Open Access

On Type of Periodicity and Ergodicity to a Class of Fractional Order Differential Equations

Advances in Difference Equations20102010:179750

https://doi.org/10.1155/2010/179750

Received: 22 October 2009

Accepted: 27 January 2010

Published: 11 February 2010

Abstract

We study several types of periodicity to a class of fractional order differential equations.

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., [18]). 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, 915]). 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. [1719], 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 [2429]). 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:

(1.1)

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 [3639] in the early nineties. Since then, such notion became of great interest to several mathematicians (see [4049]). 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

(1.2)
(1.3)

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

(2.1)

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.
  1. (a)

    and for all ,

     
  2. (b)

    Let and . Then

     
  3. (c)

    Let and . Then

     
(2.2)

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 [5861]).

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

(2.3)

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

(2.4)

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.
  1. (i)

    and is bounded for every bounded subset .

     
  2. (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
(2.5)

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
(2.6)

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
(2.7)

then .

Proof.

If , where and then we have that , where
(2.8)
(2.9)
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
(2.10)

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
(2.11)

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.

(c-1) The set is relatively compact in for each .

(c-2) The set is equicontinuous.

(c-3) 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.

(c-1) The set is relatively compact in for all .

(c-2) 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, 7274] 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
(3.1)

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
(3.2)

Then equation (1.1) has a unique pseudo-almost periodic mild solution.

Proof.

We define the operator by
(3.3)
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
(3.4)

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
(3.5)

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
(3.6)

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 , , where is given by Lemma 2.15. Set

(3.7)

where and are constants given in (2.1).

(H4) For each there is such that, for every , implies that

(3.8)

for all .

(H5) For all , , and , the set is relatively compact in .

(H6) .

Then equation (1.1) has a pseudo-almost periodic mild solution.

Proof.

We define the operator on as in (3.3). We show that has a fixed point in .

  1. (1)

    For , we have that

    (3.9)
     
It follows from condition (H3) that . From condition (H4) it follows that is a continuous map.
  1. (ii)

    We next show that is completely continuous. The argument comes from Lemma 2.15. In fact, let and for . Initially, we will prove that is a relatively compact subset of for each . It follows from condition (H3) that the function is integrable on . Hence, for , we can choose such that . Hence , where denotes the convex hull of . Using that is strongly continuous and the property (H5), we infer that is relatively compact set, and , which establishes our assertion.

    We next show that the set is equicontinuous. In fact, we can decompose

    (3.10)

    For each , we can choose and such that

    (3.11)
     

for . Moreover, since is relatively compact set and is strongly continuous, we can choose such that for . Combining these estimates, we get for small enough and independent of .

Finally, applying condition (H3), we can show that

(3.12)
and this convergence is independent of . Taking into account Lemma 2.15, is a relatively compact set in .
  1. (iii)

    If is a solution of equation for some , then we can check that and, combining with condition (H6), we conclude that the set is bounded.

     
  2. (iv)

    It follows, from Lemmas 2.11 and 2.14, that and, consequently, is completely continuous. Since is bounded and using Leray-Schauder alternative theorem, we infer that has a fixed point . Let be a sequence in that converges to . We see that converges to uniformly in . This implies that and completes the proof.

     

It is particularly interesting to note that the next result is not covered by the results by Cuevas and Lizama [34].

Corollary 3.6.

Assume that conditions (H1)–(H6) hold and that is sectorial of type . If is almost periodic in uniformly for , then equation (1.1) has an almost periodic mild solution.

Proof.

It is a consequence of Lemmas 2.6 and 2.12.

4. Asymptotically Almost Periodic Mild Solutions

We recall the following definition.

Definition 4.1 (see [50]).

Suppose that generates an integrable solution operator . A function satisfying
(4.1)

is called a mild solution of the problem (1.2)-(1.3).

Theorem 4.2.

Assume that is sectorial of type . Let be a function asymptotically almost periodic in uniformly in and assume that there exists an integrable bounded function satisfying
(4.2)

Then the problem (1.2)-(1.3) has a unique asymptotically almost periodic mild solution.

Proof.

We define the operator on the space by
(4.3)
We show initially that . In fact, we observe that the estimate (2.1) implies that . It follows from Lemma 2.9 that the function is asymptotically almost periodic; then by Lemma 2.13, and hence is well defined. Let be in and define . We have the following estimate:
(4.4)

which is responsible for the fact that has a unique fixed point in .

Corollary 4.3.

Assume that is sectorial of type . Let be a function asymptotically almost periodic in uniformly in that satisfies the Lipschitz condition (4.2) with . If , where and are the constants given in (2.1), then the problem (1.2)-(1.3) has a unique asymptotically almost periodic mild solution.

Taking with and in (1.2), the above result produces the following corollary.

Corollary 4.4.

Let be a function asymptotically almost periodic in uniformly in that satisfies the Lipschitz condition (4.2) with . Then problem (1.2)-(1.3) has a unique asymptotically almost periodic solution whenever .

Remark 4.5.

A similar result as that of the previous corollary was obtained by Cuevas and de Souza [50] for obtaining an S-asymptotically -periodic mild solution for problem (1.2)-(1.3) (see [50, Remark 3.6] for complementary comments).

Next, we establish a version of Theorem 4.2 which enable us to consider locally Lipschitz perturbations for equation (1.2). We have the following result.

Theorem 4.6.

Assume that is sectorial of type . Let be a function asymptotically almost periodic in uniformly in and assume that there is a continuous and nondecreasing function such that for each positive number , and , , , one has
(4.5)

where and for ; then there is such that for each with there exists a unique asymptotically almost periodic mild solution of (1.2)-(1.3).

Proof.

Let and be such that . We affirm that the assertion holds for . In fact, we consider such that . We set
(4.6)

endowed with the metric . We define the operator on the space by (4.3). Let we next show that We have the estimate that is, .

On the other hand, for we obtain

(4.7)

To conclude, we note that , which means that is a -contraction. This completes the proof.

Theorem 4.7.

Assume that is sectorial of type . Let be an asymptotically almost periodic in uniformly in that satisfies the following conditions.

(H*1) There is a continuous nondecreasing function such that for all and .

(H*2) is uniformly continuous on bounded sets of uniformly in .

(H*3) For each , , where is given by Lemma 2.16. Set

(4.8)

where and are constants given in (2.1).

(H*4) For each there is such that, for every , implies that

(4.9)

for all .

(H*5) For all , and , the set is relatively compact in .

(H*6) .

Then problem (1.2)-(1.3) has an asymptotically almost periodic mild solution.

Proof.

We define the operator on as in (4.3). We show that has a fixed point in .
  1. (i)
    For , we have that
    (4.10)

    It follows from (H*3) that . From condition (H*4) it follows that is a continuous map.

     
  2. (ii)

    We next show that is completely continuous. Let and for . Initially, we can infer that is a relatively compact subset of for each . In fact, using condition (H*5) we get that is relatively compact. It is easy to see that , which establishes our assertion. From the decomposition of given by , it follows that the set is equicontinuous. We can show that uniformly for all . From Lemma 2.16, we deduce that is relatively compact set in .

     

We note that the set is bounded. In fact, it follows from condition (H*6) and the estimate . It follows, from Lemmas 2.9 and 2.13, that . The remaining of proof makes use of a similar argument already done in the proof of Theorem 3.5.

5. Asymptotically Almost Automorphic Solutions of Fractional Integro Differential Neutral Equations

This section is mainly concerned with the existence and uniqueness of an asymptotically almost automorphic mild solution to the fractional integro differential neutral equation

(5.1)

where , is a linear densely defined operator of sectorial type, and are functions subject to some additional conditions.

Definition 5.1.

Suppose that generates an integrable solution operator . A function satisfying the integral equation
(5.2)

is called a mild solution of problem (5.1).

We have the following result.

Theorem 5.2.

Assume that is sectorial of type . Let be two functions asymptotically almost automorphic in uniformly for in compact subsets of such that
(5.3)

If , where and are the constants given in (2.1), then problem (5.1) has a unique asymptotically almost automorphic mild solution.

Proof.

We define the operator on the space by
(5.4)
Applying Lemma 2.22, we infer that and belong to . By Lemma 2.21, we obtain that is -valued. Furthermore, we have the estimate
(5.5)

which proves that is a contraction we conclude that has a unique fixed point in . This completes the proof.

Next, we establish a local version of the previous result.

Theorem 5.3.

Assume that is sectorial of type . Let be two functions asymptotically almost automorphic in uniformly for in compact subsets of and assume that there are continuous and nondecreasing functions such that for each positive number , and , , , one has
(5.6)

for all , where and for every . Then there is such that satisfies then there is a unique asymptotically almost automorphic mild solution of (5.1).

Proof.

Let and be such that
(5.7)
where and are the constants given in (2.1). We consider such that , with ; we define the space endowed with the metric . We also define the operator on the space by (5.4). Let be in in a similar way as that of proof of Theorem 5.2; we have that . Moreover, we obtain the estimate
(5.8)
Therefore . On the other hand, for , we see that
(5.9)

which shows that is a contraction from into . The assertion is now a consequence of the contraction mapping principle.

Remark 5.4.

A similar result was obtained by Diagana et al. [76] for the existence of asymptotically almost automorphic solutions to some abstract partial neutral integrodifferential equations.

Theorem 5.5.

Assume that is sectorial of type and that conditions (H*1),(H*3), (H*4) and (H*5) hold. In addition, suppose that the following properties hold.

(A1) The functions are asymptotically almost automorphic in and uniformly for in compact subsets of and uniformly continuous on bounded sets of uniformly in .

(A2) There is a constant such that for all and (here is given in Lemma 2.16). Set

(5.10)

where and are the constants given in (2.1).

(A3) .

Then problem (5.1) has an asymptotically almost automorphic mild solution.

Proof.

We define the operator on as in (5.4); we consider the decomposition , where
(5.11)
For , we have that
(5.12)
Hence is -valued. On the other hand, is an -contraction. It follows from the proof of the Theorem 4.7 that is completely continuous. From Lemmas 2.21 and 2.22, we have that
(5.13)
Hence and is completely continuous. Putting we claim that there is such that . In fact, if we assume that this assertion is false, then for all we can choose and such that . We observe that
(5.14)

Thus ), which is contrary to assumption (A3). We have that is a contraction on and is a compact set. It follows from [77, Corollary ] that has a fixed point . More precisely, , and this finishes the proof.

6. Applications

To illustrate our results, initially we examine sufficient conditions for the existence and uniqueness of pseudo-almost periodic mild solutions to the fractional relaxation-oscillation equation given by

(6.1)
(6.2)

where . To study this system in the abstract form (1.1), we choose the space and the operator defined by , with domain . It is well known that is generator of an analytic semigroup on . Hence, is sectorial of type . (6.1) can be formulated by the inhomogeneous problem (1.1), where . Let us consider the nonlinearity for all and , with , . We observe that . Hence . We observe that is pseudo-almost periodic in , uniformly in such that (3.6) holds for . If we assume that , then by Corollary 3.4, the fractional relaxation-oscillation equation (6.1) has a unique pseudo-almost periodic mild solution.

Taking and , we define the function as

(6.3)

We consider the following fractional relaxation-oscillation equation given by

(6.4)

Equation (6.4) can be expressed as an abstract equation of the form (1.1), where

(6.5)

Proposition 6.1.

Problem (6.4) has a pseudo-almost periodic mild solution.

Proof.

Let us briefly discuss the proof of this proposition. We get without difficulties the following two estimates:
(6.6)

which are responsible for the fact that and that is uniformly continuous on bounded sets of uniformly in .

It is straightforward to verify that

(6.7)
and . Hence, we can define in (H1) by . Taking , ; . From the discussion above, we see that
(6.8)
which means that conditions (H3) and (H4) of Theorem 3.5 are satisfied. An easy computation leads to . An argument involving Simon's theorem (see [78, Theorem , pages 71–74]) proves that the set is relatively compact in . In fact, we can verify that , . Hence, for , is bounded uniformly in and . On the other hand, we can infer the following estimate:
(6.9)
Therefore,
(6.10)

uniformly in and . Finally, Simon's theorem leads to the conclusion that is relatively compact. Using Theorem 3.5, equation (6.4) has a pseudo-almost periodic mild solution.

Next, we examine the existence and uniqueness of an asymptotically almost automorphic mild solution to the fractional differential equation

(6.11)

where , are appropriate functions, and . From Corollary 4.3, we can deduce the following result.

Proposition 6.2.

Assume that is an asymptotically almost periodic function and that there exists a constant such that for all . If , then (6.11) has a unique asymptotically almost periodic mild solution.

We consider the fractional differential equation

(6.12)
(6.13)
(6.14)

From Theorem 4.6, we deduce the following result.

Proposition 6.3.

Assume that is an asymptotically almost periodic function, then there is such that for each with there exists a unique asymptotically almost periodic mild solution of (6.12)–(6.14).

Proof.

The proof is straightforward. Indeed, (6.12) can be expressed as an abstract equation of form (1.2), where , , . We observe that , for all and . Hence the perturbation is locally Lipschitz. We remark that is asymptotically almost periodic in uniformly in , as we mentioned before, by using Theorem 4.6.

Take and . We define the function by

(6.15)

We examine asymptotically almost periodic mild solution to the fractional relaxation-oscillation equation given by

(6.16)

where .

Proposition. 6.4

Problem (6.16) has an asymptotically almost periodic mild solution.

Proof.

We briefly recall some argument of the proof. Problem (6.16) can be written as an abstract problem of the form (1.2)-(1.3) in , where the perturbation associated is
(6.17)
We can choose the function in (H*1) by . From the estimate
(6.18)

we get conditions (H*2) and (H*4), the latter being considered with , .

We can infer that

(6.19)
Hence condition (H*3) is fulfilled. By looking at the estimates
(6.20)

and using Simon's theorem, we conclude that condition (H*5) holds. Consequently, by Theorem 4.7 we can assert that problem (6.16) has an asymptotically almost periodic mild solution. This completes the proof of Proposition 6.4.

Remark 6.5.

It is easy to check that results in Section 5 are applicable to similar fractional differential equations as those treated in this section. For the sake of shortness, the details are left to the reader.

Declarations

Acknowledgment

Claudio Cuevas is partially supported by CNPQ/Brazil under Grant no. 300365/2008-0.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals
(3)
Departamento de Matemática, Universidade Federal de Pernambuco

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