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

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.


Introduction
The subject of fractional calculus (calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous fields of science and engineering. Historically, the first appearance of the concept of a fractional derivative was found in a letter by the famous mathematician Gottfried Leibniz (1646 -1716) in 1695 to a French mathematician Guillaume de L'Hospital (1661 -1704). Leibniz introduced the following symbol d n dt n f (t) which denotes the n th order derivative of a function f with the hypothesis that n ∈ N and reported this to L'Hospital. So L'Hospital posed a question; what will be the derivative if n = 1 2 ? Leibniz replied to him on September 13 th , 1695 and wrote: "This is an apparent paradox from which, one day useful consequences will be drawn" [8]. [104]

Akbar Zada, Mohammad Yar and Tongxing Li
In the last few decades, fractional differential equations have gained much attention due to extensive applications of these equations in the mathematical modeling of physical, engineering, biological phenomena and viscoelasticity etc, [13]. Several interesting and important results concerning the existence and uniqueness of solutions, stability properties of solutions, analytic and numerical methods of solutions for fractional differential equations can be found in the recent literature. Fractional-order operators are nonlocal in nature and take care of the hereditary properties of many phenomena and processes. Fractional calculus has also emerged as a powerful modeling tool for many real world problems, see [2,6,9,17].
The study of coupled systems involving fractional differential equations is also important because these systems occur in various problems of applied nature. Coupled systems of fractional differential equations have also been investigated by many authors. Such systems appear naturally in many real world situations, for example, see [4]. Some recent results on the topic can be found in [5,7,19,18,23,24].
Moreover, the theory of fractional order differential equations, involving different kinds of boundary conditions has been a field of interest in pure and applied sciences. Nonlocal conditions are used to describe certain features of applied mathematics and physics such as blood flow problems, cellular systems [1], chemical engineering, thermo-elasticity, underground water flow, population dynamics [10], and so on. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to [3,21,22,26,27,29,31,30,36,37].
In this paper, we study the nonlinear sequential coupled system of Caputo fractional differential equations with Riemann-Liouville fractional integral boundary conditions of the following form where c D q , c D p denote the Caputo fractional derivatives of order p, q, I ρi , I γj are the Riemann-Liouville fractional integral of order ρ i , γ j > 0, η i , θ j ∈ (0, T ), k ∈ R + , f, g : [0, T ] × R 2 → R and α i , β j ∈ R, i = 1, 2, . . . , n, j = 1, 2, . . . , m are real constants such that Existence and stability analysis of nonlinear sequential coupled system Here, we emphasize that the integral boundary conditions (1) can be understood in the sense that the value of the unknown function at an arbitrary position η i , θ j ∈ (0, T ) is proportional to the Riemann-Liouville fractional integral of the unknown functions where ρ i , γ j > 0. Further, for η i = θ j = 1, the integral boundary conditions reduce to the usual form of a nonlocal integral conditions We show the existence of solutions for problem (1) by applying Leray-Schauder alternative criterion while uniqueness of solutions for (1) relies on Banach contraction mapping principle. The rest of the paper is organized as follows: In Section 2 we recall some preliminary concepts which we will need in the sequel. Section 3 contains the main results for problem (1). In Section 4, we present the Hyers-Ulam stability of problem (1).

Preliminaries and background materials
In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs.
The Riemann-Liouville fractional integral of order q > 0 of a function f : (0, ∞) → R is defined by provided the right-hand side is point-wise defined on (0, ∞).  has a unique solution where c i ∈ R, i = 1, 2, . . . , m − 1. is and Existence and stability analysis of nonlinear sequential coupled system where and Proof. Writing the linear sequential fractional differential equations in (2) as and then applying the Riemann-Liouville integral operator I q−1 and I p−1 on both sides, followed by integration from 0 to t, we get and where A 0 , A 1 , A 2 , B 0 , B 1 and B 2 are arbitrary constants and Taking the Riemann-Liouville fractional integral of order ρ i > 0 for (5) and γ j > 0 for (6) and using the property of the Riemann-Liouville fractional integral, we get Akbar Zada, Mohammad Yar and Tongxing Li Substituting the values of A 1 , A 2 , B 1 and B 2 in (5) and (6), we obtain the solutions (3) and (4).

Main results
Throughout this paper, for convenience, we use the following expression where

y(s))(θj)ds
Existence and stability analysis of nonlinear sequential coupled system For the sake of convenience, we set [110]

Akbar Zada, Mohammad Yar and Tongxing Li
and The first result is concerned with the existence and uniqueness of the solution for the problem (1) and is based on Banach contraction principle.

y(s))(t)ds
Existence and stability analysis of nonlinear sequential coupled system In the same way, we can obtain that Consequently, |T (x, y)(t)| ≤ r. Now for (x 2 , y 2 ), (x 1 , y 1 ) ∈ X × Y , and for any t ∈ [0, T ], we get
Since (M 1 + M 3 )(m 1 + m 2 ) + (M 2 + M 4 )(n 1 + n 2 ) < 1, therefore, T is a contraction operator. So, By Banach fixed point theorem, the operator T has a unique fixed point, which is the unique solution of problem (1). This completes the proof.
In the next result, we prove the existence of solutions for the problem (1) by applying Leray-Schauder alternative.

Lemma 3.2 (Leray-Schauder alternative, [11]) Let F : E → E be a completely continuous operator (i.e. a map that restricted to any bounded set in E is compact). Let
Then either the set ℵ(F ) is unbounded, or F has at least one fixed point. Assume that there exist real constants k i , λ i > 0, i = 1, 2 and k 0 > 0, λ 0 > 0 such that for all x i ∈ R, i = 1, 2 we have In addition, it is assumed that where M i for i = 1, 2, 3, 4 are given by (7) - (10). Then there exists at least one solution for the boundary value problem (1).
Proof. First we show that the operator T : X × Y → X × Y is completely continuous. By continuity of functions f and g the operator T is continuous.
Let Θ ⊂ X × Y be bounded. Then there exist positive constants L 1 and L 2 such that for all (x, y) ∈ Θ, Then for any (x, y) ∈ Θ, we have Existence and stability analysis of nonlinear sequential coupled system

[115]
which implies that Similarly, we get Thus, it follows from the above inequalities that the operator T is uniformly bounded.
Next, we show that T is equicontinuous. Let t 1 , t 2 ∈ [0, T ] with t 1 < t 2 . Then we have  Analogously, we can obtain

y(s))|(T )ds
Existence and stability analysis of nonlinear sequential coupled system Obviously, the right-hand sides of the above inequalities tend to zero independently of f, g ∈ B r as t 2 − t 1 → 0. Therefore, the operator T (x, y) is equicontinuous, and thus it is completely continuous. Finally, it will be verified that the set Then [118]

Akbar Zada, Mohammad Yar and Tongxing Li
Hence we have which imply that where M 0 is defined by (5), which proves that ℵ is bounded. Thus, by Lemma 3.2 the operator T has at least one fixed point. Hence, the boundary value problem (1) has at least one solution.

Hyers-Ulam stability of system (1)
This section is devoted to the investigation of Hyers-Ulam stability for our proposed system. Consider the following inequality: where ε 1 , ε 2 are given two positive real numbers.
Existence and stability analysis of nonlinear sequential coupled system [119] Remark 4.3 If (x, y) represent a solution of inequality (13), then (x, y) is a solution of following inequality As from Remark 4.2, we have With the help of Definition 4.1 and Remark 4.2 we verified Remark 4.3, in the following lines By the same method we can obtain that where M i , i = 1, 2, 3, 4 are given by (7)- (10). Hence Remark 4.3 is verified, with the help of (14) and (15). Thus the nonlinear sequential coupled system of Caputo fractional differential equations is Hyers-Ulam stable and consequently, the system (1) is Hyers-Ulam stable.