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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 179525, 15 pages
http://dx.doi.org/10.1155/2012/179525
Research Article

Weighted Pseudo Almost-Periodic Functions and Applications to Semilinear Evolution Equations

1School of Mathematics and Statistics, Central China Normal University, Hubei, Wuhan 430079, China
2Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 24 December 2011; Accepted 9 January 2012

Academic Editor: Josef Diblík

Copyright © 2012 Jun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first give a solution to a key problem concerning the completeness of the space of weighted pseudo almost-periodic functions and then establish a new composition theorem with respect to these functions. Some important remarks with concrete examples are also presented. Moreover, we prove an existence theorem for the weighted pseudo almost-periodic mild solution to the semilinear evolution equation: , , where is the infinitesimal generator of an exponentially stable -semigroup. An application is also given to illustrate the abstract existence theorem.

1. Introduction

It is well known that periodicity and almost periodicity are natural and important phenomena in the real world. In 2006, Diagana [1] introduced the concept of weighted pseudo almost-periodic functions, which is a generalization of the classical almost-periodic functions of Bohr as well as the vector-valued almost-periodic functions of Bochner (cf., e.g., [14]). Recently, weighted pseudo almost-periodic functions are widely investigated and used in the study of differential equations. Many basic properties and applications to several classes of differential equations were established, see, for example, Blot et al. [5] and Diagana [6]. On the other hand, the properties of weighted pseudo almost-periodic functions are more complicated and changeable than the almost-periodic functions and the pseudo almost-periodic functions because the influence of the weight is very strong sometimes. Very recently, we constructed examples to show that the decomposition of weighted pseudo almost-periodic functions is not generally unique [3]. Hence, the space of the weighted pseudo almost-periodic functions may not be a Banach space under the usual supremum norm. Actually, the completeness of the space of these functions is worthy to be studied deeply by new ideas. In this paper, we will present a solution to the fundamental problem. Then, we will study the corresponding composition problem of weighted pseudo almost-periodic functions, as well as the existence of weighted pseudo almost-periodic mild solution to the following semilinear evolution equation: where is the infinitesimal generator of an exponentially stable -semigroup.

The paper is organized as follows. In Section 2, we prove a completeness theorem for the space of weighted pseudo almost-periodic functions by introducing a new norm on the space. Moreover, some remarks on weighted pseudo almost-periodic functions are given. In Section 3, we first establish a composition theorem for weighted pseudo almost-periodic functions and then give an existence theorem for the weighted pseudo almost-periodic mild solution to the evolution equation (1.1). An example is presented to illustrate the abstract existence theorem.

2. Completeness Theorem

Throughout this paper, we let be Banach spaces and the Banach space of all -valued bounded continuous functions equipped with the supremum norm. denotes the space of all locally integrable functions on , and stands for If , then we set Define the space as and define as the set of all such that is bounded with . Obviously,

In what follows we recall some definitions and notations needed in this paper.

Definition 2.1 (S. Bochner). A continuous function is called almost periodic if for each there exists an such that every interval of length contains a number with the property that The set of all such functions will be denoted by in this paper.
A continuous function is said to be almost periodic if is almost periodic in uniformly for all , where is any bounded subset of . Denote by the set of all such functions.

The set of bounded continuous functions with vanishing mean value is denoted by , that is, Let , and we define the set of continuous functions with vanishing mean value under weight by and define For simplicity of notation, we write instead of respectively.

The following definition is a slight modification of that given by Diagana [1].

Definition 2.2. A function is called weighted pseudo almost periodic (or -pseudo almost periodic) if it can be expressed as , where and . The set of weighted pseudo almost-periodic functions from into is denoted by here.

Remark 2.3. (i) Clearly, and are closed linear subspaces of under the supremum norm.
(ii) There are a lot of weighted pseudo almost-periodic functions. For example, the following function: is weighted pseudo almost periodic under weight , see Example 3.6 for another example.
(iii) If one set , then weighted pseudo almost-periodic functions are exactly the so-called pseudo almost-periodic functions. Furthermore, when the weight , Diagana’s work showed that the class of functions in coincides with . So, the concept of weighted pseudo almost-periodic functions is a generalization of the concept of pseudo almost-periodic functions. From the following example given by us, one will see that this concept is a real generalization.

Example 2.4. Let Then the bounded continuous function is pseudo almost-periodic under this weight but it is not the usual pseudo almost periodic function.
(iv) Suppose is a weighted pseudo almost-periodic function. From [3] we know that the decomposition of is not generally unique, that is, the function space cannot be decomposed as Therefore, we do not know whether is a Banach space under the norm although and are closed linear subspaces of under the supremum norm, and hence Banach spaces.

In order to obtain a completeness theorem for the space by overcoming the trouble showed in (iii) of the above remark, we use the “modular" idea and endow the weighted pseudo almost-periodic function in with a new norm as follows.

Let be all the possible decomposition of . We define Clearly, is a norm on . Moreover, we have the following result.

Theorem 2.5 (completeness theorem). is a Banach space under the norm .

Proof. Suppose is a Cauchy sequence in , relative to the norm . Then we can choose a subsequence with Set Then by the definition of the norm , we can decompose each as , where and with Observing that we claim that converges to a weighted pseudo almost-periodic function. Actually, it follows from (2.17) and (2.19) that So Therefore This, together with the fact that is a Banach space and , implies that exists and . The same arguments indicate that exists and . Thus, by (2.20) we have Therefore, converges to , which belongs to . In other words, is a Cauchy sequence having a convergent subsequence. Consequently, is a convergent sequence under the norm . This means that is a Banach space under the norm .

Remark 2.6. Clearly, the norm coincides with the sup norm when is a pseudo almost-periodic function, as a result of the uniqueness of decomposition of pseudo almost-periodic functions.

3. Composition Theorem and Existence Theorem

To obtain the existence of weighted pseudo almost periodic mild solutions to the semilinear evolution equation (1.1), we first establish a new composition theorem as follows.

Theorem 3.1 (composition theorem). Let and with and . Assume that is uniformly continuous in every bounded subset uniformly for . If and , with , and , then and furthermore .

Proof. Write Then, we claim that(a), (b), (c). Actually, by the theorem of composition of almost-periodic functions, it is easy to see that (a) holds.
Next, let us prove (b).
Clearly, is bounded and continuous. By the uniformly continuity of , we know that for any , there exists such that implies that Set Then and are measurable sets and . In addition, on the set . Observe Using , we get Moreover, for any , Therefore, we have That means that (b) is true.
Finally, we prove (c).
Since is continuous in , it is uniformly continuous in . Set . Then is compact in . So one can find finite open balls with center and radius small enough such that and The set is open in and . Let Then when , and On the other hand, by , we see that there exists such that Hence for , Thus, (c) holds.
Since is a linear space, we have , by (a), (b), and (c).

The following assumption on the weight will be used in our next investigation.

For a , for some constant and is sufficiently large.

Remark 3.2. (i) Since , condition (3.15) is satisfied when Condition (3.16) is satisfied when is nonincreasing. Therefore, if the weight is bounded and nonincreasing, then both (3.15) and (3.16) are satisfied.
(ii) Besides the weight mentioned in (i) above, there are lots of weights satisfying the assumption . By giving the following example, we present two unbounded weights satisfying the assumption .

Example 3.3. (1) Take . Then we have Therefore satisfies condition (3.15). Furthermore, where . So (3.16) is satisfied.
(2) Take . Then for every , we obtain Moreover, where is a constant.

Definition 3.4. A mild solution to (1.1) is a continuous function satisfying for all and all .

Theorem 3.5. Let generate an exponentially stable - semigroup , that is, for some positive constants and . Let the weight satisfy for the in (3.23). Assume that() is weighted pseudo almost periodic, with and .() for every , for some constants and with .Then (1.1) admits a unique weighted pseudo almost-periodic mild solution.

Proof. Define a nonlinear operator on by Fix . Then with By Theorem 3.1, we know that there exist two functions such that Therefore can be expressed as , where From (3.28) and the fact that is exponentially stable, it follows that .
Next, we show that .
By (3.29), we see that is bounded on , that is, , and is continuous. Hence, for any , we have where Clearly, This, together with the assumption , implies that Moreover, by Fubini’s theorem and assumption , we get Thus, (3.29) implies that So, .
Consequently, is a weighted pseudo almost-periodic function on . This means that maps into .
Now, let with where are periodic and vanishing mean value components of weighted pseudo almost-periodic functions . Then by Theorem 3.1, we have Observe We know from the arguments in the above paragraph that the two integrals on the right side are in, respectively, and . Therefore, Since the inequality is true for any decomposition of , we can take the infimum and then obtain Hence is a contraction under the assumption that .
By the contraction mapping theorem, the mapping has a unique fixed point , and this fixed point satisfies the integral equation for all . Fixing , we have Since it follows that satisfies (3.22). Hence is a mild solution to (1.1).
On the other hand, let be a pseudo almost-periodic mild solution to (1.1). Then satisfies (3.22), with replaced by . Letting yields Since is bounded on and is exponentially stable, hence on .
In conclusion, is the unique mild solution to (1.1), which ends the proof.

Example 3.6. Let , and Then, generates a -semigroup on with Let Then, , and satisfy () with By Theorem 3.5, the corresponding equation (1.1) has a unique weighted pseudo almost-periodic mild solution under the weight .

Acknowledgments

This work was supported by the NSF of China (11071042, 11026100, 11171210), the Chinese Academy of Sciences, the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Laboratory of Mathematics for Nonlinear Science at the Fudan University.

References

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