Existence and Stability for a Nonlinear Coupled p-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 θ1 and θ2 and Riemann–Liouville derivative of orders 91 and 92 with the p-Laplacian operator, where n − 1< θ1, θ2, 91, 92 ≤ n, and n≥ 3. With the help of two Green’s functions (Gρ1(w,I), Gρ2(w,I)), 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.


Introduction
e theoretical development of fractional calculus and its applications is more important to model nonlinear complex problems with the arbitrary fractional order. e 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][2][3][4][5][6][7][8][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. e 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. e spread of the SIQR model is investigated by [12] using the Caputo derivative. ey have justified the stability and uniqueness of the nonvirus equilibrium and virus equilibrium point.
For this problem, different authors proposed different numerical solution techniques. e 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. e 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][16][17][18]). Also, the boundary value problems with various boundary conditions for many ordinary differential equations are studied [19][20][21][22][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][25][26][27][28]). e 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 c D ϱ denotes the Caputo fractional derivative with order ϱ ∈ (0, 1) and Z: C([0, w], R) ⟶ R and π 0 ∈ R are continuous. e nonlocal term h: C([0, w], R) ⟶ R 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: Khan et al. [32] used the above-mentioned method to study the following coupled system in the sense of Caputo derivatives with p-Laplacian: where ϱ i , θ i ∈ (1, 2] and δ i , η i ∈ (0, 1], for i � 1 and 2. e 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 (0, 1) or (1,2]. For the uniqueness and existence analysis of nonlinear fractional differential equations, the case only Caputo fractional derivative is used frequently.
us, 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. e fractional order in our study is expanded to 2 Journal of Mathematics (n − 1, n], 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 9 1 , 9 2 , θ 1 , θ 2 ∈ (n − 1, n], n ≥ 3, 1 < λ, σ ≤ 2, c D θ 1 and c D θ 2 denote the Caputo fractional derivatives, R D ϱ 1 and R D ϱ 2 are the Riemann-Liouville fractional derivatives, and Z 1 , Z 2 : [0, 1] × IR ⟶ IR are nonlinear functions, and the boundary functions φ, ϖ ∈ L[0, 1]. ϕ p represents the p-Laplacian operator such that ϕ p (]) � ]|]| p− 2 , and ϕ q � ϕ − 1 p denotes the inverse of p-Laplacian, where (1/p) + (1/q) � 1. 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. erefore, 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][34][35][36][37][38]). e 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. e theoretical results are demonstrated by providing an example in Section 5, and finally, we have drawn the conclusion in Section 6.

Preliminaries
In this section, we introduce some basic notions, definitions, and important lemmas which are used in this article.Let Π � C([0, 1], IR) be a Banach space for all continuous functions π: e family of each bounded set of P(Ω) symbolized by B.
where n − 1 < ϱ < n, the integral in the right side is pointwise defined on (0, ∞), and Z(w) is a continuous function.
e Kuratowski measure of ϑ satisfies the following properties: Moreover, if η < 0, then F will be a strict ϑ− contraction.

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

Journal of Mathematics
By the help of (54) and (55), we have obtained (57) □ Theorem 5. e operator G: Ω ⟶ Ω is ϑ− Lipschitz with constant zero and is compact.
Proof. Take a bounded set E and a sequence (π 1 n , π 2 n ) such that E ⊂ B r ⊆ Ω. en, using (46), we have which means that G is bounded. Now, for all (π 1 n , π 2 n ) ∈ E, we have, for 0 ≤ w 1 < w 2 ≤ 1, Hence, it follows that Similarly, we have Both the right sides of (55) and (61) tend to be zero as w 1 ⟶ w 2 . erefore, the operators G 1 and G 2 are equicontinuous, and hence, G � (G 1 , G 2 ) is equicontinuous on E. us, G(E) is compact by the theorem of Arzela-Ascoli. Moreover, through Proposition 3, G is ϑ− Lipschitz with constant zero. □ Theorem 6. Suppose that (H 1 )-(H 3 ) are satisfied with Λ + C F ≤ 1. en, the toppled system (5) has at least one solution (π 1 , π 2 ) ∈ Ω. Furthermore, the set of solutions of (5) is bounded in Ω.
Proof. With the help of eorem 3, F is ϑ− Lipschitz with constant 0 ≤ C F < 1, and G is ϑ− Lipschitz with constant zero by eorem 5. us, T is strictly ϑ− condensing with constant η by Proposition 2. Now, let us set that we have to show that B is bounded in Ω. In fact, where q 3 � max q 1 , q 2 . (62) Obviously, ‖π 1 , π 2 ‖ is bounded. If not correct, take ‖π 1 , π 2 ‖ � S such that S ⟶ ∞ and q 3 ∈ (0, 1). Consequently, which is a contradiction. So, B is bounded. us, by eorem 7, we conclude that T has at least one fixed point and that is a solution of system (5), and the set of solutions is bounded in Ω.

Conclusion
In this study, we analyzed the stability and uniqueness solution of Caputo and Riemann-Liouville fractional derivatives with fractional orders n − 1 < θ 1 , θ 2 , ϱ 1 , ϱ 2 ≤ n, and n ≥ 3. 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 p-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
e authors declare that they have no conflicts of interest.