A Various Types of Almost Periodic Functions on Banach Spaces: … · 2011-02-23 · Bochner introduced the concept of almost automorphic functions in his papers [22], [23] and ...

International Mathematical Forum, Vol. 6, 2011, no. 20, 953 - 985

A Various Types of Almost Periodic

Functions on Banach Spaces: Part II

Farouk Cherif

SCANN-EPAM-SOUSSEAvenue du marechal Tito

[email protected]

Abstract

The existence of almost periodic, asymptotically almost periodic,almost automorphic, asymptotically almost automorphic, and pseudoalmost periodic solutions is one of the most attracting topics in thequalitative theory of differential equations due to their significance andapplications in physics, mechanics, mathematical biology, control the-ory, and others. In this paper, which is the second part of the varioustypes of almost periodic functions, we give some extentions of the notionof almost periodicity. All of these notions are studied from the point ofview of suitable applications to differential equations.

Mathematics Subject Classification: 42A75, 43A60, 35B15

Keywords: Almost automorphic, asymptotically almost periodic, Pseudoalmost periodic, Weakly almost periodic

1 Introduction

In 1955, Bochner [21] suggested another generalization of the concept of al-most periodicity that to say, almost automorphy. In the beginning of sixties,Bochner introduced the concept of almost automorphic functions in his papers[22], [23] and [24] in relation to some aspects of differential geometry. Thisconcept became a generalization of almost periodicity which is one of the mostattractive topics in the qualitative theory of differential equations because oftheir significance and applications in physics, mathematical biology, controltheory, and other related fields. Recently, the concept of almost automorphicfunctions has widely been used in the investigation of the existence of almostautomorphic solutions of various kinds of evolution equations by N’Guerekata

954 Farouk Cherif

and others. This notion was also used extensively in the theory of differen-tial equations. Some fundamental properties of almost periodic functions arenot verified by the almost automorphic functions, as example the property ofuniform continuity. Consequently, the research for the solutions almost auto-morphic for dynamic systems are more complicated.

Recently, Zhang defined a new generalization of almost periodic functions,the so-called pseudo almost periodic functions [103]. Thus, this new concept iswelcome to implement another existing generalization of the (Bochner) almostperiodicity, the notion of asymptotically almost periodicity due to Frechet.

2 The Asymptotically Almost Periodic Func-

tions

In the modern theory of differential equations, many authors apply the asymp-totic property of the functions to determine the existence and uniqueness ofalmost periodic solutions of evolution partial differential equations, retardedfunctional differential equations, and so forth, (see for example [73],[101] ).Let Ω be a closed subset of X, and let BC(R, X) (respectively, BC(R×Ω, X))be the space of bounded, continuous functions from R (resp., R×Ω) to X withsupremum norm.For f ∈ BC(R × Ω, X) and s ∈ R, the translate of f by s is the functionRsf (t, x) = f(t + s, x), t ∈ R and x ∈ Ω. Recall that f ∈ AP 0 (R × Ω, X)if and only if {Rsf, s ∈ R} is relatively compact in BC(R × Ω, X). DenotePc (Ω) the set of the compact subset of Ω.

Definition 2.1 A function f ∈ BC(R × Ω, X) is called asymptoticallyalmost periodic in t and uniform on Pc (Ω) if for every ε > 0 and everyK ∈ Pc (Ω) , there exists a relatively dense subset P and a bounded subset Cof R such that

‖f(t+ h, x) − f(t, x)‖ < ε, (h ∈ P, t, t+ h ∈ R \ C, x ∈ K)

The collection of such functions will be denoted by AAP (R×Ω,X) .

Remark 2.2 Clearly, if C = ∅ then the asymptotically almost periodicitywill reduce to the almost periodicity.

Let us give some properties of the asymptotically almost periodic functions

• Let f1, f2, f ∈ AAP (R×Ω,X) . Then, f1 + f2 ∈ AAP (R×Ω,X) andλf ∈ AAP (R×Ω,X) for any λ ∈ R.

A various types of AP functions II 955

• Let f : [0,+∞[ −→ X, then f ∈ AAP ([0,+∞[ , X) if and only iff = g|[0,+∞[ + ϕ, where g ∈ AP (R, X) and ϕ ∈ BC

(R × R+, X

)with

limt→+∞ ϕ (t) = 0.

• Let f : R×Ω −→ X, then f ∈ AAP (R×Ω,X) if and only if there existsg ∈ AP (R×Ω, X) and ϕ ∈ BC

(R × R+, X

)with

lim|t|→+∞

supx∈K

|ϕ (t, x)| = 0

for all K ∈ Pc (Ω) such that

f = g + ϕ.

Moreover this decomposition is unique. The functions g and ϕ are calledthe principal and the corrective terms of f respectively. We denote g =fap and ϕ = fe.

• If f ∈ AAP (R×Ω,X) and F ∈ AAP (R,Ω) , then the compositionf (·, F (·)) is in AAP (R, X) .

• If f ∈ AAP (R×Ω,X) , then f is uniformly continuous on R×Ω.

• If f ∈ AAP (R×Ω,X) , then the range f (R×Ω) is relatively compact.

Example 2.3 The function φ(t) = cos2 t+cos2√

3t+ 1t2+1

is in AAP (R,R)

Remark 2.4 One can extend the notion above to asymptotically C(n)-almostperiodic functions [34].

3 The Pseudo Almost Periodic Functions

The concept of pseudo almost periodicity (p.a.p.) was introduced by Zhang[103], [105], and [106] in the early nineties. It is a natural generalization of theclassical almost periodicity in the sense of Bochner. Thus, this new conceptis welcome to implement another existing generalization of almost periodicity,the so-called asymptotically almost periodicity (a.a.p) due to Frechet [56], and[57]. Moreover, one should mention that the concept of p.a.p. is a special caseof the well-known Besicovitch almost periodicity of functions of order 1 denotedby B1 (R, X) . Let (X, ‖·‖) , (Y, ‖·‖Y ) be two Banach spaces. Let BC (R, X)(respectively, BC (R × Y,X)) denote the collection of all X−valued boundedcontinuous (respectively, the class of jointly bounded continuous functions F :R × Y −→ X). It is clear that the space BC (R, X) equipped with the supnorm defined by

‖u‖∞ = supt∈R

‖u(t)‖

956 Farouk Cherif

is a Banach space. Furthermore, C (R, Y ) (respectively, C (R × Y,X)) denotesthe class of continuous functions from R into Y (respectively, the class ofjointly continuous functions F : R × Y −→ X). Define the class of functionsPAP0 (R, X) and PAP0 (R × Y,X) respectively as follows:

PAP0 (R, X) =

⎧⎨⎩f ∈ BC (R, X) / lim

T→+∞1

2T

T∫−T

‖f(t)‖ dt = 0

⎫⎬⎭ ,

and

PAP0 (R ×X) =

⎧⎨⎩F ∈ BC (R × Y,X) / lim

T→+∞1

2T

T∫−T

‖F (t, y)‖ dt = 0

⎫⎬⎭

uniformly in y ∈ Y.

Definition 3.1 A function f ∈ BC (R, X) is called pseudo almost periodicif it can be expressed as

f = h+ ϕ,

where h ∈ AP (R, X) and ϕ ∈ PAP0 (R, X) . The collection of such func-tions will be denoted by PAP (R, X) .The functions h and ϕ in above definition are respectively called the almostperiodic component and the ergodic perturbation of the pseudo almost periodicfunction f.

Definition 3.2 A function F ∈ BC (R×Y,X) is called pseudo almost pe-riodic in t ∈ R uniformly in y ∈ Y if it can be expressed as

F = G+ Φ,

where G ∈ AP (R×Y,X) and Φ ∈ PAP0 (R×Y,X) . The collection of suchfunctions will be denoted by PAP (R×Y,X) .

Now we give a list of the principal properties of the pseudo almost periodicfunctions.

• The mean value, the Bohr-Fourier coefficients and the Bohr-Fourier ex-ponents of every pseudo almost periodic function are the same of itsalmost periodic component.

• For all a ∈ R one has PAP ([a,+∞[ , X) is a Banach space [104].

• The range of a pseudo almost periodic function is bounded, but notrelatively compact.


• Let f = h+ϕ ∈ PAP (R, X) , where (h, ϕ) ∈ AP (R, X)×PAP0 (R, X)then

{h(t), t ∈ R} ⊂ {f(t), t ∈ R}.• Let (fn)n ⊂ PAP (R, X) be a sequence of functions. If (fn)n converges

uniformly to some f , then f ∈ PAP (R, X) [41].

• Let u : Ω −→ Y bounded and uniformly continuous function and suchthat f(R) ⊂ Ω ⊂ X . Then u ◦ f ∈ PAP (R, Y ) .

• Let F ∈ BC (R×Y,X) , then F ∈ PAP (R×Y,X) if it satisfies the twofollowing conditions:

i) ∀y ∈ Y, f(·, y) ∈ PAP (R, X) ,

ii) f is uniformly continuous on each compact K of Y with respect tothe second variable y.

In [103], Zhang gave a composition theorem of pseudo almost periodicfunctions in finite dimensional Banach spaces. More explicitly, suppose f :R × Y −→ X; h : R −→ Y are pseudo almost periodic functions under theassumption that f is lipschitzian with respect to the second variable; thenf(·, h(·)) : R −→ X is a pseudo almost periodic function, where X and Yare Banach spaces with finite dimensions. Then this result was generalizedto general Banach spaces by Amir and Maniar [4] under the same conditionas in [103]. On the other hand, Hong-xu Li and al [68], gave the same resultunder the essential condition that f is uniformly continuous and bounded inR × h(R). Here, we give the result of [4].

• Let f ∈ PAP (R × Y,X) satisfy the Lipschitz condition

‖f(t, y)− f(t, z)‖ ≤ L ‖y − z‖ , for all y, z ∈ Y and t ∈ R.

If h ∈ PAP (R, Y ) , then the function f(·, h (·)) ∈ PAP (R, X) [4].

Example 3.3 The function

φ(t) = cos2 t+ cos2√

3t+ exp(−t2 cos2 t

)is pseudo almost periodic function.


ψ(t, y) = sin t+ sinπt+

+∞∫−∞

K (y − t)

t2 + ω2dt

is pseudo almost periodic function, where K (·) ∈ L1 (R) and ω, y ∈ R.

958 Farouk Cherif

Example 3.5 Let N ∈ N, N > 6, the function

h(t, y) = cos y(sin t+ sinπt+ t |sinπt|tN

)is pseudo almost periodic function in t ∈ R uniformly in y.

3.1 The Sp-Pseudo almost periodic Functions

Let (X, ‖·‖) be a Banach space and let p ≥ 1. In a recent paper by N’Guerekataand Pankov [61], the concept of The Stepanov-like pseudo almost periodicfunctions (or Sp-pseudo almost periodic functions) was introduced to extendthe concept of pseudo almost periodicity.

Definition 3.6 The Bochner transform f b (t, s) , t ∈ R, s ∈ [0, 1], of afunction f ∈ C (R, X) , is defined by

f b (t, s) := f(t+ s).

Definition 3.7 The Bochner transform F b (t, s, u) , t ∈ R, s ∈ [0, 1] , u ∈ Xof a function F ∈ C (R ×X,X) , is defined by

F b (t, s, u) := F (t+ s, u)

for each u ∈ X.

Definition 3.8 Let p ∈ [1,+∞[. The space BSp(R, X) of all Stepanovbounded functions, with the exponent p, consists of all measurable functions fon R with values in X such that

f b ∈ L∞ (R, Lp(]0, 1[ , X)) . This is a Banach space with the norm

‖f‖Sp =∥∥f b

∥∥L∞(�,Lp)

= supt∈�

(∫ t+1

t

‖f(s)‖p ds

) 1p

.

Definition 3.9 A function f ∈ BSp(R, X) is called Sp-pseudo almost pe-riodic (or Stepanov-like pseudo almost periodic) if it can be expressed as f =h + ϕ; where hb ∈ AP (Lp ((0, 1), X)) and ϕb ∈ PAP0 (Lp ([0, 1] , X)) . Thecollection of such functions will be denoted by PAPSp(R, X).

In other words, a function f ∈ Lp(R;X) is said to be Sp-pseudo almostperiodic if its Bochner transform f b : R −→ Lp([0, 1] ;X) is pseudo almostperiodic in the sense that there exists two functions h;ϕ : R −→ X such thatf = h + ϕ, where hb ∈ AP (Lp([0, 1] , X)) and ϕb ∈ PAP0(L

p([0, 1] , X)), thatis

limT→+∞

1

2T

∫ T

−T

(∫ t+1

t

‖ϕ(s)‖p ds

) 1p

dt = 0.

Similarly, one gets the following definition.


Definition 3.10 A function F ∈ BSp(R×X,X) , (t, u) −→ F (t, u) withF (·, u) ∈ Lp (R, X) for each u ∈ X, is called Sp-pseudo almost periodic int ∈ R uniformly in u ∈ X if t −→ F (t, u) is Sp-pseudo almost periodic foreach u ∈ X.

This means, there exists two functions H,Φ : R×X −→ X such that

F = H + Φ,

where Hb ∈ AP (R× Lp ((0, 1), X)) and Φb ∈ PAP0 (R× Lp ((0, 1), X)) , thatis

limT→+∞

1

2T

∫ T

−T

(∫ t+1

t

‖Φ(s, u)‖p ds

) 1p

dt = 0.

uniformly in u ∈ X. We will denote by PAPSp(R × X, X) the collection ofsuch functions. Now, we give a list of the principal properties of the Sp-pseudoalmost periodic functions [40].

• If 1 ≤ p < q < +∞ and f ∈ Lq (R, X) is Sq-pseudo almost periodic,then f is Sp-pseudo almost periodic.

• Obviously, the following inclusions hold:

AP (R, X) ⊂ PAP (R, X) ⊂ PAPSp(R × X, X)

• If f ∈ PAP (R, X), then f is Sp-pseudo almost periodic for any p ∈[1,+∞[ .The next result is very useful in the theory of differential equations.

• Let F ∈ PAPSp(R × X, X). Suppose that F (t, u) is Lipschitzian in u ∈X uniformly in t ∈ R, that there exists L > 0 such

‖F (t, u) − F (t, v)‖ ≤ L ‖u− v‖

for all t ∈ R, (u, v) ∈ X ×X. If φ ∈ PAPSp(R, X), then

Γ : R −→ Xt −→ F (t, φ (t))

belongs to PAPSp(R, X).

960 Farouk Cherif

3.2 Pseudo almost periodic Functions of class p

In order to study the existence and uniqueness of pseudo almost periodic so-lutions to some abstract partial neutral functional–differential equations, Di-agana and Hernandez [42] introduced the new space of functions defined foreach p > 0 by

PAP0 (R, X, p) =

⎧⎨⎩f ∈ BC (R, X) / lim

T→+∞1

2T

T∫−T

(sup

θ∈[t−p,t]

‖f(θ)‖X

)dt = 0

⎫⎬⎭

and PAP0 (R × Y,X, p) by⎧⎨⎩F ∈ BC (R × Y,X) / lim

T→+∞1

2T

T∫−T

(sup

θ∈[t−p,t]

‖F (θ, u)‖)dt = 0

⎫⎬⎭ ,

where in both cases the limit (as T → ∞) is uniform in compact subset ofY . In view of the previous definitions it is clear that PAP0 (R, X, p) andPAP0 (R × Y,X, p) are continuously embedded in the spaces PAP0 (R, X)and PAP0 (R × Y,X), respectively. Furthermore, it is not hard to see thatPAP0 (R, X, p) and PAP0 (R × Y,X, p) are closed in the spaces PAP0 (R, X)and PAP0 (R × Y,X), respectively.

Remark 3.11 The spaces PAP0 (R, X, p) and PAP0 (R × Y,X, p) endowedwith the uniform convergence topology are Banach spaces.

Definition 3.12 A function f ∈ BC (R, X) is called pseudo almost peri-odic of class p if it can be expressed as

f = h+ ϕ,

where h ∈ AP (R, X) and ϕ ∈ PAP0 (R, X, p) . The collection of such functionswill be denoted by PAP (R, X, p) .

Definition 3.13 A function F ∈ BC (R × Y, X) is called pseudo almostperiodic of class p in t ∈ R uniformly in y ∈ Y if it can be expressed as

F = G+ Φ,

where G ∈ AP (R×Y,X) and Φ ∈ PAP0 (R×Y,X, p) . The collection of suchfunctions will be denoted by PAP (R×Y,X, p) .

Now, we give a composition theorem in PAP (R×Y,X, p) .


• Let F ∈ PAP (R×Y,X, p) and let h ∈ PAP (R, Y, p). Assume that thereexists a continuous function L

F: R −→ [0,+∞[ satisfying

‖F (t, u) − F (t, v)‖X ≤ LF ‖u− v‖Y

for all t ∈ R, (u, v) ∈ Y × Y. If for each ξ ∈ PAP0(R,R) we have

lim supT→+∞

1

2T

T∫−T

supθ∈[t−p,t]

LF (θ) dt = limT→+∞

1

2T

T∫−T

ξ (t) supθ∈[t−p,t]

LF (θ) dt = 0

thenΓ : R −→ X

t −→ F (t, h (t))

belongs to PAP (R, X, p).

3.3 Pseudo almost periodic Functions of class ∞To deal with unbounded delays, Diagana [43] introduced the following newspaces of functions:

PAP0 (R, X,∞) :=⋂p>0

PAP0 (R, X, p)

and

PAP0 (R×Y,X,∞) :=⋂p>0

PAP0 (R×Y,X, p)

Obviously, PAP0 (R, X,∞) and PAP0 (R × Y, X,∞) are respectively closedsubspaces of PAP0 (R, X, p) and PAP0 (R × Y, X, p), and hence both are Ba-nach spaces. Furthermore, it is not hard to see that:

PAP0 (R, X,∞) ∩ AP (R, X) = {0} .In view of the above, we introduce in the same way as the above section thefollowing new classes of functions.

Definition 3.14 A function f ∈ BC (R, X) is called pseudo almost peri-odic of class infinity if it can be expressed as

f = h+ ϕ,

where h ∈ AP (R, X) and ϕ ∈ PAP0 (R, X,∞) . The collection of such func-tions will be denoted by PAP (R, X,∞) .

962 Farouk Cherif

Definition 3.15 A function F ∈ BC (R×Y,X) is called pseudo almostperiodic of class infinity if it can be expressed as

F = G+ Φ,

where G ∈ AP (R×Y,X) and Φ ∈ PAP0 (R×Y,X,∞) . The collection of suchfunctions will be denoted by PAP (R×Y,X,∞) .

We give here some properties of PAP (R×Y,X,∞) .

• Let F ∈ PAP (R×Y,X,∞) and let h ∈ PAP (R, Y,∞). Assume thatthere exists a continuous function LF : R −→ [0,+∞[ satisfying

‖F (t, u) − F (t, v)‖X ≤ LF ‖u− v‖Y

for all t ∈ R, (u, v) ∈ X × X. If for each ξ ∈ PAP0(R,R) and for allp > 0 we have

limT→+∞

1

2T

T∫−T

(sup

θ∈[t−p,t]

LF (θ)

)ξ (t) dt = 0

thenΓ : R −→ Y

t −→ F (t, h (t))

belongs to PAP (R, Y,∞)[43].

• We denote by (Z, ‖·‖Z) a Banach space continuously embedded into Xsuch that:

(H1) : The function s −→ A(s)U(t, s) defined from ]−∞, t[ into L(Z,X)is strongly measurable and there exists a nonincreasing function H :[0,+∞[ −→ [0,+∞[ and δ > 0 with e−δsH(s) ∈ L1([0,+∞[) such that:

‖A(s)U(t, s)‖L(Z,X) ≤ e−δsH(t− s), t > s,

where A(t) : D(A(t)) ⊂ X −→ X is a family of densely defined closedlinear operators on a common domain D = D(A(t)) and {U(t, s) : t ≥ swith t, s ∈ R} the associated evolution family of operators associated tothe linear differential equation u′ = Au. Then for all u ∈ PAP0 (R,Z,∞)the function defined by

v(t) :=

∫ t

−∞A(s)U(t, s)u(s)ds, t ∈ R

belongs to PAP0 (R, X,∞)[43].


3.4 Weighted Pseudo-almost periodic Functions

In Diagana [46], a new generalization of Bohr almost periodic functions wasintroduced. This new concept is called weighted pseudo-almost periodicityand implements in a natural fashion the notion of pseudo-almost periodicityintroduced in the literature by Zhang [105], and [106]. To construct thosenew spaces, the main idea consists of enlarging the so-called ergodic compo-nent, utilized in Zhang’s definition of pseudo-almost periodicity, with the helpof a weighted measure dμ (x) = � (x) dx, where � : R −→ ]0,+∞[ is a lo-cally integrable function over R, which is commonly called weight. In otherwords weighted pseudo almost periodic functions are good generalizations ofthe Zhang’s pseudo almost periodic functions. Let U denote the collection ofall functions (weights) � : R −→ ]0,+∞[ which are locally integrable over R

such � (x) > 0 for almost each x ∈ R. For � ∈ U and for r > 0, we set

m(r, �) :=

r∫−r

ρ(x)dx

Throughout this section , the sets of weights U∞ and UB stand respectively for

U∞ :={� ∈ U, lim

r→∞m(r, �) = +∞

}and

UB :=

{� ∈ U, � is bounded and inf

x∈R�(x) > 0

}.

Obviously, UB ⊂ U∞ ⊂ U , with strict inclusions. To introduce those weightedpseudo-almost periodic functions, we need to define the ”weighted ergodic”space PAP0 (X, �) Hence, weighted pseudo-almost periodic functions willthen appear as perturbations of almost periodic functions by elements ofPAP0 (X, �) . Let � ∈ U∞. Define

PAP0 (R, X, �) =

⎧⎨⎩f ∈ BC (R, X) , lim

T→+∞1

m(T, �)

T∫−T

‖f(t)‖ � (t) dt = 0

⎫⎬⎭ .

Clearly, the spaces PAP0 (R, X, �) are richer than PAP0 (R, X) and give riseto an enlarged space of pseudo-almost periodic functions. In the same way,we define PAP0 (R × Y,X, �) as the collection of jointly continuous functionsF : R × Y −→ X such that F (., y) is bounded for each y ∈ Y and

limT→+∞

1

m(T, �)

T∫−T

‖f(t, y)‖ � (t) dt = 0

uniformly in y ∈ Y.

964 Farouk Cherif

Definition 3.16 Let � ∈ U∞. A function f ∈ BC (R, X) is called weightedpseudo-almost periodic (or -pseudo-almost periodic) if it can be expressed asf = g + φ , where g ∈ AP (R, X) and φ ∈ PAP0 (R, X, �). The collection ofsuch functions will be denoted by PAP (R, X, �) .

Remark 3.17 The functions g and φ appearing in definition above are re-spectively called the almost periodic and the weighted ergodic perturbation com-ponents of f .

We give here some properties of the space PAP (R, X, �) .

• The space PAP (R, X, �) is a closed subspace of (B (R, X) , ‖·‖∞). Thisyields PAP (R, X, �) is a Banach space [53].

• Let � ∈ U∞ and assume that for each τ ∈ R

sups∈R

(� (s+ τ)

� (s)

)<∞ and sup

T>0

(m (T + τ, �)

m(T, �)

)<∞.

Then the space PAP (R, X, �) is translation invariant.

• Fix � ∈ U∞. The decomposition of a weighted pseudo-almost periodicfunction f = g + φ , where g ∈ AP (R, X) and φ ∈ PAP0 (R, X, �) , isunique [44].

• The functions g and φ appearing in definition above are unique. This ismainly based upon the fact that g(R) ⊂ f(R). Hence,

PAP (R, X, �) = AP (R, X) ⊕ PAP0 (R, X, �)

.

• Let �1,∈ �2 U∞ such that �1

�2∈ UB, then

PAP (R, X, �1) = PAP (R, X, �2) .

• Let � ∈ UB, then PAP (R, X, �) = PAP (R, X) .

• Let h ∈ PAP (R, Y, �) and f ∈ PAP (R×Y,X, p) satisfying the Lipschitzcondition

‖f(t, u) − f(t, v)‖ ≤ Kf ‖u− v‖Y for all u, v ∈ Y, t ∈ R,

then f(·, h (·)) ∈ PAP (R, X, �) .


Example 3.18 [46] Let �1(x) = 1+x2 for each x ∈ R. It is easy to see thatμ (T, �1) = 2T + 2

3T 3, and hence �1 ∈ U∞. Set f(x) = cosx+ cos

√3x + e−|x|.

Obviously, the function(x −→ cosx+ cos

√3x) ∈ AP (R,R) and

limT→+∞

1

2(T + T 3

3

)T∫

−T

e−|x|�1(x)dx = limT→+∞

1

2(T + T 3

3

)T∫

−T

e−|x|(1 + x2)dx = 0.

Hence f ∈ PAP (R,R, �) .

Remark 3.19 Recently, Diagana and al [46] introduce and study a newclass of functions called Stepanov-like weighted pseud-oalmost periodic func-tion or Sp -weighted pseudo -almost periodic functions which generalize in anatural manner weighted pseudo-almost periodic functions.

4 The Almost Automorphic Functions

Definition 4.1 A function f ∈ C(R, X) is said to be almost automorphicif for every sequence of real numbers (s′n)n∈� there exists a subsequence (sn)n∈�such that

g(t) := limn→∞

f(t+ sn)

is well defined for each t ∈ R, and

limn→∞

g(t− sn) = f(t)

for each t ∈ R.

Remark 4.2 The function g in definition above is measurable but not nec-essarily continuous. Moreover, if g is continuous, then f is uniformly contin-uous. If the convergence above is uniform in t ∈ R, then f is almost periodic.Denote by AA(R, X)the collection of all almost automorphic functions R → X.We will denote by AAu(X) the closed subspace of all functions f ∈ AA(X) withg ∈ C(R, X). Equivalently, f ∈ AAu(X) if and only if f is almost automorphic,and the convergences in definition above are uniform on compact intervals, thatis, in the Frechet space C(R, X). Indeed, if f is almost automorphic, then itsrange is relatively compact. Obviously, the following inclusions hold:

AP (R, X) ⊂ AAu(R, X) ⊂ AA(R, X) ⊂ BC(R, X). (1)

Among others things, almost automorphy functions satisfy the following prop-erties:

966 Farouk Cherif

• If f1, f2 ∈ AA(R, X), then f1, f2 ∈ AA(R, X).

• If f ∈ AA(R, X) and λ ∈ R then λf ∈ AA(R, X).

• If f ∈ AA(R, X), then fα ∈ AA(R, X) where fα : R −→ X is defined byfα (·) = f(· + α).

• Let’s f ∈ AA(R, X),the range Rf := {f(t), t ∈ R} is relatively compactin X, thus f is bounded in norm.

• If fn −→ f uniformly on R where fn ∈ AA(R, X), then f ∈ AA(R, X).

• (AA(R, X), ‖ · ‖∞) is a Banach space.

• If f ∈ AA(R, X) and g ∈ L1 (R) , then the convolution f ∗g ∈ AA(R, X).

• Let K ⊂ X and Ω ⊂ R. we denote by CK (Ω ×X,X) the set of allfunctions f : Ω × X −→ X satisfying f(t, .) is uniformly continuouson K uniformly for t ∈ Ω. Let x ∈ AA(R, X), K = {x(t), t ∈ R} andf ∈ AA (R×X,X) ∩ CK (Ω ×X,X) , then f(., x (.)) ∈ AA (R, X) .

Similarly, we have

Definition 4.3 A function f ∈ C(R×X,X) is said to be almost automor-phic in t ∈ R for each x ∈ X if for every sequence of real numbers (s′n)n∈Nthere exists a subsequence (sn)n∈N such that

g(t, x) := limn→∞

f(t+ sn, x)

is well defined for each t ∈ R, x ∈ X and

limn→∞

g(t− sn, x) = f(t, x)

for each t ∈ R, x ∈ X. The collection of such functions will be denote byAA (R×X,X) .

Example 4.4 A classical example of an almost automorphic function whichis not almost periodic, as it is not uniformly continuous, is the function definedby

f(t) = cos

(1

2 + sin t+ sin√

2t

), t ∈ R.

Example 4.5 [102] The function

f(t, x) = sin1

2 + cos t+ cos√

2tcos x,

is almost automorphic in t ∈ R for each x ∈ X, where X = L2 ([0, 1]) .


4.1 Asymptotically almost automorphic functions

It is known that a lot of parabolic partial differential equations with almostperiodic coefficients, including Fisher type equations that have biological signif-icance as well as some quasi-periodically forced ordinary differential equations,have solutions being almost automorphic but not almost periodic. Therefore,in recent years, the theory of almost automorphic functions and asymptoticallyalmost automorphic functions has been developed extensively (see, [34],[58]and [59]). Recently, the asymptotically almost periodic and almost periodicsolutions for the integrodifferential equations with a nonlocal initial conditionwas studied in [64].

We denote by C0 ([0,+∞[ , X) the space of all continuous functions h :[0,+∞[ −→ X such that lim

t→+∞h(t) = 0, and by C0 ([0,+∞[ ×X,X) the space

of all continuous functions h : [0,+∞[ ×X −→ X such that limt→+∞

h(t, x) = 0

uniformly for all x ∈ X.

Definition 4.6 A function f ∈ C([0,+∞[ , X) (resp. f ∈ C([0,+∞[ ×X,X)) is said to be asymptotically almost automorphic (resp. asymptoticallyalmost automorphic in t uniformly for x ∈ X) if it admits a decomposition

f = g + h, t ∈ [0,+∞[ ,

where g ∈ AA (R, X) (resp. ∈ AA (R ×X,X))and h ∈ C0 ([0,+∞[ , X) (resp.∈ C0 ([0,+∞[ ×X,X)). Denote by AAA (R, X) (resp. AAA (R ×X,X)) theset of all such functions.

Remark 4.7 The functions g and h defined above are called, respectively,the principal and corrective terms of the function f and the decoposition isunique.

Now we give a list of basic results about asymptotically almost automorphicfunctions. For the proofs we refer the reader to see ( [47], [58],[76] and thereferences therein).

• Any asymptotically almost automorphic function is bounded in norm.

• (AAA (R, X) , |·|) is a Banach space, where

|f | = supt∈�

‖f(t)‖ + supt∈[0,+∞[

‖h(t)‖

with f = g + h ∈ AA (R, X) + C0 ([0,+∞[ , X) .

• Assume f ∈ AAA ([0,+∞[ , X) admits a decomposition f = g+h, whereg ∈ AA (R, X) and h ∈ C0 ([0,+∞[ , X) , then

{g(t), t ∈ R} ⊂ {f(t), t ∈ [0,+∞[}.

968 Farouk Cherif

• Let f ∈ AAA ([0,+∞[ , X) . The range Rf = {f(t), t ∈ [0,+∞[} is rela-tively compact in X.

• Suppose that f (t, x) = g(t, x)+h(t, x) is an asymptotically almost auto-morphic function with g (t, x) ∈ AA (R ×X,X), h (t, x) ∈ C0 (R ×X,X)and g (t, x) is uniformly continuous on any bounded subset Ω ⊂ Xuniformly for t ∈ R. Then x(t) ∈ AAA (R, X) implies f (·, x (·)) ∈AAA (R, X) .

4.2 Pseudo-almost automorphic functions

The new concept of pseudo almost automorphy generalizes the one of pseudoalmost periodicity, in fact, a pseudo almost automorphic function is the sumof an almost automorphic function and of an ergodic perturbation. Thesefunctions were introduced recently by Liang, Xiao and Zhang in [100] and [76].In the literature, many works are devoted to the existence of almost periodicand almost automorphic solutions for differential equations, but results aboutpseudo almost automorphic solutions are rare. The notationAA0 (R, X) standsfor the spaces of functions

AA0 (R, X) =

⎧⎨⎩f ∈ BC (R, X) / lim

T→+∞1

2T

T∫−T

‖f(t)‖ dt = 0

⎫⎬⎭ .

We also define

AA0 (R ×X,X) =

⎧⎨⎩F ∈ BC (R ×X,X) / lim

T→+∞1

2T

T∫−T

‖F (t, u)‖ dt = 0

⎫⎬⎭ .

uniformly for u in any bounded subset of X.

Definition 4.8 A function f : R → X (resp.R×X −→ X) is called pseudo-almost automorphic if it can be decomposed as f = g + ϕ, where g ∈ AA(X)(resp. AA(R×X,X)) and ϕ ∈ AA0(X)(resp. AA0(R×X, X)). The class ofall such functions will be denote by PAA(R, X) (resp. PAA(R ×X,X)).


f(t) = sin

(1

2 − sin t− sinπt

)+

1√1 + t2

belongs to PAA(R,R).


Example 4.10 [100] A classical example of a pseudo-almost automorphicfunction which is the following

f(t, x) = x sin

(1

2 + cos t+ cos√

2t

)+ max

k∈�

(e−(t±k2)

2)cosx, t ∈ R, x ∈ X ,

where X = L2 ([0, 1]) .

Among others things, pseudo-almost automorphic functions satisfy the follow-ing properties:

• (PAA(R, X), ‖ · ‖∞) is a Banach space.

• Let h ∈ AA(R, X), K = {h(t), t ∈ R} and f ∈ AA(R×X,X)∩CK(R×X,X). Then f (., h (.)) ∈ AA(R, X) [102].

• If f ∈ AA(R, X), then fα ∈ AA(R, X) where fα : R −→ X is defined byfα (·) = f(· + α).

• Let’s f ∈ AA(R, X),the range Rf := {f(t), t ∈ R} is relatively compactin X, thus f is bounded in norm.

• If fn −→ f uniformly on R where fn ∈ AA(R, X), then f ∈ AA(R, X).

• PAA(R, X) is a translation invariant closed subspace of BC(R, X) con-taining the constant functions. Furthermore,

PAA(R, X) = AA(R, X) ⊕ PAP0(R, X)

.

4.3 Weighted Pseudo-almost periodic Functions

Recently, N’Guerekata introduced the concept of weighted pseudo almost au-tomorphic functions, which generalizes the one of weighted pseudo almost pe-riodicity, and the author proved some interesting properties of the space ofweighted pseudo almost automorphic functions like the completeness and thecomposition theorem, which have many applications in the context of dif-ferential equations. Define as in [46], the sets of weights U∞ and UB standrespectively for

U∞ :={� ∈ U, lim

r→∞m(T, �) = +∞

}and

UB :=

{� ∈ U, � is bounded and inf

x∈R�(x) > 0

}

970 Farouk Cherif

Obviously, UB ⊂ U∞ ⊂ U , with strict inclusions.To introduce those weighted pseudo-almost automorphic functions, we need

to define the “weighted ergodic” space PAA0 (R, X, �) Weighted pseudo-almostautomorphic functions will then appear as perturbations of almost periodicfunctions by elements of PAP0 (X, �) . Let � ∈ U∞. Define

PAA0 (R, X, �) ==

⎧⎨⎩f ∈ BC (R, X) / lim

T→+∞1

m(T, �)

T∫−T

‖f(t)‖ � (t) dt = 0

⎫⎬⎭ .

Clearly, the spaces PAP0 (X, �)are richer than PAP0 (X) and give rise toan enlarged space of pseudo-almost automorphic functions.

Similarly, we define PAA0 (R ×X,X, ρ) as the collection of jointly conti-nuous functions F : R×X −→ X such that F (., y) is bounded for each y ∈ Xand

limT→+∞

1

m(T, �)

T∫−T

‖f(t, y)‖ ρ (t) dt = 0. (2)

uniformly in y ∈ X.

Definition 4.11 Let � ∈ U∞. A function f ∈ B (R, X) is called weightedpseudo-almost automorphic if it can be expressed as f = g + φ , where g ∈AA (R, X) and φ ∈ PAA0 (R, X, �). The collection of such functions will bedenoted by WPAA (R, X, �) .

Similarly, one has the space WPAA (R ×X,X, �) defined by

{f = g + φ ∈ BC (R ×X,X) , (g, φ) ∈ AA (R ×X,X) × PAA0 (R ×X,X, ρ)} .Among others things, pseudo-almost automorphic functions satisfy the fol-

lowing properties:

• When ρ = 1, we obtain the spaces PAA (R, X) and PAA (R ×X,X) .

• If ρ ∈ UB and f = g+φ with g ∈ AA (R ×X,X) and φ ∈ PAA0 (R, X, ρ) ,then g(R) ⊂ f(R).

• If ρ ∈ UB, then the decomposition f = g + φ with g ∈ AA (R, X) andφ ∈ PAA0 (R, X, ρ) is unique.

• Let ρ1, ρ2 ∈ U∞ such that ρ2

ρ1∈ UB, then

WPAA (R, X, �1) = WPAA (R, X, �2) .


• If ρ ∈ UB, then (WPAA(R, X, ρ), ‖ · ‖∞) is a Banach space.

• Let f ∈ BC (R, X) and ρ ∈ UB. Then f ∈ PAA0 (R, X) if and only iffor every ε > 0

limT→+∞

1

m(T, ρ)mes ({t ∈ [−T, T ] , ‖f(t)‖ ≥ ε}) = 0.

• Let f = g+φ ∈WPAA (R, X, ρ) with ρ ∈ U∞, g (t, x) ∈ AA (R ×X,X)and φ ∈ PAA0 (R ×X,X, ρ) . Assume both f and g are lipschitzianin x uniformly in t ∈ R. If x(t) ∈ WPAA (R, X, ρ) , then f(·, x (·)) ∈WPAA (R, X) .

For the proofs and further properties of WPAA (R, X, ρ) one can see [17].

4.4 The Sp-Almost Automorphy Functions

Recently, N’Guerekata and Pankov introduced the concept of Stepanov-like al-most automorphy and applied this concept to study the existence and unique-ness of an almost automorphic solution to the nonautonomous evolution equa-tion in [61]. Moreover, Diagana and N’Guerekata used this concept to estab-lish the existence and uniqueness of an almost automorphic solution to theautonomous semilinear equation in [62]. The notion of Sp-Almost Automor-phy functions generalizes in a natural fashion the classical almost automorphyin the sense of Bochner. Moreover, the concept of Sp-almost automorphy was,subsequently, utilized to study the existence of weak Sp-almost automorphicsolutions to some parabolic evolution equations.

Definition 4.12 The Bochner transform f b (t, s) , t ∈ R, s ∈ [0, 1], of afunction f ∈ C (R, X) , is defined by

f b (t, s) := f(t+ s).

Definition 4.13 [85]. Let p ∈ [1,+∞[. The space BSp(R, X) of all Stepa-nov bounded functions, with the exponent p, consists of all measurable functionsf on R with values in X such that

f b ∈ L∞ (R, Lp((0, 1), X)) .This is a Banach space with the norm

‖f‖Sp =∥∥f b

∥∥L∞(�,Lp)

= supt∈R

(∫ t+1

t

‖f(s)‖p ds

) 1p

.

Definition 4.14 A function f ∈ BSp(R, X) is called Sp-almost automor-phic (Sp − a.a for short) if f b ∈ AA (Lp ((0, 1), X)). The collection of suchfunctions will be denoted by ASp(R, X).

972 Farouk Cherif

In other words, a function f ∈ Lploc(R;X) is said to be Sp- almost automor-

phic if its Bochner transform f b : R −→ Lp((0; 1);X) is almost automorphicin the sense that in the sense that for every sequence of real numbers (s′n)n∈�there exists a subsequence (sn)n∈� and a function g ∈ Lp

loc (R,X) such that

[∫ 1

0

‖ f(t+ sn + s) − g(t+ s) ‖p ds

] 1p

→ 0,

and [∫ t+1

t

‖ g(t+ s− sn) − f(s) ‖p ds

] 1p

→ 0

as n→ +∞ pointwise on R. Similarly one gets the following definition.

Definition 4.15 A function F ∈ BSp(R×X,X) , (t, u) −→ F (t, u) withF (·, u) ∈ Lp (R,X) for each u ∈ X, is called Sp-pseudo almost automorphic int ∈ R uniformly in u ∈ X if t −→ F (t, u) is Sp-pseudo automorphic for eachu ∈ X.

The collection of such functions will be denoted by ASp(R × X, X).Now we give a list of the principal properties of the Sp- almost automorphic

functions. For the proofs one can see [40],[85],[59],[60].

• If 1 ≤ p < q < +∞ and f ∈ Lq (R,X) is Sq- almost automorphic, thenf is Sp- almost automorphic.

• If p ≥ 1, then a linear combinaison of Sp- almost automorphic functionsis Sp- almost automorphic function.

• The operator J : ASp(R, X) −→ ASp(R, X), (Jx) (t) = x(−t) is well-defined and linear. Moreover it is an isometry and J2 = I.

• Let a a fixed real. The operator Ta : ASp(R, X) −→ ASp(R, X) suchthat (Tax) (t) = x(t+ a) leaves ASp(R, X) invariant.

• Let (fn)1≤n<∞ be a sequence of Sp- almost automorphic functions suchthat ‖fn − f‖Sp −→ 0, as n→ +∞, then f ∈ ASp(R, X).

Example 4.16 [47]. For ε ∈ ]0, 1

2

[,the function

f(t) =

{sin

(1

2+cos n+cos√

2n

)if t ∈ ]n− ε, n+ ε[ , t ∈ Z,

0 othewise

belongs to ASp(R,R).


5 The Weakly Almost Periodic Functions

Let H a Hilbert space. Recall that a function f ∈ C (R, H) is almost periodicif and only if the orbit τ(f) = {τr(f) = f(· + r), r ∈ R}of f is relatively com-pact in C (R, H) . If we require the orbit is relatively weakly compact, thenwe will get the concept of weakly almost periodicity another generalization ofalmost periodicity. This type of functions was first introduced by E. Eberlein[52] in 1949. This category of functions permits to obtain a more adequatedescription of the physical phenomena as turbulence, noise . . . Hence, the no-tion of weakly almost periodic functions is not a formal generalization of almostperiodic functions. Things will be much more complicated if one changes com-pactness to weak compactness. The complication lies in the obscure structureof the weak topology and the fact that the operators τr(·), although equicon-tinuous in the norm topology, are not equicontinuous in the weak topology.Let BC(R, H) denote the set of bounded continued functions from R to H.Note that (BC(R, H), ‖.‖∞) is a Banach space where ‖.‖∞ denotes the supnorm

‖f‖∞ := supt∈�

‖f (t)‖ .

For x ∈ H and μ ∈ H∗, ≺ μ, x � will denote the value of μ in x. Here≺ ., . � denotes duality ≺ H∗, H � . Since weak compactness is equivalent tosequential weak compactness in a Hilbert space we will say:

Definition 5.1 A function f : R −→ H is said to be weakly almost pe-riodic (Eberlein w.a.p.) if for every sequence of real numbers (s′n)n∈� thereexists a subsequence (sn)n∈� such that (f(t+ sn))n∈� is convergent in the weaksense; uniformly in t ∈ R. In other words, for every μ ∈ H∗; the sequence(≺ μ, f(t+ sn) �)n is uniformly convergent in t ∈ R.

Denote by WAP (R, H) the collection of weakly almost periodic function fromR into H . For A ⊂ H we denote A the closure of A for the strong topologyand A

wthe closure of A for the weak topology. Among other things, weakly

almost periodic functions satisfy the following properties([52], [108]) .

• WAP (R, H) is a Banach space for the norm norm ‖·‖∞ which containsthe constant functions.

• One can easily remark that

f ∈WAP (R, H) ⇐⇒{f ∈ BC(R, H)/ τ(f) w is compact

}.

• The WAP (R, H) is an algebra for the standard inner product.

974 Farouk Cherif

• A weakly almost periodic function is uniformly continuous.

• If f ∈ WAP (R, H),then

M{f} = M{f(t)}t =

⎛⎝ lim

T−→+∞1

2T

+T∫−T

f(t)dt

⎞⎠ ∈ H.

• If f ∈ WAP (R, H), then ∀r ∈ R, τr(f) ∈WAP (R, H).

• If f ∈ WAP (R, H), then ‖f‖p ∈WAP (R, H), for each p ≥ 0.

• If f, g ∈WAP (R, H), then fg ∈ WAP (R, H).

• Every f ∈WAP (R, H) admits the unique decomposition

f = g + ϕ

where g ∈ AP (R, H) and ϕ is such that M{ϕe−iλ

}= 0, for all λ ∈ R

[35].

• If f, g ∈WAP (R, H), then the scalar valued-function F (·) = (f (·) , g (·))is in WAP (R,R).

• Let f, g ∈ WAP (R, H). The mean convolution f∗g of f and g definedby

f∗g (t) = M{(f(·), g (·))}t = limT−→+∞

1

2T

+T∫−T

(f(t), g (t− s))ds

belongs to AP (R, H) [107].

• (WAP (R, H), ((., .))) is prehilbertian space where the scalar product((., .)) is defined by

((u, v)) := M{< u(t)/v(t) >

}t.

Remark 5.2 One can remark that this space is not complete and hence nota Hilbert space.

For f ∈WAP (R, H) and λ ∈ R, we asset formally the Bohr-Fourier series

f(t) ∼∑λ∈R

a(f, λ)eiλt


where

a(f, λ) = M{f(t)e−iλt

}t=

⎛⎝ lim

T−→+∞1

2T

+T∫−T

f(t)e−iλtdt

⎞⎠ .

As is known there is an at most countable set of values λ (called the Bohr-Fourier exponents or frequencies) such that the above limit differs from zero.This set will be denoted by ∧(f) = {λ ∈ R, a(f, λ) �= 0} and called Bohr spec-trum of f. The approximation Theorem says that for every almost periodicfunction f there exists a sequence of trigonometric polynomials

Pn (t) =

Nn∑k=1

ak,n exp (λk,nt)

where λk,n ∈ ∧(f) for all k, n, that converges uniformly in t ∈ R to f asn → +∞. Hence, as in the case of almost periodic, an f ∈ WAP (R, H) isalso completely defined by its Fourier series. Furthermore, It is still an openquestion whether or not the Parseval’s equality also holds true for Banach -valued weakly almost periodic functions. However, the answer is affirmativefor the Banach -valued weakly almost periodic functions∑

λ∈R

‖a(f, λ)‖2 = M{‖f‖2} .Remark 5.3 Clearly the following hierarchy hold

AP (R, X) ⊂ AAP (R, X) ⊂WAP (R, X) ⊂ PAP (R, X),

and

AP (R×Ω, X) ⊂ AAP (R×Ω,X) ⊂WAP (R×Ω,X) ⊂ PAP (R×Ω, X) .

Furthermore, for each p ≥ 1, one has [52]

WAP (R, H) ⊂W pap (R,X) and WAP (R, H) ∩ Sp

ap (R, X) = AP (R, H)

Example 5.4 The set of pseudo-random functions developed by [15] to givea model of a some physical phenomena, as turbulence, is a subset of the weaklyalmost periodic function.

Example 5.5

f0 (t) =

{e2iπαn2

if 0 ≤ n < t < n+ 10 if t < 0

where α ∈ R − Q. Let c ∈ N, then the convolution product

f =sin (4παct)

t∗ f0 ∈WAP (R,R)

is weakly almost periodic [10] and not almost periodic.

976 Farouk Cherif

5.1 The Mp−weakly almost periodic functions

For p ≥ 1, we set Mp (R, X) the space⎧⎪⎨⎪⎩f ∈ Lp

loc(R, X) such that ‖f‖p =

⎛⎝lim sup

T→+∞

1

2T

+T∫−T

‖f(t)‖p dt

⎞⎠

1p

<∞

⎫⎪⎬⎪⎭ .

Hence

‖f‖p =

⎧⎪⎨⎪⎩

(lim supT→+∞

12T

+T∫−T

‖f(t)‖p dt

) 1p

if p ≥ 1

‖f‖L∞ if p = +∞,

and(Mp (R, X) , ‖·‖p

)is a seminormed space.

Definition 5.6 ( [15]) A continuous function f ∈ Mp (R, X) is called Mp-weakly almost periodic (WMp

ap) if and only if

τ(f)w

= {τr(f) = f(· + r), r ∈ R}w

is compact in Mp (R, X) .

Example 5.7 ( [15]) For p ∈ ]1,+∞[ the function

f(t) =

{(n)

np if nn ≤ t ≤ nn + 1, (n = 1, 2, · · · )

0 else

is in WMpap.

We have the following properties [15] :

• The space WMpap is complete subspace of Mp (R, X).

• Let p1, p2 ∈ [1,+∞[ such that p1 < p2 then WMp2ap ⊂ WMp1

ap.

• For all p ∈ [1,+∞[ , we have Mpap ⊂ WMp

ap.

• For each f ∈ WMpap can be expressed as

f = h+ ϕ,

where h ∈ Mpap (R, X) and ϕ is a pseudo-random functions defined by

Bass [10]. Furthermore, this decomposition is unique and the Fourier-Bohr series of f and h coincide.


6 Other Notions of Almost Periodicity

Many authors have furthermore generalized in different directions the notion ofalmost-periodicity; for example, R. Doss ([49], in terms of Diophantine approx-imations), S. Stoinski ([92], , in terms of -almost-periods), K. Urbanik ([93], interms of polynomial approximations), etc. Besides that, the contribution byA. S. Kovanko to the theory of generalized a.p. functions is significant in thiscontext: namely in [70]and [71], Kovanko introduced ten different definitionsof a.p. functions in terms of ε-almost periods. Later, he extended the theoryof a.p. functions to non-integrable functions, in terms of polynomial approx-imations. In particular, he introduced the space of α-a.p. functions. Here,we focus our attention on the generalizations given by C. Ryll-Nardzewski, S.Hartman, which are related to the Bohr-Fourier coefficients.,

Definition 6.1 ([69]) A function f ∈ Lploc(R,R) is called almost-periodic

in the sense of Hartman (shortly, H1ap) if, for every λ ∈ R the number

af (λ) = limT→+∞

1

2T

∫ T

−T

f(x)e−iλxdx

exists and is finite.

Definition 6.2 ([69]) A function f ∈ Lploc(R,R) is called almost-periodic

in the sense of Ryll-Nardzewski (shortly, R1ap) if, for every λ ∈ R the number

bf (λ) = limT→+∞

1

2T

∫ X+T

X

f(x)e−iλxdx

exists uniformly with respect to X ∈ R, and is finite.

Remark 6.3 Obviously, R1ap(R,R) ⊂ H1

ap(R,R)

Definition 6.4 A function f : R −→ X is called limit periodic when thereexists a sequence (fn)n of periodic functions such that

limn→+∞

supt∈R

‖fn (t) − f (t)‖ = 0.

It is clear that when f is limit periodic then f is almost periodic. Variousnon trivial characterizations and properties of the limit period functions canbe founded in chapter 4 of [37] and [84].

Definition 6.5 A function f : [0,+∞[ −→ X is called S-asymptoticallyperiodic if there exists ω > 0 such that functions such that

limt→+∞

(fn (t + ω) − f (t)) = 0.

In this case, we say that ω is an asymptotic period of f and the function f isS−asymptotically ω−periodic. For more on this concept and see, e.g. [65].

978 Farouk Cherif

Remark 6.6 J. Bass [7],[8],[9] and J.-P. Bertrandias [15] introduced thespaces of pseudo-random functions, in terms of correlation functions, showingthat these spaces are included in the subset of continuous functions of Bp

ap andin the space of Hartman functions whose spectrum is empty (see [15]).

ACKNOWLEDGEMENTS. This is a text of acknowledgements.

References

[1] M. Allais, frequence, Probabilite et hasard, Journal de la societe deStatisque de Paris, tome 124, n2 , (1983), 829-832.

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Received: September, 2010

A Various Types of Almost Periodic Functions on Banach Spaces: … · 2011-02-23 · Bochner introduced the concept of almost automorphic functions in his papers [22], [23] and ...

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