Existence and Uniqueness for Coupled Systems of Hilfer Type Sequential Fractional Differential Equations Involving Riemann–Stieltjes Integral Multistrip Boundary Conditions

Samadi, Ayub; Ntouyas, Sotiris K.; Asawasamrit, Suphawat; Tariboon, Jessada

doi:https://doi.org/10.1155/2022/7454388

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Abstract Introduction Preliminaries Examples Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fixed Points and Computational Optimization

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Research Article | Open Access

Volume 2022 | Article ID 7454388 | https://doi.org/10.1155/2022/7454388

Existence and Uniqueness for Coupled Systems of Hilfer Type Sequential Fractional Differential Equations Involving Riemann–Stieltjes Integral Multistrip Boundary Conditions

Ayub Samadi,¹Sotiris K. Ntouyas,^2,3Suphawat Asawasamrit,⁴and Jessada Tariboon⁴

Academic Editor: Sun Young Cho

Received15 Sept 2021

Accepted25 Mar 2022

Published31 May 2022

Abstract

In this paper, we study a coupled system of Hilfer type sequential fractional differential equations supplemented with Riemann–Stieltjes integral multistrip boundary conditions. The standard tools of the fixed point theory are employed to prove the existence and uniqueness results for the considered problem. Examples are constructed for the illustration of the obtained results.

1. Introduction

Fractional calculus is a generalization of the classical calculus. Fractional differential equations become another necessary tool in solving real-life problems in different research areas such as physics, biology, engineering, and mechanics, see for example the monographs and papers [1–11].

Boundary value problems of fractional differential equations represent an important and interesting branch of applied analysis. Usually, the researchers have given attention in studying fractional differential equations involving Caputo or Riemann–Liouville fractional derivative. But, Caputo or Riemann–Liouville derivative was not considered appropriate in studying some new models in engineering for example. To avoid the difficulties, some new type fractional order derivative operators were introduced in the literature such as Hadamard, Erdeyl-Kober, and Katugampola. Hilfer in [12] introduced a new derivative, which generalizes both Riemann–Liouville and Caputo derivatives. For some applications involving Hilfer fractional derivative, the interested reader is referred to [13–16] and references cited therein.

In [17], Nuchpong et al. investigated a new class of boundary value problems for fractional differential equations for involving sequential Hilfer type fractional derivative and subject to Riemann–Stieltjes integral multistrip boundary conditions of the formwhere denotes the Hilfer fractional derivative operator of order , , and parameter , , is a continuous function; is the Riemann–Stieltjes integral with respect to the function , , . Existence and uniqueness results are established by using basic tools from fixed point theory.

The study of system of Hilfer type was initiated by Wongcharoen et al. [18], by presenting the following system of fractional differential equations:in which and indicate the Hilfer fractional derivatives of orders and , , and parameters , , are continuous functions, , , and are the Riemann–Liouville fractional integrals of order , , .

Inspired by the forenamed studies, this article considers the existence and uniqueness of solutions for the following coupled system of Hilfer type fractional differential equations with Riemann–Stieltjes integral multistrip boundary conditions of the formin which and are the Hilfer fractional derivatives of orders and parameters , , are continuous functions, , are the Riemann–Stieltjes integrals with respect to the functions , , , , , , , .

The remaining of this article has been regulated as follows: In section 2 some concepts, lemmas and theorems are recalled which will be applied throughout this paper. In Section 3, an auxiliary lemma has been proved which concerns a linear variant of system (3) and it is used to convert the coupled system (3) into a fixed point problem. The classical fixed point theorems have been applied in order to obtain the results regarding the existence/uniqueness in Section 4. Thus, the classical Banach fixed point theorem is applied to obtain uniqueness result, while Leray-Schauder alternative and Krasnosel’skiĭ’s fixed point theorems are applied to present existence results. Examples are also constructed to illustrate the obtained results.

2. Preliminaries

Now, the following items are reminded which will be applied to fulfil the main results in the next steps.

Throughout the paper, the Banach space of all continuous mappings from to are denoted by which is equipped with the norm . It is clear that the space , equipped with norm defined by , is a Banach space.

Also, is the -times absolutely continuous functions defined as

For a real valued function , the Riemann–Liouville fractional integral of order is defined by , in which the right-hand side is defined point-wise on , see [2]. Besides, for the function , the Riemann–Liouville fractional derivative of order is defined by , in which , where denotes the integer part of a real number , see [2], while the Caputo fractional derivative is defined by , provided that the right-hand side exists.

Also, the Hilfer fractional derivative of order and parameter of a function is defined bywhere , see [12]. Note that if and , the Hilfer derivative is reduced to the Riemann–Liouville and Caputo fractional derivatives, respectively.

The following lemma will be applied to prove a lemma in the next section which presents a pattern of existence of solutions for system (1.3).

Lemma 1 (see [13]). Let , , , . Then, we have the following relation:

Finally, we collect the fixed point theorems applied to prove the main results in this paper.

Lemma 2 (Banach fixed point theorem, [19]). Let be a closed set in and satisfies

Then T admits a unique fixed point in .

Lemma 3 (Leray–Schauder alternative [20]). Let the set be closed bounded convex in and an open set contained in with . Then, for the continuous and compact , either(1) admits a fixed point in , or(2) there exists and with .

Lemma 4 (Krasnosel’skiĭ fixed point theorem, [21]). Let indicate a closed, bounded, convex, and nonempty subset of a Banach space and be operators such that (i) where , (ii) is compact and continuous, and (iii) is a contraction mapping. Then, there exists such that .

3. An Auxiliary Result

Lemma 5. Let , , , , , and . Then, the solution of the systemis given bywhere

Proof . Let be a solution of system (5). By Lemma 2, we havewhere are the arbitrary constants, since and . Applying and , we deduce that . Thus, the previous equations becomeNow, applying the boundary conditions and , we getConsequently, we have the systemwhereBy solving the above system, we haveSubstituting the values of and into (10) and (14), respectively, we obtain the solutions (9) and (10). We can obtain the converse by direct computation. The proof is finished.

4. Existence and Uniqueness Result

Due to Lemma 5, we define an operator bywhere

For convenience, we set

Now, Banach’s fixed point theorem is applied to present the following uniqueness result.

Theorem 1. Let and be two continuous functions such that for all and , we havewhere are positive constants and . Then, there exists a unique solution of system (3) on provided that

Proof . It suffices to display that the operator has a unique fixed point. For this aim, Banach’s theorem will be applied. Put and . Now, we locate , in whichFirst, we indicate that . Assume that and . Due to (), we haveSimilarly, we haveHence, we infer thatConsequently,In the same way, we haveHence,which yields that .
Now, it is proved is a contraction. Applying condition (25), for any and for each , we haveand henceFurthermore, we deduce thatUsing (25) and (33), we concluded thatAs , so the operator is a contraction and by applying Lemma 2, the operator has a unique solution which is the solution of the problem (3). The proof is finished.

5. Existence Results

Two existence results are proved in this section.

5.1. Existence Result via Leray–Schauder Alternative

The Leray–Schauder alternative (Lemma 3) is used in the proof of our first existence result.

Theorem 2. Let and , be continuous functions. Assume that There exist real constants for and such that for all , we have

If and , where , for are given by (23) and (24), respectively, then system (1.3) admits at least one solution on .

Proof . The functions are continuous on . Thus, the operator is continuous. Now, we will show that the operator is completely continuous. Let be a bounded set, where . Then, for any , there exist positive real numbers and such that and .
Thus, for each , we havewhich yieldsSimilarly, we obtain thatHence, from the above inequalities, we get that the operator is uniformly bounded, sinceNext, we are going to prove that the operator is equicontinuous. Let with . Then, we haveTherefore, we obtainAnalogously, we can get the following inequality:Hence, the set is equicontinuous. Accordingly, Arzelá–Ascoli theorem implies that the operator is completely continuous.
Finally, we shall show the boundedness of the set . Let any , then . We have, for all ,Then, we getwhich imply thatThus, we obtainwhere , which shows that the set is bounded. Therefore, by Leray–Schauder alternative (Lemma 3), the operator has at least one fixed point. Hence, we deduce that problem (3) admits a solution on , which completes the proof.

5.2. Existence Result via Krasnosel’skiĭ’s Fixed-Point Theorem

Now, Krasnosel’skiĭ’s fixed-point theorem (Lemma 4) is applied to prove our second existence result.

Theorem 3. Assume that and are continuous functions satisfying condition (4.8) in Theorem 4.1. Furthermore, suppose that there exist positive constants and such that for all and , we have

If , and , then problem (1.3) admits a solution on .

Proof . First, we decompose the operator defined by (1) into four operators asAccordingly, and . Let withFirst, it is showed that for all . According to the proof of Theorem 1, we getConsequently, and we conclude the condition of Lemma 4. Now, it is indicated that the operator is a contraction mapping. For , we infer thatAs , the operator is a contraction and the condition of Lemma 4 is concluded. In the final step, the condition of Lemma 4 is verified for the operator . As the functions are continuous, one can see that the operator is continuous. Furthermore, for , as in the proof of Theorem 1, we haveHence, , which implies that is uniformly bounded. Now, we claim that the set is equicontinuous. For this aim, let with . For any , similar to the proofs of equicontinuous for the operators and in 2, we can show that , as . Consequently, the set is equicontinuous and by applying Arzelá-Ascoli theorem, the operator will be compact on . Therefore, by applying Lemma 4, problem (3) has at least one solution on . This completes the proof.

Remark 1. In Theorem 2, the functions are bounded by linear planes in three-dimension space. While, in Theorem 3, are bounded by fixed constants and also satisfied Lipschitz condition in (25).

6. Examples

Now, we present some examples to show the benefits of our results.

Example 1. Consider the following coupled system of Hilfer type sequential fractional differential equations involving Riemann–Stieltjes integral multistrip boundary conditions of the formHere, , , , , , , , , , , , , , , , , , , , , , , , , , , and . Then, we can compute that , , , , , , , , , , , , , , and .(i)The Lipschitzian functions are given by From direct computation to (59)-(60), we get for . Setting and , we obtain . By application of our Theorem 1, the problem of Hilfer type sequential fractional differential system involving Riemann–Stieltjes integral multistrip boundary conditions (58) with (59)-(60) has a unique solution on .(ii)Let the nonlinear functions be defined by We remark that and . Now, we choose the constants as in Theorem 2 by , , , , , and . Then, we can find that and . The benefit of Theorem 2 can be used to conclude that the coupled system of Hilfer type sequential fractional differential equations subject to boundary conditions (58) with (62)-(63) has at least one solution on .(iii)Suppose that two Lipschitzian functions are stated byActually, we can compute the bounds of the above two functions by , , for all . In addition, we can find that and , and thus we can set and satisfying condition (4.8) in Theorem 1. Then, we obtain , , and that all conditions in Theorem 3 are fulfilled. In this step, we conclude that the problem (58) with (64) has at least one solution on . Finally, we observe that the uniqueness result cannot be obtained because .

7. Conclusions

In the present research, we studied a coupled system of Hilfer type sequential fractional differential equations supplemented with Riemann–Stieltjes integral multistrip boundary conditions. First, an auxiliary lemma, concerning a linear variant of the considered problem, has been proved which is pivotal to converting the coupled system into a fixed point problem. Then, existence and uniqueness results are established via standard fixed point theorems. Thus, the classical Banach fixed point theorem is applied to obtain a uniqueness result, while Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem are applied to present the existence results. Numerical examples are also constructed to illustrate the obtained results. The obtained results are new and enrich the existing literature on coupled systems of Hilfer type sequential fractional differential equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded by the King Mongkut’s University of Technology North Bangkok, Contract No. KMUTNB-62-KNOW-29.

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Copyright © 2022 Ayub Samadi 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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