C n-almost periodic and almost periodic solutions for some nonlinear integral equations ∗

In this paper, we investigate the existence of C n -almost periodic solution for a class of nonlinear Fredholm integral equation, and the existence of almost periodic solution for a class of more general nonlinear integral equation. Our existence theorems extend some earlier results. Two examples are given to illustrate our results.


Introduction
In [1], the author initiated the study on C n -almost periodic functions, which turns out to be one of the most important generalizations of the concept of almost periodic functions in the sense of Bohr.C n -almost periodic functions are very interesting since their properties are better than almost periodic functions to some extent as well as they have wide applications in differential equations.Recently, C n -almost periodic functions has attracted more and more attentions.We refer the reader to [6,7,11,12,14] and references therein for some recent development in this topic.
On the other hand, the existence of almost periodic type solutions for various kinds of integral equations has been of great interest for many authors (see, e.g., [2-5, 8, 10, 15] and references therein).Especially, in [15], the authors studied the existence of almost periodic solutions for the following Fredholm integral equation: Stimulated by the above works, we will make further study on these topics, i.e., we will study the existence of C n -almost periodic solutions for Eq.(1.1), and we will also investigate the existence of almost periodic solutions for the following more general integral equation: It is easy to see that Eq. (1.1) is a special case of Eq. (1.2).
In fact, to the best of our knowledge, there is no results in the literature concerning the existence of C n -almost periodic solutions for Eq.(1.1) and the existence of almost periodic solutions for Eq.(1.2).Therefore, in this paper, we will extend the results in [15] to the C n -almost periodic case and to a more general integral equation, i.e., Eq. (1.2).
Throughout the rest of this paper, if there is no special statement, we denote by R the set of real numbers, by X a Banach space, by C n (R, X) (briefly C n (X)) the space of all functions R → X which have a continuous n − th derivative on R, and by where f (i) denote the i − th derivative of f and f (0) := f.Clearly C n b (X) turns out to be a Banach space with the norm First, let us recall some definitions and notations about almost periodicity and C nalmost periodicity (for more details, see [6,7,9,13]).EJQTDE, 2012 No. 6, p. 2 Definition 1.1.A continuous function f : R → X is called almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that We denote by AP (R, X) (briefly AP (X)) the set of all such functions.Definition 1.2.F ⊆ AP (X) is said to be equi-almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that for all f ∈ F and t ∈ R, there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that We denote by AP (R × Ω, X) the set of all such functions.
there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that We denote by AP n (R, X) (briefly AP n (X)) the set of all such functions.Remark 1.5.By [7], we know that AP n (X) turns out to be a Banach space equipped with the • n norm.In addition, we usually denote AP 0 (X) by AP (X), which is the classical Banach space of all X-valued almost periodic functions in Bohr's sense.
Proof.By noting that it is not difficult to get the conclusion.
Combining Lemma 2.1 and the compactness criteria for AP (R) (cf.[9, Theorem 6.10]), we get the following compactness criteria for AP n (R): Theorem 2.2.The necessary and sufficient condition that F ⊆ AP (n) (R) be precompact is that the following properties hold true: Now, let 1 ≤ p ≤ ∞ and q be such that 1 p + 1 q = 1.For convenience, we list some assumptions.
(H1) f : R × R → R is a L p -Carathéodory function, i.e., the following two conditions hold: (i) the map t → f (t, y) is measurable for all y ∈ R, and the map y → f (t, y) is continuous for almost all t ∈ R; (ii) for each r > 0, there exists a function µ r ∈ L p (R) such that |y| ≤ r implies that |f (t, y)| ≤ µ r (t) for almost all t ∈ R. is measurable for each EJQTDE, 2012 No. 6, p. 4 (H3) there exists a constant r 0 > 0 such that Now, we are ready to establish one of our main results.Proof.We give the proof by three steps.
Step 1. F : AP n (R) → L p (R) is bounded and continuous, where Let E ⊂ AP n be a bounded subset and Then, by (H1), there exists a function µ r ∈ L p (R) such that for almost all t ∈ R and all y ∈ E, which yields that Then By using (H1) again, we know that for almost all t ∈ R, and for almost all t ∈ R.Then, by the Lebesgue's dominated convergence theorem, we get EJQTDE, 2012 No. 6, p. 5 i.e., F y k → F y in L p (R).
Step 2. K : L p (R) −→ AP n (R) is continuous and compact, where First, let us show that K is well-defined, i.e., Ky ∈ AP n (R) for y ∈ L p (R).Noting Similarly, one can show that Now, fix y ∈ L p (R) and m ∈ {0, 1, 2, . . ., n}.We have for all t 1 , t 2 ∈ R. By the almost periodicity of k m t , the map t −→ k m t is uniformly continuous on R. Combining this with (2.1), we conclude that (Ky) (m) is uniformly continuous on R. Again by the almost periodicity of k m t , for each ε > 0 there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that which and (2.1) yields that Thus, (Ky) (m) ∈ AP (R).By the definition of AP n (R), we know that Ky ∈ AP n (R).
Next, we show that K : In fact, the property (a) follows directly from for all y ∈ E and t ∈ R. In addition, by some direct calculations, it follows from (2.1) that the properties (b) and (c) hold.Thus, by Theorem 2.2, we know that K(E) is precompact in AP n (R).Moreover, noting that K is linear, we conclude that K is continuous.
Step Next, we present an example to illustrate our result.

2
C n -almost periodic solution for nonlinear Fredholm integral equation Lemma 2.1.The following two statements are equivalent:

(
H2) Let k : R × R → R be such that ∂ m k(t,s) ∂t m exists for m = 1, 2, . . ., n; and (i) there exist functions a m ∈ L q (R) such that |k m t (s)| ≤ a m (s), m = 0, 1, 2, . . ., n, for all t ∈ R and almost all s ∈ R, where k m t (s) := ∂ m k(t,s) ∂t m

3 .a m q • µ r 0 p ≤ r 0 ,
Eq. (1.1) has a C n -almost periodic solution.We denote(Sy)(t) = h(t) + [K(F y)](t) = h(t) + R k(t, s)f (s, y(s))ds, y ∈ AP n (R), t ∈ R.Noting that h ∈ AP n (R), it follows from Step 1 and Step 2 that S is from AP n (R) to AP n (R), and S : AP n (R) → AP n (R) is continuous and compact.Now, letE = {y ∈ AP n (R) : y n ≤ r 0 }.Then, for all y ∈ E, we haveSy n ≤ h n + K(F y) n ≤ h n +which means that S(E) ⊆ E. Noting that S : E → E is continuous and S(E) is precompact, by the classical Schauder's fixed point theorem, S has a fixed point in E, i.e., Eq. (1.1) has a C n -almost periodic solution.