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

This paper deals with the existence and uniqueness of periodic solutions, S-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.


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][2][3][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][6][7][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  ∈ R.
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][11][12][13][14]. Let  ∈ (0, ∞) \ N and  = [] + 1.If  : (0, ∞) → R 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],    0 sin  =  1−  2,2− (− 2 ) for  ∈ (0, 1), while   + sin  = sin( + (/2)) for  ∈ (−1, ∞), where    0 is the Caputo fractional derivative with order  and the lower terminal 0, 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][18][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][21][22]. Ponce [22] studied the existence and uniqueness of bounded solutions for semilinear fractional integrodifferential equation where  is a closed linear operator defined on a Banach space ,   + is Weyl fractional derivative of order  > 0 with the lower limit −∞,  ∈  1 (R + ) is a scalar-valued kernel, and  : R ×  →  satisfies some Lipschitz type conditions.Assume that  is the generator of an -resolvent family {  ()} ≥0 which is uniformly integrable.The mild solutions of (1) was given by By 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 1 <  < 2.
In this paper, we study the fractional evolution equations in an ordered Banach space where   + is the Weyl-Liouville fractional derivative of order  ∈ (0, 1) which is defined in Section 2 and − : () ⊂  →  is the infinitesimal generator of a  0 -semigroup {()} ≥0 .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  : R ×  →  satisfies some ordered or Lipschitz conditions.
Compared with the earlier related existence results which appeared in [21][22][23], there are at least four essence differences: (i) the order of the fractional derivative is  ∈ (0, 1); (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.

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  fl { ∈  |  ≥ } ( is the zero element of ) is normal with normal constant .Denote by L() the space of all linear bounded operator on Banach space , with the norm ‖ ⋅ ‖ L() .The notation   () stands for the Banach space of all bounded continuous functions from R into  equipped with the sup norm ‖ ⋅ ‖ ∞ ; that is, For , V ∈   (),  ≤ V if () ≤ V() for all  ∈ R.   () stands for the subspace of   () consisting of all -valued continuous -periodic functions.Set   () = { ∈   () | there exists  > 0 such that      ( + ) −  ()     → 0 as  → ∞} . ( These functions in   () are called -asymptotically periodic (see [16]).We note that   () and   () are Banach spaces (see [16]), and   () ⊂   ().
For convenience, we set  0 () fl { ∈   () | lim ||→∞ ‖()‖ = 0}.We regard the direct sum of   () and  0 () as the space of asymptotically periodic functions   (), the direct sum of () and  0 () as the space of asymptotically almost periodic functions (), the direct sum of   () and  0 () as the space of asymptotically compact almost automorphic functions, and the direct sum of () and  0 () as the space of asymptotically almost automorphic functions.
Then we set and regard the direct sum of   () and  0 () as the space of pseudoperiodic functions, the direct sum of () and  0 () as the space of pseudo-almost periodic functions, the direct sum of   () and  0 () as the space of pseudocompact almost automorphic functions, and the direct sum of () and  0 () as the space of pseudo-almost automorphic functions.

Then the results shown in
Now we recall the definitions of some fractional derivatives and integrals which are used in this paper (see [15]).Definition 2. Let  ∈   (R/2Z) (1 ≤  < ∞) be periodic with period 2 and such that its integral over a period vanishes.The Weyl fractional integral of order  is defined as where for 0 <  < 1.
The above Weyl definition is accordant with the Riemann-Liouville definition [1] for 2 periodic functions whose integral over a periodic vanishes.
The Weyl-Liouville fractional derivative is defined as for 0 <  < 1.
Let us recall the definitions and properties of operator semigroups; for details see [24].Assume that − is the infinitesimal generator of a  0 -semigroup {()} ≥0 .If there are  ≥ 0 and ] ∈ R such that ‖()‖ L() ≤  ] , then The growth bound of the semigroup {()} ≥0 is defined as Furthermore, ] 0 could also be expressed by For  0 -semigroup {()} ≥0 , if there exists a constant  > 0 such that then it is called uniformly bounded.A  0 -semigroup satisfying () ≥  for all  ≥  and  ≥ 0. For the positive operators semigroup, one can refer to [25].
In the following part, we shall recall the definitions and properties of Mittag-Leffler functions (see [1]).Note These functions have the following properties for  ∈ (0, 1) and  ∈ R.

Results for Linear Equations
Theorem 8.If {()} ≥0 is exponentially stable and satisfies (18), ℎ belongs to one of M(), and where  is defined by (24); then ℎ and ℎ belong to the same space.
Set  ∈  0 ().For  > 0 we get We can find that the set  0 () is translation-invariant and get by Lebesgue dominated convergence theorem.Then   (), (),   (), and () have the maximal regularity property under the convolution defined by (33).
Proof.The proof is similar to that in [9].
By ( 63) and (64) and the definition and the positivity of  1 , we obtain Since  1 ∘  is an increasing operator on [,  0 ], in view of (62) we can show that Therefore, By induction, Combining (66) and (70), by the nested interval method, there is a unique Since operator  1 ∘  is continuous, by (62) we have It follows from the definition of  1 and (66) that  * is a positive mild solution of (61) for ℎ() = (,  * ()) +  * ().