scholarly journals Existence and Stability of the Solution to a Coupled System of Fractional-order Differential with a $p$-Laplacian Operator under Boundary Conditions

2021 ◽  
Vol 53 ◽  
Author(s):  
Wadhah Ahmed Alsadi
Keyword(s):  
Boundary Conditions ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Laplacian Operator ◽  
Integral Boundary ◽  
Nonlinear Coupled System ◽  
Existence And Stability ◽  
Uniqueness Of The Solution ◽  
Uniqueness And Existence ◽  
Stability Of The Solution

This paper is devoted to studying the uniqueness and existence of the solution to a nonlinear coupled system of (FODEs) with p-Laplacian operator under integral boundary conditions (IBCs). Our problem is based on Caputo fractional derivative of orders $ \sigma,\lambda $, where $ k-1\leq\sigma,\lambda<k, k\geq3$. For these aims, the nonlinear coupled system will be converted into an equivalent integral equations system by the help of Green function. After that, we use Leray-Schauder's and topological degree theorems to prove the existence and uniqueness of the solution. Further, we study certain conditions for the Hyers-Ulam stability of the solution to the suggested problem. We give a suitable and illustrative example as an application of the results.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Merfat Basha ◽  
Binxiang Dai ◽  
Wadhah Al-Sadi
Keyword(s):  
Degree Theory ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Laplacian Operator ◽  
Integral Boundary ◽  
Nonlinear Dynamical ◽  
Nonlinear Coupled System ◽  
Liouville Derivative ◽  
Existence And Stability ◽  
Fractional Differential

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of orders θ 1  and  θ 2 and Riemann–Liouville derivative of orders ϱ 1  and  ϱ 2 with the p -Laplacian operator, where n − 1 < θ 1 , θ 2 , ϱ 1 , ϱ 2 ≤ n , and n ≥ 3 . With the help of two Green’s functions G ϱ 1 w , ℑ , G ϱ 2 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.

scholarly journals Ulam–Hyers Stability and Uniqueness for Nonlinear Sequential Fractional Differential Equations Involving Integral Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 235
Author(s):  
Areen Al-khateeb ◽  
Hamzeh Zureigat ◽  
Osama Ala’yed ◽  
Sameer Bawaneh
Keyword(s):  
Boundary Conditions ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Integral Boundary ◽  
Governing System ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equation ◽  
Banach’S Contraction Principle ◽  
Sequential Fractional Differential Equations

Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and uniqueness for the solution of the coupled system of the Caputo-type sequential fractional differential equation, involving integral boundary conditions, was discussed, and investigated. Leray–Schauder’s alternative was applied to derive the existence of the solution, while Banach’s contraction principle was used to examine the uniqueness of the solution. Moreover, Ulam–Hyers stability of the presented system was investigated. It was found that the theoretical-related aspects (existence, uniqueness, and stability) that were examined for the governing system were satisfactory. Finally, an example was given to illustrate and examine certain related aspects.

scholarly journals On Coupledp-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Aziz Khan ◽  
Yongjin Li ◽  
Kamal Shah ◽  
Tahir Saeed Khan
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Laplacian Operator ◽  
Uniqueness Of Solutions ◽  
Integral Boundary ◽  
Fractional Differential

This paper is related to the existence and uniqueness of solutions to a coupled system of fractional differential equations (FDEs) with nonlinearp-Laplacian operator by using fractional integral boundary conditions with nonlinear term and also to checking the Hyers-Ulam stability for the proposed problem. The functions involved in the proposed coupled system are continuous and satisfy certain growth conditions. By using topological degree theory some conditions are established which ensure the existence and uniqueness of solution to the proposed problem. Further, certain conditions are developed corresponding to Hyers-Ulam type stability for the positive solution of the considered coupled system of FDEs. Also, from applications point of view, we give an example.

scholarly journals Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions

2018 ◽  
Vol 17 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Akbar Zada ◽  
Mohammad Yar ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Uniqueness Of Solutions ◽  
Integral Boundary ◽  
Caputo Fractional Differential Equations ◽  
Existence And Stability ◽  
Fractional Differential

Abstract In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to Riemann–Liouville fractional integral boundary conditions. The uniqueness of solutions is established by Banach contraction principle, while the existence of solutions is derived by Leray–Schauder’s alternative. We also study the Hyers–Ulam stability of mentioned system. At the end, examples are also presented which illustrate our results.

scholarly journals On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 223 ◽  
Author(s):  
Kamal Shah ◽  
Poom Kumam ◽  
Inam Ullah
Keyword(s):  
Fixed Point ◽  
Boundary Conditions ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Existence Theory ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Nonlinear Coupled System ◽  
Cone Type ◽  
Stability Results

This manuscript is devoted to establishing existence theory of solutions to a nonlinear coupled system of fractional order differential equations (FODEs) under integral boundary conditions (IBCs). For uniqueness and existence we use the Perov-type fixed point theorem. Further, to investigate multiplicity results of the concerned problem, we utilize Krasnoselskii’s fixed-point theorems of cone type and its various forms. Stability analysis is an important aspect of existence theory as well as required during numerical simulations and optimization of FODEs. Therefore by using techniques of functional analysis, we establish conditions for Hyers–Ulam (HU) stability results for the solution of the proposed problem. The whole analysis is justified by providing suitable examples to illustrate our established results.

scholarly journals Existence and Uniqueness Results for a Coupled System of Caputo-Hadamard Fractional Differential Equations with Nonlocal Hadamard Type Integral Boundary Conditions

2020 ◽  
Vol 4 (2) ◽  
pp. 13 ◽  
Author(s):  
Shorog Aljoudi ◽  
Bashir Ahmad ◽  
Ahmed Alsaedi
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Fractional Differential

In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented.

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