1. Introduction

Fractional order differential equations is a very important subject matter. These orders can be complex in viewpoint of pure mathematics. During the last few decades fractional order differential equations have emerged vigorously (cf., [1–8]). We observe that there is much interest in developing the qualitative theory of such equations. Indeed, this has been strongly motivated by their natural and widespread applicability in several fields of sciences and technology. Many real phenomena in those fields can be described very successfully by models using mathematical tools of fractional calculus, such as dielectric polarization, electrode-electrolyte polarization, electromagnetic wave, modeling of earthquake, fluid dynamics, traffic model with fractional derivative, measurement of viscoelastic material properties, modeling of viscoplasticity, Control Theory, and economy (cf., [3, 4, 9–15]). Very recently, some basic theory for initial value problem of fractional differential equations involving the Riemann-Liouville differential operators was discussed by Benchohra et al. [16], Agarwal et al. [17–19], Lakshmikantham [20], and Lakshmikantham and Vatsala [21, 22]. Mophou and N'Guérékata [23] have studied existence of mild solution for fractional semilinear differential equations with nonlocal conditions (more details can be found in [24–29]). El-Sayed and Ibrahim [30] and Benchohra et al. [31] initiated the study of fractional multivalued differential inclusions. In this direction, we refer to the article by Henderson and Ouahab [32] concerning the existence of solutions to fractional functional differential inclusions with finite delay, and existence of solutions for these types of equations in the infinite delay framework (see [16, 31]). In the case that fractional order is , existence results for fractional boundary value problems of differential inclusions were studied by Ouahab [33].

We study in this work some sufficient conditions for the existence and uniqueness of pseudo-almost periodic mild solutions to the following semilinear fractional differential equation:

where , is a linear densely defined operator of sectorial type on a complex Banach space and is a pseudo-almost periodic function (see Definition 2.10) satisfying suitable conditions in . The fractional derivative is understood in the Riemann-Liouville sense. Type (1.1) equations are attracting increasing interest. For example, anomalous diffusion in fractals by Eidelman and Kochubei [10] or in macroeconomics by Ahn and McVinisch [1] has been recently studied in the setting of fractional differential equations like (1.1). The study of almost automorphic mild solutions of (1.1) was studied by Cuevas and Lizama in [34] (see also [35]).

As for almost periodic functions, pseudo-almost periodic functions have many applications in several problems, for example, in theory of functional differential equations, integral equations, and partial differential equations. The concept of pseudo-almost periodic was introduced by Zhang [36–39] in the early nineties. Since then, such notion became of great interest to several mathematicians (see [40–49]). To the knowledge of the authors, no results yet exist for pseudo-almost periodic mild solution of (1.1).

We also discuss sufficient conditions for the existence and uniqueness of an asymptotically almost periodic mild solution of the fractional Cauchy problem

In a work by Cuevas and de Souza [50] the authors proved existence and uniqueness of an S-asymptotically -periodic solution of problem (1.2)-(1.3) (see also [51]). On the other hand, we give results on existence and uniqueness of an asymptotically almost automorphic mild solution to a class of fractional integrodifferential neutral equations.

We now turn to a summary of this work. The second section provides the definitions and preliminaries results to be used in theorems stated and proved in this article. In particular, we review some of the standard properties of the solution operator generated by a sectorial operator (see Proposition 2.2). We also recall the notion of almost periodicity, asymptotically almost periodicity, asymptotically almost automorphy, and pseudo-almost periodicity. In the third section, we obtain very general results on the existence of pseudo-almost periodic mild solution to equation (1.1). The fourth section is concerned with the existence of an asymptotically almost periodic mild solution to problem (1.2)-(1.3). While in the fifth section we use the machinery developed in the previous sections to obtain new results on existence and uniqueness of an asymptotically almost automorphic solution to a class of fractional integrodifferential neutral equation. To build intuition and throw some light on the power of our results and methods, we give, in the sixth section, a few applications.

2. Preliminaries and Basic Results

Let and be two Banach spaces. The notation and stand for the collection of all continuous functions from into and the Banach space of all bounded continuous functions from into endowed with the uniform convergence topology, respectively. Similarly, and stand, respectively, for the class of all jointly continuous functions from into and the collection of all jointly bounded continuous functions from into . The notation stands for the space of bounded linear operators from into endowed with the uniform operator topology, and we abbreviate it to whenever . We set for the closed ball with center at radius in the space . A closed and linear operator is said to be sectorial of type if there exist , and such that its resolvent exists outside the sector and , . Sectorial operators are well studied in the literature. For a recent reference including several examples and properties we refer the reader to [52]. In order to give an operator theoretical approach, we recall the following definition (cf., [50, 51]).

Definition 2.1. Let be a closed and linear operator with domain defined on a Banach space . Recall the generator of a solution operator if there exist and a strongly continuous function such that and , , . In this case, is called the solution operator generated by .

We note that if is sectorial of type with , then is the generator of a solution operator given by , where is a suitable path lying outside the sector (cf., Cuesta's paper [53]). Very recently, Cuesta [53, Theorem ] has proved that if is a sectorial operator of type for some and then there exists such that

Note that is, in fact, integrable. The concept of a solution operator, as defined above, is closely related to the concept of a resolvent family (see Prüss [54, Chapter 1]). For the scalar case, where there is a large bibliography, we refer the reader to the monography by Gripenberg et al. [55] and references therein. Because of the uniqueness of the Laplace transform, in the border case the family corresponds to a -semigroup, whereas in the case a solution operator corresponds to the concept of a cosine family; see Arendt et al. [56] and Fattorini [57]. We note that solution operators, as well as resolvent families, are a particular case of -regularized families introduced by Lizama [58]. According to [58] a solution operator corresponds to a -regularized family. The following result is a direct consequence of [58, Proposition and Lemma ].

Proposition 2.2. Let be a solution operator on with generator . Then, one has the following. (a) and for all , (b)Let and . Then (c)Let and . Then

A characterization of generators of solution operators, analogous to the Hille-Yosida Theorem for -semigroup, can be directly deduced from [58, Theorem ]. Results on perturbation, approximation, representation as well as ergodic type theorems can be deduced from the more general context of -regularized resolvents (see [58–61]).

Let us recall the notions of almost periodicity, asymptotically almost periodicity, asymptotically almost automorphy, and pseudo-almost periodicity which shall come into play later on.

Definition 2.3 (see [62]). Let be a Banach space. Then is called almost periodic if is continuous, and for each there exists such that for every interval of length it contains a number with the property that for each . The number above is called an -translation number for , and the collection of such functions will be denoted by .

Remark 2.4 (see [63]). Note that each almost periodic function is bounded and uniformly continuous. It is well known that the range of an almost periodic function is relatively compact. endowed with the norm of uniform convergence on is a Banach space.

Definition 2.5. Let and be two Banach spaces. Then is called almost periodic in uniformly for if is continuous, and for each and any compact there exists such that every interval of length it contains a number with the property that for all , . The collection of such functions will be denoted by .
It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. We begin with the following standard result in the theory of almost periodic function (see [39, 63]).

Lemma 2.6. Let and . Then the function .

Definition 2.7. A continuous function (resp., ) is called asymptotically almost periodic (resp., asymptotically almost periodic in uniformly in ) if it admits a decomposition , where (resp., ) and (resp., ). Here denotes the subspace of such that and denotes the space of all continuous functions such that uniformly for in any compact subset of . Denote by (resp., ) the set of all such functions. is a Banach space with the sup norm.

Definition 2.8. A continuous function is called uniformly continuous on bounded sets uniformly for if for every and every bounded subset of there exists such that for all and all so that .

Lemma 2.9. Let and let be uniformly continuous on bounded sets uniformly for . If , then .

Let denote the space of all bounded continuous functions such that

and denotes the space of all continuous functions such that is bounded for all and

uniformly in .

Definition 2.10 (see [36, 64]). A function (resp, ) is called pseudo-almost periodic (resp., pseudo-almost periodic in uniformly in ) if where () and ().

The functions and are called the almost periodic component and, respectively, the ergodic perturbation of the function . The set of all such functions will be denoted by (resp., ). Obviously is a subspace of . Furthermore, we have that is a closed subspace of hence, it is a Banach space with the supremum norm (see [65]).

Lemma 2.11 (see [65]). Let satisfy the following conditions. (i) and is bounded for every bounded subset . (ii) is uniformly continuous in each bounded subset of uniformly in . More explicitly, given and bounded, there exists such that and imply that for all .
If , then .

Lemma 2.12. Assume that is sectorial of type . If is an almost periodic function and is given by then .

Proof. For , we take involved in Definition 2.3, then for every interval of length contains a number such that for each . The estimate is responsible for the fact that .

Lemma 2.13. Assume that is sectorial of type . If is an asymptotically almost periodic function and is given by then .

Proof. If , where and then we have that , where By the previous lemma . Next, let us show that . Since , for each there exists a constant such that for all . Then for all , we deduce Therefore, , that is, . This completes the proof.

Lemma 2.14. Assume that is sectorial of type . If is pseudo-almost periodic function and is the function defined in (2.5). Then .

Proof. It is clear that . In fact, we get where and are given by (2.1). If , where and , then from Lemma 2.12, . To complete the proof, we show that . For we see that where , .
It is not hard to check that as . Next, since is bounded and is integrable in , using the Lebesgue dominated convergence theorem, it follows that . The proof is now completed.

Let be a continuous function such that as . We consider the space endowed with the norm

Lemma 2.15 (see [66]). A subset is a relatively compact set if it verifies the following conditions. The set is relatively compact in for each .The set is equicontinuous. For each there exists such that for all and all .
Let be a continuous function such that as . Consider the space endowed with the norm

Lemma 2.16 (see [67]). A subset is a relatively compact set if it verifies the following conditions. The set is relatively compact in for all . uniformly for all .

Definition 2.17. A continuous function is called almost automorphic if for every sequence of real numbers there exists a subsequence such that is well defined for each , and for each . Denote by the set of all such functions; it constitutes a Banach space when it is endowed with the sup norm.
Almost automorphic functions were introduced by Bochner [68] as a natural generalization of the concept of almost periodic function. A complete description of the properties and further applications to evolution equations can be found in the monographs [69] and [70] by N'Guérékata.

Definition 2.18. Let and be two Banach spaces. A continuous function is called almost automorphic in uniformly for in compact subsets of if for every compact subset of and every real sequence there exists a subsequence such that is well defined for each , and for each , . Denote by the set of all such functions.

Lemma 2.19 (see [34]). Assume that is sectorial of type . If is an almost automorphic function and is given by (2.5), then .

In 1980s, N'Guérékata [71] defined asymptotically almost automorphic functions as perturbation of almost automorphic functions by functions vanishing at infinite. Since then, those functions have generated lots of developments and applications; we refer the reader to [69, 72–74] and the references therein.

Definition 2.20 (see [75]). A continuous function (resp., ) is called asymptotically almost automorphic (asymptotically almost automorphic in uniformly for in compact subsets of ) if it admits a decomposition , , where (resp., ) and (resp., ). Denote by (resp., ) the set of all such functions. is a Banach space with the sup norm (see [75, Lemma ]). We note that the range of an asymptotically almost automorphic function is relatively compact [75].

Lemma 2.21. Assume that is sectorial of type . If is an asymptotically almost automorphic function and is given by (2.7), then .

Proof. , where and . We have that , where and are the functions given by (2.8) and (2.9), respectively. By previous lemma and by the proof of Lemma 2.13. This ends the proof.

Lemma 2.22 (see [75]). Let and let be uniformly continuous on bounded sets uniformly for . If , then .

3. Pseudo-Almost Periodic Mild Solutions

We recall the following definition that will be essential for us.

Definition 3.1 (see [34]). Suppose that generates an integrable solution operator . A continuous function satisfying the integral equation is called a mild solution to the equation (1.1).

The following are the main results of this section.

Theorem 3.2. Assume that is sectorial of type . Let be a function pseudo-almost periodic in , uniformly in and assume that there exists an integrable bounded function satisfying Then equation (1.1) has a unique pseudo-almost periodic mild solution.

Proof. We define the operator by Given , in view of Lemma 2.11, we have that is a pseudo-almost periodic function, and hence bounded in . Since the function is integrable on (), we get that exists. Now, by Lemma 2.14, we obtain that and hence is well defined. It suffices to show that the operator has a unique fixed point in . For this, consider . We can deduce that where . Since for sufficiently large, by the contraction principle, has a unique fixed point . This completes the proof.

We can establish the following existence result.

Proposition 3.3. Assume that is sectorial of type . Let be a function pseudo-almost periodic in uniformly in that satisfies the Lipschitz condition (3.2) with . Let . If , where and are the constants in (2.1), then equation (1.1) has a unique pseudo-almost periodic mild solution.

Proof. Let be the map defined in the previous theorem. For we can estimate that which finishes the proof.

Corollary 3.4. Assume that is sectorial of type . Let be a function pseudo-almost periodic in uniformlies in that satisfy the Lipschitz condition If , where and are the constants given in (2.1), then equation (1.1) has a unique pseudo-almost periodic mild solution.

To establish our next result we consider perturbations of (1.1) that satisfy the following boundedness condition.

(H1) There exists a continuous nondecreasing function such that for all and .

We have the following result.

Theorem 3.5. Assume that is sectorial of type . Let be a function pseudo-almost periodic in uniformly in that satisfies assumption (H1) and the following conditions.
(H2) is uniformly continuous on bounded subset of uniformly in .
(H3) For each