Existence of Solutions for a Coupled System of p-Laplacian Caputo–Hadamard Fractional Sturm–Liouville–Langevin Equations with Antiperiodic Boundary Conditions

Ni, Jinbo; Zhang, Jifeng; Zhang, Wei

doi:https://doi.org/10.1155/2022/3346115

Journal of Mathematics

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

Research Article | Open Access

Volume 2022 | Article ID 3346115 | https://doi.org/10.1155/2022/3346115

Existence of Solutions for a Coupled System of p-Laplacian Caputo–Hadamard Fractional Sturm–Liouville–Langevin Equations with Antiperiodic Boundary Conditions

Jinbo Ni,¹Jifeng Zhang,¹and Wei Zhang¹

Academic Editor: Ji Gao

Received20 Dec 2021

Accepted26 Feb 2022

Published27 Apr 2022

Abstract

Fractional Sturm–Liouville and Langevin equations have recently attracted much attention. In this paper, we investigate a coupled system of fractional Sturm–Liouville–Langevin equations with antiperiodic boundary conditions in the framework of Caputo–Hadamard fractional derivative. Comparing with the existing literature, we study fractional differential equations with a p-Laplacian operator, which will enrich and generalize the previous work. Based on the Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, some interesting existence results are obtained. Finally, an example is constructed to illustrate our main results.

1. Introduction

On the one hand, the Sturm–Liouville problem plays an important role in various fields of sciences and engineering [1]. A standard form of the second-order Sturm–Liouville differential equation is given bywhere the function is a continuous function.

On the other hand, Langevin equation was first formulated by Langevin in 1908 which is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [2]. The classical Langevin equation as follows:where is the coordinate of the particle at time , is the resistance coefficient per unit mass, is the mass of particle, and represents lifting force per unit mass.

During the past two decades, fractional differential equations have gained prominence and attention by many researchers due to their wide applications as a modeling tool in various research fields, such as physics, biomedical, signal processing, control theory, electric circuits, viscoelasticity, diffusion equations, and electromagnetic waves. For further details, refer [3–6].

Recently, by means of different tools, such as the Banach contraction principle, Schaefer’s fixed point, Leray–Schauder nonlinear alternative, Krasnoselskii’s fixed point theorem, coupled fixed-point theorems in partially ordered boundary value problems (BVPs) for fractional Langevin equations have extensively been studied in the papers [7–15].

Due to the comprehensive practical application background of Sturm–Liouville and Langevin equations, the Sturm–Liouville–Langevin differential equations have been under consideration by many researchers; for example, in 2016, Kiataramkul et al. [16] discussed the following fractional Sturm–Liouville–Langevin equations with antiperiodic boundary conditions:where denotes the Caputo–Hadamard fractional derivative of order with , and , . The existence and uniqueness results are obtained by using the Banach contraction mapping principle, Leray–Schauder nonlinear alternative, and Krasnoselskii’s fixed point theorem, respectively.

More recently, coupled systems of fractional differential equations are of significant importance as such systems appear in a variety of problems of interdisciplinary fields such as synchronization phenomena, nonlocal thermoelasticity, and bioengineering [17], thus attracted scholars to research on the fractional coupled system of Sturm–Liouville–Langevin equations with various boundary conditions [18–20].

In 2020, Muensawat et al. [18] investigated the following fractional coupled Sturm–Liouville–Langevin equations with antiperiodic boundary value conditions.where denotes Caputo fractional derivative of order and . The existence and uniqueness results are obtained by using the Banach contraction mapping principle and Leray–Schauder nonlinear alternative.

In [19], Berhail et al. discussed the following fractional coupled Sturm–Liouville–Langevin equations with the help of the Banach contraction mapping principle and Leray–Schauder nonlinear alternative:where denotes the Hadamard fractional derivative of order that are continuous functions for .

Motivated by the above results, in this paper, we study the existence results for the following fractional p-Laplacian coupled Sturm–Liouville–Langevin equations with antiperiodic boundary conditions:where denotes Caputo–Hadamard fractional derivative of order with , , is a p-Laplacian operator, and its inverse is , where , , are two continuous functions.

We note that system (6) is a generalization of Sturm–Liouville and Langevin fractional differential systems. For the special case:(1)If for , then the problem (6) is reduced to fractional p-Laplacian coupled Sturm–Liouville equations with antiperiodic boundary conditions.(2)If , then the problem (6) is reduced to fractional p-Laplacian coupled Langevin equations with antiperiodic boundary conditions.

The rest of the paper is organized as follows. In Section 2, we recall some useful preliminaries. Section 3 contains two existence results which are obtained by applying Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, respectively. In Section 4, we present a concrete example to illustrate the theoretical results. Finally, some concluding remarks are given in Section 5.

2. Preliminaries

In this section, we recall some preliminary definitions, lemma, and theorems that will be used to prove our main results.

Definition 1. (see [3]). The Hadamard fractional integral of order is defined asprovided that the integral exists.

Definition 2. (see [21]). For an at least -times differentiable function , the Caputo–Hadamard derivative of fractional order is defined aswhere denotes the integer part of the real number .

Lemma 1 (see [21]). Let or and , where . Then,where .

Theorem 1 (see [22]). (Leray–Schauder nonlinear alternative) Let be a Banach space, be a closed and convex subset of , be a relatively open subset of such that , and be completely continuous. Then, either(i) has a fixed point in , or(ii)there exist (the boundary of in ) and such that .

Theorem 2 (see [23]). (Krasnoselskii’s fixed point theorem). Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let be two operators such that(i), where (ii) is compact and continuous(iii) is a contraction mappingThen, there exists such that .

3. Main Results

It is defined that the space endowed with the norm . It is obvious that is a Banach space. Then, the product space is also a Banach space with the norm .

Lemma 2. Let . Then, the solution of the coupled systemis given by

Proof. Applying the operator on both sides of the first equation in (12) and then applying Lemma 1, we obtainSince , we can find thatBy using the boundary conditions , we getCombining formulas (17) and (18), we find thatSubstituting the value of into (16), we obtainTaking the operator to the both sides of (20), and using Lemma 1 yieldsApplying the boundary condition in (21), it follows thatSubstituting the value into (21), we obtain the solution (13). Similarly, using the same way in second equation of the system (12), we obtain (14). The converse follows by direct computation. This completes the proof.
Based on Lemma 2, we define the operator as follows:whereWe observe that the solutions of system (6) are equivalent operator that has fixed points.
For the sake of computational convenience, we make the following notations:Now, we give an existence result for BVP (6) via Leray–Schauder nonlinear alternative.

Theorem 3. We assume that are two continuous functions and the following conditions hold:(i). there exist continuous nondecreasing function and functions such that(ii). There exists a constant such that where . Then, coupled system (6) has at least one solution on , provided that .

Proof. We divide the proof in to the following two steps:

Step 1. We prove the operator is completely continuous. From the continuity of the functions and , we can easily obtain that the operator is continuous. We now show that the operator is compact. To this end, let , and we define . Firstly, we claim that is uniformly bounded on . In fact, by using condition , for any and , we haveThus, . In a similar way, we can obtainHence,Consequently, the operator is uniformly bounded on . Secondly, we claim that is equicontinuous on . In fact, for any , with , we haveThis yieldsas . In a similar manner, we deduce thatas . Thus, the operator is equicontinuous. From the Arzelá–Ascoli theorem, it follows that the operator is completely continuous.

Step 2. We shall show that the operator has fixed points. To this end, we define a set . First, we show that . In fact, for any and , we shall adopt the same procedure as in the proof of formula (30), and it followsand then combined with yields thatwhich means that . Next, we claim that for any and , then . By using reduction to absurdity, we suppose that there exists such that for some . Then, we haveThis leads to a contradiction. Consequently, the operator has a fixed point by the virtue of the Leray–Schauder nonlinear alternative theorem. Hence, the given problem (6) has at least one solution on . The proof is completed.

Corollary 1. We suppose that the condition holds. If there exists constant such thatthen the problem (7) has at least one solution on .

Corollary 2. We assume that the condition is satisfied. If there exists constant such thatthen the problem (8) has at least one solution on .

Our last existence result is derived by Krasnoselskii’s fixed point theorem. To this end, we define the operators and bywithwhere the operators are defined by

Theorem 4. We assume that are continuous functions and the following condition holds:(i). There exists a function such that .Then, BVP (6) has at least one solution, provided that .

Proof. We define a set bywhereClearly, is a closed, convex, and nonempty subset. Now, the proof will be given in three steps:

Step 1. For any . In fact, taking into account assumption , we havewhich on taking the norm for , yieldsIt follows thatThus, .

Step 2. It is to show that is a contraction mapping. In fact, for each and , we haveEvaluating the maximum values, for , one hasUsing the definition of the norm in the Banach space , we obtainIn view of , we can conclude that is a contraction mapping.

Step 3. On the one hand, we prove that is compact and continuous. Indeed, the continuity of , , and implies that the operator is continuous. On the other hand, by Step 1, it is not hard to see that is uniformly bounded on . Finally, we have to show that the operator is equicontinuous; for with and , we havewhich is independent of and tends to zero as . Thus, are equicontinuous, which implies that is equicontinuous. By Arzelá–Ascoli theorem, is compact on . Hence, by Theorem 2, there exists a fixed point such that , which is a solution to the BVP (6) on . This completes the proof.

Corollary 3. We suppose that the condition is satisfied. Then, the BVP (7) has at least one solution.

Corollary 4. We suppose that the condition is satisfied. Ifthen the BVP (8) has at least one solution.

4. Example

Example 1. Considering the following coupled Strum–Liouville–Langevin equations via Caputo–Hadamard derivative with antiperiodic boundary condition involving p-Laplacian operatorCorresponding to problem (6), here,We chooseOne can easily find that is satisfied. A simple calculation givesMoreover,This implies that is satisfied. By Theorem 3, the problem (52) has at least one solution.
Let ; then, the assumption holds. In view of , then from Theorem 4, it follows that problem (52) has at least one solution.

5. Conclusion

In the present article, we investigated the existence results for a kind of coupled system of p-Laplacian fractional Sturm–Liouville–Langevin equations involving Caputo–Hadamard factional derivative and supplemented with antiperiodic boundary conditions. By using the Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, the existence of solutions for the underlying problem is established under certain assumptions on the righthand side functions and . In the end, we provided an example to illustrate the applicability of our results. Comparing our work with the available literature on the topic [16, 18–20], we considered the fractional Sturm–Liouville–Langevin equations with p-Laplacian operator (, ). We notice that is a quasilinear operator, which brings difficulties to the study of the problem (6). For instance, since ϕ_p(s) is a quasilinear operator, which brings the difficulties to prove the compactness and the priori estimations of the operators S and G in Theorems 3 and 4, etc. In the special case if , then , which means that (6) can be reduced to the problems studied in [18–20]. Therefore, our results are new, and we extend some previous results. Last but not least, an interesting extension of our studies would be to consider the fractional p-Laplacian Sturm–Liouville–Langevin equations on graph.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0291).

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Copyright © 2022 Jinbo Ni 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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