Some Existence Results for a System of Nonlinear Fractional Differential Equations

Ameer, Eskandar; Aydi, Hassen; Işık, Hüseyin; Nazam, Muhammad; Parvaneh, Vahid; Arshad, Muhammad

doi:https://doi.org/10.1155/2020/4786053

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Fractional Calculus and Related Inequalities

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Volume 2020 | Article ID 4786053 | https://doi.org/10.1155/2020/4786053

Some Existence Results for a System of Nonlinear Fractional Differential Equations

Eskandar Ameer,¹Hassen Aydi,^2,3Hüseyin Işık,⁴Muhammad Nazam,⁵Vahid Parvaneh,⁶and Muhammad Arshad⁷

Academic Editor: Serkan Araci

Received16 Jun 2020

Revised02 Oct 2020

Accepted22 Oct 2020

Published19 Nov 2020

Abstract

In this paper, we show that a sequence satisfying a Suzuki-type JS-rational contraction or a generalized Suzuki-type Ćirić JS-contraction, under some conditions, is a Cauchy sequence. This paper presents some common fixed point theorems and an application to resolve a system of nonlinear fractional differential equations. Some examples and consequences are also given.

1. Introduction

The application of Banach contraction principle (BCP) [1] is wide spread. Recently, fixed point theory is being applied to show the existence of solutions of different mathematical models expressed in the forms of differential, integral, functional, fractional differential, and matrix equations (both linear and nonlinear). There are several common fixed point theorems, in the literature, which generalize BCP and have been applied to show the existence of solutions of different mathematical models involving two or more functions (see [2–19], for details).

Suzuki [20] presented a new generalization of BCP, so called a Suzuki-contraction, and established an existence theorem which characterized the metric completeness. Piri and Kumam [21, 22] investigated results in [20] under the -contraction structure both in metric and -metric spaces, respectively. Further generalizations of Suzuki contractions have been studied by Aydi et al. [23] who introduced a Suzuki-type multivalued contraction to obtain fixed point theorems in the setting of weak partial metric spaces. Abbas et al. [2] discussed generalized Suzuki-type multivalued contractions under the effect of a binary relation and obtained some fixed point results.

On the contrary, Jleli et al. [24, 25] introduced another generalization of BCP, known as a JS-contraction (also known as a -contraction) and Li and Jiang [26] modified the JS-contraction to JS-quasi contractions in order to obtain more general fixed point results, as compared to [24]. Altun et al. [27] obtained some fixed point theorems for JS-contraction type mappings via a binary operation. Several honorable researchers have published their valuable investigations on JS-contractions in well ranked journals (see [28–31]).

In this paper, we structure two common fixed point theorems comprising four self-mappings involving generalized Suzuki-type -rational contractions and generalized Suzuki-type Ćirić -contractions. The existence of the solution of a mathematical model in terms of fractional differential equations is shown through a common fixed point theorem.

2. Advances on -Contractions

Let , where is nondecreasing. For each sequence , There exist and such that .

Jleli and Samet [24] introduced the following.

Definition 1. Let be a metric space. The mapping is called a -contraction (or a JS-contraction) if there exist a constant and such thatThe following examples show that the set is not empty.

Example 1. Let be defined by, for all ,Then, and are in .
Jleli and Samet [24] established the following fixed point theorem.

Theorem 1 (see [24]). Let be a complete metric space and be a -contraction. Then, has a unique fixed point.
Consistent with [28], let , where is continuous on Note that and are independent of each other. For this, see examples in [28]. The set of functions is not empty, as shown in the following.

Example 2 (see [28]). Let be defined by, for all ,Here, .
Hussain et al. [29] modified the class of mappings as follows:where is non decreasing. . For each sequence : There exist and such that . .The functions and (for all ) are in . : for all with , there exists such that .
Let denote the set of all functions satisfying the condition .

Example 3 (see [31]). If , where and , then .

Example 4 (see [31]). If , where and , then
The following class of mappings was considered in [32] to develop some fixed point theorems. We also use this class of mappings in our investigations:

Definition 2 (see [5]). Let be a metric space. The pair is said to be compatible if and only ifwhenever is a sequence in so thatThe following lemma is important in the sequel.

Lemma 1 (see [5]). Let be a metric space. If there exist two sequences and such thatthen

3. Generalized Suzuki-Type Contractions and Fixed Point Results

Jleli and Samet [24, 25] have employed -contractions to obtain some fixed point theorems. Suzuki [20] extended Edelstein contraction to develop a new generalization of Banach contraction called Suzuki contraction. The contractions developed in [20, 24] were then followed by Liu et al. [33] to introduce a Suzuki-type -contraction. In this section, we introduce more general forms of Suzuki-type -contractions involving four self-mappings. We construct some conditions under which a sequence, whose terms satisfy generalized Suzuki-type -rational contractions or generalized Suzuki-type Ćirić -contractions, is a Cauchy sequence. We obtain two common fixed point theorems, which then are applied to show the existence of the solution of a system of fractional differential equations. We start with the following definition.

Definition 3. Let be a metric space, and be four single-valued maps. These maps form a new generalized Suzuki-type -rational contraction if, for all , with , for some , , and ,whereOur first result is as follows.

Theorem 2. Let be a complete metric space, and be four single-valued maps. Suppose that the mappings form a new generalized Suzuki-type -rational contraction with and . If and are compatible pair and and are continuous on , then , and have a unique common fixed point in .

Proof. Let be an arbitrary point. As , there exists such thatSince , we can choose such that In general and are chosen in such thatWe construct a sequence in such thatfor . Assume that and sinceWe haveHence, from the contractive condition (12),For every , whereAs , so by , there exists such thatSinceone writesIf, for some , , then from (19), (21), and (24), we havea contradiction. Therefore,Thus, for all , we haveIt implies thatfor all . Taking the limit as in (28) and using the fact that , we haveBy , we obtainNow, we will show that is a Cauchy sequence. We suppose on the contrary that is not a Cauchy sequence; then, there exist and subsequences and of natural numbers such that, for , we have . Then, for all . Therefore,By taking the upper limit as in (31) and using (30), we obtainUsing the triangular inequality, we haveBy taking upper limit as in (33) and (34) and applying (30) and (32),Thus,Similarly, we can obtainAssume that , and sincewe obatinHence, from (12),whereTaking limit at in (41), we obtainBy , there exists such that . Using the continuity of and (41),Also,Taking and using (30), (32), (36), and (37), we obtainThus, from , (37), (40), (43), and (45), we havewhich is a contradiction. Thus, is a Cauchy sequence. Since is a complete metric space, there exists such that andNow, we shall prove that is a common fixed point of , , and . As is continuous, soSince is a compatible pair,From Lemma 1, we havePut and in (12) and if , we obtainHence,and by (12), we obtainwhereSetting the upper limit in (53), we obtaina contradiction. Thus, and . Again since is continuous,Since is a compatible pair,From Lemma 1, we havePut and in (12) and if , we obtainTherefore,By (12), one obtainswhereSetting the upper limit, we obtaina contradiction. Thus, and so . Suppose , we obtainHence,and by (12), we obtainwhereTaking the upper limit in (66), and as , so we obtaina contradiction. Thus, and . Finally, suppose , and as , so, we obtainThus, from (12)a contradiction. Thus, and . Therefore, is a common of the four mappings and . Next, assume that is another common of the four mappings , and such that , one obtainsThen, by (12), we obtainwhereIt implies thata contradiction. Thus, . Therefore, is the unique common fixed point of , and .

Corollary 1. Let be a complete metric space, and be four single-valued maps. Suppose that if, for all with for some , and ,wherewith and . If and are compatible pairs and and are continuous on , then , , and have a unique common fixed point in .

Example 5. Let and define the function by . Clearly, is a complete metric space. Let , , and , then and . Define the mappings byClearly, and , if is a sequence in such thatthenand equivalently,Thus,We conclude (by uniqueness of limit); hence, . Using continuity of and , one obtainsfor . Hence, the pair is compatible. Similarly, the pair is compatible. Define as . Next, for all with ,and soHence, all hypotheses of Theorem 2 are satisfied. Thus, , , and have a unique common fixed point, which is .
We can easily obtain the following results by chasing the proof of Theorem 2.

Corollary 2. Let be a complete metric space, and be four single-valued maps. Suppose that if, for all , with , for some , and ,wherewith and . If and are compatible pairs, and and are continuous on , then , , and have a unique common fixed point in .

The following corollary generalizes the result introduced by Sehgal [34].

Corollary 3. Let be a complete metric space, and be four single-valued maps. Suppose that if, for all , with , for some , and , implies thatwherewith and . If and are compatible pairs, and and are continuous on , then , , and have a unique common fixed point in .

4. On Suzuki-Type Ćirić JS-Contractions

We start this section with the following definition.

Definition 4. Let be a metric space and be four single-valued maps. These maps form a new generalized Suzuki-type Ćirić JS-contraction if, for all , for some , , implies thatwhere with .
Our second result is as follows.

Theorem 3. Let be a complete metric space and be four single-valued maps. Suppose that the mappings form a generalized Suzuki-type Ćirić JS-contraction with and . If and are compatible pairs and and are continuous on , then , , and have a unique common fixed point in .

Proof. Let be an arbitrary point. As , there exists such thatSince , we can choose such that In general, and are chosen in such thatWe construct a sequence in such thatfor . Sincewe haveThen, from (89),By , we have, for every ,Hence, (95) becomesIt implies thatTherefore,and soIt implies thatfor all . Taking the limit as in (101) and knowing that , we haveBy , we obtainFrom condition , there exist and such thatSuppose that . Let . From the definition of the limit, there exists such thatThis impliesThen,where . Suppose now that . Let be an arbitrary positive number. From the definition of the limit, there exists such thatwhich implieswhere . Thus, in all cases, there exist and such thatBy using (101), we obtainSetting in inequality (111), we obtainThus, there exists such thatTo prove is a Cauchy sequence, we use (113) and for ,The convergence of the series entails . Thus, is a Cauchy sequence. Since is a complete metric space, there exists such that andNow, we shall prove that is a common fixed point of , , and . As is continuous, soSince is a compatible pair,From Lemma 1, we havePut and in (89) and if , we obtainHence,Then, from (89), we obtainSetting the upper limit, we obtaina contradiction. Thus, , and so . Again, since is continuous,Since is a compatible pair,From Lemma 1, we havePut and in (89) and if , we obtainTherefore,and from (89), one obtainsPassing to the upper limit in (128), we obtaina contradiction. Thus, and hence . Suppose that , we obtainand soBy (89), we obtainTaking the upper limit in (132) and as , we obtaina contradiction. Thus, and hence . Finally, suppose and as , we haveFrom (89), we deducea contradiction. Thus, and . Therefore, is a common of the four mappings , and . Next, assume that is another common of the four mappings , and such that , one obtainsThen, by (89),It implies thata contradiction. Thus, , that is, is the unique common fixed point of , and .

Example 6. Let and define the function by . Clearly, is a complete metric space. Let , and , then and . Define the mappings byClearly, if is a sequence in such thatThen,and equivalently,Thus,It gives that (by uniqueness of limit); hence, . Using continuity of and , one obtainsfor . Hence, the pair is compatible. Similarly, the pair is compatible. Next, for all withSo, for and , we haveHence, all hypotheses of Theorem 3 are satisfied. Thus, , , and have a unique common fixed point.

5. An Application

We apply the result given by Theorem 2 to study the existence of a solution for a system of nonlinear fractional differential equations. Let be the space of all continuous functions on . The metric on is given by

Then, is complete metric space.

Consider the following system of fractional differential equations:with boundary conditions

Note that denotes the Caputo fractional derivative of order , defined bywhere we considerand and denote the Riemann–Liouville fractional integral of order of continuous functions and , given by

System (148) can be written in the following integral form:

Define the mappings by

Theorem 4. Assume that the following conditions hold:(i) are continuous functions.(ii) are increasing functions.(iii)For all with and , we have where(iv)There exists such that, for all , we have(v)If there exists a sequence in such thatwheneverfor some .
Then, system (148) has a solution.

Proof. Following assumptions (iii) and (iv), we havewhere is the beta function. From the above inequality, we obtain thatIt implies thatwhere and . Since this inequality holds for all with , so it is true for any ,Hence, we haveThus, , and are generalized Suzuki-type -rational contraction mappings. Therefore, all hypotheses of Theorem 2 are satisfied. Hence, , and have a common fixed point, that is, system (148) has at least one solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

Vahid Parvaneh dedicates this article to the first teacher, Mr. Allah Karam Parsaei, of the village where he was born, for his hard work and efforts in education during the years 1975‐1977.

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Copyright © 2020 Eskandar Ameer 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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