Complexity

Complexity / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5730853 | https://doi.org/10.1155/2021/5730853

Hasanen A. Hammad, Hassen Aydi, Manuel De la Sen, "Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions", Complexity, vol. 2021, Article ID 5730853, 13 pages, 2021. https://doi.org/10.1155/2021/5730853

Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions

Academic Editor: Atila Bueno
Received16 Jun 2020
Revised19 Jan 2021
Accepted08 Feb 2021
Published23 Feb 2021

Abstract

This paper involves extended metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.

1. Introduction and Preliminaries

In the last years, the fractional calculus branch [1, 2] has attracted great interest. There exist many kinds of proposed fractional operators, for instance, we have the well-known Caputo, Riemann–Liouville, Grunwald–Letnikov derivative etc. Among all the papers dealing with fractional derivatives, fractional differential equations as an important research field have attained great deal of attention from many researchers (see [38]).

There are many applications of the fractional topic in complex analysis, such as, in the sense of conformable derivatives and integrals, interesting results for fractional formulations of complex-valued functions of a real variable have been successfully introduced, which in turn open the door to the researchers to construct the theory of conformable integration by studying functions of a complex variable [9]. On the other hand, the standard definition for the Atangana–Baleanu fractional derivative involves an integral transform with a Mittag–Leffler function, where the kernel can be rewritten as a complex contour integral, which can be used to provide an analytic continuation of the definition to complex orders of differentiation [10]. These lines are very important due to their applications in the field of natural science or engineering.

In the last few decades and in the branch of fractional differential equations, Riemann–Liouville and Caputo derivative ones are the mostly used. Note that several fractional differential equations have been resolved by using fixed point techniques. This paper is concerned with this fact when considering the class of extended -metric spaces.

Let be a metric space. Denote by a set of nonempty closed bounded subsets of . Define the function bywhere , for .

Then, ¥ is called the Hausdorff–Pompeiu metric. Consider

The following can be deduced from the definition of . For all , we have the following:(a)(b) iff (c)(d)

In 2017, the concept of extended -metric spaces has been initiated by Kamran et al. [11], by considering a control function at the right-hand side of the triangular inequality.

Definition 1 (see [11]). Letbe a nonempty set andbe a given function. An extended-metric is a functionsuch that, for all, we have the following:(1)(2)(3)This (generalized) metric space has attracted many researchers where many real applications have been resolved. For more details, see [1215]. Some of the related topological concepts are as follows.

Definition 2. Letbe an extendedmetric space. Letbe a sequence in.(1) converges to some in , if for each , there is so that for each (2) is Cauchy, if for each , there is so that for all (3) is called complete if each Cauchy sequence is convergent

Lemma 1 (see [12]). Letbe an extendedmetric space. If the sequenceinis such that, whereand, thenis Cauchy.

Example 1 (see [16]). Take. Givenas. Consider the extended-metricso thatand for .

Definition 3 (see [17]). Denote bythe set of functionssuch that(i)(ii)For any sequence,implies thatas

Example 2. GivenasClearly, .
The manuscript is organized as follows. In Section 2, some fixed point results in the class of extended -metric spaces have been provided. We also present some useful examples. By using fixed point techniques, we solve in Section 3 a fractional nonlinear differential equation, we ensure in Section 4 the existence of a unique solution of a system of nonlinear fractional differential equations, and in Section 5, we establish that a two-dimensional linear Fredholm integral equation has a unique solution. At the end, in Section 6, we give a conclusion.

2. Main Theorems

In this section, refers to an extended metric space equipped with the distance . We begin with the following lemmas.

Lemma 2. Letbe an extendedmetric space with the function. For anyandthe following assertions are valid:(i) for (ii)(iii)(iv)(v)(vi)(vii)

Proof. The assertions (i)–(v) follow immediately by Czerwik [18] in metric spaces and (vi)-(vii) follow immediately by the definition of an extended metric space and (1) with (2).

Lemma 3. Letbe an extendedmetric space. Then, for all,and, there existssuch that.

Proof. By a similar way as in the proof of Lemma 4 in [19], we get the result.
Now, we state and prove our main theorems.

Theorem 1. Letbe a complete extendedmetric space andbe multivalued mappings satisfying, for all,where(1)is continuous and nondecreasing function so thatiff(2)is a continuous function so thatiffIffor, thenandhave a unique common fixed point (cfp).

Proof. Let be a fixed element. Define , and let , by Lemma 3, there exists such that For , there is such that
Continuing with the same manner, we have , . If , then the sequence is Cauchy. Suppose that . Then, by (3), we havewhereandApplying (8) and (9) in (7), one can write By definition of , we conclude thatSimilarly, if we replace with and with , we haveFrom (10) and (11), we getNow, by Lemma 1, we observe that is Cauchy sequence. Since is complete, then there is such that . Assume that , then we havewhereTaking as in the above inequalities, we conclude thatIt follows from definition of and (11) that or
Using Lemma 2, we getAt the limit, we have . Thus, . Similarly, we can show that . Hence, is a cfp of the two mappings and . For the uniqueness, let be another cfp of and , then, by our contractive condition, one can writeThis leads to , or ; since , then . Thus, , i.e., the uniqueness holds. Then, the proof is completed.
If we consider in the above theorem, we get the important below result.

Corollary 1. Letbe a complete extendedmetric space andbe a multivalued mapping such that, for all, the following hypothesis is fulfilled:where(i)is a nondecreasing and continuous function such thatif(ii)is a continuous function such thatifIfwith, thenhas a unique fixed point.

Example 3. Assume thatand. Definebyfor all, then the pairis an extendedmetric space with the functiondefined by, see [20]. Defineandbyand and , for any .
Now, we haveThus, all required conditions of Theorem 1 are fulfilled. Hence, and have a unique cfp, which is 0.

Theorem 2. Suppose thatare multivalued mappings defined on a complete extendedmetric space. Let for all,whereandare defined in the above theorem andsuch that. If,, thenand have a unique cfp.

Proof. For a fixed element , define and let , by Lemma 3 (for , there exists such that For , there exists such that
With the same scenario, we obtain that , . If for some , , then is a Cauchy sequence. Assume that, for each , . Then, by (22), we getwhere and according to the above theorem.
It follows from (23) thatSimilarly, replacing with and with , we can writeFrom (24) and (25), we haveFrom Lemma 1, we obtain that is a Cauchy sequence. The completeness of leads to the conclusion that there is such that . Let , then, by Theorem 1, one can writePassing to the upper limit, we can getBy the fact , one can see . Thus, we conclude thatFrom Lemma 2,By taking the upper limit, we have . Thus, . Similarly, we can show that . Hence, is a cfp of and . The uniqueness comes immediately in a similar way as in Theorem 1.
If we put in Theorem 2, we have the following result.

Corollary 2. letbe a multivalued mapping defined on a complete extendedmetric space. Let for all,whereandare defined in Corollary 1andsuch that. If,, thenhas a unique fixed point.

Example 4. Suppose that. Definebyfor all, then the pairis an extendedmetric space with the function, which takes the form. Defineandas follows:and, for all. ConsiderHence, the conditions managed by Theorem 2 are fulfilled, thereby concluding is the unique cfp of and .

Example 5. Suppose that all data of Example 4are fulfilled. Define the multivalued mappingsandbyand , for all . ConsiderHence, the conditions managed by Theorem 2 are fulfilled, thereby concluding is the unique cfp of and .

3. Solving a Fractional Nonlinear Differential Equation

Recently, by the technique of nonlinear analysis such as fixed-point results, the Leray–Schauder theorem and stability, there are some papers dealing with the existence of solutions of nonlinear initial-value problems of fractional differential equations (see [2123]). The main advantage of using fractional nonlinear differential equations is to describe the dynamics of complex nonlocal systems with memory. This part is devoted to obtain an existence solution of the subsequent nonlinear differential equation of fractional order:with boundary conditions

The Caputo fractional derivative with ordered is defined as follows:where , , and is a continuous function. Let be the set of all real-valued continuous functions on . Define and byfor all , . Then, the pair is a complete extended metric space [20]. Here, we need to be reminded that the Riemann–Liouville fractional integral of order is as follows:

Now, our main theorem of this section is.

Theorem 3. The problem (36) with boundary conditions (37) has a unique solution if the following assumptions are fulfilled:(i) is a continuous function satisfying(ii)There is a constantsuch that, where(iii), where.

Proof. Define the mapping byFor . The function is a unique solution of problem (36) if , i.e., is a unique fixed point of the multivalued mapping . To get that, we shall prove that satisfies the contractive condition of Corollary 1. Consider