , h ( k / n , q k / n )

m+l , = E I T n"J(DJu)(k/n)S(m,j)

j=0 J-

(m + 1)! 1 -m-1

n m T— i ,\m-i (rcn .m+l, ,. , , >, E Q ( - D l i J i M k / n , e i k / n )

where S(m,j) are known as the Stirling numbers of the second kind (see [AS], 24.1.4). For these numbers we have

(3.5.6)

S(m,j) = 0 if j < m - 1,

S(m,m) = 1,

S(m,m+1)= r t 1 ) .

Now from (3.5.4)-(3.5.5) we obtain

175

(DmBnu)(x) = ( n _"m), E {n_ m(Dmu)(k/n) +

+ (nTTT) (m+2

1)n"m" ,(Dn,+ lu)(k/ll)}-Pn-m,kW

(3.5.7) + (R u)(x), v m,n /v "

where

(R u ) ( x ) = , n ! ,, n^^ttm^l)!)-1 E E <-l)m_1 P ) i * m,n A ' (n - m)! vv k=0 i=0

h ( k / n ^ i , k / n ) P n - m , k « -Thus

n-m m_, .. . n! -m ^ /T^m (D"'B u)(x) = 7 — ~ r - n E (D u)(k/n).p . (x) + n /v ' (n - m)! . ^ v 'y ' ' n-m,kv '

, , n-m . m n! -m-1 v-. ,_.m+l ... . , , .

+ T • "7 ü" n E (D u)(k/n).p . (x) 2 (n - m)! . ^ - /v ' ' n-m,k '

(3.5.8) + (R u)(x). m,n /v '

Next, we will investigate each of these terms. Clearly

( n _n!m)! n " m - 1(1 - l/n)(l - 2/n) (1 - (m - l)/n)

= 1 - m(m - l)/(2n) + o(n~ ), n —► oo

= 1 - ( m )n~ + o(n" ), n ■2-(3.5.9) = 1 +o(l) , n — oo.

Since for all x e [0,1]

n-m E P , (x) = (B e«(x) s 1 £Q n-m,kv ' v n-m 0V '

we have

176

n-m *m £ (D11 u)(k/n).p (x) k=0 n m , K

< IIDmu||

and

. n-m n! -m ,-. /T^m 7 — ^ n T (D u)(k/n).p . (x) =

(n - m)! *-•- v A ' ' Kn-m,kv ' k=0 rTïT\ - 1 n-m

= (1 - ( m ) n"1 + oOT1)). £ (Dmu)(k/n).p (x), n - oo 2 k=0 '

n-m (3.5.10) = (1 - ( m ) n _ 1 ) . E (Dmu)(k/n).pn (x) + o(n"J), n - oo,

z k = Q n-m,K

uniformly in x G [0,1]. Similarly

, . n-m . , _ _ . . . m n; -m- i *-■ /r^m+l .,, . . , . ( 3 5 1 1 ) yTiT^jT n

k?0 (D U ) ( k / n ) pn-m,k ( x )

_. n-m = m(2n)" £ ( ° m + u)(k/n).p (x) + o(n" ), n - o o

k=0 n-m,K

uniformly in x G [0,1]. Since u G C [0,1] the function h is continuous on V, and thus also uniformly continuous on V. Then, if e > 0 is arbitrarily chosen, there exists a number 6(e) > 0 such that for all (x,t) G [0,l]x(-6,5) n V

|h(x,t)| < e.

In particular, if n > mS~ , then for all k G {0,l,...,n-m} and all i G {0,1,...,m}

|h(k/n,e u / n) | < ,

With 3.5.9 it follows that for all x G [0,1]

177

n-m m |(R u)(x)| < n _ 1 ( l + o(l))((m + l)!)"1 E E ( ? ) ™+1 *** m k ( x ) ' m,n R = 0 i = Q i n-m,K

n —»oo n-m

= n" ( l + o( l ) ) . e .Q m . E P n _ m > k W. » - oo, k=0

w h e r e Q m - ( ( m + l ) ! ) - , i : ( m ) . i m + 1 , m i=0

= n _ 1 ( l + o(l)).€.Qm(Bn_me0)(x), n - oo

= n" (1 +o(l)).€.Qm, n - K »

uniformly in x e [0,1]. This implies

(3.5.12) n(R m yO(x) = o(l), n - o o ,

uniformly in x e [0,1]. Returning to (3.5.8) we obtain with (3.5.9)-(3.5.12) for n > m

n(DmB u - B Dmu)(x) = v n n-m /v ' n-m

= n. E {(Dmu)(k/n) - (Dmu)(k/(n - m))}.p . (x) k=0 m '

n-m - (m3- E o (Dmu)(k/n).pn.m)k(x)

n-m . + (m/2). E (D m uXk/n).Pn (x)

k=0 '

+ 0(1), n -» oo,

and by the Mean Value Theorem

178

n-m = n. E ( k / n - k / ( n - m ) ) ( D m + 1 u ) ( x ).p (x)

n-m - ( ? ) E (Dmu)(k/n).P (x)

z k = Q n-m,K n-m .

+ (m/2) £ (Dm+1u)(k/n).p (x) k=0 '

+ o(l), n — oo,

with k/n < x, < k/(n - m)

n-m = -m E o ( k / ( n - m ) ) ( D m + 1 u ) ( x M ) . p n _ m J c ( x )

n-m - I ? ] E (Dmu)(k/n).p (x)

z k = Q n-m,K n-m

(3-5.13) + (m/2) E (Dm+1u)(k/n).pn (x) k=0 n-m,K

+ o(l), n - K » ,

uniformly in x. By the well-known convergence properties of the Bernstein operators ([Lo, 1 ]) we finally obtain

lim n{(DmBûXx) - (B„ _DmU)(x)} =

n_KX) n 'v ' v n-m

- mx(Dm+1u)(x) - (m)(Dmu)(x)

+ (m/2)(Dm+1u)(x)

= (m/2)(l - 2x)(Dm+1u)(x) - ( m ) (Dmu)(x),

uniformly in x. This concludes Lemma 3.5.6. D

179

If f G D(A), and u = D~mf then clearly Dmu G D(A). Therefore, the next

lemma is a consequence of Step 3 of the proof of Theorem 3.1 in section 3.2.

Lemma 3.5.7 For all u G Cm+2(0,1), such that Dmu G D(A) we have

Um n(B D u - D u) = 6D u nôo n-m


D

After these preparations we are able to prove the following proposition.

Proposition 3.5.8 For all f G D(C ) , m> 1, we have 1 m

Um C f = CJ „-►oo m,nJ nf


Proof

Assume f G D(C m ) . Set u = D~mf. By Proposition 3.5.5 we have f e c ' t O J l n

D(A), or u G C m + l [ 0 , l ] and D m u G D(A). Using the Lemmata 3.5.6 and 3.5.7

we see

lim C f = lim n(F f - f)

= hm n(D B D f - D D f) n—>oo n

. . /T^nir, _ . m . = lim n(D B u - D u)

n—»oo n

180

= lim n(D m B u - B D m u ) + n-ôo n n " m

+ lim n(n - m) . lim (n - m)(B D u - D u) n—*oo n—»oo n-m , , _ jn+ l rmi _ m ,T~wm+2 = m<£'D u - L 2 J D u + <£D u

= <^D2f + m<f> 'D f - ( m ) f

= C f m


D

We arrive to the saturation theorem for the operators F , which is an m,n

application of Theorem 3.2.1.

Theorem 3.5.9 Let in the Banach space C[0,1 ], equipped with the supremum norm, for m > 1 the sequence of operators (F ) , strongly convergent to the identity operator I, be defined by 3.5.1. Then for f G Cf 0,1 ] the conditions (') \\F f ~ A\< Mn'1 for some M > 0, n = 1,2,... " m,n " (ii) f£Fav(T )

(iii) f e C](0,l), ƒ ' e AC[oc(0,l), <f>f" + m<l>,f> e L°°(0,l)

(iv) f G C](0,1), ƒ ' G ACloc(0,l) n L°°(0,1), <j>f" G L°°(0,1)

are equivalent.

Proof ,

If A := C = n(F - I) and A := C , then it follows from the Lemmata n m,n m,n m 3.5.4-3.5.8 that the conditions of Theorem 3.3.1 are satisfied and the equivalence of (i) and (ii) follows from the second part of the theorem. Further, the equivalence of (ii) and (iii) will be proved by applying of the theory of the sections 2.8-2.9 and by using the fact that

181

Fav(T^ ) = D(C®*)nC[0,l] 'm m

Since D(C ) = D(G ) and nr v nr

G -C = 0 1 ,

it follows that D(C°*) = D(G0*).

From Theorem 2.8.1 we have

X =NBV[0,1], x Q = 1/2,

D(Gm) = {w G X* | w e NAC[0,1], <£w' G AC l oc(0,l),

( 0 w ' ) ' - m ^ ' w 1 - ( ( ^ w ' ) ' - m ^ ' w ' X x ^ e NBV[0,1],

lim (^ I " m w' ) (x )= lim ( ^ 1 " mw

, ) ( x ) = 0 } , x-+0 x-*l

G m w = ( < ^ w ' ) ' " m<^ ' w '* f o r all w G D(G* ).

Moreover we obtain from (2.8.7)

XQ = NAC[0,1],

D(G m ) 0 ) = ( w G X * 0 | ^ w . G A C l o c ( 0 , l ) ,

( ^ w ' ) ' - m ^ ' w ' G AC[0,1],

lim (^ 1 " m w') (x)= lim ( / ~ m w ' ) ( x ) = 0}, x-»0 x-»l

G m 0W = ^ w ' ) ' " m ^ ' w ' f o r a11 w € D(G* 0 ) .

182

and from (2.8.8)

X® = L1(0,1),

D(G®) = ( v e X ® | ^ v e A C l o c ( 0 , l ) ,

( W - n t f ' v e AC(0,1),

lim ( / " m v ) ( x ) = lim (^1_mv)(x) = 0}, x-+0 x->l

G®v = ((0v) • - m<j> ' v ) ' , for all v G D(G®).

Finally from Theorem 2.9.1 we obtain

f X®* = L°°(0,1),

D(G®*) = {z G X®* | z 6 CÔ.l), z ' G AC, (0,1), m 'locv

oo. <f>z" + m<f>'z' G L ( 0 , 1 ) } ,

G®*z = <j>z" + m0 ' z ' , for all z G D(G®*).

It follows from (3.3.1) that

Fav(T^ ) = D(G®*) n X 'm m

= {f G C[0,1] | f G C'tf),!), f • G AC l oc(0,l),

^ r + m^'f» G L (0,1)}.

Next we prove the equivalence of (iii) and (iv). The implication (iv) =>■ (iii) is trivial.

Let f e C[0,1] n CJ(0,1) with f' e AC Joc(0,l) and let g := <j>f" + m^ ' f ' G L°°(0,1), where 0(x) = x(l - x)/2. Then

183

Amr)> = g<t,m-1 e 1^(0,1) and

(*mf ')(x) = f g^ m _ 1 dt + K, K £ IR, x e (0,1), J 0 or ,x

f ( x ) = ^ m ( x ) f g ^ d t + K ^ x ) . 0

Since g s L (0,1) we have for x e (0,1)

x < llg||^""'(x) I At)dt. x) f g^m_1dt < ||g|| ^ m ( x ) f

J 0 J 0

In a similar way as in the proof of Proposition 3.5.5 we can show that

x - ^ _ m (x) f g<f>m~l dt e AC(0,1 /2), J 0

TV» 1

and that since <j> £ L (0,1/2) the constant K has to be zero. It follows that f1 e L°°(0,l/2). Analogously we can obtain f • e L°°(l/2,1). Thus f ' e L ° ° ( 0 , l ) and then also </>(" e L°°(0,1). This completes the proof of Theorem 3.5.9.

Remark 1. From (iv) we see the saturation class of the sequence F . ,F -,...,

where m = 1,2,..., is independent of m.

Remark 2. The class of functions Z for which the convergence of F f to f j m,n is faster than 0(n ) is the so-called "trivial class" of the sequence (F ). Assume f e Z , then we have

n ( F m,n f " 0 - o ( l ) , n - o o

and since Z C D(A) (by Theorem 3.2.1) we have

lim n(F f - f) = C f = 0, n-Kx, m,n m

184

and thus f belongs to the null space of C , denoted by N(C ). It follows that Z c N(C ). Since N(C ) c Z we have v m m

Z = N(C ). v m'

Thus f e Z if and only if

(3.5.14) ^ r + m^'f ' - ( m ) f = 0.

This equation is a hypergeometric differential equation (cf. [Bie], p. 210). Two independent solutions are the functions f. and f- defined by f. (x) = x , f2(x) = (1 - x ) 1 _ m if m > 2, and fj(x) = 1, f2(x) = log(x/(l - x)) if m = 1. It

follows that f G Z if and only if f is a linear combination of f. and f». Since f. and f~ are unbounded if m > 2 and f_ is unbounded if m = 1 we have: f G Z if and only if f is constant for m = 1 and f is zero if m > 2.

We conclude this section by solving the problem of uniform saturation for Bernstein operators in C [0,1]. Let f G Cm[0, l ] , m G UN, n > m, such that

sup ||Dm(n(Bnf - f))|| < oo. n>m

We can rewrite f as

f = p . + D u, m-1 '

where Pm_i is a polynomial of degree less or equal m-1 and u e C[0,1]. It follows

Hence follows from Lemma 3.5.1 with p = P = 0 that DmB p , = 0

nKm-1

sup ||Dm(n(B D"mu - D - mu) | | < oo n>m

or equivalently

185

sup ||C u|| < oo. n>m '

Then u satisfies (iv) of Theorem 3.5.9, and thus D f satisfies (iv). Conversely, let f G C [0,1] such that Dmf satisfies (iv). Then

sup ||C Dmf|| < oo 1 m,n " n>m

by Theorem 3.5.9, or equivalently

sup | |D m (n(B n f - f))|| < oo. n>m

Finally, if

Jim | |Dm(n(B f - f))|| = O n—»oo n

then D f belongs to the null space of C . Thus the trivial class Z„ consists of all f G Cm[0, l ] , such that u := Dmf satisfies equation (3.5.14). Thus f G Cm[0,l] belongs to Z~ if and only if Df is constant for m = 1 and D f = 0 for m > 2. If C [0,1] is normed by

m k 11*11:= E IID fll

k=0

we have the following saturation result:

Theorem 3.5.10 For m> 1 and f G C^fOJJ, d) \\Bnf-I\\m = o(n'1),nôo,

if and only if ƒ is a linear function, (ii) \\Bnf-j\\m = 0(n-1),n^°o,

if and only if D f satisfies condition (iv) of Theorem 3.5.9. a

186

REFERENCES

[ACK] Arendt, W., P.R. Chernoff and T. Kato, A generalization of dissi-pativity and positive semigroups, J. Operator Theory 8(1982), pp. 167-180.

[AS] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover Publ., N.Y.

[B] Becker, M , Über den Satz von Trotter mit Anwendungen auf die Approximationstheorie, Forschungsberichte des Landes Nordrhein-Westfalen Nr. 2577, Westdeutscher Verlag (1975), pp. 1-36.

[BB] Butzer, P.L. and H. Berens, Semigroups of operators and approximation, Springer Verlag (1967).

[Be] Berens, H., Interpolationsmethoden zur Behandlung von Approxi-mationsprozessen auf Banachraümen, Lecture Notes in Mathematics 64, Springer, Berlin (1968).

[Bi] Bieberbach, L., Theorie der gewöhnlichen Differentialgleichungen, Springer Verlag (1965).

[BL] Berens, H. and G.G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J. 21 (1972), pp. 693-708.

[BN] Becker, M. and R.J. Nessel, Iteration von Operatoren und Saturation in lokal konvexe Raümen, Forschungsberichte des Landes Nord-rhein-Westfalen Nr. 2470, Westdeutscher Verlag Opladen, pp. 27-49.

[Bs] Bernstein, S., Demonstration du theorème de Weierstrass, fondée sur Ie calcul de probabilités, Comm. Soc. Math. Kharkow 13 (1912-'13), pp. 1-2.

[CH] Clément, Ph., H.J.A.M. Heymans, S. Angenent, C.J. van Duijn and B. de Pagter, One-parameter semigroups, CWI-Monogr. 5, North-Holland Publ. Comp. (1987).

[CL] Coddington, E.A. and N. Levinson, Theory of ordinary differential equations, McGraw Hill, N.Y.

[CT,1] Clément, Ph. and CA. Timmermans, On Cg-semigroups generated by differential operators satisfying Ventcel's boundary conditions, Indag. Math. 89 (1986), pp. 379-387.

[CT,2] Clément, Ph. and CA. Timmermans, A unified approach to point-wise and uniform saturation for Bernstein polynomials, Rep. 86-45, Delft Univ. of Techn., Fac. of Math, and Inf. (1986).

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[Da] Davis, P.J., Interpolation and approximation, Blaisdell Publ. Comp. (1963).

[Fa] Favard, J., Sur 1'approximation d'une variable réelle, (Colloque Intern. d'Analyse Harmonique, 1947), Publ. Centre Nat. de la Recherche Scientifique, Paris, 15, (1949), pp. 97-100.

[Fe,l] Feller, W., The parabolic differential equations and the associated semigroups of transformations, Ann. of Math., (2), 55 (1952), pp. 468-519.

[Fe,2] Feller, W., On differential operators and boundary conditions, Comm. on Pure and Applied Math., Vol. Vin, (1955), pp. 203-216.

[Fb] Felbecker, G., Linearkombinationen von iterierten Bernsteinopera-toren, Manuscripta Mathematica 29 (1979), pp. 229-248.

[G] Goldstein, J.A., Semigroups of linear operators and applications, Oxford Univ. Press (1985).

[HP] Hille, E. and R.S. Phillips, Functional analysis and semigroups (revised edition), Amer. Math. Soc. Colloq. Publ., Vol. 31, Providence, R.I. (1957).

[Iv] Ivanov, K.G., Approximation by Bernstein polynomials in Lp metric, Constructive theory of functions '84, Sofia (1984), pp. 421- 429.

[K] Kato, T., Perturbation theory for linear operators, Springer Verlag (1966).

[Lo,l] Lorentz, G.G., Bernstein polynomials, Univ. of Toronto Press (1953). [Lo,2] Lorentz, G.G., Inequalities and saturation classes of Bernstein poly

nomials, in "On Approximation Theory", Proc. Conf. Oberwohlfach, 1963, Birkhaüser Verlag (1964).

[LP] Lumer, G. and R.S. Phillips, Dissipative operators in a Banach space, Pacific Journ. Math. 11 (1961), pp. 679-698.

[LS] Lorentz, G.G. and L.L. Schumaker, Saturation of positive operators, Journ. Approx. Theory 5 (1972), pp. 223-232.

[Ma,l] Martini, R., A relation between semigroups and sequences of approximation operators, Indag. Math. 35 (1973), pp. 456-465.

[Ma,2] Martini, R., Differential operators degenerating at the boundary as infinitesimal generators, Thesis, TU Delft (1975).

[MB] Martini, R. and W.L. Boer, On the construction of semigroups of operators, Indag. Math. 36 (1974), pp. 392-405.

[Mi] Micchelli, C.A., The saturation class and iterates of the Bernstein polynomials, Journ. Approx. Theory 8 (1973), pp. 1-18.

[P] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sc., Vol. 44, Springer-Verlag (1983).

[PW] Protter, M. and H. Weinberger, Maximum principles in partial differential equations, Prentice Hall, Englewood Cliffs, New Jersey (1967).

[R] Rudin, W., Functional analysis, McGraw-Hill (1973). [Ta] Taira, K., Semigroups and boundary value problems II, Proc. Japan

Acad. 58, Ser. A, (1982), pp. 277-280. [Tay] Taylor, A.E., General theory of functions and integration, Blaisdell,

(N.Y.),(1965). [Ti] Timmermans, C.A., On'Co-semigroups in a space of bounded con

tinuous functions in the case of entrance boundary points, Rep. 87-24, Delft Univ. of Techn., Fac. of Math, and Inf. (1987).

[Tr] Trotter, H.F., Approximation of semigroups of operators, Pacific Journ. Math. 8 (1958), pp. 887-919.

[Ve] Ventcel, A.D., On boundary conditions for multidimensional diffusion processes, Th. of Probl. Appl. 4 (1959), pp. 164-177.

[Vo] Voronowskaya, E., Determination de la forme asymptotique d'ap-proximation des fonctions par des polynömes de Bernstein, CR. Acad. Sci. URSS, (1932), pp. 79-85.

[Y] Yosida, K., Functional analysis, Springer-Verlag (1978).

SAMENVATTING

189

In dit proefschrift houden we ons bezig met de theorie van halfgroepen van operatoren in Banach ruimten alsmede met saturatieproblemen in de approximatietheone.

Saturatie is een interessant verschijnsel in de approximatietheorie. Indien (L ) een rij lineaire operatoren is in een Banach ruimte X, sterk convergent naar de identiteit-operator in X, dan betreft het saturatiep rob leem de bepaling van de optimale approximatie-orde en de (niet-triviale) klasse van elementen in X die met deze optimale orde geapproximeerd kunnen worden.

In hoofdstuk 3 worden saturatieproblemen voor Bernstein operatoren in C[0,1] en C [0,1], m = 1,2,... beschouwd. De resultaten voor C [0,1] zijn nieuw, (Theorem 3.5.10). De methode van het gebruik van halfgroepen van operatoren om saturatieproblemen te onderzoeken werd eerder toegepast, onder andere door Becker en Nessel in 1975. Het aantal toepassingen op concrete approxi-matieprocessen was echter zeer beperkt.

De hier ontwikkelde methode is nieuw en bouwt voort op de door Butzer en Berens (1967) ontwikkelde theorie van duale halfgroepen. Sterk continue halfgroepen van operatoren kunnen op injectieve wijze geassocieerd worden met hun infinitesimale generator. De hier gehanteerde methode maakt slechts gebruik van de infinitesimale generator en laat de halfgroep zelf buiten beschouwing. Daardoor is er een goed hanteerbare theorie voor het oplossen van saturatieproblemen ontstaan.

In hoofdstuk 3 worden de toepassingen gegeven. De algemene theorie wordt behandeld in hoofdstuk 2. De Banach ruimte waarin we werken is C(J), waarbij J een open (niet noodzakelijk begrensd) interval op de reële as is. Het opzetten van de theorie is mogelijk geworden door de randpunten van J te onderscheiden in reguliere, exit, entrance en natuurlijke randpunten. Dit is

190

een verdeling die Feller in 1952 maakte bij een onderzoek naar het verband tussen parabolische differentiaalvergelijkingen en halfgroepen van operatoren. Mede door deze classificatie is het mogelijk om noodzakelijke en voldoende voorwaarden te geven voor a en f) opdat de operator A, gedefinieerd door

' D(A) = {f G C[r j , r2] | f e C2(r {,r2),

lim (aD2f + 0Df)(x) = 0, i = 1,2} x-+ri i

Af = aD2f + £Df voor f E D(A)

een infinitesimale generator is van een halfgroep in C[r . , r-] (met supremum-norm).

De classificatie van de randpunten en de voorbereidende theorie wordt gegeven in hoofdstuk 1.

ACKNOWLEDGEMENT

191

The completion of this thesis and the development to its final form would not have been possible without the help of others. In particular I wish to express my sincere gratitude to Professor Philippe Clément for his valuable and constructive criticisms. He also provided the needed expertise and encouragement during the preparation of this manuscript. Further I wish to thank Angelina de Wit for her very efficient typing of the manuscript. Finally, I wish to thank my wife Hanny and children Ellen and Henkjaap for their tolerance and understanding throughout the entire development of this thesis.

Kr raia

p.5 1.2 and 1.21: ü(av) for aDv

p.50 lines 1,3,7.11.18: D(av) for aDv

p.60 after (1.3.1): and h (x) - h„(x) = c (constant) at all points of

continuity of h and thus also of h„.

p.SO (2.4.4): A*(w w,w ) = (O.(uw')' - pw' - ((aw1)' - Pw'))(x ).0)

p.90 (2-1.5): B(wrw.w2) = (Oâw')' - |3w" - ({«')' - Pw))(xQ). 0)

p.161 last line: ... I defined as in (2.6.14), a second representation

0* tor A is obtained, namely ...

SEMIGROUPS OF OPERATORS, APPROXIMA TION AND SA ...

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