Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

Lachouri, Adel; Ardjouni, Abdelouaheb; Jarad, Fahd; Abdo, Mohammed S.

doi:https://doi.org/10.1155/2021/8162890

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Advances in Fractional Functional Analysis

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Volume 2021 | Article ID 8162890 | https://doi.org/10.1155/2021/8162890

Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

Adel Lachouri,¹Abdelouaheb Ardjouni,²Fahd Jarad,^3,4and Mohammed S. Abdo⁵

Academic Editor: Andrea Scapellato

Received08 Aug 2021

Revised09 Oct 2021

Accepted11 Oct 2021

Published03 Nov 2021

Abstract

In this paper, we study the existence of solutions to initial value problems for a nonlinear generalized Caputo fractional differential inclusion with Lipschitz set-valued functions. The applied fractional operator is given by the kernel and the derivative operator . The existence result is obtained via fixed point theorems due to Covitz and Nadler. Moreover, we also characterize the topological properties of the set of solutions for such inclusions. The obtained results generalize previous works in the literature, where the classical Caputo fractional derivative is considered. In the end, an example demonstrating the effectiveness of the theoretical results is presented.

1. Introduction

Fractional calculus is a field of mathematical analysis that embraces the integrals and derivatives of functions of any real or complex order. For the past few decades, this field has been one of the hand-over-fist sprawling fields of mathematics by virtue of the amazing findings obtained when researchers enrolled the fractional operators in their attempts to construe some problems that arise in nature (see [1–6]). As a matter of fact, the classical fractional calculus consisted of one main integral operator, namely, the Riemann-Liouville fractional integral, and two fractional derivatives, namely, the Riemann-Liouville and Caputo derivatives. Because of the penurious number of operators, researchers were compelled to discover and develop new fractional operators that allow them to better comprehend the world around them. In the last 10 years or so, new fractional operators have been introduced and wielded on many occasions. One can touch upon the operators in [7–13]. It should be noted that some of these operators are an extension or generalization of the Riemann-Liouville integrals and Riemann-Liouville and Caputo derivatives that are nonlocal but singular kerneled. The others are brand-new ones, and they are nonlocal and contain nonsingular kernels.

One of the main applications of advanced fractional calculus is the theory of fractional evolution inclusions since they are abstract formulations for numerous problems arising in physics and engineering. Evolution equations and inclusions are commonly applied to describe the systems and models that change or evolve over time. Many studies have investigated the existence and uniqueness of solutions for fractional evolution problems based on semigroup and fixed point theories; e.g., Bedi et al. [14] studied the existence of mild solutions for Hilfer-type fractional neutral impulsive evolution problems in a Banach space. The same author, with others in [15], discussed the stability and controllability results for Hilfer-type fractional evolution equations in a Banach space. The existence of saturated mild (and global) solutions of Caputo-type fractional semilinear evolution problems with noncompact semigroups has been obtained in Banach spaces by Chen et al. [16]. The authors in [17] studied the existence of local (and global) solutions and the uniqueness of a mild solution to fractional semilinear evolution problems with compact (and noncompact) semigroups in Banach spaces. Zhou and Jiao [18] considered a class of nonlocal Cauchy problems for a Caputo-type fractional neutral evolution problem to investigate the existence and uniqueness of mild solutions. For differential equations-inclusions governed by Cauchy conditions or boundary conditions, the issue of importance is to tackle the existence, uniqueness, and stability of their solutions. These properties constitute the most essential parts of the analysis of these equations. There are heaps of papers that went about the existence, uniqueness, and stability results of differential equations-inclusions within the scope of a variety of fractional derivatives based on different types of fixed point techniques. For instance, Abdo et al. [19] have proven the existence and different types of stability of solutions for -Hilfer-type fractional integro-problems. The authors in [20] investigated some existence results of Caputo-type fractional neutral inclusions by using weak topology. The existence of solutions for a generalized Caputo-type fractional differential inclusion problem with integral boundary conditions has been studied by Belmor et al. [21]. In [22], the authors discussed the existence of solutions for a Caputo-type fractional for higher-order fractional inclusion problems. Lachouri et al. [23] considered a nonlinear Hilfer-type nonlocal fractional inclusion problem to prove the existence results, taking into account the convex and nonconvex values of the nonlinear term.

Recently, Chen et al. [16] proved the existence of saturated mild solutions and global mild solutions for fractional evolution equations of the type where is the Caputo fractional derivative of order , is a closed linear operator and generates a uniformly bounded -semigroup in , and is the given function.

In [17], Suechoei and Ngiamsunthorn studied the following fractional evolution equations: where is the -Caputo FD of order , is the infinitesimal generator of a -semigroup of uniformly bounded linear operators on , and is the given function.

Motivated by the aforementioned works and inspired by [17], we consider the following fractional evolution inclusion involving -Caputo FD:where is a set-valued map () from to the family of and is a Banach space with the norm .

The novelty of the present work is choosing a more general operator than the classical fractional operators. More precisely, problem (3) is reduced to a Caputo-type problem, for ; Hadamard-Caputo-type problem, for ; and Katugampola-Caputo-type problem, for , . In addition, we describe the topological properties of the considered solution in the present work.

In this paper, we aim to prove the existence of mild solutions for problem (3) involving the generalized Caputo derivative using the fixed point theorem (FPT) of Nadler and Covitz and to characterize the topological properties of the set of solutions for such inclusions.

The acquired results are more general and cover many of the parallel problems that contain special cases of function , such as [16, 18].

2. Preliminary Notions

2.1. Fractional Calculus (FC)

In this portion, we introduce several basic notions of FC and necessary lemmas that will be needed in such a study.

Let . Denoted by , we have the Banach space of continuous functions with

Let such that is increasing and , , and

Definition 1 (see [24]). The -Riemann-Liouville FI of a function of order is defined by

Definition 2 (see [24]). The -Riemann-Liouville FD of order for a function is given bywhere ,

Definition 3 (see [13]). The -Caputo FD of a function of order is described bywhere and ,

Lemma 4 (see [11, 25]). Let . Then,

Lemma 5 (see [13]). If , , and , thenIn particular, given , we have

Definition 6 (see [11]). Let be real-valued functions. The generalized Laplace transform of is defined by

Theorem 7 (see [11]). Let and be a piecewise continuous function on each interval and -exponential order. Then,

Definition 8 (see [26, 27]). Let and . The Wright-type function is defined by

Proposition 9 (see [26, 27]). The Wright function is an entire function and has the properties listed below:(1) for and (2), for (3), (4),

Next, we recall some conceptions concerning the semigroups of linear operators. For more details, see [28, 29].

For strongly continuous semigroups (i.e., C-semigroup on ), we define the generator

By , we denote the domain of , that is,

Lemma 10 (see [28, 29]). Let be a -semigroup; then, there exist constants and such that

Lemma 11 (see [28, 29]). A linear operator is the infinitesimal generator of a -semigroup iff(1) is closed, and (2)The resolvent set of contains , and for every , we havewhere .

In relation to problem (3), we need the following lemma which was proven in [17].

Lemma 12. The mild solution of IVPsis obtained aswherefor .

Lemma 13 (see [18]). The operators and have the following properties:(1)For any fixed , and are bounded linear operators with(2) and are strongly continuous ; that is, for every and , we haveas .(3)If is a compact operator , then and are compact for all .(4)The operators and are continuous in the uniform operator topology.

3. Main Results

In what follows, we will utilize the notation for a normed space . For more information about the svm, we refer the reader to [30, 31].

Definition 14. A function is a solution of (3), if with fulfilling the initial conditionThe first outcome treats the nonconvex based on the theorem of Covitz and Nadler [32].

Let be a metric space induced from the normed space . Consider defined bywhere and . Then, is a metric space (see [33]).

Definition 15. A svm is -Lipschitz iff such thatParticularly, if , then is a contraction.

Theorem 16. Letand assume that
(As1) such that is measurable for any .
(As2) for (a.e.) all and with and for (a.e.) all .
Then, (3) possesses at least one solution on if

Proof. At first, to switch problem (3) into a FP, we formulate asfor . The solution of (3) is obviously an FP of . The following are the steps in the proofing process.
By virtue of assumption (As1) and [34] (Theorem III.6), has a measurable selection, and thus, . In the sequel, we demonstrate that defined in (29) fulfills the assumptions of FPT of Covitz and Nadler. First, we show that is closed for every . Let such that in . Then, and there is such thatAccordingly, there is a subsequence that converges to in because has compact values. As an outcome, , and we getHence, .
It remains to demonstrate that there is a such thatLet and . Then, there exists such thatBy (As2), we haveSo, there exists such thatAs follows, we build a svm , whereFrom [34] (Proposition III.4), the svm is measurable. We now select the function such thatWe defineAs a consequence, we getTherefore,By the analogous relation, obtained by interchanging the roles of and , we getBecause is a contraction, we conclude that it has a FP, which is a solution of (3) based on the Covitz and Nadler theorem.

Next, we study the topological properties of the set of solutions of problem (3).

Theorem 17. Assume that
(As3) is a compact operator for each .
(As4) is a -Carathéodory set-valued map.
(As5) and a nondecreasing such thatThen, the set of solutions of (3) is convex and relatively compact.

Proof. We consider the operator defined in (29) and demonstrate in the following steps that the set of solutions of (3) is convex and relatively compact in .(Step 1)The svm is convex for every .Let . Then, there exist such thatLet . Then, for every ,Since has convex values, is convex and . Thus, .(Step 2) is bounded on bounded sets of .For a constant , let be a bounded set in . Then, for each and , there exists such thatUnder hypothesis (As4) and Lemma 13 (1), for any , we attainThus,where is defined in (27).(Step 3) sends bounded sets of into equicontinuous sets.Let and . Then, there is a function such thatLet , . Then,From Lemma 13, it is obvious that as and we getas , which yields to and . For and , it is clear that . Then, for any , we getas and , which yields to ; thus,Hence, is equicontinuous.(Step 4)We show that for each , is relatively compact in .Clearly, is relatively compact in . Let be fixed, and for any and , defineThen, by the compactness of for , we see that the set is relatively compact in for all . Moreover, for each , we haveHence, there are relatively compact sets arbitrarily close to for , Therefore, is relatively compact in .
From the Arzela-Ascoli theorem, is relatively compact in . Consequently, is relatively compact in .

4. An Example

Let . Consider the following fractional partial differential inclusions with Caputo derivative :where and , and we define an operator by for , where . Then, generates a strongly continuous semigroup which is compact. Furthermore, , . Let defined by

Clearly, , where and for (a.e.) all . Besides, we obtain which implies . Consequently, all items of Theorem 16 are satisfied. Then, there exists at least one solution of (10) on .

Remark 18. Our current results for problem (3) remain true for the following cases:(i)Caputo-type inclusion problem: .(ii)Caputo-Hadamard-type inclusion problem: .(iii)Generalized Caputo-type inclusion problem: .

Remark 19. The obtained results for problem (3) include the results of Suechoei and Ngiamsunthorn [17]; i.e., problem (3) is reduced to the considered problem in [17] when .

5. Concluding Remarks

We have studied the class of IVPs for generalized Caputo FDIs with Lipschitz set-valued functions. We obtained the existence result by using the FPT of Covitz and Nadler. Also, we characterized the topological properties of the set of solutions. We confirm that our acquired outcomes are new in the frame of generalized Caputo FDIs with initial conditions and it greatly contributes to enriching the existing literature on this theme.

In future works, many cases can be established when one takes a more generalized operator, for example, -Hilfer fractional operator [13]. Further, it will be of interest to study the existing problem in this article for the Mittag-Leffler power law [8] and for fractal fractional operators [35].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors made equal contributions and read and supported the last original copy.

Acknowledgments

The authors are grateful for the support provided by their institutions.

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Copyright © 2021 Adel Lachouri 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.

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