ON SOME CLASSES OF ALMOST PERIODIC FUNCTIONS IN ABSTRACT SPACES

We deal with C(n)-almost periodic functions taking values in a Banach space. We give several properties of such functions, in particular, we investigate their behavior in view of differentiation as well as integration. The superposition operator acting in the space of such functions is also under consideration. Some applications to ordinary as well as partial differential equations are presented. Moreover, we introduce the class of the so-called asymptotically C(n)-almost periodic functions and give some of their properties.


Introduction. C
-almost periodic functions are one of the most important generalizations of the concept of almost periodic functions in the sense of Bohr.This generalization relies on the requirement that a given function as well as its derivatives up to the nth order inclusively are almost periodic in the sense of Bohr.Many properties of such functions with real values are given in [1,2].
In the present paper, we deal with C (n) -almost periodic functions taking values in a Banach space.This work is organized in the following way: in Section 2, we give the definition of vector-valued C (n) -almost periodic functions and we prove that the linear space of such functions considered with a suitable norm turns out to be a Banach space.Moreover, we give an application to an ordinary linear homogeneous differential equation, where the corresponding operator A on the right side is linear and compact.In Section 3, we investigate the behavior of C (n) -almost periodic functions with respect to differentiation as well as integration.Further, we prove that the C (n) -almost periodicity of the initial functions f and g imply that the solutions of the classical wave equations in view of the first variable are C (n) -almost periodic.Next, in Section 4, we investigate the superposition operator acting in the space of C (n) -almost periodic functions.In particular, we give examples of such functions and we consider the case of functions of two variables.Further, in Section 5, we introduce a new class of the so-called asymptotically C (n) -almost periodic functions and give several of their properties.

Definitions and basic properties.
Let (E, • ) be a Banach space.We will use the following notation: f h (x) = f (h+ x), where f : R → E and h, x ∈ R.
Denote by C (n) (R,E) the linear space of all functions R → E which have a continuous nth derivative on R. By C (n) B (R,E), we will denote the subspace of C (n) where f (i) denotes the ith derivative of f .In the space B (E)) we introduce the norm 2) It can be proved easily using the continuity of f (i) , i = 1, 2,...,n, that this set of ( • n ,ε)-almost periods is a closed set.
Denote by AP (n) (E) the set of all C (n) -a.p. functions.Directly from the above definitions it follows that AP (n+1) (E) ⊂ AP (n) (E).Moreover, putting n = 0, we have AP (0) (E) = AP (E), where AP (E) denotes the classical Banach space of all almost periodic functions R → E, in the sense of Bohr.Now, we prove the following lemma.
Proof.Let f ∈ AP (n) (E) and let ε > 0. For t ∈ R, we have , where l = l(ε) > 0 is a number which characterizes the relative density of E (n) where s = t + τ.Thus, f n < +∞, which completes the proof.
In what follows, we will need the following well-known lemma.
An immediate consequence of Lemma 2.4 is the following criterion.
By the above result, we infer that if f ∈ AP (n) (E), then the range of f (i) is relatively compact for each i = 0, 1,...,n (see [7, almost periodic if and only if for every sequence of real numbers (s n ), there exists a subsequence (s n ) such that (f (i) Corollary 2.8.Let (T (t)) t∈R be a C 0 -group of bounded linear operators on E and for an e ∈ E, the function x e : R → E defined by x e (t) = T (t)e satisfies the assumptions: (1) x e ∈ C (n) (E) for some n; (2) R xe is relatively compact in E, where R xe denotes the range of the function x e .Then, x e ∈ AP (n)

(E).
Proof.It is an immediate consequence of Theorem 2.7 and the group property.Application 2.9.Consider in a Banach space E the abstract differential equation where A ∈ L(E) is a compact linear operator.Let x(t) be a solution of the above equation.
Then, the following hold: , for all n = 0, 1, 2,... , since A is continuous and defined on all E.
To prove assertion (1), we suppose that x(t) is bounded; then Ax(t), and consequently, x (t) is relatively compact since A is compact.Thus, for all n = 1, 2,... , the function x (t) = Ae tA x(0) = e tA Ax(0) is C n -almost periodic by Corollary 2.8.
We have also the following simple proposition.Proposition 2.10.A linear combination of C (n) -a.p. functions is a C (n) -a.p. function.Moreover, let E be a Banach space over the field , then the following functions are also in AP (n) Proof.It is a simple consequence of Theorem 2.7.
Corollary 2.12.AP (n) (E) considered with the norm (2.2) turns out to be a Banach space.
Proof.In view of Proposition 2.10 and Lemma 2.3, AP (n)

Differentiation and integration.
The first result in this Section gives a sufficient condition which guarantees that the derivative of a function f ∈ AP (n) Proof.Fix ε > 0. In view of Theorem 2.5 and the uniform continuity of f (n+1) , there exists δ > 0 such that for h ∈ R with |h| < δ we have Further, for t ∈ R and h ≠ 0, we have Thus, by (3.1), we obtain One can extend this result in the following way.Theorem 3.3.If f ∈ AP (E) and the range R F is relatively compact, then F ∈ AP (1)

(E).
Theorem 3.3 is a special case of the following theorem.
Proof.Since f is continuous on R, so, In view of Theorem 3.4, f ∈ AP (n+1) (E).
Example 3.6.Let f ∈ AP (n) (E), and for a fixed a ∈ R, define a function F : R → E by F a (t) := t+a t f (s)ds for each t ∈ R.
Then, F ∈ AP (n) (E).Moreover, f and F have the same ( • n ,ε)-almost periods.First, we note that by a suitable change of variable we get Now, let τ ∈ E (n) ( /|a|,f ); then, for any t ∈ R, we have that is τ ∈ E (n) ( /|a|,F), which proves our assertion.
Example 3.7.Consider the wave equation Assume that both f ,g ∈ AP (n) (R).Then, for each t 0 ∈ R, the solution u(x, t 0 ) ∈ AP (n) .Indeed, we have g(s)ds, we deduce that it is in AP (n) (R) by the previous example.

Superposition operator.
Let E 1 , E 2 be Banach spaces.We begin this section with the following proposition.
For every f ∈ AP (n) (E 1 ), define the E 2 -valued function G as follows: where A : E 1 → E 2 is a bounded linear operator.
We have the following corollary.
Corollary 4.2.Let A : E 1 → E 2 be a bounded linear operator with relatively compact range, then the E 2 -valued function G defined above is in AP (n+1) (E 2 ).
Proof.According to Proposition 4.1, Af ∈ AP (n) (E 2 ) for every f ∈ AP (n) (E 1 ).We also note that R G ⊂ R(A), where R(A) is the range of the operator A, hence R G is relatively compact.We now complete the proof using Theorem 3.4.
In the Corollary 4.2, note that if the operator A is compact (or of finite rank), the stated result holds.Now, we will consider the superposition operator (the autonomous case) acting on the space AP (n) (E 1 ).
We will utilize the fact that the range R f = {f (t) | t ∈ R} of a C (n) -a.p. function is relatively compact to prove the following theorem.
The function f can be utilized to construct an example for arbitrary n ≥ 2 (see [1]).Now, we will deal with functions of two variables f (t,x).First, recall the following definition from [4] in a Banach space setting.Definition 4.6.A continuous function f : R × E → E is said to be almost periodic in t for each x ∈ E if for each ε > 0 there exists l > 0 such that every interval [a, a + l] contains at least one point s such that In view of Bochner's criterion (Theorem 2.6), it is clear that the above definition is equivalent to the following one.Definition 4.7.A continuous function f : R×E → E is almost periodic in t for each x ∈ E if for every sequence of real numbers (s n ), there exists a subsequence (s n ) such that (f (t + s n ,x)) converges uniformly in t ∈ R and x ∈ E.
Based on Theorem 2.5, we can introduce the following definition.

Definition 4.8. A function
An analogue of Theorem 2.7 is the following one.Theorem 4.9.Let f : R × E → E be such that (∂ i f /∂t i )(t, x) are continuous for i = 0, 1,...,n.Then, f is (n)-almost periodic if and only if for every sequence of real numbers (s n ), there exists a subsequence (s n ) such that (D (i) 1 f denotes the ith partial derivative of f in view of the first variable.Remark 4.10.We consider the nonautonomous composition operator acting in the space AP (n) (E).First, we recall the corresponding result for AP (E) (see [4,Lemma 3.8] in Fréchet space setting).
Let f : R × E → E be almost periodic in t for each x ∈ R and assume that f satisfies a Lipschitz condition in x uniformly in t ∈ R, that is, f (t,x) − f (t,y) ≤ L x − y for all t ∈ R and x, y ∈ E. Let ϕ : R → E be almost periodic.Then, the function We notice that the assumptions of the above result imply that F satisfies a (global) Lipschitz condition as a superposition operator acting in the space C(R,E).
Recall that in the case of the space [3,Theorem 8.4, page 212] or [6]).If we assume additionally that f : R × E → E is (n)-almost periodic in t for each x ∈ E, then it is equivalent to assume that a, b ∈ AP (n)  (R).In this situation, F acts in AP (n) (E), obviously.

Asymptotically C (n) -almost periodic functions
Definition 5.1.A function f ∈ C (n) (R + ,E) is said to be asymptotically C (n) -almost periodic if it admits a decomposition where g : R → E is C (n) -almost periodic function and h ∈ C (n) (R + ,E) with lim t→+∞ h(t) = 0. Thus, g and h are called, respectively, the principal and corrective terms of the function f .
As an obvious consequence of the above definition and Proposition 2.10, we have the following proposition.Proof.The idea of the proof is similar to that from the proof of [7,Theorem 2.5.4].Let f : R + → E be C (n) -almost periodic with two decompositions: with principal terms g 1 , g 2 and corrective terms h 1 , h 2 , respectively.Then, for t ∈ R + we have uniformly in t ∈ R. Thus, by (5.4), ) In view of Proposition 5.2, we can consider the set of all C (n) -a.p. functions R + → E as a linear space.We will denote it by AAP (n) (E).Moreover, by Theorem 5.3, the formula where g and h are the principal and corrective terms of the function f , respectively, defines the norm on the space AAP (n) (E) (obviously in the case of the norm of the corrective term h, the supremum is taken over R + ).We have the following theorem.
Proof.Let (f n ) be a Cauchy sequence in AAP (n) (E).Then, the principal terms of the functions f n : (f n ) g form a Cauchy sequence of C (n) -almost periodic functions with respect to the norm of the space AP (n) (E).Thus, by Corollary 2.8, there exists a C (n)almost periodic function g such that (f n ) g −g n → 0, as n → ∞.Further, the corrective terms of the functions f n : (f n ) h form a Cauchy sequence in the space C (n) (R + ,E).Hence, there exists a continuous function h ∈ C (n) Theorem 5.5.Let E 1 , E 2 be Banach spaces and let f : R + → E 1 be an asymptotically C (n) -almost periodic function.Let φ : E 1 → E 2 be a mapping of the class C (n) .Moreover, assume that there exists a compact set B which contains the closure of the set {h(t) : t ∈ R + }, where h is the corrective term of f .Then, the function φ Proof.Let f (t) = g(t)+h(t) for t ∈ R + , where g and h are principal and corrective terms of f , respectively.In view of Theorem 4.3, the function φ(g(t)) is C (n) -almost periodic.Set Γ (t) = φ(f (t)) − φ(g(t)) for t ∈ R + .Obviously, Γ is a C (n) -mapping.Let ε > 0. By assumption, there exists a compact set C which contains the closures of {f (t) : t ∈ R + } and {g(t) : t ∈ R + }, so φ restricted to C is uniformly continuous.Thus, there exists δ = δ(ε) > 0 such that φ(x) − φ(y) 2 < ε if x − y 1 < δ, x,y ∈ E 1 , ( where • 1 , • 2 denote the norms of E 1 and E 2 , respectively.Since lim t→+∞ h(t) = 0, there exists t 0 > 0 such that f (t)− g(t) 1 = h(t) 1 < δ for t > t 0 .
The last result of this section concerns the integration of asymptotically C (n) -almost periodic functions.(5.11) The proof is complete.

Theorem 5 . 6 .
Let E be a Banach space and f : R + → E an asymptotically C(n) -almost periodic function.Consider the function F : R + → E defined by F(t) = t 0 f (s)ds and G : R → E defined by G(t) = t 0 g(s)ds, where g is the principal term of f .Assume G has a relatively compact range in E and that +∞ 0 h(t) dt < +∞, where h is the corrective term of f .Then, F is asymptotically C (n) -almost periodic; its principal term is G(t) + +∞ 0 h(s)ds and its corrective term is H(t) = − +∞ t h(s)ds.Proof.In view of Theorem 3.4, G(t) is C (n) -almost periodic.Since +∞ 0 h(s)ds exists, G(t) + +∞ 0 h(s)ds is almost periodic.Moreover, the function H is C (n) -mapping and lim t→+∞ H(t) = 0. Since F(t) = G(t) + +∞ 0 h(s)ds + H(t) for t ∈ R + .