Abstract and Applied Analysis

Volume 2018 (2018), Article ID 5947393, 17 pages

https://doi.org/10.1155/2018/5947393

## Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

Correspondence should be addressed to G. M. N’Guérékata

Received 13 October 2017; Accepted 10 December 2017; Published 22 January 2018

Academic Editor: Khalil Ezzinbi

Copyright © 2018 G. M. N’Guérékata and Marko Kostić. 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

The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.

#### 1. Introduction and Preliminaries

Almost periodic and asymptotically almost periodic solutions of differential equations in Banach spaces have been considered by many authors so far (for the basic information on the subject, we refer the reader to the monographs [1–10]). Concerning almost automorphic and asymptotically almost automorphic solutions of abstract differential equations, one may refer, for example, to the monographs by Diagana [4], N’Guérékata [5], and references cited therein.

Of concern is the following abstract multiterm fractional differential inclusion:where , are bounded linear operators on a Banach space , is a closed multivalued linear operator on , , , is an -valued function, and denotes the Caputo fractional derivative of order ([11, 12]). In this paper, we provide the notions of -regularized -existence and uniqueness propagation families for (1) and -regularized -propagation families for (1). In Section 4, we profile these solution operator families in terms of vector-valued Laplace transform, while in Section 5 we consider asymptotical behaviour of analytic integrated solution operator families for (1). The main result of paper, Theorem 18, enables one to consider asymptotically periodic solutions, asymptotically almost periodic solutions, and asymptotically almost automorphic solutions of certain classes of abstract integrodifferential equations in Banach spaces. In a similar way, we can give the basic information about the following abstract semilinear multiterm fractional differential inclusion:where , are bounded linear operators on a Banach space , is a closed multivalued linear operator on , , , and is an -valued function satisfying certain assumptions.

Since we essentially follow the method proposed by Kostić et al. [13] (see also [12, Subsection 2.10.1]), the boundedness of linear operator is crucial for applications of vector-valued Laplace transform and therefore will be the starting point in our work.

The organization and main ideas of this paper can be briefly described as follows. In Section 2, we present the basic information about Stepanov and Weyl generalizations of asymptotically almost periodic functions and asymptotically almost automorphic functions (Proposition 4 is the only new contribution in this section). The main aim of third section is to give a brief recollection of results and definitions about multivalued linear operators in Banach spaces; in a separate Section 3.1, we analyze degenerate -regularized -resolvent families subgenerated by multivalued linear operators. Section 4, which is written almost in an expository manner, is devoted to the study of -regularized -propagation families for (1). The main result of fifth section is Theorem 18, where we investigate the asymptotic behaviour of -regularized -propagation families for (1). In the proof of this theorem, we use the well-known results on analytical properties of vector-valued Laplace transform established by Sova in [14] (see, e.g., [2, Theorem 2.6.1]) in place of Cuesta’s method established in the proof of [15, Theorem 2.1]. The proof of Theorem 18 is much simpler and transparent than that of [15, Theorem ] because of the simplicity of contour in our approach. We will essentially use this fact for improvement of some known results on the asymptotic behaviour of solution operator families governing solutions of abstract two-term fractional differential equations, established recently by Keyantuo et al. [16] and Luong [17]. Contrary to a great number of papers from the existing literature, Theorem 18 is applicable to the almost sectorial operators, generators of integrated or -regularized semigroups, and multivalued linear operators employing in the analysis of (fractional) Poisson heat equations in -spaces ([18, 19]). For more details, see Section 6.

We use the standard notation throughout the paper. By we denote a complex Banach space. If is also such a space, then by we denote the space of all continuous linear mappings from into ; If is a linear operator acting on , then the domain, kernel space, and range of will be denoted by , , and , respectively. The symbol denotes the identity operator on . By we denote the space consisted of all bounded continuous functions from into ; the symbol denotes the closed subspace of consisting of functions vanishing at infinity. By we denote the space consisted of all bounded uniformly continuous functions from to This space becomes one of Banach’s spaces when equipped with the sup-norm. Let us recall that a subset of is said to be total in iff its linear span is dense in .

Let . Consider the Laplace integralfor . If exists for some , then we define the abscissa of convergence of byotherwise, . It is said that is Laplace transformable or equivalently that belongs to the class (P1)-, iff ; in scalar-valued case, we write - and .

If , then we define , ; the Dirac delta distribution. Here, denotes the Gamma function. Set (), , and ().

During the past few decades, considerable interest in fractional calculus and fractional differential equations has been stimulated due to their numerous applications in many areas of physics and engineering. A great number of important phenomena in electromagnetics, acoustics, viscoelasticity, aerodynamics, electrochemistry, and cosmology are well described and modelled by fractional differential equations. For basic information about fractional calculus and nondegenerate fractional differential equations, one may refer, for example, to [11, 12, 20–25] and the references cited therein.

We will use only the Caputo fractional derivatives. Let Then the Caputo fractional derivative ([11, 12]) is defined for those functions for which , byAssuming that the Caputo fractional derivative exists, then for each number the Caputo fractional derivative exists, as well.

The Mittag-Leffler function (, ) is defined bySet ,

The asymptotic behaviour of the Mittag-Leffler function is given in the following lemma (see, e.g., [12]):

Lemma 1. *Let Then, for every and ,where is defined by and the first summation is taken over all those integers satisfying *

If , , and , then the following special cases of Lemma 1 hold good:where

For further information about the Mittag-Leffler functions, compare [11, 12] and the references cited there.

#### 2. Stepanov and Weyl Generalizations of (Asymptotically) Almost Periodic and Almost Automorphic Functions

The class of almost periodic functions was introduced by H. Bohr in 1925 and later generalized by many other mathematicians. Let or , and let be continuous, where is a Banach space with the norm . For any number given in advance, we say that a number is an -period for iff , The set consisting of all -periods for is denoted by We say that is almost periodic, a.p. for short, iff for each the set is relatively dense in , which means that there exists such that any subinterval of of length meets . For basic information about various classes of almost periodic functions and their generalizations, we refer the reader to [4–8, 10, 12, 13, 16, 19, 21, 26–34]. The space consisting of all almost periodic functions from the interval into will be denoted by

It is well known that the vector space consisting of all bounded continuous -periodic functions, denoted by , , is a vector subspace of Set .

Suppose that , , and , where or Define the Stepanov “metric” byThen, in scalar-valued case, there existsin The distance appearing in (11) is called the Weyl distance of and The Stepanov and Weyl “norm” of are introduced byrespectively. We say that a function is Stepanov -bounded, -bounded shortly, iffThe space consisting of all -bounded functions becomes a Banach space when equipped with the above norm. A function is called Stepanov -almost periodic, -almost periodic shortly, iff the function , defined by , , is almost periodic. It is said that is asymptotically Stepanov -almost periodic, asymptotically -almost periodic for short, iff is asymptotically almost periodic.

It is a well-known fact that if is an almost periodic (resp., a.a.p.) function then is also -almost periodic (resp., asymptotically -a.a.p.) for The converse statement is not true, in general.

By we denote the space consisted of all -almost periodic functions A function is said to be asymptotically Stepanov -almost periodic, asymptotically -almost periodic for short, iff is asymptotically almost periodic. By and we denote the vector spaces consisting of all Stepanov -almost periodic functions and asymptotically Stepanov -almost periodic functions, respectively.

Let us recall that any asymptotically almost periodic function is also asymptotically Stepanov -almost periodic (). The converse statement is clearly not true because an asymptotically Stepanov -almost periodic function need not be continuous.

We are continuing by explaining the basic definitions and results about the (asymptotically) Weyl-almost periodic functions.

*Definition 2 (see [35]). *Assume that or Let and .(i)It is said that the function is equi-Weyl--almost periodic, for short, iff for each we can find two real numbers and such that any interval of length contains a point such that(ii)It is said that the function is Weyl--almost periodic, for short, iff for each we can find a real number such that any interval of length contains a point such that

We know that in the set theoretical sense and that any of these two inclusions can be strict ([26]).

We refer the reader to [35] for basic definitions and results about asymptotically Weyl-almost periodic functions.

*Definition 3. *We say that is Weyl--vanishing iff

It is clear that for any function we can replace the limits in (16). It is said that is equi-Weyl--vanishing iff

If and is equi-Weyl--vanishing, then is Weyl--vanishing. The converse statement does not hold, in general ([35]). By and we denote the vector spaces consisting of all Weyl--vanishing functions and equi-Weyl--vanishing functions, respectively.

It can be simply proved that the limit of any uniformly convergent sequence of bounded continuous functions that are (asymptotically) almost periodic or automorphic, respectively (asymptotically), Stepanov almost periodic or automorphic, has again this property. The following result holds for the Weyl class.

Proposition 4. *Let be a uniformly convergent sequence of functions from , respectively, , where If is the corresponding limit function, then , respectively, .*

*Proof. *We will prove the part (i) only for the equi-Weyl--almost periodic functions. It is clear that Let be given in advance. Then there exists an integer such that for each we have thatBy definition, we know that there exist two real numbers and such that any interval of length contains a point such thatThen, for the proof of equi-Weyl--almost periodicity of function , we can choose the same and , and the same from any subinterval ; speaking-matter-of-factly, we havefor all , so that a simple calculation involving (18) gives the existence of a finite constant such thatThen the final result simply follows from (19).

And, just a few words about (generalized) automorphic extensions of introduced classes, where our results clearly apply. Let be continuous. As it is well known, is called almost automorphic, a.a. for short, iff for every real sequence there exist a subsequence of and a map such thatpointwise for If this is the case, then it is well known that and that the limit function must be bounded on but not necessarily continuous on Furthermore, it is clear that the uniform convergence of one of the limits appearing in (22) implies the convergence of the second one in this equation and that, in this case, the function has to be almost periodic and the function has to be continuous. If the convergence of limits appearing in (22) is uniform on compact subsets of , then we say that is compactly almost automorphic, c.a.a. for short. The vector space consisting of all almost automorphic, respectively, compactly almost automorphic functions, is denoted by , respectively, By Bochner’s criterion [4], any almost periodic function has to be compactly almost automorphic. The converse statement is not true, however [36]. It is also worth noting that P. Bender proved in doctoral dissertation that that a.a. function is c.a.a. iff it is uniformly continuous (1966, Iowa State University).

It is well-known that the reflexion at zero keeps the spaces and unchanged and that the function from (22) satisfies and , later needed to be a compact subset of An interesting example of an almost automorphic function that is not almost periodic has been constructed by W. A. Veech

A continuous function is called asymptotically (compact) almost automorphic, a.(c.)a.a. for short, iff there exist a function and a (compact) almost automorphic function such that , Using Bochner’s criterion again, it readily follows that any asymptotically almost periodic function is asymptotically (compact) almost automorphic. It is well known that the spaces of almost periodic, almost automorphic, compactly almost automorphic functions and asymptotically (compact) almost automorphic functions are closed subspaces of when equipped with the sup-norm.

We refer the reader to [28] for the notion of Stepanov-like almost automorphic functions. The concepts of Weyl-almost automorphy and Weyl pseudo almost automorphy, more general than those of Stepanov almost automorphy and Stepanov pseudo almost automorphy, were introduced by Abbas [37] in 2012. Besides the concepts of Stepanov-like almost automorphic functions, our results apply also to the classes of Weyl-almost automorphic functions and Besicovitch almost automorphic functions, introduced in [38] (cf. [7, 39] for more details).

#### 3. Multivalued Linear Operators in Banach Spaces

In this section, we will present some necessary definitions and auxiliary results from the theory of multivalued linear operators in Banach spaces. For further information in this direction, the reader may consult the monographs by Cross [40] and Favini and Yagi [18].

Let and be two Banach spaces over the field of complex numbers. A multivalued mapping is said to be a multivalued linear operator (MLO) iff the following two conditions hold:(i) is a linear subspace of ;(ii), , and , , .

In the case that , then we say that is an MLO in It is well-known that the equality holds for every and for every with If is an MLO, then is always a linear subspace of and for any and Put Then the set is called the kernel of The inverse of an MLO is generally defined by and . It is checked at once that is an MLO in , and that and If , that is, if is single-valued, then is called injective. If are two MLOs, then we define its sum by and , It is evident that is likewise an MLO. We write iff and for all

Let and be two MLOs, where is a complex Banach space. The product of and is defined by and A simple proof shows that is an MLO and The scalar multiplication of an MLO with the number , for short, is defined by and , Then is an MLO and , .

It is said that an MLO is closed iff for any two sequences in and in such that ; for all we have that and imply and .

We need the following lemma from [19].

Lemma 5. *Let be a locally compact, separable metric space, and let be a locally finite Borel measure defined on Suppose that is a closed MLO. Let and be -integrable, and let , Then and *

Henceforward, will always be an appropriate subspace of and will always be the Lebesgue measure defined on

Denote by (P1)- the vector space consisting of all Laplace transformable functions ; by we denote the Laplace transform of , defined as in [2]. We need also the following lemma from [19].

Lemma 6. *Assume that is a closed MLO and that ( P1), (P1) and , for Then for any which is a point of continuity of both functions and .*

Suppose that is an MLO in and that is possibly noninjective operator satisfying Then the -resolvent set of , for short, is defined as the union of those complex numbers for which(i);(ii) is a single-valued linear continuous operator on .

The operator is called the -resolvent of ; the resolvent set of is defined by , .

We will use the following extension of [19, Theorem 1.2.4(i)], whose proof can be left to the reader as an easy exercise (see also the proof of [18, Theorem 1.7, p. 24]).

Lemma 7. *Let and let be an MLO. If , , , and , then one has*

Suppose that is an MLO in Then is said to be an eigenvalue of iff there exists an element such that ; we call an eigenvector of operator corresponding to the eigenvalue Let us recall that, in purely multivalued case, an element can be an eigenvector of operator corresponding to different values of scalars The point spectrum of , for short, is defined as the union of all eigenvalues of

##### 3.1. Degenerate -Regularized -Resolvent Operator Families

If it is not stated otherwise, we assume that , , , , , is an MLO, , is injective, is injective, and .

We need the following notions from [19].

*Definition 8. *Suppose , , , , , is an MLO, , and is injective.(i)Then it is said that is a subgenerator of a (local, if ) mild -regularized -existence and uniqueness family iff the mappings , , and , , are continuous for every fixed and , and the following conditions hold: (ii)Let be strongly continuous. Then it is said that is a subgenerator of a (local, if ) mild -regularized -existence family iff (25) holds.(iii)Let be strongly continuous. Then it is said that is a subgenerator of a (local, if ) mild -regularized -uniqueness family iff (26) holds.

*Definition 9. *Suppose that , , , , , is an MLO, is injective, and Then it is said that a strongly continuous operator family is an -regularized -resolvent family with a subgenerator iff is a mild -regularized -uniqueness family having as subgenerator, , and .

If , is said to be exponentially bounded (bounded) iff there exists () such that the family is bounded. If , where , then it is also said that is an -times integrated -resolvent family; -times integrated -resolvent family is further abbreviated to -resolvent family. We accept a similar terminology for the classes of mild -regularized -existence families and mild -regularized -uniqueness families.

The integral generator of a mild -regularized -uniqueness family (mild -regularized -existence and uniqueness family ) is defined throughthe integral generator of an -regularized -regularized family is defined in a similar fashion. The integral generator is a closed MLO in which is, in fact, the maximal subgenerator of () with respect to the set inclusion. We refer the reader to [19] for the notion of an exponentially bounded, analytic -regularized -resolvent operator family.

Unless stated otherwise, we will always assume henceforth that the function is a scalar-valued kernel on and that the operator is injective. For more details about abstract degenerate differential equations, the reader may consult the monographs [18, 41–43].

#### 4. -Regularized -Propagation Families for (1)

Recall that , are bounded linear operators on a Banach space , is a closed multivalued linear operator on , , , and is an -valued function. Henceforth, we always assume that are scalar-valued kernels and in Set , , , , and .

We will use the following definition.

*Definition 10. *A function is called a (strong) solution of (1) iff for , , and (1) holds.

Integrating both sides of (1) -times and employing the closedness of , Lemma 5, and the equality [11, ], it readily follows that any strong solution , of (1) satisfies the following:

If , then we define Plugging , , , in (28), we getwhere appears in the th place () starting from Proceeding as in nondegenerate case [12], this inclusion motivates us to introduce the following extension of [12, Definition 2.10.2] (cf. also [34, Definition ] and [32, Definitions and ] for similar notions).

*Definition 11. *Suppose that , , , and and are injective. A sequence of strongly continuous operator families in is called a (local, if ):(i)-regularized -existence propagation family for (1) iff the following holds: for any (ii)-regularized -uniqueness propagation family for (1) iff the following holds: provided and .(iii)-regularized -resolvent propagation family for (1), in short -regularized -propagation family for (1), iff is a -regularized -uniqueness propagation family for (1), and if for every , , and , one has , , and .

In the case that , where , then we also say that is a -times integrated -resolvent propagation family for (1); -times integrated -resolvent propagation family for (1) is simply called -resolvent propagation family for (1). For a -regularized -existence and uniqueness family , it is said that is exponentially bounded iff each single operator family is. The above terminological agreement is accepted for all other classes of -regularized -propagation families introduced so far.

If , where for , then it is also said that is a subgenerator of The notion of integral generator of is introduced as in nondegenerate case [12].

Hereafter, the following equality will play an important role in our analysis:for any The basic properties of subgenerators and integral generators continue to hold, with appropriate changes, in degenerate case; compare [12] and [19, Section ] for more details. We leave to the interested reader the problem of transferring the assertions of [12, Propositions 2.10.3–2.10.5, Theorem 2.10.7] to degenerate case.

The following is a degenerate version of [12, Definition 2.10.6].

*Definition 12. *Let Consider the following inhomogeneous Cauchy inclusion:A function is said to be(i)a strong solution of (33) iff there exists a continuous function such that for all and (ii)a mild solution of (33) iff

Clearly, every strong solution of (33) is also a mild solution of the same problem while the converse statement is not true, in general. We similarly define the notion of a strong (mild) solution of problem (28).

We have the following:(a)If is a -existence propagation family for (1), then the function , , is a mild solution of (28) with for (b)If is a -uniqueness propagation family for (1), and