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Merfat Basha, Binxiang Dai, Wadhah Al-Sadi, "Existence and Stability for a Nonlinear Coupled -Laplacian System of Fractional Differential Equations", Journal of Mathematics, vol. 2021, Article ID 6687949, 15 pages, 2021. https://doi.org/10.1155/2021/6687949
Existence and Stability for a Nonlinear Coupled -Laplacian System of Fractional Differential Equations
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.
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 , 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  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  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 , in which the numerical solution is found using the fractional variational iteration method with convergence analysis. The nonlinear garden equation is studied in  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  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 , 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  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.  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.
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 ). Letand. Then, the general solution of the fractional differential equationis given byfor
Lemma 4. (see ). 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 ). 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 ). LetbeLipschitz operators with constants, respectively, thenisLipschitz with constants.
Proposition 3. (see ). The operatoris compact if and only ifisLipschitz with.
Proposition 4. (see ). The operatoris Lipschitz with constantif and only ifisLipschitz with constant.
Lemma 5. (see ). Letbe a nonlinear p-Laplacian operator.(1)If , and , then(2)If and , then
Theorem 1 (see ). 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
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