\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 211, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/211\hfil 
Existence of solutions for Riemann-Liouvillle systems]
{Existence of solutions for  Riemann-Liouvillle type coupled systems of
fractional integro-differential equations and  \\ boundary conditions}

\author[A. Alsaedi, S. Aljoudi, B. Ahmad \hfil EJDE-2016/211\hfilneg]
{Ahmed Alsaedi, Shorog Aljoudi, Bashir Ahmad}

\address{Ahmed Alsaedi \newline
Department of Mathematics,
Faculty of Science,
King Abdulaziz University, P.O. Box 80203,
 Jeddah 21589, Saudi Arabia}
\email{aalsaedi@hotmail.com}

\address{Shorog Aljoudi \newline
Department of Mathematics,
Faculty of Science,
King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia}
\email{sh-aljoudi@hotmail.com}

\address{Bashir Ahmad \newline
Department of Mathematics,
Faculty of Science,
King Abdulaziz University, P.O. Box. 80203,
 Jeddah 21589, Saudi Arabia}
\email{bashirahmad\_qau@yahoo.com}

\thanks{Submitted June 3, 2016. Published August 10, 2016.}
\subjclass[2010]{34A08, 34B10, 34B15}
\keywords{Riemann-Liouvillle; fractional derivative; coupled system; 
\hfill\break\indent nonlocal integral conditions; existence of solutions}

\begin{abstract}
 In this article, we study a boundary value problem of
 coupled systems of nonlinear Riemann-Liouvillle fractional
 integro-differential equations supplemented with nonlocal
 Riemann-Liouvillle fractional integro-differential boundary
 conditions. Our results rely on some standard tools of the fixed
 point theory. An illustrative example is also discussed.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

 Fractional calculus is regarded as an important mathematical modelling 
tool for describing dynamical systems involving phenomena such as fractal
and chaos. The subject started with the speculations of Leibniz
(1697) and Euler (1730) about fractional-order derivatives and
developed into an important branch of mathematical analysis with
the passage of time. It deals with differential and integral
operators of arbitrary (non-integer) order. An important and
useful feature characterizing fractional-order differential and
integral operators (in contrast to integer-order operators) is
their nonlocal nature that accounts for the past and hereditary
behavior of materials and processes involved in the real world
problems. In addition to the extensive applications of
fractional-order differential equations in various disciplines of
technical and applied sciences, there has been a great focus on
developing the theoretical aspects, and analytic and numerical
methods for solving fractional order differential equations. For
applications of fractional calculus in engineering and physics, we
refer the reader to the texts \cite{B1, B2, B3}, while some recent
results on fractional differential equations can be found in
\cite{f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13}.


Coupled systems of fractional-order differential equations appear
in the mathematical formulation of several real world phenomena
and processes. Examples of the occurrence of fractional systems
include disease models \cite{S1, S2, S3, S5, S6}, anomalous
diffusion \cite{pI-3, pI-4}, ecological models \cite{S4},
synchronization of chaotic systems \cite{Fai, Ge1, Zha}, nonlocal
thermoelasticity \cite{Povstenko3}, etc. For details concerning
the theoretical development of coupled systems of fractional-order
differential equations supplemented with a variety of boundary
conditions, for instance, see  \cite{s-1, s-2, s-3, s-4, s-5, s-6,
s-7, s-8, s-9}.


In this article, we study the existence of solutions for a
Riemann-Liouville coupled system of nonlinear fractional
integro-differential equations given by
\begin{equation} \label{1}
\begin{gathered}
D^{\alpha}u(t)=f(t,u(t),v(t),(\phi_1 u) (t),(\psi_1 v)(t)),\quad
t\in[0,T], \\
D^{\beta}v(t)=g(t,u(t),v(t),(\phi_2 u) (t),(\psi_2
v)(t)),\quad 1<\alpha,\beta\leq 2.
\end{gathered}
\end{equation}
subject to coupled Riemann-Liouville  integro-differential
boundary conditions:
\begin{equation} \label{2}
\begin{gathered}
D^{\alpha-2} u(0^{+})
=0, \quad D^{\alpha-1} u(0^{+})=\nu I^{\alpha-1}v(\eta),\quad 0<\eta<T, \\
D^{\beta-2} v(0^{+})=0,\quad D^{\beta-1} v(0^{+})=\mu
I^{\beta-1}u(\sigma),\quad 0<\sigma<T,
\end{gathered}
\end{equation}
where $ D^{(\cdot)},I^{(\cdot)}$ denote the Riemann-Liouville derivatives
and integral of fractional order $(\cdot)$, respectively,
$f,g:[0,T]\times\mathbb{ R}^{4}\to \mathbb{R}$ are given
continuous functions,  $\nu, \mu $ are real constants, and
\begin{gather*}
(\phi_1 u)(t)=\int_0^{t}\gamma_1(t,s)u(s)ds,\quad
(\phi_2 u)(t)=\int_0^{t}\gamma_2(t,s)u(s)ds, \\
(\psi_1 v)(t)=\int_0^{t}\delta_1(t,s)v(s)ds,\quad
(\psi_2 v)(t)=\int_0^{t}\delta_2(t,s)v(s)ds,
\end{gather*}
with $\gamma_i$ and $\delta_i$ $(i=1,2)$ being continuous function
on $ [0,T]\times[0,T]$.

The rest of the article is organized as follows. In Section 2, we
recall some preliminary concepts of Riemann-Liouville calculus and
prove an auxiliary lemma. Section 3 contains the existence and
uniqueness results. Though we use the standard methodology
(Leray-Schauder alternative to prove the existence of solutions
and Contraction mapping principle to obtain the uniqueness
result), yet its exposition to the given problem is new. Indeed
our results are new and contribute to the existing literature on
fully Riemann-Liouville type nonlinear nonlocal coupled systems of
fractional integro-differential equations and boundary conditions.

\section{Preliminaries}

This section is devoted to some basic concepts of fractional
calculus concerning Riemann-Liouville derivatives and integrals
\cite{B4}. We also present an auxiliary lemma related to linear
variant of the given problem.

 \begin{definition} \label{def2.1} \rm
The Riemann-Liouville fractional
integral of order $\rho>0$ for a continuous function 
$u:(0,\infty)\to\mathbb{R}$ is defined as
$$
I^{\rho}u(t)=\frac{1}{\Gamma(\rho)}\int_{0}^{t}(t-s)^{\rho-1}u(s)ds,
$$
provided the integral exists.
\end{definition}

 \begin{definition} \label{def2.2} \rm
 For a continuous function $u:(0,\infty)\to\mathbb{R}$, the Riemann-Liouville
derivative of fractional order $\rho,~n=[\rho]+1$ ( $[\rho]
$denotes the integer part of the real number $\rho$) is defined as
$$
D^{\rho}u(t)=\frac{1}{\Gamma(n-\rho)}\big(\frac{d}{dt}\big)^n
\int_{0}^{t}(t-s)^{n-\rho-1}u(s)ds=\big(\frac{d}{dt}\big)^n
I^{n-\rho}u(t),
$$
provided it exists.
\end{definition}

For $\rho<0$, we use the convention that $D^\rho u=I^\rho u$. 
Also, for $\beta \in [0, \rho]$, we have $D^\beta I^\rho u=I^{\rho-\beta}$.
Note that for $\lambda>-1$, $\lambda \neq \rho-1,\rho-2,  \dots,
 \rho-n$, we have
$$
D^\rho
t^\lambda=\frac{\Gamma(\lambda+1)}{\Gamma(\lambda-\rho+1)}t^{\lambda-\rho}
\, \, \mbox{and} \, \, D^\rho t^{\rho-i}=0,~~i=1,2,\dots,n.
$$
In particular, for the constant function $u(t)=1$, we obtain
$$
D^{\rho}1=\frac{1}{\Gamma(1-\rho)}t^{-\rho},\quad 
\rho\notin \mathbb{N}.
$$
For $\rho\in \mathbb{N}, $ we obtain, of course, $ D^{\rho}1=0$
because of the poles of the gamma function at the points $0,-1,-2,\dots$.
For $\rho>0$, the general solution of the homogeneous equation
$D^{\rho}u(t)=0$ in $C(0,T)\cap L(0,T)$ is
$$
u(t)=c_0
t^{\rho-n}+c_1t^{\rho-n-1}+\dots+c_{n-2}t^{\rho-2}+c_{n-1}t^{\rho-1},
$$
where $c_i$, $i=1,2,\dots,n-1$, are arbitrary real constants.
Further,  we always have $D^\rho I^\rho u=u$, and
\begin{equation}
   I^\rho D^\rho u(t)=u(t)+c_0 t^{\rho-n}+c_1t^{\rho-n-1}
+\dots+c_{n-2}t^{\rho-2}+c_{n-1}t^{\rho-1}.\label{formula}
\end{equation}

To define the solution for problem \eqref{1}-\eqref{2}, we
consider the following lemma.

 \begin{lemma} \label{lem1} 
Let $ h_1, h_2 \in C[0,T]\cap L[0,T]$. Then the integral solution
for the linear system of fractional differential equations: 
\begin{equation}
D^{\alpha}u(t)=h_1(t), \quad  D^{\beta}v(t)=h_2(t),\label{3}
\end{equation}
supplemented with the boundary conditions \eqref{2} is given by
\begin{gather} \label{s1}
\begin{aligned}
u(t)&= \frac{\nu \Gamma(\beta)
t^{\alpha-1}}{\Delta}\Big\{\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}
{\Gamma(\alpha-1)}\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
h_2(\tau)d\tau\Big)ds  \\
&\quad + \frac{\mu\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
\int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}h_1
(\tau)d\tau\Big)ds\Big\} \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}h_1(s)ds,
\end{aligned} \\
\label{s2} \begin{aligned}
v(t)&= \frac{\mu\Gamma(\alpha) t^{\beta-1}}{\Delta}\Big\{\int_{0}^{\sigma}
\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
h_1(\tau)d\tau\Big)ds\\
&\quad + \frac{\nu\sigma^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}h_2
(\tau)d\tau\Big)ds\Big\}  \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}h_2(s)ds,
\end{aligned}
\end{gather}
where
\begin{equation}
 \Delta=\Gamma(\alpha)\Gamma(\beta)-
\frac{\nu\mu\Gamma(\alpha)\Gamma(\beta)(\eta\sigma)^{\alpha+\beta-2}}
{(\Gamma(\alpha+\beta-1))^2}\ne 0. \label{del}
\end{equation}
\end{lemma}

\begin{proof}
 Using the formula \eqref{formula}, the general
solution of the system \eqref{3} can be written as
\begin{gather}
u(t)=a_{0}t^{\alpha-2}+a_1t^{\alpha-1}+I^{\alpha}h_1(t),\label{5-1} \\
v(t)=b_{0}t^{\beta-2}+b_1t^{\beta-1}+I^{\beta}h_2(t),\label{5-2}
\end{gather}
where $ a_{i}, b_{i}$, $(i=0,1) $ are unknown arbitrary constants.
From \eqref{5-1} and \eqref{5-2}, we have
\begin{gather}
D^{\alpha-1}u(t)= a_1\Gamma(\alpha)+Ih_1(t),\label{5-3}\\
D^{\beta-1}v(t)= b_1\Gamma(\beta)+Ih_2(t),\label{5-4}\\
D^{\alpha-2}u(t)= a_0\Gamma(\alpha-1)+a_1\Gamma(\alpha)t+I^2 h_1(t),\label{5-5}\\
D^{\beta-2}v(t)= b_0\Gamma(\beta-1)+b_1\Gamma(\beta)t+I^2
h_2(t).\label{5-6}
\end{gather}
Using the given conditions: $D^{\alpha-2} u(0^{+})=0=D^{\beta-2}
v(0^{+})$ in \eqref{5-5}-\eqref{5-6}, we find that $a_{0}=0$,
$b_{0}=0$. Thus \eqref{5-1} and \eqref{5-2} take the form
\begin{gather}
u(t)= a_1t^{\alpha-1}+I^{\alpha}h_1(t),\label{6-1}\\
v(t)= b_1t^{\beta-1}+I^{\beta}h_2(t).\label{6-2}
\end{gather}
Using the coupled integral boundary conditions: $D^{\alpha-1}
u(0^{+})=\nu I^{\alpha-1}v(\eta)$ and $ D^{\beta-1} v(0^{+})=\mu
I^{\beta-1}u(\sigma)$ in \eqref{5-3} and \eqref{5-4}, we obtain
\begin{equation}
\begin{gathered} 
\Gamma(\alpha)
a_1-\frac{\nu\Gamma(\beta)\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
b_1=\nu I^{\alpha+\beta-1}h_2(\eta),  \\
\frac{\mu\Gamma(\alpha)\sigma^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
 a_1-\Gamma(\beta)b_2=-\mu
I^{\alpha+\beta-1}h_1(\sigma).
\end{gathered}\label{7}
\end{equation}
Solving the system \eqref{7}, we find that
\begin{gather}
a_1= \frac{\nu}{\Delta}\Big\{\Gamma(\beta)I^{\alpha+\beta-1}h_2(\eta)
+\frac{\mu\Gamma(\beta)\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}I^{\alpha
+\beta-1}h_1(\sigma)\Big\},\label{8}\\
b_1= \frac{\mu}{\Delta}\Big\{\Gamma(\alpha)I^{\alpha+\beta-1}h_1(\sigma)
+\frac{\nu\Gamma(\alpha)\sigma^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
I^{\alpha+\beta-1}h_2(\eta)\Big\},\label{9}
\end{gather}
where $ \Delta$ is given by  \eqref{del}. Substituting the values
of $a_1$ and $b_1$ (from \eqref{8} and \eqref{9}) in \eqref{6-1} and \eqref{6-2}, 
we obtain the solution \eqref{s1}-\eqref{s2}. Note that the converse follows by
direct computation. This completes the proof.
\end{proof}

The following lemma contains certain estimates that we need in the
sequel. We do not provide the proof as it is based on simple
computation.


\begin{lemma}\label{lem2}
 For $ h_1, h_2 \in C[0,T]\cap L[0,T]$ with 
$ \| h_1 \| = \sup_{t\in [0,T]}| h_1(t) |  $ and
$ \| h_2 \| =\sup_{t\in [0,T]}| h_2(t) |  $, we have
\begin{align*}
&(i)\quad \big|\int_{0}^{\sigma}\frac{(\sigma-s)^{\beta -2}}{\Gamma(\beta-1)}
 \Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha -1}}{\Gamma(\alpha)}h_1(\tau)
 d\tau\Big)ds\big|
\leq\frac{\sigma^{\alpha+\beta -1}}{\Gamma(\alpha+\beta)} \| h_1\|.\\
&(ii)\quad \big|\int_{0}^{\eta}\frac{(\eta-s)^{\alpha -2}}{\Gamma(\alpha-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta -1}}{\Gamma(\beta)}h_2(\tau)d\tau\Big)ds
\big| \leq\frac{\eta^{\alpha+\beta -1}}{\Gamma(\alpha+\beta)}\| h_2\|.\\
&(iii)\quad \big|\int_{0}^{t}\frac{(t-s)^{\alpha -1}}{\Gamma(\alpha)}h_1(s)ds
\big|\leq\frac{T^{\alpha}}{\Gamma(\alpha+1)}\| h_1\|.\\
&(iv)\quad \big|\int_{0}^{t}\frac{(t-s)^{\beta
-1}}{\Gamma(\beta)}h_2(s)ds\big| 
\leq\frac{T^{\beta}}{\Gamma(\beta+1)}\| h_2\|.
\end{align*}
\end{lemma}

\section{Existence and uniqueness of solutions}

Denote by $X=\{x:x\in C([0,T],\mathbb{R})\}$ and
$Y=\{y:y\in C([0,T],\mathbb{R})\}$ the spaces equipped
respectively with the norms 
$\| x\|_{X}=\sup_{t\in[0,T]}| x(t)|$ and
$\| y\|_{Y}=\sup_{t\in[0,T]}| y(t)|$. Observe that
$(X,\|\cdot\|_{X})$ and  $ (Y,\| \cdot\|_{Y}) $ are Banach
spaces. In consequence,  the product space 
$(X\times Y,\|\cdot\|_{X\times Y})$ is a Banach space endowed with the norm
$\|(x,y)\|_{X\times Y}=\| x\|_{X}+\| y\|_{Y}$
for $(x,y)\in X \times Y$.

By Lemma \ref{lem1}, we define an operator $ F : X\times
Y\to X\times Y$ associated with the problem
\eqref{1}-\eqref{2} as follows:
\begin{equation}
F(u,v)(t):= (F_1(u,v)(t),F_2(u,v)(t)),
\label{10}\end{equation}
where
\begin{equation}
\begin{aligned}
F_1(u,v)(t)  
&= \frac{\nu\Gamma(\beta)
t^{\alpha-1}}{\Delta}\Big\{\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}
 {\Gamma(\alpha-1)}\\ 
&\quad \times \Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
g(\tau,u(\tau),v(\tau),(\phi_2 u)(\tau),(\psi_2
v)(\tau))d\tau\Big)ds  \\
&\quad + \frac{\mu\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
 \int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}\\ 
&\quad \times \Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)} 
 f(\tau,u(\tau),v(\tau),(\phi_1 u)(\tau),(\psi_1 v)(\tau))d\tau\Big)ds\Big\} \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
f(s,u(s),v(s),(\phi_1 u)(s),(\psi_1 v)(s))ds,
\end{aligned}\label{11}
\end{equation}
\begin{equation}
\begin{aligned}
F_2(u,v)(t)  &= \frac{\mu\Gamma(\alpha)
t^{\beta-1}}{\Delta}\Big\{\int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}
 {\Gamma(\beta-1)}\\ 
&\quad \times \Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
f(\tau,u(\tau),v(\tau),(\phi_1
u)(\tau),(\psi_1 v)(\tau))d\tau\Big)ds   \\
&\quad + \frac{\nu\sigma^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}\\ 
&\quad \times \Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
 g(\tau,u(\tau),v(\tau),(\phi_2 u)(\tau),(\psi_2 v)(\tau))d\tau\Big)ds\Big\} \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}g(s,u(s),v(s),(\phi_2
u)(s),(\psi_2 v)(s))ds. 
\end{aligned}\label{12}
\end{equation}
For computational convenience, we set
\begin{gather}
\lambda=1+\gamma_0,\quad \bar{\lambda}=1+\bar{\gamma}_0,\label{lambda}\\
\theta=1+\delta_0,\quad \bar{\theta}=1+\bar{\delta}_0,\label{theta}\\
\varepsilon_1 = T^{\alpha-1}\Big\{\frac{|\nu\mu|\Gamma(\beta)
 (\eta\sigma)^{\alpha+\beta-1}}
{\eta|\Delta|\Gamma(\alpha+\beta-1)\Gamma(\alpha+\beta)}
+\frac{T}{\Gamma(\alpha+1)}\Big\},  \label{e3.6} \\
\varepsilon_2 =\frac{|\nu|\Gamma(\beta)\eta^{\alpha+\beta-1}T^{\alpha-1}}
 {|\Delta|\Gamma(\alpha+\beta)}, \label{14}\\
\bar{\varepsilon}_1 =\frac{|\mu|\Gamma(\alpha)\sigma^{\alpha+\beta-1}T^{\beta-1}
}{|\Delta|\Gamma(\alpha+\beta)}, \label{e3.8}  \\ 
\bar{\varepsilon}_2 
= T^{\beta-1}\Big\{\frac{|\nu\mu|(\eta\sigma)^{\alpha+\beta-1}
 \Gamma(\alpha)}{\sigma|\Delta|
\Gamma(\alpha+\beta)\Gamma(\alpha+\beta-1)}+\frac{T}{\Gamma(\beta+1)}\Big\},
\label{16}\\
\Omega_1=\varpi_0(\varepsilon_1+\bar{\varepsilon}_1)
 +\kappa_0(\varepsilon_2+\bar{\varepsilon}_2),\label{omg1}\\
\Omega_2=\lambda \max \{\varpi_1,\varpi_3\}
 (\varepsilon_1+\bar{\varepsilon}_1)+\bar{\lambda}\max 
\{\kappa_1,\kappa_3\}(\varepsilon_2
+\bar{\varepsilon}_2),\label{omg2}\\
\Omega_3=\theta
\max \{\varpi_2,\varpi_4\}(\varepsilon_1+\bar{\varepsilon}_1)
+\bar{\theta}\max \{\kappa_2,\kappa_4\}(\varepsilon_2+\bar{\varepsilon}_2).
\label{omg3}
\end{gather}
Observe that  problem \eqref{1}-\eqref{2} has solutions if and
only if the operator equation $F(u,v)=(u,v)$ has a fixed point.

Now we are ready to present our first existence result, which is
based on Leray-Schauder alternative.

\begin{lemma}[Leray-Schauder alternative \cite{GrDu}] \label{LSL}
 Let $ F : E\to E $ be a completely continuous operator.
 Let $V(F)=\{x\in E: x=\lambda F(x) \text{ for some}0<\lambda<1\}$. 
Then either the set $ V(F) $ is unbounded or $F$ has at least one fixed point.
\end{lemma}

\begin{theorem}\label{thm1}
Let $f,g:[0,T]\times\mathbb{R}^4\to\mathbb{R} $ be continuous 
functions and there exist real constants $\varpi_i,\kappa_i\geq0~(i=1,\dots,4)$ 
and $\varpi_0,\kappa_0>0$ such that
\begin{itemize}
\item[(H1)] $|f(t,u(t),v(t),(\phi_1 u) (t),(\psi_1 v)(t))| \le
\varpi_0+\varpi_1 |u| +\varpi_2 |v| + \varpi_3 |\phi_1 u|+
\varpi_4 |\psi_1 v|$, 
$| g(t,u(t),v(t),(\phi_2 u)(t),(\psi_2 v)(t))|\leq\kappa_0+\kappa_1 |u| 
+\kappa_2 |v| + \kappa_3 |\phi_2 u|+\kappa_4 |\psi_2 v|$, for all
$(u,v)\in X\times Y$.
\end{itemize}
Further it is assumed that $\max\{\Omega_2,\Omega_3\}<1$, where
$\Omega_2$ and $\Omega_3$ are given by \eqref{omg2} and
\eqref{omg3} respectively.
 Then  problem \eqref{1}-\eqref{2} has at least one
solution on  $[0,T]$.
\end{theorem}

\begin{proof} 
In the first step,  we show that the operator 
$F:X\times Y\to X\times Y$ defined by \eqref{10} is completely continuous. By
continuity of the functions $f$ and $g$, we deduce that the
operators $F_1$ and $F_2$ respectively given by \eqref{11} and \eqref{12} 
are continuous. In consequence, the
operator $F$ is continuous. Next we show that the operator  $F$ is
uniformly bounded. For that, let $ \mathcal{A}\subset X\times Y $
be a bounded set. Then, for any $(u,v) \in \mathcal{A}$, there
exist positive constants $ L_1$ and $L_2$ such that 
$ |f(t,u(t),v(t),(\phi_1 u)(t),(\psi_1 v)(t))|\leq L_1$, 
$| g(t,u(t),v(t),(\phi_2 u)(t),(\psi_2 v)(t))|\leq L_2$,
for all $(u,v)\in\mathcal{A}$. Then, for any $(u,v)\in\mathcal{A}$,
we have
\begin{align*}
& | F_1(u,v)(t)|   \\  
&\leq \frac{\Gamma(\beta) | \nu|
t^{\alpha-1}}{|\Delta|}\Big\{\int_{0}^{\eta}
\frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}  \\
&\quad \times
\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} |
g(\tau,u(\tau),v(\tau),(\phi_2 u)(\tau),(\psi_2 v)(\tau))|
d\tau\Big)ds  \\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}\int_{0}^{\sigma}
\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}  \\
&\quad \times
\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}| 
 f(\tau,u(\tau),v(\tau),(\phi_1 u)(\tau),(\psi_1 v)(\tau))| d\tau\Big)ds\Big\} \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} 
| f(s,u(s),v(s),(\phi_1 u)(s),(\psi_1 v)(s))| ds,\\
&\leq \frac{|\nu|\Gamma(\beta)
T^{\alpha-1}}{|\Delta|}\Big\{L_2\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}
{\Gamma(\alpha-1)}\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
d\tau\Big)ds   \\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}L_1}{\Gamma(\alpha+\beta-1)}
 \int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
d\tau\Big)ds\Big\}
+L_1\int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}  ds,\\
&\leq T^{\alpha-1}L_1\Big\{\frac{|\nu\mu|(\eta\sigma)^{\alpha+\beta-1}
 \Gamma(\beta)}{\eta|\Delta|\Gamma(\alpha+\beta-1)\Gamma(\alpha+\beta)}
 +\frac{T}{\Gamma(\alpha+1)}\Big\}+\frac{|\nu|\Gamma(\beta)
 \eta^{\alpha+\beta-1}T^{\alpha-1}L_2}{|\Delta|\Gamma(\alpha+\beta)},
\end{align*}
which, on taking the norm for $t\in[0,T]$ and using the notation
\eqref{14},  yields
\begin{equation}\label{17}
\| F_1(u,v)\| \leq \varepsilon_1 L_1+\varepsilon_2
L_2.
\end{equation}
In a similar manner, we can find that
\begin{align*}
| F_2(u,v)(t)| \leq
\frac{|\mu|\Gamma(\alpha)T^{\beta-1}}{|\Delta|}
\Big\{\frac{L_1 \sigma^{\alpha+\beta-1}}{\Gamma(\alpha+\beta)}+
\frac{|\nu|
L_2(\eta\sigma)^{\alpha+\beta-1}}{\sigma\Gamma(\alpha+\beta)\Gamma(\alpha+\beta-1)}\Big\}+\frac{L_2
T^\beta}{\Gamma(\beta+1)},
\end{align*}
which together with \eqref{e3.8} and \eqref{16} implies 
\begin{equation}
\| F_2(u,v)\| \leq  \bar{\varepsilon}_1
L_1+\bar{\varepsilon}_2 L_2.\label{18}
\end{equation}
From the inequalities \eqref{17} and \eqref{18}, we infer that 
$F_1 $ and $ F_2 $ are uniformly bounded, and hence
the operator  $  F$ is  uniformly bounded.

Next, we show that $  F$ is equicontinuous. Let $
t_1,~t_2\in[0,T] $ with $ t_1<t_2$. Then we have
\begin{align*}
& | F_1(u,v)(t_2)-F_1(u,v)(t_1)|\\
&\leq \frac{|\nu|\Gamma(\beta) |
t_2^{\alpha-1}-t_1^{\alpha-1}|}{|\Delta|} \Big\{\frac{L_2
\eta^{\alpha+\beta-1}}{\Gamma(\alpha+\beta)}
 +\frac{|\mu|(\eta\sigma)^{\alpha+\beta-1}L_1}
{\eta\Gamma(\alpha+\beta-1)\Gamma(\alpha+\beta)}\Big\}\\
&\quad +  \frac{L_1}{\Gamma(\alpha+1)}\big(2(t_2-t_1)^\alpha+|
t_2^\alpha-t_1^\alpha|\big).
\end{align*}
Obviously $ | F_1(u,v)(t_2)-F_1(u,v)(t_1)|\to 0 $ as $ t_2\to t_1 $.

In a similar manner, one can show  that $ |F_2(u,v)(t_2)-F_2(u,v)(t_1)|\to 0$ as 
$t_2\to t_1$. Thus the operator $F$ is equicontinuous
in view of equicontinuity of $F_1$ and $F_2$. Therefore, by
Arzela-Ascoli's theorem, we
deduce that the operator $F$ is compact (completely continuous).

Finally, we consider a set $V(F)=\{(u,v)\in X\times Y:
(u,v)=\lambda F(u,v)~;~0\leq \lambda\leq 1\}$ and show that it is
bounded. Let $ (u,v)\in V$. Then $ (u,v)=\lambda F(u,v)$. For any
$ t\in[0,T]$, we have $u(t)=\lambda F_1(u,v)(t),~v(t)=\lambda
F_2(u,v)(t)$. Using the assumption $(H1)$ together with the notation
 \eqref{lambda} and  \eqref{theta}, we obtain
\begin{align*}
&| u(t)| \\
&\leq  | \lambda| | F_1(u,v)(t)|\leq| F_1(u,v)(t)| \\
&\leq \frac{\Gamma(\beta) | \nu|
t^{\alpha-1}}{|\Delta|}\Big\{\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
[\kappa_0+\kappa_1 |u(\tau)| +\kappa_2 |v (\tau)|\\
&\quad +  \kappa_3 |(\phi_2 u)(\tau)|+ \kappa_4 |(\psi_2 v)(\tau)|]
d\tau\Big)ds\\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
 \int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}[\varpi_0+\varpi_1
|u (\tau)| +\varpi_2 |v (\tau)| \\ &\quad +  \varpi_3 |(\phi_1 u)
(\tau)|+\varpi_4 |(\psi_1 v) (\tau)|] d\tau\Big)ds\Big\}\\
&\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
[\varpi_0+\varpi_1 |u (s)| +\varpi_2 |v (s)| +\varpi_3 |(\phi_1
u)(s)| + \varpi_4 |(\psi_1
v) (s)|] ds\\
 &\leq \frac{\Gamma(\beta) | \nu|
t^{\alpha-1}}{|\Delta|}\Big\{\int_{0}^{\eta}
 \frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
[\kappa_0+(\kappa_1 +\bar{\gamma}_0\kappa_3)|u(\tau)|\\
&\quad + (\kappa_2 +\bar{\delta}_0 \kappa_4)|v (\tau)|  ] d\tau\Big)ds  \\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
 \int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
 [\varpi_0+(\varpi_1+\gamma_0 \varpi_3)|u (\tau)| \\ 
&\quad +  (\varpi_2 + \delta_0\varpi_4 )|v (\tau)| ] d\tau \Big)ds\Big\} \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} 
 [\varpi_0+(\varpi_1+\gamma_0\varpi_3) |u (s)|  
 +(\varpi_2 + \delta_0\varpi_4) |v (s)| ] ds,\\
&\leq \frac{|\nu|\Gamma(\beta) T^{\alpha-1}}{|\Delta|}
\Big\{[\kappa_0+\bar{\lambda}\max \{\kappa_1,\kappa_3\}\|
u\|_X \\
&\quad + \bar{\theta}\max \{\kappa_2 , \kappa_4 \}\| v\|_Y
]\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} d\tau\Big)ds \\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
 \int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}
\Big([\varpi_0+\lambda\max \{\varpi_1, \varpi_3\}\|u\|_X\\
&\quad + \theta\max \{\varpi_2 , \varpi_4 \}\|
v\|_Y]\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
d\tau\Big)ds\Big\}\\
&\quad + [\varpi_0+\lambda\max \{\varpi_1, \varpi_3\}\| u\|_X 
 +\theta\max \{\varpi_2 , \varpi_4 \}\| v\|_Y]\int_{0}^{t}
 \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}  ds,\\
&\leq T^{\alpha-1}\Big\{\frac{|\nu\mu|(\eta\sigma)^{\alpha+\beta-1}
 \Gamma(\beta)}{\eta|\Delta|\Gamma(\alpha+\beta-1)\Gamma(\alpha+\beta)}
+\frac{T}{\Gamma(\alpha+1)}\Big\}[\varpi_0+\lambda\max \{\varpi_1, \varpi_3\}
 \| u\|_X \\
&\quad + \theta\max \{\varpi_2 , \varpi_4 \}\| v\|_Y ]\\
&\quad + \frac{|\nu|\Gamma(\beta)\eta^{\alpha+\beta-1}
 T^{\alpha-1}}{|\Delta|\Gamma(\alpha+\beta)}[\kappa_0
 +\bar{\lambda}\max \{\kappa_1,\kappa_3\}\|
u\|_X +\bar{\theta}\max \{\kappa_2 , \kappa_4 \}\|v\|_Y   ],
\end{align*} 
which, on taking the norm for $t \in [0, T]$ and using \eqref{14},
yields
\begin{equation}
\begin{aligned}
 \| u\|_X 
&\leq \varepsilon_1[\varpi_0+\lambda\max \{\varpi_1, \varpi_3\}\| u\|_X 
+\theta\max \{\varpi_2 , \varpi_4 \}
 \| v\|_Y ]+\varepsilon_2[\kappa_0   \\ 
&\quad + \bar{\lambda}\max \{\kappa_1,\kappa_3\}\| u\|_X
 +\bar{\theta}\max \{\kappa_2 , \kappa_4 \}\| v\|_Y ].
\end{aligned}\label{19} 
\end{equation}
Similarly, with the aid of notation \eqref{lambda}, \eqref{theta} and \eqref{16}, 
 one can obtain
\begin{equation}
\begin{aligned}
\| v\|_{Y}&\leq \bar{\varepsilon}_1[\varpi_0+\lambda\max \{\varpi_1,
\varpi_3\}\| u\|_X +\theta\max \{\varpi_2 , \varpi_4 \}
\| v\|_Y ]+\bar{\varepsilon}_2[\kappa_0  \\ 
&\quad + \bar{\lambda}\max \{\kappa_1,\kappa_3\}\| u\|_X 
 +\bar{\theta}\max \{\kappa_2 , \kappa_4 \}\| v\|_Y ]. 
\end{aligned} \label{20}
\end{equation}
From \eqref{19} and \eqref{20}, we find that
\begin{equation}
\begin{aligned}
&\| u\|_{X}+\| v\|_{Y} \\
&\leq \varpi_0(\varepsilon_1+\bar{\varepsilon}_1)+\kappa_0(\varepsilon_2+\bar{\varepsilon}_2) \\
&\quad + \Big[\lambda\max \{\varpi_1,\varpi_3\}(\varepsilon_1+\bar{\varepsilon}_1)+\bar{\lambda}\max \{\kappa_1,\kappa_3\}(\varepsilon_2+\bar{\varepsilon}_2)\Big]\| u\|_X \\
&\quad + \Big[\theta\max \{\varpi_2,\varpi_4\}(\varepsilon_1+\bar{\varepsilon}_1)+\bar{\theta}\max \{\kappa_2,\kappa_4\}(\varepsilon_2+\bar{\varepsilon}_2)\Big]\| v\|_Y, \\
&\leq \Omega_1+\max \{\Omega_2,\Omega_3\}\|
(u,v)\|_{X\times Y},
\end{aligned} \label{21}
\end{equation}
which, in view of $\| (u,v)\|_{X\times Y}=\| u\|_X+\| v\|_Y$, yields
$$
\| (u,v)\|_{X\times Y}\leq\frac{\Omega_1}{1-\max \{\Omega_2,\Omega_3\}},
$$
where $\Omega_1, \Omega_2, \Omega_3$ are respectively given  by
\eqref{omg1}, \eqref{omg2} and \eqref{omg3}.
This shows that $ V(F) $ is bounded. Thus, by Lemma \ref{LSL}, the
operator $F$ has at least one fixed point. Consequently, the
problem \eqref{1}-\eqref{2} has at least one solution on $ [0,T].
$ This completes the proof.
\end{proof}

Our next result deals with the uniqueness of solutions for 
problem \eqref{1}-\eqref{2} and relies on Banach's contraction
mapping principle. For computational convenience, we introduce the
notation:
\begin{gather}
\begin{aligned}
\Lambda &= \frac{|\nu|\Gamma(\beta)T^{\alpha-1}}{|\Delta|}
\Big\{\xi_1(1+\bar{\gamma}_0)+\xi_2(1+\bar{\delta}_0) 
+\frac{| \mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}\Big[\zeta_1(1+\gamma_0) \\
&\quad +\zeta_2(1+\delta_0)\Big]\Big\}+\bar{\zeta_1
}(1+\gamma_0)+\bar{\zeta_2 }(1+\delta_0),
\end{aligned} \label{32}\\
\begin{aligned} 
\Lambda_1
&= \frac{|\mu|\Gamma(\alpha)T^{\beta-1}}{|\Delta|}
\Big\{\zeta_1(1+\gamma_0)+\zeta_2(1+\delta_0)+\frac{|
\nu|\sigma^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}\Big[\xi_1(1+\bar{\gamma_0})
 \\
&\quad + \xi_2(1+\bar{\delta_0})\Big]\Big\}+\bar{\xi_1
}(1+\bar{\gamma_0})+\bar{\xi_2 }(1+\bar{\delta_0}),
\end{aligned}\label{33}
\end{gather}
where
\begin{gather}
\zeta_1 =\max \Big\{|
I^{\alpha+\beta-1}M_1(\sigma)|,|
I^{\alpha+\beta-1}M_3(\sigma)|\Big\},  \nonumber \\
 \zeta_2
=\max \Big\{| I^{\alpha+\beta-1}M_2(\sigma)|,|
I^{\alpha+\beta-1}M_4(\sigma)|\Big\},\label{23}
\\
 \bar{\zeta_1}=\sup_{t\in[0,T]}\Big\{|
I^{\alpha}M_1(t)|,| I^{\alpha}M_3(t)|\Big\}, \nonumber\\
\bar{\zeta_2}=\sup_{t\in[0,T]}\Big\{| I^{\alpha}M_2(t)|,
| I^{\alpha}M_4(t)|\Big\},\label{25}
\\
 \xi_1 =\max \Big\{| I^{\alpha+\beta-1}N_1(\eta)|,
 |I^{\alpha+\beta-1}N_3(\eta)|\Big\},   \nonumber \\
\xi_2 =\max \Big\{| I^{\alpha+\beta-1}N_2(\eta)|,
 | I^{\alpha+\beta-1}N_4(\eta)|\Big\},\label{27}
\\
 \bar{\xi_1}=\sup_{t\in[0,T]}\Big\{|
I^{\beta}N_1(t)|,| I^{\beta}N_3(t)|\Big\}, \nonumber \\
 \bar{\xi_2}=\sup_{t\in[0,T]}\Big\{|I^{\beta}N_2(t)|,|
I^{\beta}N_4(t)|\Big\},\label{29}
\\
\gamma_0=\sup_{t\in[0,T]}\Big|\int_{0}^{t}\gamma_1(t,s)ds\Big|,\quad
\bar{\gamma_0}=\sup_{t\in[0,T]}\Big|\int_{0}^{t}\gamma_2(t,s)ds\Big|,
\label{30}
\\
\delta_0 =\sup_{t\in[0,T]}\Big|\int_{0}^{t}\delta_1(t,s)ds\Big|,\quad
\bar{\delta_0}=\sup_{t\in[0,T]}\Big|\int_{0}^{t}\delta_2(t,s)ds\Big|,
\label{31} 
\\
\varepsilon=\varepsilon_1 \varrho_1+\varepsilon_2\varrho_2,\quad 
\bar{\varepsilon}=\bar{\varepsilon}_1
\varrho_1+\bar{\varepsilon}_2 \varrho_2, \quad
\varrho_1=\sup_{t\in[0,T]}| f(t,0,0,0,0)|, \nonumber\\
\varrho_2=\sup_{t\in[0,T]}| g(t,0,0,0,0)|,\label{35}
\end{gather}
where $\varepsilon_i, \bar{\varepsilon}_i (i=1,2)$ are respectively given 
by \eqref{e3.6}--\eqref{16}.


\begin{theorem} \label{thm2}
 Let $f,g: [0,T]\times \mathbb{R}^{4}\to \mathbb{R}$ be continuous functions 
and there exist positive functions $M_{i}(t)$, $N_{i}(t)\geq 0~(i=1,\dots,4)$ 
 such that
\begin{gather*}
| f(t,u_1,u_2,u_{3},u_4)-f(t,v_1,v_2,v_{3},v_4)|
\leq \sum_{i=1}^4 M_i(t)| u_{i}-v_{i}|, \\
g(t,u_1,u_2,u_{3},u_4)-g(t,v_1,v_2,v_{3},v_4)| 
\leq \sum_{i=1}^4 N_i(t)| u_{i}-v_{i}|,
\end{gather*}
for all $t\in[0,T]$, $u_{i},v_{i}\in \mathbb{R}$.
In addition, assume that $\Lambda<\frac{1}{2} $ and 
$\Lambda_1<\frac{1}{2}$,
where  $\Lambda$ and $\Lambda_1$ are given by  \eqref{32}and \eqref{33} respectively.
Then  boundary value problem \eqref{1}-\eqref{2} has a unique
solution on $[0, T]$.
\end{theorem}

\begin{proof} 
Let us fix $r \ge \max \{2\varepsilon/(1-2\Lambda),
  2\bar{\varepsilon}/(1-2\Lambda_1)\}$, where 
$ \Lambda,\Lambda_1$, and  $\varepsilon,  \bar{\varepsilon}$ are respectively 
given by \eqref{32}, \eqref{33} and \eqref{35}. 
Firstly, we show that $ FB_{r}\subset B_{r}$, where  
$ B_{r}=\{(u,v)\in X\times Y:\|(u,v)\|_{X\times Y}\leq r\}$ and  
$F$ is given by \eqref{10}.  For $ (u,v)\in B_{r} $, note
that
\begin{align*}
& | f(t,u(t),v(t), (\phi_1 u)(t),(\psi_1 v)(t))|\\
&\leq | f(t,u(t),v(t), (\phi_1 u)(t),(\psi_1 v)(t))-
f(t,0,0,0,0)| + | f (t,0,0,0,0) |
\\
&\leq  M_1(t)| u(t)| +M_2(t) | v(t) | 
 +M_3(t) | (\phi_1 u)(t)| +M_4(t) | (\psi_1 v)(t)|  + \varrho_1 \\
&\leq  M_1(t)| u(t)| +M_2(t) | v(t) | +\gamma_0 M_3(t) |  u(t)| 
 +\delta_0 M_4(t) |  v(t)|  + \varrho_1 \\
&\leq \Big[ M_1(t) +\gamma_0 M_3(t)\Big]| u(t)| +\Big[M_2(t)
 +\delta_0 M_4(t)\Big] | v(t) |   + \varrho_1 \\
&\leq  \Big(M_1(t) +M_2(t)  +\gamma_0 M_3(t)  
 +\delta_0 M_4(t)\Big)\|(u,v)\|_{X\times Y}  + \varrho_1\\
&\leq  \Big(M_1(t) +M_2(t)  +\gamma_0 M_3(t)  +\delta_0
M_4(t)\Big)r  + \varrho_1.
\end{align*}
Similarly, one can obtain
$$
| g(t,u(t),v(t), (\phi_2 u)(t),(\psi_2 v)(t))| \leq
\Big(N_1(t) +N_2(t)  +\bar{\gamma_0} N_3(t)  +\bar{\delta_0}
N_4(t)\Big)r  + \varrho_2.
$$
Then, using the notation \eqref{23}, \eqref{25}, \eqref{27} and  \eqref{35}, 
we have
\begin{align*}
&| F_1(u,v)(t)| \\
&\leq  \frac{\Gamma(\beta)
T^{\alpha-1}}{|\Delta|}\Big\{\int_{0}^{\eta}
 \frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}
\Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
\Big(N_1(\tau) +N_2(\tau)  \\  
&\quad +  \bar{\gamma_0} N_3(\tau) +\bar{\delta_0} N_4(\tau)\Big)r
+\varrho_2\Big)d\tau\Big)ds   \\
&\quad + \frac{|\nu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}\int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}
{\Gamma(\beta-1)}\Big(\int_{0}^{s}\frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
\Big( M_1(\tau) +M_2(\tau)  +\gamma_0 M_3(\tau)   \\
&\quad + \delta_0 M_4(\tau)\Big)r  + \varrho_1\Big)d\tau\Big)ds\Big\}  \\
&\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} \Big(
M_1(s) +M_2(s) +\gamma_0 M_3(s)  +\delta_0 M_4(s)\Big)r  +
\varrho_1\Big)ds \\
&\leq  \frac{|\nu|\Gamma(\beta)
T^{\alpha-1}}{|\Delta|}\Big\{I^{\alpha+\beta-1}\Big(N_1(\eta)
+\bar{\gamma_0} N_3(\eta) \Big)r +I^{\alpha+\beta-1}\Big(
N_2(\eta) +\bar{\delta_0} N_4(\eta)\Big)r
  \\
&\quad + \frac{\eta^{\alpha+\beta-1} \varrho_2}{\Gamma(\alpha+\beta)}
  \\
&\quad +
\frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}\Big[I^{\alpha+\beta-1}
\Big( M_1(\sigma)
  +\gamma_0 M_3(\sigma) \Big)r +I^{\alpha+\beta-1} \Big(  M_2(\sigma)+\delta_0 M_4(\sigma)\Big)r \Big]
   \\
&\quad + \frac{
\rho_1|\mu|(\eta\sigma)^{\alpha+\beta-1}}{\eta\Gamma(\alpha+\beta)\Gamma(\alpha+\beta-1)}\Big\}
+I^{\alpha} \Big( M_1(t) +\gamma_0 M_3(t)\Big)r  +I^{\alpha} \Big( M_2(t)  +\delta_0 M_4(t)\Big)r   \\
&\quad + \frac{ \varrho_1T^{\alpha}}{\Gamma(\alpha+1)}\\
&\leq  r\Big\{
\frac{|\nu|\Gamma(\beta)T^{\alpha-1}}{|\Delta|}\Big(\xi_1(1+\bar{\gamma_0})
+\xi_2(1+\bar{\delta_0})+\frac{|
\mu|\zeta_1\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}(1+\gamma_0
)   \\
&\quad + \frac{|
\mu|\zeta_2\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}(1+\delta_0)\Big)+\bar{\zeta_1
}(1+\gamma_0)+\bar{\zeta_2}\Big(1+\delta_0\Big)\Big\}  \\
&\quad + \varrho_1\Big[\frac{|\mu\nu|\Gamma(\beta)
(\eta\sigma)^{\alpha+\beta-1}T^{\alpha-1}}{\eta|\Delta|\Gamma(\alpha+\beta)\Gamma(\alpha+\beta-1)}
+\frac{T^{\alpha}}{\Gamma(\alpha+1)}\Big]
+\varrho_2\frac{|\nu|\Gamma(\beta)\eta^{\alpha+\beta-1}
T^{\alpha-1}}{|\Delta|\Gamma(\alpha+\beta)},
\end{align*}
which, in view of \eqref{32} and \eqref{35}, implies
\begin{equation} 
\| F_1(u,v)\|_X \leq \Lambda r+\varepsilon\leq \frac{r}{2}.\label{36}
\end{equation}
 Analogously, using \eqref{33} and \eqref{35}, we can obtain
 \begin{equation}
 \| F_2(u,v)\|_{Y} = \Lambda_1 r+\bar{\varepsilon}\leq  \frac{r}{2}. \label{37}
 \end{equation}
 From the estimates \eqref{36} and \eqref{37}, it clearly follows
 that
\[
 \| F(u,v)\|_{X\times Y} =  \| F_1(u,v)\|_{X}+
 \| F_2(u,v)\|_{Y} \le r, 
\]
 and hence  $ FB_{r} \subset B_{r}$.

 Now we show that the operator $F$ is a contraction. For that, let  
$ u_{i},v_{i}\in \mathbb{R}$, $i=1,2$. Then, for each $t\in[0,T]$,
it follows by \eqref{23}-\eqref{27} that
 \begin{align*}
 & | F_1(u_1,v_1)(t)-F_1(u_2,v_2)(t)| \\
 &\leq \frac{|\nu|\Gamma(\beta) T^{\alpha-1}}{|\Delta|}\Big\{\int_{0}^{\eta}\frac{(\eta-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \Big(\int_{0}^{s}\frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)}
| g(\tau,u_1(\tau),v_1(\tau),(\phi_2 u_1)(\tau),\\
&\quad (\psi_2 v_1)(\tau))  -g(\tau,u_2(\tau),v_2(\tau),(\phi_2 u_2)(\tau),(\psi_2
 v_2)(\tau))| d\tau\Big)ds
   \\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
\int_{0}^{\sigma}\frac{(\sigma-s)^{\beta-2}}{\Gamma(\beta-1)}\Big(\int_{0}^{s}
 \frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)}
 | f(\tau,u_1(\tau),v_1(\tau),(\phi_1 u_1)(\tau),\\
&\quad (\psi_1 v_1)(\tau)) 
  - f(\tau,u_2(\tau),v_2(\tau),(\phi_1 u_2)(\tau),(\psi_1 v_2)(\tau))|  
 d \tau\Big)ds\Big\} \\
 &\quad + \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
| f(s,u_1(s),v_1(s),(\phi_1 u_1)(s),(\psi_1 v_1)(s)) \\
&\quad -f(s,u_2(s),v_2(s),(\phi_1 u_2)(s),(\psi_1 v_2)(s))| ds\\
 &\leq  \frac{|\nu|\Gamma(\beta)
 T^{\alpha-1}}{|\Delta|}\Big\{I^{\alpha+\beta-1}\Big(N_1(\eta)
 +\bar{\gamma_0} N_3(\eta) \Big)\| u_1- u_2\|_{X}
   \\
&\quad + I^{\alpha+\beta-1}\Big( N_2(\eta) +\bar{\delta_0}
 N_4(\eta)\Big)\| v_1- v_2\|_{Y}
   \\
&\quad + \frac{|\mu|\eta^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}
 \Big[I^{\alpha+\beta-1} \Big( M_1(\sigma)
  +\gamma_0 M_3(\sigma) \Big)\| u_1- u_2\|_{X} \\
&\quad + I^{\alpha+\beta-1} \Big(  M_2(\sigma)+\delta_0 M_4(\sigma)\Big)
 \| v_1- v_2\|_{Y} \Big]\Big\}  \\
&\quad + I^{\alpha} \Big( M_1(t) +\gamma_0 M_3(t)\Big)\| u_1- u_2\|_{X}  
+I^{\alpha} \Big( M_2(t)
 +\delta_0 M_4(t)\Big)\| v_1- v_2\|_{Y} \\
&\leq  \Big\{\frac{|\nu|\Gamma(\beta) T^{\alpha-1}}{|\Delta|}
 \Big[\xi_1(1+\bar{\gamma_0})
 +\frac{|\mu|\eta^{\alpha+\beta-2}\zeta_1}{\Gamma(\alpha+\beta-1)}
(1+\gamma_0)\Big]+\bar{\zeta_1}(1+\gamma_0)\Big\}\| u_1-u_2\|_X\\
&\quad + \Big\{\frac{|\nu|\Gamma(\beta)
 T^{\alpha-1}}{|\Delta|}\Big[\xi_2(1+\bar{\delta_0})
 +\frac{|\mu|\eta^{\alpha+\beta-2}\zeta_2}{\Gamma(\alpha+\beta-1)}
(1+\delta_0)\Big]+\bar{\zeta_2}(1+\delta_0)\Big\}\| v_1-v_2\|_Y,
 \end{align*}
which yields
 \begin{equation}
\| F_1(u_1,v_1)-F_1(u_2,v_2)\|_{X}\leq
 \Lambda[\| u_1-u_2\|_{X} + \| v_1
 -v_2\|_{Y}], \label{38}
 \end{equation}
where we have used \eqref{32}.  Similarly, we can find that
 \begin{equation}
\| F_2(u_1,v_1)-F_2(u_2,v_2)\|_{Y}
  \leq  \Lambda_1[\| u_1-u_2\|_{X}
  + \| v_1-v_2\|_{Y}], \label{39}
 \end{equation}
where we have used \eqref{33}. Thus,  from \eqref{38}  and
 \eqref{39}, we have
 \begin{align*}
 &\| F(u_1,v_1)-F(u_2,v_2)\|_{X\times Y} \\ 
&=  \| F_1(u_1,v_1)- F_1(u_2,v_2)\|_{X} + \| F_2(u_1,v_1)- F_2(u_2,v_2)\|_{Y} \\
&\leq  (\Lambda +\Lambda_1 )[\| u_1-
 u_2\|_{X} + \| v_1- v_2\|_{Y}],
 \end{align*}
which implies that $F$ is a contraction in view of the given
 condition $\Lambda +\Lambda_1<1$. Hence, by Banach's fixed point
 theorem, the operator $F$ has a unique fixed point which
  corresponds to the unique solution of the problem \eqref{1}-\eqref{2} on $[0, T]$. 
This completes the proof. 
\end{proof}

\subsection*{Example}
  Consider the  boundary-value problem
  \begin{equation}\label{40}
\begin{gathered}
\begin{aligned}
  D^{3/2}u(t)&=\frac{\sqrt{ t^2+1}}{10}+\frac{t^2}{15}| u(t)|
+\frac{t^2}{10}\tan^{-1}v(t)\\
  &\quad +\frac{t}{25}\int_{0}^{t}\frac{(t-s)^{1/2}}{\Gamma(\frac{3}{2})}u(s)ds
  +\frac{1}{25}\int_{0}^{t}\frac{(t-s)^{1/3}}{\Gamma(\frac{4}{3})}v(s)ds,
\end{aligned}\\
\begin{aligned}
  D^{5/4}v(t)&=\frac{e^t}{80}(|\sin u(t)|+1)+\frac{1}{20}| v(t)|\\
&\quad +\frac{1}{80}\int_{0}^{t}\frac{e^{-(s-t)}}{50}u(s)ds
 +\frac{t}{20}\int_{0}^{t}\frac{e^{-(s-t)/2}}{140}v(s)ds,\quad 0<t<1
\end{aligned}\\
  D^{-1/2}u(0^+)=0,\quad D^{1/2}u(0^+)=-I^{1/2}(1/2),\\
  D^{-3/4}v(0^+)=0,\quad D^{1/4}v(0^+)=-2I^{1/4}(1/4).
\end{gathered}
  \end{equation}
Here,  $\alpha=3/2$, $\beta=5/4$, $v=-1$, $\mu=-2$, $\eta=1/2$,
$\sigma=1/4$, $T=1$, $\gamma_1=(t-s)^{1/2}/\Gamma(3/2)$,
$\delta_1=  (t-s)^{1/3}/\Gamma(4/3)$, $\gamma_2=  e^{-(s-t)}/50$,
$\delta_2= e^{-(s-t)/2}/140$,  $M_1(t)=t^2/15$,
$M_2(t)=  t^2/10$, $M_3(t)=t/25$, $M_4(t)= 1/25$,
$N_1(t)=e^t/80$, $N_2(t)= 1/20$,
$N_3(t)=1/80$, $N_4(t)=  t/20$.
 Using the given   data, we find that $\gamma_0\simeq0.75225$,
$\delta_0\simeq0.83988$, $\bar{\gamma}_0\simeq0.03437$,
$\bar{\delta}_0\simeq0.00927$,
$\Delta\simeq1.20312$, $\zeta_1\simeq0.00489$, 
$\zeta_2\simeq  0.00733$, $\bar{\zeta}_1\simeq0.05015$,
$\bar{\zeta}_2\simeq0.07523$, $\xi_1\simeq  0.01006$,
$\xi_2\simeq0.01480$, $\bar{\xi}_1\simeq0.11841$, $\bar{\xi}_2\simeq0.04413$.
Further,   $\Lambda\simeq0.26689<1/2$, and $\Lambda_1\simeq0.21388<1/2$.
Thus, by Theorem \ref{thm2},  problem \eqref{40} has a unique
solution on $[0,1]$.

\begin{thebibliography}{00}

\bibitem{f1} B. Ahmad, J. J. Nieto;
 Riemann-Liouville fractional integro-differential equations with fractional 
nonlocal integral boundary conditions,
\emph{Bound. Value Probl.}  (2011), 2011:36, 9 pp.

\bibitem{f2} B. Ahmad, J. J. Nieto;
 Riemann-Liouville fractional differential equations with fractional boundary 
conditions, \emph{Fixed Point Theory} \textbf{13} (2012),  329-336.

\bibitem{f3} B. Ahmad, S. K. Ntouyas;
 Initial value problems of fractional order Hadamard-type functional 
differential equations,
 \emph{Electron. J. Differential Equations}  2015, No. 77, 9 pp.

\bibitem{s-1} B. Ahmad, J. J. Nieto;
Existence results for a coupled system of
nonlinear fractional differential equations with three-point
boundary conditions, \emph{Comput. Math. Appl.} 58 (2009),
1838-1843.

\bibitem{s-2} B. Ahmad, S. K. Ntouyas;
Existence results for a coupled system of Caputo type sequential
fractional differential equations with nonlocal integral boundary
conditions, \emph{Appl. Math. Comput.} \textbf{266} (2015), 615-622.

\bibitem{s-3} S. Aljoudi, B. Ahmad, J. J. Nieto, A. Alsaedi;
 A coupled system of Hadamard type sequential fractional
differential equations with coupled strip conditions, Chaos
Solitons Fractals \textbf{91} (2016), 39-46.

\bibitem{S1} A. A. M. Arafa, S. Z. Rida, M. Khalil;
Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection, 
Nonlinear Biomed Phys.   (2012),   6:1, 7 pages.

\bibitem{f4} Z. B. Bai, W. Sun;
Existence and multiplicity of positive solutions for singular fractional
 boundary value problems,
 \emph{Comput. Math. Appl.}  \textbf{63} (2012), 1369-1381.

\bibitem{S2} A. Carvalho, C. M. A. Pinto;
A delay fractional order model for the
co-infection of malaria and HIV/AIDS, Int. J. Dynam. Control,
(2016) DOI 10.1007/s40435-016-0224-3.

\bibitem{S3} Y. Ding, Z. Wang, H. Ye;
Optimal control of a fractional-order HIV-immune system with memory,  
IEEE Trans. Contr. Sys. Techn. \textbf{20} (2012), 763-769.

\bibitem{f5} Y. Ding, Z. Wei, J. Xu, D. O'Regan;
Extremal solutions for nonlinear fractional boundary value problems with 
$p$-Laplacian, \emph{J. Comput. Appl. Math.} \textbf{288} (2015), 151-158.

\bibitem{Fai} M. Faieghi,  S. Kuntanapreeda,  H. Delavari, D. Baleanu;
LMI-based stabilization of a class of  fractional-order chaotic systems,  
\emph{Nonlinear Dynam.} \textbf{72} (2013),  301-309.

\bibitem{Ge1} Z. M. Ge, C. Y. Ou;
Chaos synchronization of fractional order modified Duffing systems with 
parameters excited by a chaotic signal, 
\emph{Chaos Solitons Fractals} \textbf{35} (2008), 705-717.

\bibitem{f6} J. R. Graef, L. Kong, Q. Kong;
Application of the mixed monotone operator method to fractional boundary 
value problems,  \emph{Fract. Calc.   Differ.  Calc.}  \textbf{2} (2011),  554-567.

\bibitem{GrDu} A. Granas,  J. Dugundji;
 \emph{Fixed Point Theory}, Springer-Verlag, New York, 2003.

\bibitem{f7} J. Henderson, N. Kosmatov;
Eigenvalue comparison for fractional boundary value problems with the
 Caputo derivative, \emph{Fract. Calc. Appl. Anal.} \textbf{17} (2014),  872-880.

\bibitem{s-4} J. Henderson, R. Luca;
Nonexistence of positive solutions for a system of coupled fractional boundary value
problems, \emph{Bound. Value Probl.} (2015),  2015:138, 12 pp.

\bibitem{S4} M. Javidi, B. Ahmad;
 Dynamic analysis of time fractional order
phytoplankton-toxic phytoplankton--zooplankton system, 
\emph{Ecological Modelling}, 318 (2015), 8-18.

\bibitem{f8}  M. Jia,  H. Zhang, Q. Chen;
Existence of positive solutions for fractional differential equation with
 integral boundary conditions on the half-line, \emph{Bound. Value Probl.} 
 (2016), 2016:104, 16 pp.

\bibitem{B4} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo;
\emph{Theory and Applications of Fractional Differential
Equations}, North-Holland Mathematics Studies, 204. Elsevier
Science B. V., Amsterdam, 2006.

\bibitem{s-5} M. Kirane, B. Ahmad, A. Alsaedi, M. Al-Yami;
Non-existence of global solutions to a system of fractional diffusion equations,
\emph{Acta Appl. Math.} \textbf{133} (2014), 235-248.

\bibitem{B1} J. Klafter, S. C Lim, R. Metzler (Editors);
 \emph{Fractional Dynamics in Physics}, World Scientific,  Singapore, 2011.

\bibitem{f9} B. Li, S. Sun, Y. Li,  P. Zhao;
 Multi-point boundary value problems for a class of Riemann-Liouville fractional
 differential equations, \emph{Adv. Difference Equ.}  (2014), 2014:151, 11 pp.

\bibitem{B2} R.L. Magin;
\emph{Fractional Calculus in Bioengineering},  Begell House Publishers,  2006.

\bibitem{pI-3} R. Metzler, J. Klafter;
 The random walk­s guide to anomalous diffusion: a fractional dynamics approach,
 \emph{Phys. Rep.}  \textbf{339} (2000),  1-77.

\bibitem{S5} N. Nyamoradi, M. Javidi, B. Ahmad;
Dynamics of SVEIS epidemic model with distinct incidence, 
\emph{Int. J. Biomath.} \textbf{8} (2015), no. 6, 1550076, 19 pp.

\bibitem{S6} I. Petras, R. L. Magin;
Simulation of drug uptake in a two
compartmental fractional model for a biological system, \emph{Commun
Nonlinear Sci Numer Simul.} \textbf{16} (2011), 4588-4595.

\bibitem{Povstenko3}  Y. Z.  Povstenko;
\emph{Fractional Thermoelasticity}, Springer, New York, 2015.

\bibitem{f10} F. Punzo,  G. Terrone;
 On the Cauchy problem for a general fractional porous medium equation
 with variable density, \emph{Nonlinear Anal.}  \textbf{98} (2014), 27-47.

\bibitem{B3} J. Sabatier, O. P. Agrawal, J. A. T. Machado (Eds.);
\emph{Advances in Fractional Calculus: Theoretical
Developments and Applications in Physics and Engineering},
Springer, Dordrecht, 2007.

\bibitem{s-6} B. Senol, C. Yeroglu;
 Frequency boundary of fractional order systems with nonlinear uncertainties,
 \emph{J. Franklin Inst.} \textbf{350} (2013), 1908-1925.

\bibitem{pI-4} I. M. Sokolov, J. Klafter, A. Blumen;
 Fractional kinetics, \emph{Phys. Today} \textbf{55} (2002), 48-54.

\bibitem{s-7} J. Sun, Y. Liu, G. Liu;
 Existence of solutions for fractional differential systems with antiperiodic boundary
conditions, \emph{Comput. Math. Appl.}  \textbf{64} (2012), 1557-1566.

\bibitem{s-8} J. Tariboon, S. K. Ntouyas, W. Sudsutad;
Coupled systems of Riemann-Liouville fractional differential equations with Hadamard
fractional integral boundary conditions, \emph{J. Nonlinear Sci. Appl.}
9 (2016),  295-308.

\bibitem{f11} P. Thiramanus, S. K. Ntouyas, J. Tariboon;
 Existence of solutions for Riemann-Liouville fractional differential
 equations with nonlocal Erdélyi-Kober integral boundary conditions 
on the half-line, \emph{Bound. Value Probl.}  (2015), 2015:196, 15 pp.

\bibitem{s-9} J. R. Wang, Y. Zhang;
 Analysis of fractional order differential coupled systems, 
\emph{Math. Methods Appl. Sci.} \textbf{38} (2015), 3322-3338.

\bibitem{f12} C. Zhai, L. Xu;
Properties of positive solutions to a class of four-point boundary 
value problem of Caputo fractional differential equations with a parameter,
 \emph{Commun. Nonlinear Sci. Numer. Simul.} \textbf{19} (2014), 2820-2827.

\bibitem{Zha} F. Zhang,  G. Chen C. Li, J. Kurths;
Chaos synchronization in fractional differential systems, 
\emph{Phil Trans R Soc A} \textbf{371} (2013), 20120155.

  \bibitem{f13}  L. Zhang, B. Ahmad, G. Wang;
Successive iterations for positive extremal solutions of nonlinear 
fractional differential equations on a half line, \emph{Bull. Aust. Math. Soc.}
 \textbf{91} (2015), 116-128.


\end{thebibliography}


\end{document}