Stability , Existence and Uniqueness of Boundary Value Problems for a Coupled System of Fractional Di ff erential Equations

The current article studies a coupled system of fractional differential equations with boundary conditions and proves the existence and uniqueness of solutions by applying Leray-Schauder’s alternative and contraction mapping principle. Furthermore, the Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. Obtained results are supported by examples and illustrated in the last section.


Introduction
Fractional calculus is undoubtedly one of the very fast-growing fields of modern mathematics, due to its broad range of applications in various fields of science and its unique efficiency in modeling complex phenomena [1,2].In particular, fractional differential equations with boundary conditions are widely employed to build complex mathematical models for numerous real-life problems such as blood flow problem, underground water flow, population dynamics, and bioengineering.As an example, consider the following equation that describes a thermostat model − x = g(t) f (t, x), x(0) = 0, βx (1) = x(η), where t ∈ (0, 1), η ∈ (0, 1] and β is a positive constant.Note that solutions of the above equation with the specified integral boundary conditions are in fact solutions of the one-dimensional heat equation describing a heated bar with a controller at point 1, which increases or reduces heat based on the temperature picked by a sensor at η.A few of the relevant studies on coupled systems of fractional differential equations with integral boundary conditions are briefly reviewed below and for further information on this topic, refer to References [3,4].
The current paper studies the following coupled system of nonlinear fractional differential equations: supplemented with boundary conditions of the form: Here, c D k denotes Caputo fractional derivative of order k (k = α, β); and f , g ∈ C [0, T] × R 2 , R are given continuous functions.Note that η, ζ are real constants such that T 2 − ηζ 0.
The rest of this paper is organized in the following manner: In Section 2, we briefly review some of the relevant definitions from fractional calculus and prove an auxiliary lemma that will be used later.Section 3 deals with proving the existence and uniqueness of solutions for the given problem, and Section 4 discusses the Hyers-Ulam stability of solutions and presents sufficient conditions for the stability.The paper concludes with supporting examples and obtained results.

Preliminaries
We begin this section by reviewing the definitions of fractional derivative and integral [1,2].Definition 1.The Riemann-Liouville fractional integral of order τ for a continuous function h is given by provided that the right-hand side is point-wise defined on [0, ∞).Definition 2. The Caputo fractional derivatives of order τ for (h − 1)-times absolutely continuous function g where [τ] is the integer part of real number τ.
Here we prove the following auxiliary lemma that will be used in the next section.Lemma 1.Let u, v ∈ C([0, T], R) then the unique solution for the problem where ∆ = T 2 − ηζ 0.
Proof.General solutions of the fractional differential equations in (3) are known [6] as where a, b, c, and d are arbitrary constants.

Considering boundary conditions
and Hence, by substituting the value of a into c, we obtain the final result for these constants as and Substituting the values of a, b, c, and d in ( 6) and (7) we get (4) and (5).The converse follows by direct computation.This completes the proof.

Existence and Uniqueness of Solutions
In view of Lemma 1, we define the operator where and Here we establish the existence of the solutions for the boundary value problem ( 1) and ( 2) by using Banach's contraction mapping principle.
Firstly, we show that GΩ ε ⊆ Ω ε .By our assumption, for (x, y) ∈ Ω ε , t ∈ [0, T], we have and g(t, x(t), y(t)) ≤ ψ x(t) + y(t) + g 0 ≤ ψ( x + y ) + g 0 , ≤ ψε + g 0 , which lead to In a similar manner: Hence, and Consequently, and we get G(x, y) ≤ ε that is Then we have and likewise From ( 11) and ( 12) we have , therefore, the operator G is a contraction operator.Hence, by Banach's fixed-point theorem, the operator G has a unique fixed point, which is the unique solution of the BVP (1) and ( 2).This completes the proof.
Proof.This proof will be presented in two steps.
Step 1: We will show that G : ) is completely continuous.The continuity of the operator G holds by the continuity of the functions f , g.
Step 2: (Boundedness of operator) Finally, we will show that This proves that Z is bounded and hence by Leray-Schauder alternative theorem, operator G has at least one fixed point.Therefore, the BVP ( 1) and ( 2) has at least one solution on [0, T].This completes the proof.

Hyers-Ulam Stability
In this section, we will discuss the Hyers-Ulam stability of the solutions for the BVP ( 1) and ( 2) by means of integral representation of its solution given by where G 1 and G 2 are defined by ( 8) and (9).
Therefore, we deduce by the fixed-point property of operator G, that is given by ( 8) and ( 9), which and similarly From ( 14) and ( 15) it follows that (x, y) − (x * , y * ) ≤ (Q Thus, sufficient conditions for the Hyers-Ulam stability of the solutions are obtained.

Examples
Example 1.Consider the following coupled system of fractional differential equations Using the given data, we find that ∆ (1.1398 + 1.554) = 0.0283 < 1.
Thus, all the conditions of Theorem 1 are satisfied, then problem ( 16) has a unique solution on [0, 1], which is Hyers-Ulam-stable.

Conclusions
In this paper, the existence, uniqueness and the Hyers-Ulam stability of solutions for a coupled system of nonlinear fractional differential equations with boundary conditions were established and discussed.
Future studies may focus on different concepts of stability and existence results to a neutral time-delay system/inclusion, time-delay system/inclusion with finite delay.