Existence and Ulam–Hyers stability of coupled sequential fractional differential equations with integral boundary conditions

Abstract

We study the existence and uniqueness of solutions for coupled sequential fractional differential equations involving Caputo fractional derivative of order $1<\alpha \leq 2$ with integral boundary conditions. Moreover, we discuss Ulam–Hyers stability for the problem at hand.

1 Introduction

The theory of differential equations of fractional order, involving various types of boundary conditions, has been the subject of interest in the pure and applied sciences. In addition to the classical two-point boundary conditions, much attention is paid to non-local multipoint and integral boundary conditions. Non-local conditions are used to describe certain features of physical, chemical, or other processes occurring in the inner positions of a given region, while integral boundary conditions provide a plausible and practical approach to modeling blood flow problems. For more information and explanation, see, for example, [1, 2]. Some recent results on the boundary value problem of fractional order can be found in the series [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and in the references cited there. Sequential differential equations with fractional derivatives also received considerable attention, for example, see [4,5,6,7,8,9].

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. Some recent results on the topic can be found in [7, 17, 22,23,24,25,26].

We study the following nonlinear sequential fractional differential equation subject to nonseparated nonlocal integral fractional boundary conditions:

$$ \textstyle\begin{cases} ( D^{\alpha }+\lambda D^{\alpha -1} ) x ( t ) =f_{1} ( t,x ( t ) ,y ( t ) ) ,\quad 1< \alpha \leq 2, 0\leq t\leq T, \\ ( D^{\beta }+\lambda D^{\beta -1} ) y ( t ) =f _{2} ( t,x ( t ) ,y ( t ) ) ,\quad 1< \beta \leq 2, 0\leq t\leq T, \\ \nu _{1}x ( \eta ) +\mu _{1}x ( T ) =\int _{0} ^{T}h_{1} ( x ( s ) ) \,ds, \qquad \nu _{1}y ( \eta ) +\mu _{1}y ( T ) =\int _{0} ^{T}h_{2} ( y ( s ) ) \,ds, \\ \nu _{2}x^{\prime } ( \eta ) +\mu _{2}x^{\prime } ( T ) =\int _{0}^{T}g_{1} ( x ( s ) , ) \,ds, \qquad \nu _{2}y^{\prime } ( \eta ) +\mu _{2}y^{\prime } ( T ) =\int _{0}^{T}g_{2} ( y ( s ) ) \,ds, \end{cases} $$

(1.1)

where $D^{\alpha }$, $D^{\beta }$ denote the Caputo derivative, $0<\eta <T$, $\lambda \in \mathbb{R}_{+}$, $\nu _{1},\nu _{2},\mu _{1},\mu _{2}\in \mathbb{R}$.

During the last few decades another part of research, which has been considered for fractional differential equations and got much attention from the researchers, is stability analysis. Numerous forms of stabilities have been studied in literature which are Mittag-Leffler stability, exponential stability, Lyapunov stability, etc. For historical background of Ulam–Hyers stability and recent results, we refer to works [27,28,29,30,31,32,33,34,35,36]. To the best of our knowledge, the Ulam–Hyers stability has been very rarely studied for coupled system of fractional differential equations. Therefore in this article we investigate existence and Ulam–Hyers stability to the considered problem. The rest of the paper is organized as follows. In Sect. 2, we recall some basic concepts of fractional calculus and obtain the integral solution for the linear variants of the given problems. Section 3 contains the existence results for problem (1.1) obtained by applying Leray–Schauder’s nonlinear alternative, Banach’s contraction mapping principle. In Sect. 4, Ulam–Hyers stability for problem (1.1) is studied. Finally, in Sect. 5, an example is provided to illustrate the theoretical results.

2 Preliminaries

We begin this section with some basic definitions of fractional calculus [2]. Later we prove an auxiliary lemma, which plays a key role in defining a fixed point problem associated with the given problem.

Definition 1

The Riemann–Liouville fractional integral of order $\alpha >0$ for a function $f: [ 0,+\infty ) \rightarrow R$ is defined as

$$ I_{0+}^{\alpha }f(t)=\frac{1}{\varGamma (\alpha )} \int _{0}^{t}(t-s)^{ \alpha -1}f(s)\,ds, $$

provided that the right-hand side of the integral is pointwise defined on $(0,+\infty )$ and Γ is the gamma function.

Definition 2

The Caputo derivative of order $\alpha >0$ for a function $f: [ 0,+\infty ) \rightarrow R$ is written as

$$ D_{0+}^{\alpha }f(t)=\frac{1}{\varGamma (n-\alpha )} \int _{0}^{t}(t-s)^{n- \alpha -1}f^{ ( n ) }(s) \,ds, $$

where $n= [ \alpha ] +1$, $[ \alpha ]$ is an integral part of α.

Lemma 1

([2])

Let $\alpha >0$. Then the differential equation $D_{0+}^{\alpha }f(t)=0$ has solutions

$$ f(t)=c_{0}+c_{1}t+c_{2}t^{2}+ \cdots+c_{n-1}t^{n-1}, $$

and

$$ I_{0+}^{\alpha }D_{0+}^{\alpha }f(t)=f(t)+c_{0}+c_{1}t+c_{2}t^{2}+ \cdots+c _{n-1}t^{n-1}, $$

(2.1)

where $c_{i}\in \mathbb{R}$ and $i=1,2,\ldots$ , $n= [ \alpha ] +1$.

Let $C ( [ 0,T ] ;\mathbb{R} )$ denote the Banach space of all continuous functions from $[ 0,T ]$ to $\mathbb{R}$ equipped with the sup-norm $\Vert x \Vert _{\infty }=\sup \{ \vert x ( t ) \vert :0 \leq t\leq T \}$. For computational convenience, in what follows we use the following notations:

$$\begin{aligned}& a_{11} :=\nu _{1}e^{-\lambda \eta }+\mu _{1}e^{-\lambda T}, \qquad a_{12}:=\nu _{1} \frac{1}{\lambda } \bigl( 1-e^{-\lambda \eta } \bigr) +\mu _{1} \frac{1}{\lambda } \bigl( 1-e^{-\lambda T} \bigr), \\& a_{21} :=-\lambda \nu _{2}e^{-\lambda \eta }-\lambda \mu _{2}e^{- \lambda T}, \qquad a_{22}:=\nu _{2}e^{-\lambda \eta }+\mu _{2}e^{-\lambda T}, \\& \Delta :=a_{11}a_{22}-a_{12}a_{21}, \qquad \Delta \neq 0, \\& \varphi _{1} ( t ) = \biggl( \frac{a_{12}}{\Delta }e^{- \lambda t}- \frac{a_{11}}{\Delta }\frac{1}{\lambda } \bigl( 1-e^{- \lambda t} \bigr) \biggr) , \\& \varphi _{2} ( t ) = \biggl( \frac{a_{21}}{\Delta }\frac{1}{ \lambda } \bigl( 1-e^{-\lambda t} \bigr) -\frac{a_{22}}{\Delta }e ^{-\lambda t} \biggr) . \end{aligned}$$

Lemma 2

Let $\rho ,\gamma _{1},\gamma _{2}\in C ( [ 0,T ] ; \mathbb{R} )$. Then the following boundary value problem

$$ \textstyle\begin{cases} ( D^{\alpha }+\lambda D^{\alpha -1} ) z ( t ) =\rho ( t ) , \quad 1< \alpha \leq 2, 0\leq t\leq T,\\ \nu _{1}z ( \eta ) +\mu _{1}z ( T ) =\int _{0}^{T}\gamma _{1} ( s ) \,ds, \\ \nu _{2} z^{\prime } ( \eta ) +\mu _{2} z^{\prime } ( T ) =\int _{0}^{T}\gamma _{2} ( s ) \,ds, \end{cases} $$

(2.2)

is equivalent to the fractional integral equation

$$\begin{aligned} z ( t ) & = \int _{0}^{t} ( t-s ) ^{\alpha -1}E _{1,\alpha } ( -\lambda ;t-s ) \rho ( s ) \,ds \\ & \quad{}+ \sum_{j=1}^{2} \nu _{j}\varphi _{j} ( t ) \int _{0}^{\eta } ( \eta -s ) ^{\alpha -j}E_{1,\alpha +j-1} ( -\lambda ;\eta -s ) \rho ( s ) \,ds \\ & \quad{}+ \sum_{j=1}^{2} \mu _{j}\varphi _{j} ( t ) \int _{0}^{T} ( T-s ) ^{\alpha -j}E_{1,\alpha +j-1} ( -\lambda ;T-s ) \rho ( s ) \,ds \\ & \quad{}- \sum_{j=1}^{2} \varphi _{i} ( t ) \int _{0}^{T}\gamma _{i} ( s ) \,ds. \end{aligned}$$

(2.3)

Proof

Applying $I^{\alpha -1}$ to both sides of (2.2) and using (2.1), we get

$$\begin{aligned} &I^{\alpha -1} D^{\alpha -1} ( D+\lambda ) z ( t ) =I^{\alpha -1}\rho ( t ) , \\ &( D+\lambda ) z ( t ) =c_{0}+I^{\alpha -1} \rho ( t ) . \end{aligned}$$

We solve the above linear ordinary differential equation:

$$\begin{aligned} z ( t ) & =c_{1}e^{-\lambda t}+c_{0}\frac{1}{\lambda }-c _{0}\frac{1}{\lambda }e^{-\lambda t}+ \int _{0}^{t}e^{-\lambda ( t-s ) }I^{\alpha -1}\rho ( s ) \,ds \\ & =c_{1}e^{-\lambda t}+c_{0}\frac{1}{\lambda } \bigl( 1-e^{-\lambda t} \bigr) + \int _{0}^{t} ( t-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;t-r ) \rho ( r ) \,dr. \end{aligned}$$

(2.4)

It is clear that

$$ z^{\prime } ( t ) =-\lambda c_{1}e^{-\lambda t}+c_{0}e^{- \lambda t}+ \int _{0}^{t} ( t-r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ;t-r ) \rho ( r ) \,dr. $$

(2.5)

The first boundary condition implies that

$$\begin{aligned} & \nu _{1}z ( \eta ) +\mu _{1}z ( T ) \\ &\quad =\nu _{1}c _{1}e^{-\lambda \eta }+\nu _{1}c_{0} \frac{1}{\lambda } \bigl( 1-e^{- \lambda \eta } \bigr) +\nu _{1} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;\eta -r ) \rho ( r ) \,dr \\ & \qquad{}+\mu _{1}c_{1}e^{-\lambda T}+\mu _{1}c_{0}\frac{1}{\lambda } \bigl( 1-e ^{-\lambda T} \bigr) +\mu _{1} \int _{0}^{T} ( T-r ) ^{ \alpha -1}E_{1,\alpha } ( -\lambda ;T-r ) \rho ( r ) \,dr \\ & \quad = \int _{0}^{T}\gamma _{1} ( s ) \,ds. \end{aligned}$$

It follows that

$$\begin{aligned} & \bigl( \nu _{1}e^{-\lambda \eta }+\mu _{1}e^{-\lambda T} \bigr) c _{1}+ \biggl( \nu _{1}\frac{1}{\lambda } \bigl( 1-e^{-\lambda \eta } \bigr) +\mu _{1}\frac{1}{\lambda } \bigl( 1-e ^{-\lambda T} \bigr) \biggr) c_{0} \\ &\quad = \int _{0}^{T}\gamma _{1} ( s ) \,ds-\nu _{1} \int _{0}^{ \eta } ( \eta -r ) ^{\alpha -1}E_{1,\alpha } ( - \lambda ;\eta -r ) \rho ( r ) \,dr \\ & \qquad{}-\mu _{1} \int _{0}^{T} ( T-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;T-r ) \rho ( r ) \,dr. \end{aligned}$$

The second boundary condition with (2.5) implies that

$$\begin{aligned}& \nu _{2} z^{\prime } ( \eta ) +\mu _{2} z^{\prime } ( T ) \\& \quad =\nu _{2} \bigl( -\lambda c_{1}e^{-\lambda \eta }+c_{0}e^{- \lambda \eta } \bigr) +\nu _{2} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ;\eta -r ) \rho ( r ) \,dr \\& \qquad {}+\mu _{2} \bigl( -\lambda c_{1}e^{-\lambda T}+c_{0}e^{-\lambda T} \bigr) +\mu _{2} \int _{0}^{T} ( T-r ) ^{\alpha -2}E_{1,\alpha -1} ( - \lambda ;T-r ) \rho ( r ) \,dr \\& \quad = \int _{0}^{T}\gamma _{2} ( s ) \,ds. \end{aligned}$$

Thus

$$\begin{aligned}& \begin{aligned} a_{11}c_{1}+a_{12}c_{0} & = \int _{0}^{T}\gamma _{1} ( s ) \,ds- \nu _{1} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -1}E_{1, \alpha } ( -\lambda ;\eta -r ) \rho ( r ) \,dr \\ & \quad{}-\mu _{1} \int _{0}^{T} ( T-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;T-r ) \rho ( r ) \,dr, \end{aligned} \\& \begin{aligned} a_{21}c_{1}+a_{22}c_{0} & = \int _{0}^{T}\gamma _{2} ( s ) \,ds- \nu _{2} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -2}E_{1, \alpha -1} ( -\lambda ;\eta -r ) \rho ( r ) \,dr \\ & \quad{}-\mu _{2} \int _{0}^{T} ( T-r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ;T-r ) \rho ( r ) \,dr. \end{aligned} \end{aligned}$$

Solving the above system of equations for $c_{0}$ and $c_{1}$, we get

$$\begin{aligned}& \begin{aligned} c_{0} & =\frac{a_{11}}{\Delta } \biggl( \int _{0}^{T}\gamma _{2} ( s ) \,ds-\nu _{2} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ; \eta -r ) \rho ( r ) \,dr \\ &\quad {}-\mu _{2} \int _{0}^{T} ( T-r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ;T-r ) \rho ( r ) \,dr \biggr) \\ &\quad {} -\frac{a_{21}}{\Delta } \biggl( \int _{0}^{T}\gamma _{1} ( s ) \,ds-\nu _{1} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ; \eta -r ) \rho ( r ) \,dr \\ &\quad {}-\mu _{1} \int _{0}^{T} ( T-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;T-r ) \rho ( r ) \,dr \biggr), \end{aligned} \\& \begin{aligned} c_{1} & =\frac{a_{22}}{\Delta } \biggl( \int _{0}^{T}\gamma _{1} ( s ) \,ds-\nu _{1} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ; \eta -r ) \rho ( r ) \,dr \\ &\quad {}-\mu _{1} \int _{0}^{T} ( T-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;T-r ) \rho ( r ) \,dr \biggr) \\ &\quad {} -\frac{a_{12}}{\Delta } \biggl( \int _{0}^{T}\gamma _{2} ( s ) \,ds-\nu _{2} \int _{0}^{\eta } ( \eta -r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ; \eta -r ) \rho ( r ) \,dr \\ &\quad {}-\mu _{2} \int _{0}^{T} ( T-r ) ^{\alpha -2}E_{1,\alpha -1} ( -\lambda ;T-r ) \rho ( r ) \,dr \biggr) . \end{aligned} \end{aligned}$$

Inserting $c_{0}$ and $c_{1}$ in (2.4), we obtain the desired formula (2.3).

Conversely, assume that u satisfies (2.3). By a direct computation, it follows that the solution given by (2.3) satisfies (2.2). □

Lemma 3

For any $g,h\in C ( [ 0,T ] ;\mathbb{R} )$, $\gamma >0$, we have

$$ \biggl\vert \int _{0}^{t} ( t-s ) ^{\gamma -1}E_{1,\gamma } ( -\lambda ;t-s ) \bigl( g ( s ) -h ( s ) \bigr) \,ds \biggr\vert \leq t^{\gamma }E_{1,\gamma +1} ( -\lambda ;t ) \Vert g-h \Vert _{\infty }. $$

Proof

Indeed,

$$\begin{aligned} & \biggl\vert \int _{0}^{t} ( t-s ) ^{\gamma -1}E_{1,\gamma } ( -\lambda ;t-s ) \bigl( g ( s ) -h ( s ) \bigr) \,ds \biggr\vert \\ &\quad \leq \sum_{k=0}^{\infty } \frac{\lambda ^{k}}{\varGamma ( k+ \gamma ) } \int _{0}^{t} ( t-s ) ^{k+\gamma -1} \bigl\vert g ( s ) -h ( s ) \bigr\vert \,ds \\ &\quad \leq \sum_{k=0}^{\infty } \frac{\lambda ^{k}t^{k+\gamma }}{\varGamma ( k+\gamma +1 ) }=t^{\gamma }E_{1,\gamma +1} \bigl( \vert \lambda \vert ;t \bigr) \Vert g-h \Vert _{\infty }. \end{aligned}$$

 □

3 Main results

By Lemma 2, we introduce a fixed point problem associated with the problem as follows:

$$ \begin{aligned} ( x,y ) &= ( T_{1},T_{2} ) ( x,y ) = \mathfrak{T} ( x,y ) :C \bigl( [ 0,T ] ; \mathbb{R} \bigr) \times C \bigl( [ 0,T ] ;\mathbb{R} \bigr) \\ &\rightarrow C \bigl( [ 0,T ] ; \mathbb{R} \bigr) \times C \bigl( [ 0,T ] ; \mathbb{R} \bigr) , \end{aligned} $$

where

$$\begin{aligned}& \begin{aligned}[b] T_{1} ( x,y ) ( t ) & = \int _{0}^{t} ( t-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;t-r ) f _{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\quad {} + \sum_{j=1}^{2} \nu _{j}\varphi _{j} ( t ) \int _{0}^{\eta } ( \eta -r ) ^{\alpha -j}E_{1,\alpha +j-1} ( -\lambda ;\eta -r ) f_{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\quad {} + \sum_{j=1}^{2} \mu _{j}\varphi _{j} ( t ) \int _{0}^{T} ( T-r ) ^{\alpha -j}E_{1,\alpha +j-1} ( -\lambda ;T-r ) f_{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\quad {} + \sum_{j=1}^{2} \varphi _{i} ( t ) \int _{0}^{T}h_{i} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr, \end{aligned} \end{aligned}$$

(3.1)

$$\begin{aligned}& \begin{aligned}[b] T_{2} ( x,y ) ( t ) & = \int _{0}^{t} ( t-r ) ^{\beta -1}E_{1,\beta } ( -\lambda ;t-r ) f _{2} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\quad {} + \sum_{j=1}^{2} \nu _{j}\varphi _{j} ( t ) \int _{0}^{\eta } ( \eta -r ) ^{\beta -j}E_{1,\beta +j-1} ( -\lambda ; \eta -r ) f_{2} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\quad {} + \sum_{j=1}^{2} \mu _{j}\varphi _{j} ( t ) \int _{0}^{T} ( T-r ) ^{\beta -j}E_{1,\beta +j-1} ( -\lambda ;T-r ) f _{2} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\quad {} + \sum_{j=1}^{2} \varphi _{i} ( t ) \int _{0}^{T}g_{i} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr. \end{aligned} \end{aligned}$$

(3.2)

Evidently, the existence of fixed points of the operator $\mathfrak{T}$ is equivalent to the existence of solutions for problem (1.1).

For $\gamma =\alpha ,\beta$, let

$$ \begin{aligned} R^{\gamma }&:=\max \Biggl\{ \Biggl( T^{\gamma }E_{1,\gamma +1} \bigl( \vert \lambda \vert ;T \bigr) + \sum _{j=1}^{2} \vert \nu _{j} \vert \Vert \varphi _{j} \Vert \eta ^{\gamma -j}E_{1,\gamma +j-1} \bigl( \vert \lambda \vert ; \eta \bigr) \\ &\quad {} + \sum_{j=1}^{2} \vert \mu _{j} \vert \Vert \varphi _{j} \Vert T^{\gamma -j}E_{1,\gamma +j-1} \bigl( \vert \lambda \vert ;T \bigr) \Biggr) , \bigl( \Vert \varphi _{1} \Vert + \Vert \varphi _{2} \Vert \bigr) T \Biggr\} . \end{aligned} $$

Here, we prove the existence and uniqueness of solutions for problem (1.1). We apply a fixed point theorem due to Banach.

Theorem 1

Let $f_{i},h_{i},g_{i}: [ 0,T ] \times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ be continuous functions such that the following conditions hold:

($\mathrm{A}_{1}$):

There exist $L_{f},L_{h},L_{g}>0$ such that

$$\begin{aligned} & \bigl\vert f_{i} ( t,x_{1},y_{1} ) -f_{i} ( t,x_{2},y _{2} ) \bigr\vert \leq L_{f} \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert \bigr) , \\ & \bigl\vert h_{i} ( t,x_{1},y_{1} ) -h_{i} ( t,x_{2},y _{2} ) \bigr\vert \leq L_{h} \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert \bigr) , \\ & \bigl\vert g_{i} ( t,x_{1},y_{1} ) -g_{i} ( t,x_{2},y _{2} ) \bigr\vert \leq L_{g} \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert \bigr) , \\ &\forall ( t,x_{1},y_{1} ) , ( t,x_{2},y_{2} ) \in [ 0,T ] \times \mathbb{R}\times \mathbb{R}. \end{aligned}$$

($\mathrm{A}_{2}$):

$1-2 ( L_{f}+L_{h} ) R^{\alpha }>0$, $1-2 ( L_{f}+L_{g} ) R^{\beta }>0$.

Then problem (1.1) has a unique solution in $C ( [ 0,T ] ,\mathbb{R} ) \times C ( [ 0,T ] , \mathbb{R} )$.

Proof

Consider a ball

$$ B_{r}:= \bigl\{ u\in C \bigl( [ 0,T ] ,\mathbb{R} \bigr) \times C \bigl( [ 0,T ] ,\mathbb{R} \bigr) : \Vert u \Vert _{\infty }\leq r \bigr\} $$

with radius

$$ r\geq \max \biggl\{ \frac{ ( M_{f}+M_{h} ) R^{\alpha }}{1-2 ( L_{f}+L_{h} ) R^{\alpha }},\frac{ ( M_{f}+M_{g} ) R^{\beta }}{1-2 ( L_{f}+L_{g} ) R^{ \beta }} \biggr\} , $$

where

$$\begin{aligned} &M_{f} :=\max \bigl( \bigl\Vert f_{1} ( t,0,0 ) \bigr\Vert _{\infty }, \bigl\Vert f_{2} ( t,0,0 ) \bigr\Vert _{ \infty } \bigr) , \\ &M_{h} :=\max \bigl( \bigl\Vert h_{1} ( t,0,0 ) \bigr\Vert _{\infty }, \bigl\Vert h_{2} ( t,0,0 ) \bigr\Vert _{ \infty } \bigr) , \\ &M_{g} :=\max \bigl( \bigl\Vert g_{1} ( t,0,0 ) \bigr\Vert _{\infty }, \bigl\Vert g_{2} ( t,0,0 ) \bigr\Vert _{ \infty } \bigr) . \end{aligned}$$

It is clear that for all $x,y\in \mathbb{R}$

$$\begin{aligned} & \bigl\vert f_{i} ( t,x,y ) \bigr\vert \leq L_{f} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{f}, \\ & \bigl\vert h_{i} ( t,x,y ) \bigr\vert \leq L_{h} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{h}, \\ & \bigl\vert g_{i} ( t,x,y ) \bigr\vert \leq L_{g} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{g}. \end{aligned}$$

Using this inequality and Lemma 3, from (3.1) it follows that

$$\begin{aligned} & \bigl\vert T_{1} ( x,y ) ( t ) \bigr\vert \\ &\quad \leq t^{\alpha }E_{1,\alpha +1} \bigl( \vert \lambda \vert ;t \bigr) \bigl\Vert f_{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert + \sum_{j=1}^{2} \vert \nu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert \eta ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;\eta \bigr) \bigl\Vert f _{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert \\ & \qquad{}+ \sum_{j=1}^{2} \vert \mu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert T^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;T \bigr) \bigl\Vert f_{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert _{\infty } \\ &\qquad {}+ \sum_{j=1}^{2} \bigl\vert \varphi _{i} ( t ) \bigr\vert \int _{0}^{T} \bigl\vert h_{i} \bigl( r,x ( r ) ,y ( r ) \bigr) \bigr\vert \,dr \\ &\quad \leq \Biggl( t^{\alpha }E_{1,\alpha +1} ( -\lambda ;t ) + \sum _{j=1}^{2} \vert \nu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert \eta ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;\eta \bigr) \\ &\qquad {} + \sum_{j=1}^{2} \vert \mu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert T^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;T \bigr) \Biggr) \\ &\qquad{} \times \bigl( L_{f} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{f} \bigr) + \bigl( \Vert \varphi _{1} \Vert + \Vert \varphi _{2} \Vert \bigr) T \bigl( L_{h} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{h} \bigr) \\ &\quad \leq \bigl( ( L_{f}+L_{h} ) r+M_{f}+M_{h} \bigr) R ^{\alpha }\leq \frac{r}{2}. \end{aligned}$$

(3.3)

In the like manner we have

$$ \bigl\vert T_{2} ( x,y ) ( t ) \bigr\vert \leq \bigl( ( L_{f}+L_{h} ) r+M_{f}+M_{h} \bigr) R^{ \beta }\leq \frac{r}{2}. $$

(3.4)

From (3.3) and (3.4) it follows that $\mathfrak{T}B_{r} \subset B_{r}$. Next, using condition ($\mathrm{A}_{1}$), we obtain

$$\begin{aligned} \bigl\vert T_{1} ( x_{1},y_{1} ) ( t ) -T_{1} ( x_{2},y_{2} ) ( t ) \bigr\vert & \leq R ^{\alpha } \bigl\Vert f_{1} \bigl( \cdot ,x_{1} ( \cdot ) ,y_{1} ( \cdot ) \bigr) -f_{1} \bigl( \cdot ,x_{2} ( \cdot ) ,y_{2} ( \cdot ) \bigr) \bigr\Vert _{\infty } \\ & \leq ( L_{f}+L_{h} ) R^{\alpha } \bigl\Vert ( x _{1},y_{1} ) - ( x_{2},y_{2} ) \bigr\Vert _{\infty }. \end{aligned}$$

(3.5)

Similarly,

$$\begin{aligned} \bigl\vert T_{2} ( x_{1},y_{1} ) ( t ) -T_{2} ( x_{2},y_{2} ) ( t ) \bigr\vert & \leq R ^{\beta } \bigl\Vert f_{2} \bigl( \cdot ,x_{1} ( \cdot ) ,y _{1} ( \cdot ) \bigr) -f_{2} \bigl( \cdot ,x_{2} ( \cdot ) ,y_{2} ( \cdot ) \bigr) \bigr\Vert _{\infty } \\ & \leq ( L_{f}+L_{g} ) R^{\beta } \bigl\Vert ( x _{1},y_{1} ) - ( x_{2},y_{2} ) \bigr\Vert _{\infty }. \end{aligned}$$

(3.6)

It follows from (3.5) and (3.6) that

$$ \bigl\Vert T ( x_{1},y_{1} ) -T ( x_{2},y_{2} ) \bigr\Vert \leq \bigl[ ( L_{f}+L_{h} ) R^{\alpha }+ ( L_{f}+L_{g} ) R^{\beta } \bigr] \bigl\Vert ( x _{1},y_{1} ) - ( x_{2},y_{2} ) \bigr\Vert _{\infty }. $$

By ($\mathrm{A}_{2}$) the operator $\mathfrak{T}$ is a contraction. Thus by the Banach fixed point theorem, $\mathfrak{T}$ has a unique fixed point in $C ( [ 0,T ] ,\mathbb{R} ) \times C ( [ 0,T ] ,\mathbb{R} )$. This completes the proof. □

In the next result, we prove the existence of solutions for problem (1.1) by applying the Leray–Schauder alternative.

Theorem 2

Let $f. [ 0,T ] \times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that the following condition holds:

($\mathrm{A}_{3}$):

There exist $\gamma _{f},\gamma _{h},\gamma _{g}\in C ( [ 0,T ] ,\mathbb{R}_{+} )$ and a nondecreasing function $\psi :\mathbb{R}_{+}\rightarrow \mathbb{R} _{+}$ such that

$$\begin{aligned} & \bigl\vert f_{i} ( t,x,y ) \bigr\vert \leq \gamma _{f} ( t ) \psi _{f} \bigl( \vert x \vert + \vert y \vert \bigr) , \quad \forall ( t,x,y ) \in [ 0,T ] \times \mathbb{R}\times \mathbb{R}. \\ & \bigl\vert h_{i} ( t,x,y ) \bigr\vert \leq \gamma _{h} ( t ) \psi _{h} \bigl( \vert x \vert + \vert y \vert \bigr) , \\ & \bigl\vert g_{i} ( t,x,y ) \bigr\vert \leq \gamma _{g} ( t ) \psi _{g} \bigl( \vert x \vert + \vert y \vert \bigr) , \quad i=1,2. \end{aligned}$$

($\mathrm{A}_{4}$):

There exists $M>0$ such that

$$ \frac{M}{ ( \Vert \gamma _{f} \Vert _{\infty } \psi _{f} ( M ) + \Vert \gamma _{h} \Vert _{\infty } \psi _{h} ( M ) ) R^{\alpha }+ ( \Vert \gamma _{f} \Vert _{\infty } \psi _{f} ( M ) + \Vert \gamma _{g} \Vert _{\infty } \psi _{g} ( M ) ) R^{ \beta }}>1. $$

Then BVP (1.1) has at least one solution.

Proof

Step 1: Show that $\mathfrak{T}:C ( [ 0,T ] , \mathbb{R} ) \times C ( [ 0,T ] ,\mathbb{R} ) \rightarrow C ( [ 0,T ] , \mathbb{R} ) \times C ( [ 0,T ] ,\mathbb{R} )$ maps bounded sets into bounded sets and is continuous.

Let $B_{r}$ be a ball in $C ( [ 0,T ] ,\mathbb{R} ) \times C ( [ 0,T ] ,\mathbb{R} )$. Then

$$ \bigl\vert f_{1} \bigl( t,x ( t ) ,y ( t ) \bigr) \bigr\vert \leq \Vert \gamma _{f} \Vert \psi _{f} ( r ) , \qquad \bigl\vert f_{2} \bigl( t,x ( t ) ,y ( t ) \bigr) \bigr\vert \leq \Vert \gamma _{f} \Vert \psi _{f} ( r ) , $$

and by Lemma 3

$$\begin{aligned} & \bigl\vert T_{1} ( x,y ) ( t ) \bigr\vert \\ &\quad \leq t^{\alpha }E_{1,\alpha +1} \bigl( \vert \lambda \vert ;t \bigr) \bigl\Vert f_{1} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert + \sum_{j=1}^{2} \vert \nu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert \eta ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;\eta \bigr) \bigl\Vert f _{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert \\ & \qquad{}+ \sum_{j=1}^{2} \vert \mu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert T^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;T \bigr) \bigl\Vert f_{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert _{\infty } \\ &\qquad {}+ \sum_{j=1}^{2} \bigl\vert \varphi _{i} ( t ) \bigr\vert \int _{0}^{T} \bigl\vert h_{i} \bigl( r,x ( r ) ,y ( r ) \bigr) \bigr\vert \,dr \\ & \quad \leq \Biggl( t^{\alpha }E_{1,\alpha +1} \bigl( \vert \lambda \vert ;t \bigr) + \sum_{j=1}^{2} \vert \nu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert \eta ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;\eta \bigr) \\ &\qquad {} + \sum_{j=1}^{2} \vert \mu _{j} \vert \bigl\vert \varphi _{j} ( t ) \bigr\vert T^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;T \bigr) \Biggr) \\ &\qquad{} \times \bigl( \gamma _{f} ( t ) \psi _{f} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{f} \bigr) + \bigl( \Vert \varphi _{1} \Vert + \Vert \varphi _{2} \Vert \bigr) T \bigl( L_{h} \bigl( \vert x \vert + \vert y \vert \bigr) +M_{h} \bigr) \\ & \quad \leq \bigl( \Vert \gamma _{f} \Vert \psi _{f} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) + \Vert \gamma _{h} \Vert \psi _{h} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) \bigr) R^{\alpha }. \end{aligned}$$

Similarly,

$$ \bigl\vert T_{2} ( x,y ) ( t ) \bigr\vert \leq \bigl( \Vert \gamma _{f} \Vert \psi _{f} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) + \Vert \gamma _{g} \Vert \psi _{g} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) \bigr) R^{\beta }. $$

It follows that $\mathfrak{T} ( B_{r} )$ is bounded.

Step 2: Next we show that $\mathfrak{T}$ maps bounded sets into equicontinuous sets of $C ( [ 0,T ] ,\mathbb{R} )$.

Let $t_{1},t_{2}\in [ 0,T ]$ with $t_{1}< t_{2}$ and $( x,y ) \in B_{r}$. Then we obtain

$$\begin{aligned} & \bigl\vert T_{1} ( x,y ) ( t_{1} ) -T_{1} ( x,y ) ( t_{2} ) \bigr\vert \\ &\quad \leq \biggl\vert \int _{0}^{t_{1}} ( t_{1}-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;t_{1}-r ) f_{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ & \qquad {} - \int _{0}^{t_{2}} ( t_{2}-r ) ^{\alpha -1}E_{1, \alpha } ( -\lambda ;t_{2}-r ) f_{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \biggr\vert \\ & \qquad{}+ \sum_{j=1}^{2} \vert \nu _{j} \vert \bigl\vert \varphi _{j} ( t_{1} ) -\varphi _{j} ( t_{2} ) \bigr\vert \eta ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ; \eta \bigr) \bigl\Vert f_{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert \\ & \qquad{}+ \sum_{j=1}^{2} \vert \mu _{j} \vert \bigl\vert \varphi _{j} ( t_{1} ) -\varphi _{j} ( t_{2} ) \bigr\vert T ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;T \bigr) \bigl\Vert f_{i} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr\Vert _{\infty } \\ & \qquad{}+ \sum_{j=1}^{2} \bigl\vert \varphi _{j} ( t_{1} ) -\varphi _{j} ( t_{2} ) \bigr\vert \int _{0}^{T} \bigl\vert h_{i} \bigl( r,x ( r ) ,y ( r ) \bigr) \bigr\vert \,dr. \end{aligned}$$

(3.7)

It is clear that the last three terms approach to zero independently of $( x,y ) \in B_{r}$ as $t_{1}\rightarrow t_{2}$. Now, we estimate the first term of (3.7):

$$\begin{aligned} & \biggl\vert \int _{0}^{t_{1}} ( t_{1}-r ) ^{\alpha -1}E _{1,\alpha } ( -\lambda ;t_{1}-r ) f_{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \\ &\qquad {}- \int _{0}^{t_{2}} ( t _{2}-r ) ^{\alpha -1}E_{1,\alpha } ( -\lambda ;t_{2}-r ) f_{1} \bigl( r,x ( r ) ,y ( r ) \bigr) \,dr \biggr\vert \\ &\quad = \biggl\vert \int _{0}^{t_{1}}e^{-\lambda ( t_{1}-s ) } \bigl( I^{\alpha -1}f_{1} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr) ( s ) \,ds- \int _{0}^{t_{2}}e^{-\lambda ( t_{2}-s ) } \bigl( I^{\alpha -1}f _{1} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr) ( s ) \,ds \biggr\vert \\ &\quad \leq \biggl\vert \int _{t_{1}}^{t_{2}}e^{-\lambda ( t_{1}-s ) } \bigl( I^{\alpha -1}f_{1} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr) ( s ) \,ds \biggr\vert \\ & \qquad{}+ \biggl\vert \int _{0}^{t_{1}} \bigl[ e^{-\lambda ( t_{2}-s ) }-e^{-\lambda ( t_{1}-s ) } \bigr] \bigl( I^{\alpha -1}f_{1} \bigl( \cdot ,x ( \cdot ) ,y ( \cdot ) \bigr) \bigr) ( s ) \,ds \biggr\vert . \end{aligned}$$

Obviously, the right-hand side of the above inequality tends to zero independently of $( x,y ) \in B_{r}$ as $t_{1}\rightarrow t_{2}$. A similar result is true for $T_{1} ( x,y )$. As $\mathfrak{T}$ is uniformly bounded and equicontinuous, therefore it follows by the Arzelá–Ascoli theorem that $\mathfrak{T}:C ( [ 0,T ] ,\mathbb{R} ) \times C ( [ 0,T ] ,\mathbb{R} ) \rightarrow C ( [ 0,T ] ,\mathbb{R} ) \times C ( [ 0,T ] ,\mathbb{R} )$ is completely continuous.

The result will follow from the Leray–Schauder nonlinear alternative once we have proved the boundedness of the set of all solutions to equations $( x,y ) =\theta \mathfrak{T} ( x,y )$ for $0\leq \theta \leq 1$.

Let $( x,y )$ be a solution. Then, using the computations employed in proving that $\mathfrak{T}$ is bounded, we have

$$\begin{aligned} \bigl\vert ( x,y ) ( t ) \bigr\vert & = \theta \bigl\vert \mathfrak{T} ( x,y ) ( t ) \bigr\vert \\ &\leq \bigl( \Vert \gamma _{f} \Vert \psi _{f} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) + \Vert \gamma _{h} \Vert \psi _{h} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) \bigr) R^{\alpha } \\ & \quad{}+ \bigl( \Vert \gamma _{f} \Vert \psi _{f} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) + \Vert \gamma _{g} \Vert \psi _{g} \bigl( \bigl\Vert ( x,y ) \bigr\Vert \bigr) \bigr) R^{\beta }. \end{aligned}$$

Consequently, we have

$$ \begin{aligned} &\bigl\Vert ( x,y ) \bigr\Vert _{\infty } \\ &\quad {}/\bigl( \bigl( \Vert \gamma _{f} \Vert _{\infty } \psi _{f} \bigl( \bigl\Vert ( x,y ) \bigr\Vert _{\infty } \bigr) + \Vert \gamma _{h} \Vert _{\infty }\psi _{h} \bigl( \bigl\Vert ( x,y ) \bigr\Vert _{\infty } \bigr) \bigr) R^{\alpha } \\ &\quad {}+ \bigl( \Vert \gamma _{f} \Vert _{\infty }\psi _{f} \bigl( \bigl\Vert ( x,y ) \bigr\Vert _{\infty } \bigr) + \Vert \gamma _{g} \Vert _{\infty }\psi _{g} \bigl( \bigl\Vert ( x,y ) \bigr\Vert _{\infty } \bigr) \bigr) R^{\beta } \bigr)\leq 1. \end{aligned} $$

In view of ($\mathrm{A}_{4}$), there exists M such that $\Vert ( x,y ) \Vert _{\infty }\neq M$. Let us set

$$ \mathfrak{U}= \bigl\{ ( x,y ) \in C \bigl( [ 0,T ] ,\mathbb{R} \bigr) \times C \bigl( [ 0,T ] , \mathbb{R} \bigr) : \bigl\Vert ( x,y ) \bigr\Vert _{ \infty }< M\text{ } \bigr\} . $$

Note that the operator $\mathfrak{T}:\overline{\mathfrak{U}}\rightarrow C ( [ 0,T ] ,\mathbb{R} )$ is continuous and completely continuous. From the choice of $\mathfrak{U}$, there is no $( x,y ) \in \partial \mathfrak{U}$ such that $( x,y ) =\theta \mathfrak{T} ( x,y )$ for some $0<\theta <1$. Consequently, by the nonlinear alternative of Leray–Schauder type, we deduce that $\mathfrak{T}$ has a fixed point $( x,y ) \in \overline{\mathfrak{U}}$ which is a solution of problem (1.1). This completes the proof. □

4 Ulam–Hyers stability

In this section, we discuss the Ulam–Hyers stability for problem (1.1) by means of integral representation of its solution given by

$$ x ( t ) =T_{1} ( x,y ) ( t ) , \qquad y ( t ) =T_{2} ( x,y ) ( t ) , $$

where $T_{1}$ and $T_{2}$ are defined by (3.1) and (3.2).

Define the following nonlinear operators $Q_{1},Q_{2}:C([0,T], \mathbb{R})\times C([0,T],\mathbb{R})\rightarrow [4] C([0,T],\mathbb{R})$:

$$\begin{aligned} &Q_{1} ( x,y ) ( t ) := \bigl( D^{\alpha }+ \lambda D^{\alpha -1} \bigr) x ( t ) -f_{1} \bigl( t,x ( t ) ,y ( t ) \bigr) , \\ &Q_{2} ( x,y ) ( t ) := \bigl( D^{\beta }+ \lambda D^{\beta -1} \bigr) y ( t ) -f_{2} \bigl( t,x ( t ) ,y ( t ) \bigr). \end{aligned}$$

For some $\varepsilon _{1},\varepsilon _{2}>0$, we consider the following inequality:

$$ \bigl\Vert Q_{1} ( x,y ) \bigr\Vert \leq \varepsilon _{1},\qquad \bigl\Vert Q_{2} ( x,y ) \bigr\Vert \leq \varepsilon _{2}. $$

(4.1)

Definition 3

The coupled system (1.1) is said to be Ulam–Hyers stable if there exist $V_{1},V_{2}>0$ such that, for every solution $( x^{ \ast },y^{\ast } ) \in C([0,T],\mathbb{R})\times C([0,T], \mathbb{R})$ of inequality (4.1), there exists a unique solution $( x,y ) \in C([0,T],\mathbb{R})\times C([0,T],\mathbb{R})$ of problem (1.1) with

$$ \bigl\Vert ( x,y ) - \bigl( x^{\ast },y^{\ast } \bigr) \bigr\Vert _{\infty }\leq V_{1}\varepsilon _{1}+V_{2} \varepsilon _{2}. $$

(4.2)

Theorem 3

Let the assumptions of Theorem 1 hold. Then problem (1.1) is Ulam–Hyers stable.

Proof

Let $( x,y ) \in C([0,T],\mathbb{R})\times C([0,T], \mathbb{R})$ be the solution of problem (1.1) satisfying (3.1) and (3.2). Let $( x^{\ast },y^{\ast } )$ be any solution satisfying (4.1):

$$\begin{aligned} \bigl( D^{\alpha }+\lambda D^{\alpha -1} \bigr) x^{\ast } ( t ) & =f_{1} \bigl( t,x^{\ast } ( t ) ,y^{\ast } ( t ) \bigr) +Q_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) , \\ \bigl( D^{\beta }+\lambda D^{\beta -1} \bigr) y^{\ast } ( t ) & =f_{2} \bigl( t,x^{\ast } ( t ) ,y^{\ast } ( t ) \bigr) +Q_{2} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) . \end{aligned}$$

So

$$\begin{aligned} x^{\ast } ( t ) & =T_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) + \int _{0}^{t} ( t-r ) ^{\alpha -1}E_{1, \alpha } ( -\lambda ;t-r ) Q_{1} \bigl( x^{\ast },y^{ \ast } \bigr) ( r ) \,dr \\ & \quad{}+ \sum_{j=1}^{2} \nu _{j}\varphi _{j} ( t ) \int _{0}^{\eta } ( \eta -r ) ^{\alpha -j}E_{1,\alpha +j-1} ( -\lambda ;\eta -r ) Q_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( r ) \,dr \\ & \quad{}+ \sum_{j=1}^{2} \mu _{j}\varphi _{j} ( t ) \int _{0}^{T} ( T-r ) ^{\alpha -j}E_{1,\alpha +j-1} ( -\lambda ;T-r ) Q_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( r ) \,dr. \end{aligned}$$

It follows that

$$\begin{aligned} & \bigl\vert T_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) -x^{\ast } ( t ) \bigr\vert \\ &\quad \leq \int _{0}^{t} ( t-r ) ^{\alpha -1}E_{1,\alpha } \bigl( \vert \lambda \vert ;t-r \bigr) \,dr\varepsilon _{1}+ \sum _{j=1}^{2} \vert \nu _{j} \vert \Vert \varphi _{j} \Vert \int _{0}^{\eta } ( \eta -r ) ^{\alpha -j}E _{1,\alpha +j-1} \bigl( \vert \lambda \vert ;\eta -r \bigr) \,dr\varepsilon _{1} \\ & \qquad{}+ \sum_{j=1}^{2} \vert \mu _{j} \vert \Vert \varphi _{j} \Vert \int _{0}^{T} ( T-r ) ^{\alpha -j}E_{1, \alpha +j-1} \bigl( \vert \lambda \vert ;T-r \bigr) \,dr \varepsilon _{1} \\ &\quad \leq \Biggl( T^{\alpha }E_{1,\alpha +1} \bigl( \vert \lambda \vert ;T \bigr) + \sum_{j=1}^{2} \vert \nu _{j} \vert \Vert \varphi _{j} \Vert \eta ^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ; \eta \bigr)\\ &\qquad {} + \sum_{j=1}^{2} \vert \mu _{j} \vert \Vert \varphi _{j} \Vert T^{\alpha -j}E_{1,\alpha +j-1} \bigl( \vert \lambda \vert ;T \bigr) \Biggr) \varepsilon _{1} \\ &\quad =:U^{\alpha }\varepsilon _{1}. \end{aligned}$$

Similarly,

$$\begin{aligned} & \bigl\vert T_{2} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) -y^{\ast } ( t ) \bigr\vert \\ &\quad \leq \Biggl( T^{\beta }E_{1,\beta +1} \bigl( \vert \lambda \vert ;T \bigr) + \sum_{j=1}^{2} \vert \nu _{j} \vert \Vert \varphi _{j} \Vert \eta ^{\beta -j}E_{1,\beta +j-1} \bigl( \vert \lambda \vert ; \eta \bigr)\\ &\qquad {} + \sum_{j=1}^{2} \vert \mu _{j} \vert \Vert \varphi _{j} \Vert T^{\beta -j}E_{1,\beta +j-1} \bigl( \vert \lambda \vert ;T \bigr) \Biggr) \varepsilon _{2} \\ &\quad =:U^{\beta }\varepsilon _{1}. \end{aligned}$$

Therefore, we deduce by the fixed point property of the operator $\mathfrak{T}$, given by (3.1) and (3.2), that

$$\begin{aligned} \bigl\vert x ( t ) -x^{\ast } ( t ) \bigr\vert & = \bigl\vert x ( t ) -T_{1} \bigl( x^{ \ast },y^{\ast } \bigr) ( t ) +T_{1} \bigl( x^{\ast },y ^{\ast } \bigr) ( t ) -x^{\ast } ( t ) \bigr\vert \\ & \leq \bigl\vert T_{1} ( x,y ) ( t ) -T_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) \bigr\vert + \bigl\vert T_{1} \bigl( x^{\ast },y^{\ast } \bigr) ( t ) -x^{\ast } ( t ) \bigr\vert \\ & \leq ( L_{f}+L_{h} ) U^{\alpha } \bigl\Vert ( x,y ) - \bigl( x^{\ast },y^{\ast } \bigr) \bigr\Vert _{ \infty }+U^{\alpha }\varepsilon _{1}, \end{aligned}$$

(4.3)

and similarly

$$\begin{aligned} \bigl\vert y ( t ) -y^{\ast } ( t ) \bigr\vert & = \bigl\vert y ( t ) -T_{2} \bigl( x^{ \ast },y^{\ast } \bigr) ( t ) +T_{2} \bigl( x^{\ast },y ^{\ast } \bigr) ( t ) -y^{\ast } ( t ) \bigr\vert \\ & \leq ( L_{f}+L_{g} ) U^{\beta } \bigl\Vert ( x,y ) - \bigl( x^{\ast },y^{\ast } \bigr) \bigr\Vert _{ \infty }+U^{\beta }\varepsilon _{2}. \end{aligned}$$

(4.4)

From (4.3) and (4.4) it follows that

$$ \bigl\Vert ( x,y ) - \bigl( x^{\ast },y^{\ast } \bigr) \bigr\Vert _{\infty }\leq \bigl( ( L_{f}+L_{h} ) U^{\alpha }+ ( L_{f}+L_{g} ) U^{\beta } \bigr) \bigl\Vert ( x,y ) - \bigl( x^{\ast },y^{\ast } \bigr) \bigr\Vert _{\infty }+U^{ \alpha }\varepsilon _{1}+U^{\beta } \varepsilon _{2}, $$

and

$$\begin{aligned} \bigl\Vert ( x,y ) - \bigl( x^{\ast },y^{\ast } \bigr) \bigr\Vert _{\infty } & \leq \frac{U^{\alpha }\varepsilon _{1}+U^{\beta } \varepsilon _{2}}{1- ( ( L_{f}+L_{h} ) U^{\alpha }+ ( L_{f}+L_{g} ) U^{\beta } ) } \\ & =V_{1}\varepsilon _{1}+V_{2}\varepsilon _{2}, \end{aligned}$$

with

$$\begin{aligned} &V_{1} =\frac{U^{\alpha }}{1- ( ( L_{f}+L_{h} ) U ^{\alpha }+ ( L_{f}+L_{g} ) U^{\beta } ) }, \\ &V_{2} =\frac{U^{\beta }}{1- ( ( L_{f}+L_{h} ) U ^{\alpha }+ ( L_{f}+L_{g} ) U^{\beta } ) }. \end{aligned}$$

Thus, problem (1.1) is Ulam–Hyers stable. □

5 Application

We consider the following fractional order coupled system:

$$ \textstyle\begin{cases} ( D^{\alpha }+\lambda D^{\alpha -1} ) x ( t ) =L_{f}\frac{ \vert y ( t ) \vert }{1+ \vert y ( t ) \vert },\quad 1< \alpha \leq 2, 0\leq t\leq T, \\ ( D^{\beta }+\lambda D^{\beta -1} ) y ( t ) =L _{f} ( \sin x ( t ) + ( \cos t ) x ( t ) ) ,\quad 1< \beta \leq 2, 0\leq t\leq T, \\ \nu _{1}x ( \eta ) +\mu _{1}x ( T ) =L_{h}\int _{0}^{T}\frac{ \vert x ( t ) \vert }{11+ \vert x ( t ) \vert }\,ds, \\ \nu _{1}y ( \eta ) +\mu _{1}y ( T ) =L_{h}\int _{0}^{T} ( \sin y ( t ) +\cos y ( t ) ) \,ds, \\ \nu _{2}x^{\prime } ( \eta ) +\mu _{2}x^{\prime } ( T ) =L_{g}\int _{0}^{T}\frac{ \vert x ( t ) \vert }{21+ \vert x ( t ) \vert }\,ds, \\ \nu _{2}y^{\prime } ( \eta ) +\mu _{2}y^{\prime } ( T ) =L_{g}\int _{0}^{T} ( \sin y ( t ) +\cos y ( t ) ) \,ds. \end{cases} $$

Here

$$\begin{aligned} &f_{1} ( t,x,y ) =L_{f}\frac{ \vert y \vert }{1+ \vert y \vert }, \qquad f_{2} ( t,x,y ) =L_{f} \bigl( \sin x+ ( \cos t ) x \bigr) ,\qquad h_{1} ( x ) =L_{h}\frac{ \vert x \vert }{11+ \vert x \vert }, \\ &h_{2} ( y ) =L_{h} ( \sin y+\cos y ) , \qquad g_{1} ( x ) =L_{g}\frac{ \vert x \vert }{21+ \vert x \vert }, \qquad g_{2} ( y ) =L_{g} ( \sin y+\cos y ). \end{aligned}$$

As

$$\begin{aligned} &\bigl\vert f_{1} ( t,x_{1},y_{1} ) -f_{1} ( t,x_{2},y _{2} ) \bigr\vert \leq L_{f} \vert y_{1}-y_{2} \vert , \qquad \bigl\vert f_{2} ( t,x_{1},y_{1} ) -f_{2} ( t,x_{2},y _{2} ) \bigr\vert \leq L_{f} \vert x_{1}-x_{2} \vert , \\ &\bigl\vert h_{1} ( t,x_{1} ) -h_{1} ( t,x_{2} ) \bigr\vert \leq L_{h} \vert x_{1}-x_{2} \vert , \qquad \bigl\vert h_{2} ( t,y_{1} ) -h_{2} ( t,y_{2} ) \bigr\vert \leq L_{h} \vert y_{1}-y_{2} \vert , \\ &\bigl\vert g_{1} ( t,x_{1} ) -g_{1} ( t,x_{2} ) \bigr\vert \leq L_{g} \vert x_{1}-x_{2} \vert , \qquad \bigl\vert g_{2} ( t,y_{1} ) -g_{2} ( t,y_{2} ) \bigr\vert \leq L_{g} \vert y_{1}-y_{2} \vert , \end{aligned}$$

therefore ($\mathrm{A}_{1}$) is satisfied. It is obvious that $L_{f},L_{h},L _{g}>0$ can be chosen so that condition ($\mathrm{A}_{2}$) is satisfied. Therefore, coupled system (1.1) has a unique solution and is Ulam–Hyers stable.

6 Conclusion

Here we have studied the existence and uniqueness of the solutions as well as the Ulam–Hyers stability for a coupled sequential fractional system with integral boundary conditions. As a future work, one can generalize different concepts of stability and existence results to an impulsive fractional system, a neutral time-delay system/inclusion, and a time-delay system/inclusion with finite delay.