Study of a Class of Generalized Multiterm Fractional Differential Equations with Generalized Fractional Integral Boundary Conditions

Shammakh, Wafa; Alzumi, Hadeel Z.; Albarqi, Zahra

doi:https://doi.org/10.1155/2020/6824672

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Fractional Difference and Differential Operators and their Applications in Nonlinear Systems

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Volume 2020 | Article ID 6824672 | https://doi.org/10.1155/2020/6824672

Study of a Class of Generalized Multiterm Fractional Differential Equations with Generalized Fractional Integral Boundary Conditions

Wafa Shammakh,¹Hadeel Z. Alzumi,¹and Zahra Albarqi¹

Guest Editor: Thabet Abdeljawad

Received31 Dec 2019

Accepted25 Apr 2020

Published11 May 2020

Abstract

The aim of this work is to study the new generalized fractional differential equations involving generalized multiterms and equipped with multipoint boundary conditions. The nonlinear term is taken from Orlicz space. The existence and uniqueness results, with the help of contemporary tools of fixed point theory, are investigated. The Ulam stability of our problem is also presented. The obtained results are well illustrated by examples.

1. Introduction

Fractional differential equations (FDEs) have a considerable interest both in mathematics and in applications. They were used in modeling in various disciplines of engineering, physics, chemical technology, population dynamics, biotechnology, economics, etc. (see, for example, [1–10], and the references cited therein). Recently, some authors explored FDEs with various types of conditions as multipoint, nonlocal, antiperiodic, and integral boundary conditions [11–15].

Furthermore, there has been a significant development in fractional derivatives and integrals due to the necessity of a better model for real phenomena. In [16], Katugampola suggested a new fractional integral that combines the Riemann-Liouville and Hadamard integrals into a single integral. In 2015, Caputo and Fabrizio [17] defined the so-called Caputo–Fabrizio fractional derivatives by imposing a nonsingular exponential kernel. Later, the exponential kernel was replaced by the Mittag–Leffler function [18, 19]. Another definition of generalized proportional fractional derivatives-generated Caputo and Riemann–Liouville involving exponential functions in their kernels was introduced by Jarad et al. [20]. Another new type of new fractional derivative can be found in [21]. An application on such derivatives can be seen in [22]. More recently, generalized fractional derivatives that contain kernels depending on the function on the space of absolutely continuous functions were investigated [23]. Following the above points, many authors established various generalized fractional boundary value problems (GFBVPs) (see, for example, [24–27]). It is worth to mention that some authors worked on the existence of solutions and studied the stability analysis for nonlinear singular fractional differential equations. For more details, we refer to [28–30].

On the other hand, Birnbaum and Orlicz [31] proposed a more general setting of function spaces in 1931, which were called Orlicz spaces. These spaces are the generalization of the classical Lebesgue spaces , where the kernel is given by a convex function instead of . Some results that deal with differential equations in the framework of Orlicz spaces can be found in [32, 33].

In this paper, we investigate the boundary value problem as follows:where is the generalized fractional derivative of Caputo type of order , denotes the Katugampola-type fractional integral of order , is defined on an Orlicz space .

The rest of the paper is organized as follows. In Section 2, we will briefly recall some preliminary materials related to our problem. In Section 3, we discuss the existence and uniqueness results of GFBVP using some fixed point theorems and support the obtained results by using examples to well illustrate. The Ulam stability of our problem is given in Section 4.

2. Preliminaries

For convenience of the reader, we present some basic definitions about generalized fractional calculus theory, which can be found in [16, 34, 35]. Also, we introduce some necessary concepts for Orlicz spaces which are used throughout this paper. For more details about Orlicz spaces, one can see [36, 37].

Definition 1. Let be a right continuous, monotone, increasing function with the following:(i)(ii)(iii) whenever Then, the function defined byis called function. Alternatively, the function is an function iff is continuous, even, and convex with the following:(i)(ii)(iii) if

Definition 2. For an N-function, we definewhere is the right inverse of the right derivative of and is called the complementary of and it satisfies the following condition:

Remark 1. Note that the function is also N-function and the complementary pairs and satisfy the following Young inequality:

Definition 3. A function is called a Young function if it is convex and satisfies the conditions and .

Definition 4. (Orlicz space). For an -function , the Orlicz space is the space of measurable functions such that . This space endowed with the Luxemburg norm, i.e.,and the pair is a Banach space.

Remark 2. For an Orlicz space, the Hölder inequality holds, that is,where and .
For generalized fractional calculus, Katugampola [16, 34, 38] introduced the following definitions.

Definition 5. (Katugampola generalized fractional integral). The generalized fractional integral of order with and for is defined bywhere belongs to the space , which denotes the space of all complex-valued Lebesgue measurable functions on for which , where the norm is defined byNote that integral (8) is called the left-sided generalized fractional integral.

Definition 6. (Katugampola generalized Caputo fractional derivative). The left generalized fractional derivative of order with and of , where , is defined bywhere and denotes the space of all absolutely continuous real valued functions on , wherefor , endowed with the norms and , respectively.
Next, we recall some basic properties of generalized fractional integral and derivative [39].

Lemma 1. Let , and , with . For ,

Theorem 1. Let , and , where . Then,(i)If (ii)If (iii)

Theorem 2. Let , and such that . Then,Our main results are based on utilizing the following fixed point theorems.

Theorem 3. (Schaefer’s fixed point theorem [40]). Let be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then, has a fixed point in .

Theorem 4. (Krasnoselskii’s fixed point theorem [40]). Let be a closed, convex, bounded, and nonempty subset of a Banach space . Let be operators such that(i) belong to whenever (ii) is a compact and continuous and is a contraction mappingThen, there exists such that .
For computational convenience, we introduce the following notations:

Lemma 2. Let and be given by (15). Then, the solution of the GFBVPis given by

Proof. By using Lemma 1, we obtainwhere are the arbitrary constants. Using the boundary conditions , we get . Then, from , it follows thatSubstituting the values of , and in (18), we get (17).

3. Main Results

In this section, we prove the existence and uniqueness of solutions of GFBVP (1). We assume that belongs to an Orlicz space . For let denote the Banach space of all continuous functions defined on into endowed with the norm

Relative to GFBVP (1), in view of Lemma 2, we define an operator as

Notice that GFBVP (1) has solutions if and only if the operator has fixed points.

The following result plays a major role in our analysis.

Lemma 3. Let be a Young function which has a Young complement satisfyingfor and . Then, the operator exists and is well defined.

Proof. Let . Define a function as follows:At the beginning, we will show that . By using appropriate substitution and properties of the Young functions, one obtainsby the assumption of the theorem, and we get . Similarly, settingone can get . Next, we show that is well defined, i.e., . Let . Thus, we havewhereThe functions , belong to with , where is a continuous and increasing function with . Using the Hölder inequality, we haveThen, and by the continuity of , we see that is continuous, which completes the proof.
Our first existence result relies on Schaefer’s fixed point theorem.

Theorem 5. Assume that there exists such thatwith. Then, the GFBVP (1) has at least one solution on .

Proof. The proof proceeds in few steps as follows: Step 1: we will prove the operator maps bounded sets into bounded sets in . Let be a bounded set. Then, for all , by using the assumption , for , we are able to obtain Thus, we obtain Now, with some efforts of computation, we have Therefore, we have In a similar way, we arrive at Step 2: from (32)–(36), we can immediately get the operator to map bounded sets into bounded sets. Let , and for all , we getThus, we haveSimilarly, it can be easily shown thatThe right side of (38)–(41) goes to zero as . In view of steps I and II, it follows that by Arzela–Ascoli theorem, the sets , and are relatively compact in . Therefore, is a relatively compact set in . We consider the set . Then, is bounded. Indeed, let . So, , for any , and it follows thatThus, all the hypotheses of Theorem 3 are satisfied. Therefore, we can conclude that the operator has at least one fixed point. Hence, the GFBVP (1) has at least one solution on [0, 1].
For computation convenience, we setNow, we make use of Theorem 4 to prove the existence of solutions of GFBVP (1).

Theorem 6. Let be a continuous function such that the following assumptions hold:Then, GFBVP (1) has at least one solution on , provided that

Proof. We define , where . We split the operator defined by (21) on as , where and are given byFor , we find thatFurthermore,Consequently, we obtainwhich shows that . In what follows, we prove that is a contraction. Let and for all , we getWith the same process, one hasHence, we getIt remains to show that is continuous and compact. The continuity of the function implies that the operator is continuous. To achieve the compactness of the operator , we first prove that is uniformly bounded on as follows:So,Now, let . Consequently, for , we haveFurthermore,Therefore, as , the right-hand sides of the above inequalities tend to zero independently of . Thus, is equicontinuous and so it is relatively compact on . According to the Arzela–Ascoli theorem, the operator is compact. By using Theorem 4, there exists at least one solution of GFBVP (1) on .
In the next result, we prove the existence of solutions for GFBVP (1) by applying Banach’s fixed point theorem.

Theorem 7. Assume that the condition holds. Then, GFBVP (1) has a unique solution on , provided thatwhere and , and are given by (43)–(45).

Proof. Consider the operator defined bySetWe shall show that , where . For , we haveTherefore,Similarly,By taking the norm from , it yields , and . Consequently, we havewhich shows that maps into itself. In order to show that the operator is a contraction, let . Then, for each , we obtainSimilarly,Consequently,Thus, in view of condition (59), it follows that the operator is a contraction. Hence, the operator has a unique fixed point which corresponds to a unique solution of GFBVP (1).
We conclude this section with some examples showing the applicability of our main results.

Example 1. Let us consider the following boundary value problem:where , andTo check that the condition holds, let and ; then, we getHence, the condition holds with and it follows that , where and and are given by (43)–(45). Then, the hypotheses of Theorem 7 are satisfied and problem (69) has a unique solution on .

Example 2. We consider the following boundary value problem:where . ChooseSet , . In particular, if , then is an N-function and satisfieswhich shows that belongs to an Orlicz space . Furthermore,with and . Thus, the condition holds and by the conclusion of Theorem 5, problem (72) has at least one solution on .
Moreover,which proves the validity of condition with , where and , and are given by (43)–(45). Hence, Theorem 7 can be applicable and so problem (72) has a unique solution on .

4. Ulam Stability

In this section, we develop the criteria for Ulam stability of GFBVP (1) by means of its equivalent integral equation. For simplicity, we set

Then,where and is a continuous function. Next, we define a continuous nonlinear operator as

For definitions of Ulam–Hyers, generalized Ulam–Hyers and Ulam–Hyers–Rassias stability, and we refer to [41].

Definition 7. GFBVP (1) is said to be Ulam–Hyers stable if there exists a real number such that, for each and for each solution ,there exists a solution of (1) satisfying the inequalitywhere is a positive real number depending on .

Definition 8. GFBVP (1) is generalized Ulam–Hyers stable if there exists such that, for each solution of (1), there exists a solution of (1) with

Definition 9. GFBVP (1) is Ulam–Hyers–Rassias stable with respect to if there exists a real number such that, for each and for each solution of (1),we can find a solution of (1) satisfying the inequalitywhere is a positive real number depending on .

Theorem 8. Assume that conditions (47) and (59) hold. Then, GFBVP (1) satisfies both Ulam–Hyers and generalized Ulam–Hyers stability criteria.

Proof. We know that is a unique solution of (1) (by Theorem 7). Let be another solution of (1) satisfying . Observe that the operators and are equivalent for every solution of (1). Therefore, by the fixed point property of the operator together with and , we havewhere and . Taking the norm for and solving for , we obtainChoosing and , the Ulam–Hyers stability condition is satisfied. More generally, defining , the generalized Ulam–Hyers stability condition is also satisfied which completes the proof.

Theorem 9. Assume that conditions (47) and (59) hold and there exists a function satisfying condition . Then, GFBVP (1) is Ulam–Hyers–Rassias stable with respect to .

Proof. Similar to the discussion in the proof of Theorem 8, we havewith , and we get the desired result.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

Wafa Shammakh, Hadeel Z. Alzumi, and Zahra Albarqi contributed to the design and implementation of the research, to the analysis of the results, and to the writing of the manuscript.

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Copyright © 2020 Wafa Shammakh 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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