Existence and Stability for a Nonlinear Coupled <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Laplacian System of Fractional Differential Equations

Basha, Merfat; Dai, Binxiang; Al-Sadi, Wadhah

doi:https://doi.org/10.1155/2021/6687949

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Volume 2021 | Article ID 6687949 | https://doi.org/10.1155/2021/6687949

Existence and Stability for a Nonlinear Coupled -Laplacian System of Fractional Differential Equations

Merfat Basha,^1,2Binxiang Dai,¹and Wadhah Al-Sadi³

Academic Editor: Ahmet Ocak Akdemir

Received28 Dec 2020

Revised04 Feb 2021

Accepted11 Feb 2021

Published26 Feb 2021

Abstract

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of orders and Riemann–Liouville derivative of orders with the -Laplacian operator, where , and . With the help of two Green’s functions , the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.

1. Introduction

The theoretical development of fractional calculus and its applications is more important to model nonlinear complex problems with the arbitrary fractional order. The subject of fractional differential equations (FDEs) has become an important area in real life because of their ability to model a lot of physical phenomena associated with rapid and concise changes with their significance in science and engineering through the past three decades, such as chemistry, physics, biology, engineering, visco-elasticity, electrotechnical, signal processing, electrochemistry, and controllability (see the details, [1–9], and the reference therein). In the near time, the nonlinear fractional partial differential equations are the most applied research area in which most authors and scientists are focused for their investigation. In this case, the Caputo derivative plays a great role to analyze the specific application of nonlinear PDEs. in [10], the authors have studied the cancer treatment model based on Caputo–Fabrizio fractional derivative. After integrating the model into the Caputo–Fabrizio fractional derivative, they have analyzed the existence of the solution as well. The Caputo–Fabrizio fractional derivative is implemented in [11] for the modeling and characterizing of the alcoholism. By applying the fixed-point theorem, they have studied the existence and uniqueness of the alcoholism model. The spread of the SIQR model is investigated by [12] using the Caputo derivative. They have justified the stability and uniqueness of the nonvirus equilibrium and virus equilibrium point.

For this problem, different authors proposed different numerical solution techniques. The analysis with the nonlinear time-fractional HIV/AIDS transmission model is considered in [13], in which the numerical solution is found using the fractional variational iteration method with convergence analysis. The nonlinear garden equation is studied in [14] based on the Atangana–Baleau Caputo derivative. He has highlighted the fixed-point theorem for proving the existence and uniqueness of the garden equation.

One of the main difficulties for the solution of the nonlinear fractional PDEs is to analyze the existence theory of solutions. Sufficient conditions for the existence and uniqueness of solutions (EUS) have been obtained by using different nonlinear analysis techniques and fixed-point theorems (for more details, read [15–18]). Also, the boundary value problems with various boundary conditions for many ordinary differential equations are studied [19–23]. However, the theory of boundary value problems for nonlinear FDEs is still not discussed more, and many problems of this theory require to be explored. On the contrary, the investigation of coupled systems of the differential equations is also significant because systems of this kind appear in various applied nature problems (refer [24–28]).

The topological degree theory is a useful tool in nonlinear analysis with numerous applications to operatorial equations, optimization theory, fractal theory, and other topics. We will see the following consideration of topological degree theory with boundary conditions based on the Caputo fractional derivative by different authors. Isaia [29] applied the topological degree theory to establish sufficient conditions for the existence of a solution for the following nonlinear integral equations:where and are continuous functions. In their study [30], Wang et al. used the topological degree method to obtain some existence conditions of the solution for the following nonlocal Cauchy problem:where denotes the Caputo fractional derivative with order and and are continuous. The nonlocal term is a given function. Proceeding on the same fashion, Shah and Khan [31] proved the EUS for a coupled system under the fractional derivatives by using the technique of degree theory given as follows:where , , and are continuous. Khan et al. [32] used the above-mentioned method to study the following coupled system in the sense of Caputo derivatives with -Laplacian:where for . The study of positive solutions to boundary value problems for fractional-order differential equations using the topological degree theory technique is rarely available in the literature, so this research field needs further elaboration. Most papers that dealt the topological degree theory with fractional orders belong to or . For the uniqueness and existence analysis of nonlinear fractional differential equations, the case only Caputo fractional derivative is used frequently.

Thus, our motivation to this study is developing a sufficient condition for the coupled nonlinear fractional derivative that is based on both Caputo and Riemann–Liouville derivatives. The fractional order in our study is expanded to , and we have used a technique of topological degree theory for the analysis of existence and uniqueness of our coupled system defined below. Besides, we have investigated Hyers–Ulam stability to the nonlinear coupled system of fractional-ordered ordinary differential equations with boundary conditions designed by the following:where denote the Caputo fractional derivatives, are the Riemann–Liouville fractional derivatives, and are nonlinear functions, and the boundary functions . represents the -Laplacian operator such that denotes the inverse of -Laplacian, where . Since it is difficult to find the exact solution of the nonlinear differential equations, stability and uniqueness have played a great role to get the approximate solution for the given nonlinear problems. Therefore, scientists and researchers have given attention to study the various forms of stability to the nonlinear problems in the sense of Ulam and their multiple types in the last few decades. We observe that the concept of Hyers–Ulam stability is fundamental in realistic problems, such as numerical analysis, biology, and economics (see [33–38]).

The remaining part of this manuscript is structured as follows. In Section 2, we have introduced some basic definitions and lemmas that we need to prove our main results. By using the topological degree theory, the results of existence and uniqueness for the solutions are obtained in Section 3. In Section 4, we investigate the stability of Hyers–Ulam to our proposed coupled system. The theoretical results are demonstrated by providing an example in Section 5, and finally, we have drawn the conclusion in Section 6.

2. Preliminaries

In this section, we introduce some basic notions, definitions, and important lemmas which are used in this article.Let be a Banach space for all continuous functions with the norm . Further, is also a Banach space under the norms and . The family of each bounded set of symbolized by .

Definition 1. For , the Caputo fractional derivative of noninteger order is known bywhere , the integral in the right side is pointwise defined on , and is a continuous function.

Definition 2. For , the Riemann–Liouville fractional derivative of noninteger order is known bywhere , the integral in the right side is pointwise defined on , and is a continuous function.

Definition 3. For , the Riemann–Liouville fractional integral of order is defined bywhere the integral on the right side is pointwise defined on and indicates the Gamma function defined as

Lemma 1. (see [39]). Letand. Then, the general solution of the fractional differential equationis given byfor

Lemma 2. (see [2, 8]). Let, andis the fractional derivative for Caputo, thenforand.

Lemma 3. (see [2, 8]). Let, andis the fractional derivative for Riemman–Liouville, thenforand.

Lemma 4. (see [22]). For, the following relations are satisfying:

Definition 4. The Kuratowski measure of noncompactness is the map known as which admits a finite cover by sets of diameter , where .

Proposition 1. (see [40]). The Kuratowski measure ofsatisfies the following properties:(a)The Kuratowski measure; for a relative compact(b)is a seminorm, i.e.,, and(c)implies(d)(e)

Definition 5. Suppose that the function is a continuous and bounded map, where . is called Lipschitz with , and if is bounded.
Moreover, if , then will be a strict contraction.

Definition 6. The function is called condensing, and if is bounded with .
In other words, implies .
We indicate that the class of each condensing mappings by and the class of each strict contractions by .

We remark that , and every is Lipschitz with . As well, we recall that is Lipschitz if such that . Also, is a strict contraction under the condition .

Proposition 2. (see [31]). LetbeLipschitz operators with constants, respectively, thenisLipschitz with constants.

Proposition 3. (see [41]). The operatoris compact if and only ifisLipschitz with.

Proposition 4. (see [31]). The operatoris Lipschitz with constantif and only ifisLipschitz with constant.

Lemma 5. (see [39]). Letbe a nonlinear p-Laplacian operator.(1)If , and , then(2)If and , then

Theorem 1 (see [29]). Letbe acontraction, andsuch that. Ifis a bounded set, there existssuch that, then the degree.

Thus, has at least one fixed point, and the set of the fixed points of lies in .

The above theorem that we mentioned plays a substantial role in obtaining our main results.

3. Main Results

In the current section, we establish some appropriate conditions for proposed coupled system (5).

Theorem 2. Let be atimes’ integrable function. Then, forand positive integer, the solution of the boundary value problem is as follows:is given bywhereis the Green’s function provided by

Proof. Applying the integral operator and using Lemma 2 on (16), we getUsing the condition for in (19), we obtain , and then, we getFrom (20), we haveApplying the operator and using Lemma 3 in (21), we getUsing the condition we get , and then, we obtainUsing the condition and Lemma 4 in (22), we getPutting the value of in (23), we getwhich can be written after rearranging as follows:where is the Green’s function defined in (18).

In view of Theorem 2, the identical coupled system of Hammerstein-kind integral equations to fractional differential equation coupled system (5) is given as follows:where is the Green’s function provided by

From and obviously,

We define the operators asas

Therefore, we have , and Thus, the equivalent operator equation for the toppled system of Hammerstein-kind integral equations (27) is provided by

Consequently, the solutions of system (27) are the fixed points of operator equation (32).

Now, we need to list the following assumptions to complete our results. For , the nonlocal functions satisfy such that With the positive constants given and , the nonlocal functions for satisfy the following growth conditions With the presence of constants , and , the nonlinear functions for satisfy the following growth conditions: For , there exists positive constants such that

Theorem 3. Assume thatandhold true. Then, the operatoris Lipschitz and satisfies the following growth condition:

Proof. By assumption , we getwhich yieldswhereTo get the growth condition, considerwhich means thatIn a similar manner, we havewhich implies thatNow,Thus,where

Theorem 4. Suppose thatis satisfied. Then, the operatoris continuous and satisfies the following growth condition:whereandsuch that

Proof. Let be a bounded set with a sequence converging to in . In order to show that is continuous, we have to prove that as . Let us choose the following:The continuity of implies that as , and then,and also, we can in the same way prove thatThe continuity of implies that as , and then,Thus, from (49) and (51), we haveFrom the continuity of and (52), we haveTo calculate (46) for , using assumption and (29), we obtainFrom assumption and (29), we getThen,By the help of (54) and (55), we have obtained

Theorem 5. The operatorisLipschitz with constant zero and is compact.

Proof. Take a bounded set and a sequence such that . Then, using (46), we havewhich means that is bounded. Now, for all , we have, for ,Hence, it follows thatSimilarly, we haveBoth the right sides of (55) and (61) tend to be zero as . Therefore, the operators are equicontinuous, and hence, is equicontinuous on . Thus, is compact by the theorem of Arzela–Ascoli. Moreover, through Proposition 3, is Lipschitz with constant zero.

Theorem 6. Suppose that–are satisfied with. Then, the toppled system (5) has at least one solution. Furthermore, the set of solutions of (5) is bounded in.

Proof. With the help of Theorem 3, is Lipschitz with constant , and is Lipschitz with constant zero by Theorem 5. Thus, is strictly condensing with constant by Proposition 2. Now, let us set that we have to show that is bounded in . In fact,Obviously, is bounded. If not correct, take such that and . Consequently,which is a contradiction. So, is bounded. Thus, by Theorem 7, we conclude that has at least one fixed point and that is a solution of system (5), and the set of solutions is bounded in .

Theorem 7. Suppose that–hold and, where

Then, toppled system (5) has a unique solution.

Proof. Let are two solutions, thenand after simplification, we obtainwhich implies thatThus, the operator is a contraction as , and by the Banach fixed-point theorem, has a unique fixed point, and then, considered toppled system (5) has a unique solution.

4. Hyers–Ulam Stability

In this section, we investigate the stability of Hyers–Ulam for the suggested toppled system.

Definition 7. We say that the toppled system of Hammerstein-kind integral equations (27) is Hyers–Ulam stable if there exists positive constants such that, for each and any solution of the systemthere exists which is the unique solution of (27) satisfying that

Theorem 8. The toppled system (5) is Hyers–Ulam stable under hypotheses–.

Proof. With the help of Definition 7 and Theorem 7, suppose that to be the correct solution and the pair be the other solution of system (27). Then, we have, from the first equation of (27),Then,whereSimilarly, we getwhereHence, by (71) and (73), integral equations’ toppled system (27) is Hyers–Ulam stable. Thus, proposed toppled system (5) is Hyers–Ulam stable.

5. Illustrative Example

In this section, we introduce an application of our results, which were proved in Sections 3 and 4.

Example 1. Consider the following toppled fractional system with the -Laplacian operator and integral boundary conditions for n = 5:where . Then, we obtain and . Via simple calculation and taking , we get . Hence, by Theorem 7, toppled system (75) has a unique solution. With comparable fashion, it is easy to verify the fulfillment of the conditions of Theorem 6. Likewise, the conditions of Theorem 8 can be easily confirmed, and consequently, the solution of system (75) is Hyers–Ulam stable.

6. Conclusion

In this study, we analyzed the stability and uniqueness solution of Caputo and Riemann–Liouville fractional derivatives with fractional orders , and . By using the topological degree theory, we have proved sufficient conditions for the EUS of the coupled system of fractional differential equations with integral boundary conditions involving the -Laplacian operator. Also, we have found appropriate conditions for Hyers–Ulam stability of the solution for the considered system. At the end, we have provided an example that supported our results as we have done in Section 5 to confirm the theoretical analysis.

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 work was supported by the National Natural Science Foundation of China (no. 11871475).

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