Discrete Dynamics in Nature and Society

Volume 2017, Article ID 1364532, 12 pages

https://doi.org/10.1155/2017/1364532

## Periodic Solutions and -Asymptotically Periodic Solutions to Fractional Evolution Equations

^{1}Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China^{2}School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730000, China^{3}Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Yong Zhou; nc.ude.utx@uohzy

Received 1 January 2017; Accepted 15 March 2017; Published 3 April 2017

Academic Editor: Can Li

Copyright © 2017 Jia Mu 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

This paper deals with the existence and uniqueness of periodic solutions, -asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

#### 1. Introduction

It is well known that fractional order differential equations provide an excellent setting for capturing in a model framework real-world problems in many disciplines, such as chemistry, physics, engineering, and biology [1–4]. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Podlubny [2], Kilbas et al. [1], Zhou [4], and the recent papers [5–8] and the references therein.

As a matter of fact, periodic motion is a very important and special phenomenon not only in natural science but also in social science such as climate, food supplement, insecticide population, and sustainable development.

In our previous paper [9], we discussed a class periodic boundary value problem to fractional evolution equations and obtained the existence and uniqueness results for positive mild solutions. However, since the fractional derivatives provide the description of memory property, the solution of periodic boundary value problems cannot be periodically extended to the time .

On the other hand, several authors have showed that for some fractional order systems the solutions do not show any periodic behavior if the lower terminal of the derivative is finite; see [10–14]. Let and . If is a nonconstant -periodic function of class , [10, 11] tell us that cannot be -periodic function, where is understood as one of the fractional derivatives (Caputo, Riemann-Liouville or Grunwald-Letnikov) with the lower terminal finite. Nevertheless, the authors also point out in [11] that fractional order derivative of other form, such as Weyl-Liouville fractional derivatives defined for periodic function [15], perhaps preserves periodicity. As indicated in [2, 15], the Weyl-Liouville derivative coincides with the Caputo, Riemann-Liouville or Grunwald-Letnikov derivative with lower limit , which is denoted by . There is essential difference between finity and infinity. As in, for instance, [2, 11], for , while for , where is the Caputo fractional derivative with order and the lower terminal , and denotes the two parameters Mittag-Leffler functions. It is obvious that the Weyl-Liouville fractional derivative is suitable for the study of periodic solutions to differential equations.

In real life, many phenomena are not strictly periodic; therefore many other generalized periodic cases need to be studied, such as almost periodic, asymptotically almost periodic, -asymptotically periodic, asymptotically periodic, pseudoperiodic, and pseudo-almost periodic. As the advantages of fractional derivatives, such as the memorability and heredity, many papers concern these types of solutions for fractional differential equations. Since -asymptotically periodic functions in Banach space were first studied by Henríquez et al. [16], there are some papers about -asymptotically periodic solutions for fractional equations; one can refer to [17–19]. For almost periodic solutions, asymptotically almost periodic and other types of bounded solutions to fractional differential equations, one can refer to [13, 17, 20–22]. Ponce [22] studied the existence and uniqueness of bounded solutions for semilinear fractional integrodifferential equationwhere is a closed linear operator defined on a Banach space , is Weyl fractional derivative of order with the lower limit , is a scalar-valued kernel, and satisfies some Lipschitz type conditions. Assume that is the generator of an -resolvent family which is uniformly integrable. The mild solutions of (1) was given byBy Banach contraction principle, existence and uniqueness results of almost periodic, asymptotically almost periodic and other types of bounded solutions are established. In addition, Lizama and Poblete [21] gave some sufficient conditions ensuring the existence and uniqueness of bounded solutions to a fractional semilinear equation of order .

In this paper, we study the fractional evolution equations in an ordered Banach space where is the Weyl-Liouville fractional derivative of order which is defined in Section 2 and is the infinitesimal generator of a -semigroup . Applying Fourier transform, we give reasonable definitions of mild solutions of (3). Then the existence and uniqueness results for the corresponding linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. Finally, some sufficient conditions are established for the existence and uniqueness of periodic solutions, -asymptotically periodic solutions, and other types of bounded solutions when satisfies some ordered or Lipschitz conditions.

Compared with the earlier related existence results which appeared in [21–23], there are at least four essence differences: (i) the order of the fractional derivative is ; (ii) in many papers concerning bounded solutions for differential equations, such as [21, 23], the solution is defined by the limit of the solution for the limit equation, without strict proofs of the solution for the given equation possessing that form, but, in this paper, we apply the Fourier transform to (3) itself and obtain the expression of the solution; naturally, we may define expression as a mild solution under suitable conditions; besides, the mild solution in this paper is expressed by the semigroups generated by , which are more convenient than some resolvent families applied in [22]; (iii) the spectral radius of resolvent operator is accurately estimated; (iv) the conditions on contain some ordered relations, and the associated method is monotone iterative technique.

The paper is organized as follows. Section 2 provides the definitions and preliminary results to be used in the article. In Section 3, the existence and uniqueness results for the linear equations are obtained. In Section 4, we consider the existence and uniqueness of the mild solutions for (3). In Section 5, we also give two examples to apply the abstract results.

#### 2. Preliminaries

This section is devoted to some preliminary facts needed in the sequel. Let be an ordered Banach space with norm and partial order , whose positive cone ( is the zero element of ) is normal with normal constant . Denote by the space of all linear bounded operator on Banach space , with the norm . The notation stands for the Banach space of all bounded continuous functions from into equipped with the sup norm ; that is,For , if for all . stands for the subspace of consisting of all -valued continuous -periodic functions. Set These functions in are called -asymptotically -periodic (see [16]). We note that and are Banach spaces (see [16]), and .

A function , if for any there is a real number and in arbitrary interval of length such that for all is said to be almost periodic (in the sense of Bohr). We denote by the set of all these functions. The space of almost automorphic functions (resp., compact almost automorphic functions) will be written as (resp., ). The bounded continuous function (resp., ) for every sequence there is a subsequence such that and for each (resp., uniformly on compact subsets of ). Clearly the compact almost automorphic function above is continuous on .

For convenience, we set . We regard the direct sum of and as the space of asymptotically periodic functions , the direct sum of and as the space of asymptotically almost periodic functions , the direct sum of and as the space of asymptotically compact almost automorphic functions, and the direct sum of and as the space of asymptotically almost automorphic functions.

Then we setand regard the direct sum of and as the space of pseudoperiodic functions, the direct sum of and as the space of pseudo-almost periodic functions, the direct sum of and as the space of pseudocompact almost automorphic functions, and the direct sum of and as the space of pseudo-almost automorphic functions.

The relationship of the above different classes of subspaces is presented in [21, p. 805]:

Denote by or simply the following function spaces: and the space of all functions satisfying uniformly for each in any bounded subset of . For fixed , then for and , the following conditions could ensure that : () is uniformly continuous with respect to on for each bounded subset of . More precisely, given and , there exists such that and imply that ;()for all bounded subset , is bounded;()if , where and , is uniformly continuous with respect to on for each bounded subset of .Then the results shown in Table 1 hold; see [21, Remark3.4 and p. 810].