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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 91, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/91\hfil Fractional-order impulsive PDEs]
{Impulsive discontinuous hyperbolic partial differential equations
of fractional order on Banach algebras}

\author[S. Abbas, R. P. Agarwal, M. Benchohra\hfil EJDE-2010/91\hfilneg]
{Sa\"{\i}d Abbas, Ravi P. Agarwal, Mouffak Benchohra}  % in alphabetical order

\address{Sa\"{\i}d Abbas \newline
Laboratoire de Math\'{e}matiques, Universit\'{e} de Sa\"{\i}da,
B.P. 138, 20000, Sa\"{\i}da, Alg\'{e}rie}
\email{abbas\_said\_dz@yahoo.fr}

\address{Ravi P. Agarwal \newline
Department of Mathematical Sciences, 
Florida Institute of Technology\\
Melboune, Florida, 32901-6975, USA\newline
KFUPM Chair Professor, Mathematics and Statistics Department\\
King Fahd University of Petroleum and Minerals, Dhahran 31261,
Saudi Arabia}
\email{agarwal@fit.edu}

\address{Mouffak Benchohra \newline
Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\`es \\
B.P. 89, 22000, Sidi Bel-Abb\`es, Alg\'erie}
\email{benchohra@univ-sba.dz}

\thanks{Submitted May 6, 2010. Published July 7, 2010.}
\subjclass[2000]{26A33}
\keywords{Impulsive differential equations; fractional order;
 upper solution; \hfill\break\indent
lower solution; extremal solutions;
 left-sided mixed Riemann-Liouville integral; \hfill\break\indent
 Caputo fractional-order derivative; fixed point; Banach algebra}

\begin{abstract}
 This article studies the existence of solutions and extremal
 solutions to partial hyperbolic differential equations of fractional
 order with impulses in Banach algebras under Lipschitz and
 Carath\'eodory conditions and certain monotonicity conditions.
\end{abstract}

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

\section{Introduction}

This article studies the existence of solutions to fractional order
initial-value problems ($IVP$ for short), for the system
\begin{gather}\label{e1}
^{c}D_{0}^{r}\Big(\frac{u(x,y)}{f(x,y,u(x,y))}\Big)
=g(x,y,u(x,y)),\quad (x,y)\in J,\; x\neq x_k, \; k=1,\ldots,m, \\
\label{e2}
u(x_k^+,y)=u(x_k^-,y)+I_{k}(u(x_k^-,y)), \quad  y\in
[0,b]; \; k=1,\dots,m, \\
\label{e3}
u(x,0)=\varphi (x), \quad u(0,y)=\psi (y), \quad x\in [0,a],
\; y\in [0,b],
\end{gather}
where $J=[0,a]\times [0,b]$, $a,b>0$, $ ^{c}D_{0}^{r}$ is the
Caputo's fractional derivative of order
$r=(r_1,r_2)\in (0,1]\times (0,1]$,
$0=x_{0}<x_{1}<\dots<x_{m}<x_{m+1}=a$,
$f:J\times\mathbb{R}^{n}\to \mathbb{R}^{n}
\backslash\{0_{\mathbb{R}^{n}}\}$,
$g:J\times\mathbb{R}^{n}\to \mathbb{R}^{n}$  and
$I_{k}: \mathbb{R}^{n}\to\mathbb{R}^{n}$,
$k=1,\ldots,m$ are given functions
satisfying suitable conditions and
$\varphi :[0,a]\to \mathbb{R}^{n}$, $\psi :[0,b]\to \mathbb{R}^{n}$
are given absolutely
continuous functions with $\varphi(0)=\psi(0)$.

There has been a significant development in ordinary and partial
fractional differential equations in recent years; see the
monographs of Kilbas \cite{KiSrJuTr}, Lakshmikantham {\em et al.}
\cite{LLV},   Podlubny \cite{Pod}, Samko \cite{SaKiMa1},
the papers by  Abbas and Benchohra \cite{AbBe1, AbBe2,
AbBe3}, Agarwal {\it et al.} \cite{ABH1}, Belarbi {\it et al.}
\cite{BBO}, Benchohra {\it et al.} \cite{BeHaGr, BeHaNt, BeHeNtOu},
Diethelm \cite{DiFo}, Vityuk and  Golushkov \cite{ViGo} and the
references therein.  We can find numerous applications of
differential equations of fractional order in viscoelasticity,
electrochemistry, control, porous media, electromagnetic, etc. (see
\cite{Hi,MeScKiNo}).

The theory of impulsive differential equations have become important
in some mathematical models  of real processes and phenomena studied
in physics, chemical technology, population dynamics, biotechnology
and economics. There has been a significant development in impulse
theory in recent years, especially in the area of impulsive
differential equations and inclusions with fixed moments; see the
monographs of Benchohra {\em et al.} \cite{BHN}, Lakshmikantham {\em
et al.} \cite{LaBaSi}, and Samoilenko and Perestyuk \cite{SaPe}, and
the references therein.

In this article, we prove the existence of extremal solutions under
discontinuous nonlinearity under certain Lipschitz and monotonicity
conditions. These results extend to the Banach algebra setting those
considered with integer order derivative
\cite{DaKu1, Kam} and those with fractional derivative
\cite{AbBe4}. Also, we extend some results considered on Banach
algebras with integer order derivative and without impulses
\cite{ABD,DhNt}. Finally, an example illustrating the abstract
results is presented in the last Section.

This paper initiates the study of fractional hyperbolic differential
equations with impulses on Banach algebras.

\section{Preliminaries}

 In this section, we introduce notation, definitions,
and preliminary facts which are used throughout this paper. Let
$L^{\infty}(J,\mathbb{R}^{n})$ be the Banach space of measurable
functions $u: J \to \mathbb{R}^{n}$ which are bounded, equipped with
the norm
$$
\|u\|_{L^{\infty}}=\inf\{c>0: \|u(x,y)\|\leq c, \text{ a.e. } (x,y)\in
J\},
$$
where $\|\cdot\|$ denotes a suitable complete norm on $\mathbb{R}^{n}$.
By $L^{1}(J,\mathbb{R}^{n})$ we denote the space of
Lebesgue-integrable functions $u:J\to \mathbb{R}^{n}$ with
the norm
$$
\|u\|_{1}=\int_{0}^{a}\int_{0}^{b}\|u(x,y)\|\,dy\,dx.
$$
$AC(J, \mathbb{R}^{n})$ is the space of absolutely continuous valued
functions on $J$. Denote by
$D ^{2}_{xy}:=\frac{\partial ^{2}}{\partial x\partial y}$
the mixed second order partial derivative. In all what follows set
$$
J_{k}:=(x_{k},x_{k+1}]\times[0,b], \quad k=0,1,\ldots,m.
$$
Consider the space
\begin{align*}
PC(J,\mathbb{R}^{n})=
&\big\{u: J\to\mathbb{R}^{n}: u \in
C(J_{k}, \mathbb{R}^{n}) ;  k=1, \ldots,m, \text{ and there exist}\\
& u(x_{k}^-,y), u(x_{k}^+,y) ;
k=1,\ldots,m, \text{ with } u(x_{k}^-,y)=u(x_{k},y) \big\}.
\end{align*}
This set is a Banach space with the norm
$$
\|u\|_{PC}=\sup_{(x,y)\in J}\|u(x,y)\|.
$$
Define a multiplication $``\,\cdot\,"$  by
$$
(u\cdot v)(x,y)=u(x,y)v(x,y) \quad \text{for } (x,y)\in J.
$$
Then
$PC(J,\mathbb{R}^{n})$ is a Banach algebra with the above norm and
multiplication.

 Let $a_1\in[0,a]$, $z^+=(a_1^+,0)\in J$,
$J_z=[a_1,a]\times[0,b]$,  $r_{1}, r_{2}>0 $ and $r=(r_{1},r_{2})$.
For $u\in L^{1}(J_z,\mathbb{R}^{n})$, the expression
$$
(I_{z^+}^{r}u)(x,y)=\frac{1}{\Gamma (r_{1})\Gamma (r_{2})}
\int_{a_1^+}^{x}
\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}u(s,t)\,dt\,ds,
$$
where $\Gamma (.)$ is the Euler gamma function, is called the
left-sided mixed Riemann-Liouville integral of order $r$.

\begin{definition}[\cite{ViGo}] \label{def2.1} \rm
For $u\in L^{1}(J_z,\mathbb{R}^{n})$, the Caputo fractional-order
derivative of order $r$ is defined by the expression
$(^{c}D_{z^+}^{r}u)(x,y)=( I_{z^+}^{1-r}D ^{2}_{xy}u) (x,y)$.
\end{definition}

  Let $X$ be a Banach algebra with norm $\|\cdot \|$. An
operator $ T:X \to X$ is called \emph{compact} if $\overline {T(S)}$
is a compact subset of $ X$ for any $S\subset X$. Similarly
$T: X \to X $ is called  \emph{totally bounded} if $T$ maps a bounded
subset of $X$ into the relatively compact subset of $X$. Finally $T:
X \to X$ is called \emph{completely continuous} operator if it is
continuous and totally bounded operator on $X$. It is clear that
every compact operator is totally bounded, but the converse may not
be true.

\begin{definition}\label{Car} \rm
A function $\gamma :J \times \mathbb{R}^{n} \to \mathbb{R}^{n}$ is
called \emph{Carath\'eodory}'s  if
\begin{enumerate}
\item [(i)] the function $(x,y)\to \gamma(x,y,u)$ is measurable
for each   $u\in \mathbb{R}^{n}$,
\item [(ii)] the function $u\to \gamma(x,y,u)$ is continuous for
almost each $(x,y) \in J$.
\end{enumerate}
\end{definition}

A non-empty closed set $K$ in a Banach algebra $X$ is called a
\emph{cone} if
\begin{itemize}
\item [(i)] $K+K\subseteq K$, \
\item [(ii)] $\lambda K\subseteq K$ for $\lambda\in \mathbb{R}, \ \lambda\ge
0$  and
\item [(iii)] $\{-K\}\cap K=0$, where $0$ is the zero element of $X$.
\end{itemize}
The cone $K$ is called to be \emph{positive} if
\begin{itemize}
\item [(iv)] $K\circ K\subseteq K$, where ``$\circ$'' is a
 multiplication composition in $X$.
\end{itemize}
We introduce an order relation, $\le$, in $X$ as follows.
Let $u,v\in X$. Then $u\le v$ if and only if $v-u\in K$.
A cone $K$ is called to
be \emph{normal} if the norm $\|\cdot\|$ is monotone increasing on
$K$. It is known that if the cone $K$ is normal in $X$, then every
order-bounded set in $X$ is norm-bounded.

\begin{lemma}[\cite{D2}]  Let $K$ be
a positive cone in a real Banach algebra $X$ and let $u_{1}, u_{2},
v_{1}, v_{2}\in K$ be such that $u_{1}\le v_{1}$ and $u_{2}\le
v_{2}$. Then $u_{1}u_{2}\le v_{1}v_{2}$.
\end{lemma}

For any $v,w\in X, v\le w$, the order interval $[v,w]$ is a set in
$X$ given by
$$
[v,w]=\{u\in X: v\le u\le w\}.
$$

The nonlinear alternative of Schaefer type proved by Dhage \cite{D1}
is embodied in the following theorem.

\begin{theorem} \label{td}
Let $X$ be a Banach algebra and let $A,B : X \to X$ be two operators
satisfying
\begin{itemize}
\item [(a)] $A$ is  Lipschitz  with a Lipschitz
constant $\alpha$,
\item [(b)] $B$ is compact and continuous, and
\item [(c)] $\alpha M< 1$,   where $M= \|B(X)\|:=\sup\{\|Bu\| : u\in
X\}$.
\end{itemize}
Then either
\begin{itemize}
\item[(i)] the equation $\lambda [Au \,Bu]=u $
 has a solution for $0<\lambda <1$, or
\item[(ii)] the set ${\mathcal  E }=\{u \in X:
\lambda [Au\,Bu] =u,\, 0<\lambda <1 \}$ is unbounded.
\end{itemize}
\end{theorem}

We use the following fixed point theorems by Dhage \cite{D2} for
proving the existence of extremal solutions for our problem under
certain monotonicity conditions.

\begin{theorem}\label{t21}
Let $K$ be a cone in a Banach algebra $X$ and let $v,w\in X$.
Suppose that $A,B: [v,w]\to K$ are two operators such that
\begin{itemize}
\item[(a)] $A$ is completely continuous,
\item[(b)] $B$ is totally bounded,
\item[(c)] $Au_1Bu_2\in [v,w]$ for all $u_1,u_2\in [v,w]$, and
\item[(d)] $A$ and $B$ are nondecreasing.
\end{itemize}
Further if the cone $K$ is positive and normal, then the operator
equation $Au\,Bu=u$ has a least and a greatest positive solution in
$[v,w]$.
\end{theorem}

\begin{theorem} \label{t22}
Let $K$ be a cone in a Banach algebra $X$ and let $v,w\in X$.
Suppose that $A,B: [v,w]\to K$ are two operators such that
\begin{itemize}
\item[(a)] $A$ is Lipschitz with a Lipschitz constant
$\alpha$,
\item[(b)] $B$ is totally bounded, \item[(c)]
$Au_1\,Bu_2\in [v,w]$ for all $u_1,u_2\in [v,w]$, and \item[(d)]
$A$ and $B$ are nondecreasing.
\end{itemize}
Further if the cone $K$ is positive and normal, then the operator
equation $Au\,Bu=u$ has least and a greatest positive solution in
$[v,w]$, whenever $\alpha M<1$, where $M=\|B([v,w])\|:=\sup\{\|Bu\|:
u\in [v,w]\}$.
\end{theorem}

\begin{remark} \label{rmk2.7} \rm
Note that hypothesis (c) of Theorems \ref{t21}  and \ref{t22} holds
if the operators $A$ and $B$ are positive monotone increasing and
there exist elements $v$ and $w$ in $X$ such that $v\le Av\,Bv$ and
$Aw\,Bw\le w$.
\end{remark}


\section{Auxiliary Results}

 Let us start by defining what we mean by a solution of
problem \eqref{e1}-\eqref{e3}. Set
$J':=J\backslash\{(x_{1},y),\dots,(x_{m},y), \;  y\in [0,b]\}$.

\begin{definition}\label{d1} \rm
A function  $u\in PC(J,\mathbb{R}^n)$ whose $r$-derivative exists on $J'$
is said to be a solution of \eqref{e1}-\eqref{e3} if
\begin{itemize}
\item[(i)] the function $(x,y)\mapsto
\frac{u(x,y)}{f(x,y,u(x,y))}$ is absolutely continuous, and
\item[(ii)] $u$ satisfies $^{c}D_{0}^{r}\big(\frac{u(x,y)}{f(x,y,u(x,y))}\big)= g(x,y,u(x,y))$ on $ J'$ and
conditions \eqref{e2}, \eqref{e3}  are satisfied.
\end{itemize}
\end{definition}

Let $f\in C([x_k,x_{k+1}]\times[0,b],\mathbb{R}^{n}\backslash
\{0_{\mathbb{R}^{n}}\})$,
$g\in L^{1}([x_k,x_{k+1}]\times[0,b],\mathbb{R}^{n}) $,
$z_k=(x_k,0)$, and
$$
\mu_{0,k}(x,y)=\frac{u(x,0)}{f(x,0)}+\frac{u(x_k^+,y)}{f(x_k^+,y)}
-\frac{u(x_k^+,0)}{f(x_k^+,0)},
\quad k=0,\dots,m.
$$
For the existence of solutions for the problem
\eqref{e1}-\eqref{e3}, we need the following lemma.

\begin{lemma}\label{L1}
A function $u\in AC([x_k,x_{k+1}]\times[0,b],\mathbb{R}^{n})$,
$k=0,\dots,m $ is a solution of the differential equation
\begin{equation}\label{e1'}
 ^{c}D_{z_k}^{r}\Big(\frac{u}{f}\Big)(x,y)=g(x,y), \quad
 (x,y)\in [x_k,x_{k+1}]\times[0,b],
\end{equation}
if and only if $u(x,y)$ satisfies
\begin{equation}\label{e2'}
u(x,y)=f(x,y)\Big(\mu_{0,k}(x,y)+(I_{z_k}^{r}g)(x,y)\Big),\quad
 (x,y)\in [x_k,x_{k+1}]\times[0,b].
\end{equation}
\end{lemma}

 \begin{proof}
 Let $u(x,y)$ be a solution of \eqref{e1'}.
Then, taking into account the definition
of the derivative $^{c}D_{z_k}^{r}$, we have
$$
I_{z_k^+}^{1-r}( D_{xy}^{2}\frac{u}{f})(x,y)=g(x,y).
$$
Hence, we obtain
$$
I_{z_k^+}^{r}(I_{z_k}^{1-r}D_{xy}^{2}\frac{u}{f}) (x,y)
=(I_{z_k^+}^{r}g)(x,y),
$$
then
$$
I_{z_k^+}^{1}(D_{xy}^{2}\frac{u}{f})( x,y) =( I_{z_k^+}^{r}g)(x,y).
$$
Since
$$
I_{z_k^+}^{1}( D_{xy}^{2}\frac{u}{f})(x,y)
 =\frac{u(x,y)}{f(x,y)} -\frac{u(x,0)}{f(x,0)}
-\frac{u(x_k^+,y)}{f(x_k^+,y)} +\frac{u(x_k^+,0)}{f(x_k^+,0)} ,
$$
we have
$$
u(x,y) =f(x,y)\Big(\mu_{0,k}(x,y)+(I_{z_k}^{r}g)(x,y)\Big).
$$
Now let $u(x,y)$ satisfies \eqref{e2'}. It is clear that $u(x,y)$
satisfies  \eqref{e1'}.
\end{proof}

\begin{corollary} \label{coro3.3}
 The function $u\in AC([x_k,x_{k+1}]\times[0,b],\mathbb{R}^{n})$,
$k=0,\dots,m $  is a solution of the differential equation \eqref{e1}
if and only if $u$ satisfies the equation
\begin{equation} \label{e32}
\begin{aligned}
u(x,y)&=\big[f(x,y,u(x,y))\big]\Big(\mu_k(x,y)\\
&\quad +\frac{1}{\Gamma (r_{1})\Gamma (r_{2})}\int_{0}^{x}
\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}g(s,t,u(s,t))\,dt\,ds\Big),
\end{aligned}
\end{equation}
for $(x,y) \in [x_k,x_{k+1}]\times[0,b]$, where
$$
\mu_k(x,y)=\frac{u(x,0)}{f(x,0,u(x,0))}
+\frac{u(x_k^+,y)}{f(x_k^+,y,u(x_k^+,y))}
-\frac{u(x_k^+,0)}{f(x_k^+,0,u(x_k^+,0))},
\quad k=0,\dots,m.
$$
\end{corollary}

Let $\mu':=\mu_{0,0}$.

\begin{lemma}\label{L2}
 Let  $0< r_1,r_2\leq 1$ and let
$ f: J \to\mathbb{R}^{n}\backslash \{0_{\mathbb{R}^{n}}\}$,
$ g: J \to\mathbb{R}^{n}$ be
continuous. A function $u$ is a solution of the fractional integral
equation
\begin{equation}\label{e4}
u(x,y)=\begin{cases}
f(x,y)\Big[\mu'(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\int_{0}^{x}
\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big],\\
\quad \text{if } (x,y)\in [0,x_{1}]\times[0,b],
\\[3pt]
f(x,y)\Big[\mu'(x,y)+\sum_{i=1}^{k}
\Big(\frac{I_{i}(u(x_{i}^{-},y))}{f(x_{i}^{+},y)}
-\frac{I_{i}(u(x_{i}^{-},0))}{f(x_{i}^{+},0)}\Big) \\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{i=1}^{k}
\int_{x_{i-1}}^{x_{i}}\int_{0}^{y}
(x_{i}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds  \\
+ \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds
\Big],\\
 \quad \text{if  } (x,y)\in (x_{k},x_{k+1}]\times[0,b],\; k=1,\dots,m,
\end{cases}
\end{equation}
 if and only if $u$ is a solution of the
fractional initial-value problem
\begin{gather}\label{e5}
^{c}D^{r}(\frac{u}{f})(x,y)= g(x,y), \quad   (x,y)\in J', \\
\label{e6} u(x_{k}^{+},y)= u(x_{k}^{-},y)+I_{k}(u(x_{k}^{-},y)), \
y\in [0,b], \  k=1,\dots,m.
\end{gather}
\end{lemma}

\begin{proof}
Assume $ u $ satisfies \eqref{e5}-\eqref{e6}. If
$(x,y)\in [0,x_{1}]\times[0,b]$, then
$$
^{c}D^{r}(\frac{u}{f})(x,y)= g(x,y).
$$
Lemma \ref{L1} implies
$$
u(x,y)=f(x,y)\Big(\mu'(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{0}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds\Big).
$$
If $(x,y)\in (x_{1},x_{2}]\times[0,b]$, then Lemma \ref{L1} implies
\begin{align*}
&u(x,y)\\
&=f(x,y)\Big(\mu_{0,1}(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}+\frac{u(x_{1}^{+},y)}{f(x_{1}^{+},y)}
-\frac{u(x_{1}^{+},0)}{f(x_{1}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}+\frac{u(x_{1}^{-},y)}{f(x_{1}^{+},y)}
-\frac{u(x_{1}^{-},0)}{f(x_{1}^{+},0)}
 +\frac{I_{1}(u(x_{1}^{-},y))}{f(x_{1}^{+},y)}-\frac{I_{1}(u(x_{1}^{-},0))}{f(x_{1}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}+\frac{u(x_{1},y)}{f(x_{1}^{+},y)}
-\frac{u(x_{1},0)}{f(x_{1}^{+},0)}
 +\frac{I_{1}(u(x_{1}^{-},y))}{f(x_{1}^{+},y)}-\frac{I_{1}(u(x_{1}^{-},0))}{f(x_{1}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}+\frac{\psi(y)}{f(0,y)}
-\frac{u(0,0)}{f(0,0)}
 +\frac{I_{1}(u(x_{1}^{-},y))}{f(x_{1}^{+},y)}-\frac{I_{1}(u(x_{1}^{-},0))}{f(x_{1}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{0}^{x_{1}}\int_{0}^{y}(x_{1}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\mu'(x,y)+\frac{I_{1}(u(x_{1}^{-},y))}{f(x_{1}^{+},y)}-\frac{I_{1}(u(x_{1}^{-},0))}{f(x_{1}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{0}^{x_{1}}\int_{0}^{y}(x_{1}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big).
\end{align*}
If $(x,y)\in (x_{2},x_{3}]\times[0,b]$, then  from Lemma \ref{L1} we
obtain
\begin{align*}
&u(x,y)\\
&=f(x,y)\Big(\mu_{0,2}(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{2}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}+\frac{u(x_{2}^{+},y)}{f(x_{2}^{+},y)}-\frac{u(x_{2}^{+},0)}{f(x_{2}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{2}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}
 +\frac{u(x_{2}^{-},y)}{f(x_{2}^{+},y)}
 -\frac{u(x_{2}^{-},0)}{f(x_{2}^{+},0)}
 +\frac{I_{2}(u(x_{2}^{-},y))}{f(x_{2}^{+},y)}-\frac{I_{2}(u(x_{2}^{-},0))}{f(x_{2}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{2}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\frac{\varphi(x)}{f(x,0)}
 +\frac{u(x_{2},y)}{f(x_{2}^{+},y)}-\frac{u(x_{2},0)}{f(x_{2}^{+},0)}
 +\frac{I_{2}(u(x_{2}^{-},y))}{f(x_{2}^{+},y)}-\frac{I_{2}(u(x_{2}^{-},0))}{f(x_{2}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{2}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big)\\
&=f(x,y)\Big(\mu'(x,y)+\frac{I_{1}(u(x_{1}^{-},y))}{f(x_{1}^{+},y)}
 -\frac{I_{1}(u(x_{1}^{-},0))}{f(x_{1}^{+},0)}
 +\frac{I_{2}(u(x_{2}^{-},y))}{f(x_{2}^{+},y)}-\frac{I_{2}(u(x_{2}^{-},0))}{f(x_{2}^{+},0)}\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{0}^{x_{1}}\int_{0}^{y}(x_{1}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{1}}^{x_{2}}\int_{0}^{y}(x_{2}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{2}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\Big).
\end{align*}
If $(x,y)\in (x_{k},x_{k+1}]\times[0,b]$ then again from Lemma
\ref{L1} we get \eqref{e4}.

Conversely, assume that $u$ satisfies the impulsive fractional
integral equation \eqref{e4}. If $(x,y)\in [0,x_{1}]\times[0,b]$ and
using the fact that $^{c}D^{r}$ is the left inverse of $I^{r}$ we
get
$$
^{c}D^{r}(\frac{u}{f})(x,y)= g(x,y), \quad \text{for each } (x,y)\in
[0,x_{1}]\times[0,b].
$$
If $(x,y)\in [x_{k},x_{k+1})\times[0,b]$, $k=1,\dots,m$ and using
the fact that $^{c}D^{r}C=0$, where $C$ is a constant, we get
$$
^{c}D^{r}(\frac{u}{f})(x,y)=g(x,y), \text{for each } (x,y)\in
[x_{k},x_{k+1})\times[0,b].
$$
Also, we can easily show that
$$
u(x_{k}^{+},y)=u(x_{k}^{-},y)+ I_{k}(u(x_{k}^{-},y)), \quad y\in
[0,b], k=1,\dots,m.
$$
\end{proof}

Let $\mu:=\mu_{0}$.
\begin{corollary}\label{c2}
 Let  $0< r_1,r_2\leq 1$ and let $ f: J\times\mathbb{R}^{n} \to\mathbb{R}^{n}\backslash \{0_{\mathbb{R}^{n}}\}$, $ g: J\times\mathbb{R}^{n} \to\mathbb{R}^{n}$ be
continuous. A function $u$ is a solution of the fractional integral
equation
\begin{equation}\label{e7}
u(x,y)=\begin{cases}
f(x,y,u(x,y))\Big[\mu(x,y)\\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\int_{0}^{x}\int_{0}^{y}
(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\Big],\\
\quad \text{if } (x,y)\in [0,x_{1}]\times[0,b],
\\[4pt]
f(x,y,u(x,y))\Big[\mu(x,y)+\sum_{i=1}^{k}
\Big(\frac{I_{i}(u(x_{i}^{-},y))}{f(x_{i}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{i}(u(x_{i}^{-},0))}{f(x_{i}^{+},0,u(x_{i}^{+},0))}\Big)\\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{i=1}^{k}
\int_{x_{i-1}}^{x_{i}}\int_{0}^{y}
(x_{i}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\\
+ \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}
 g(s,t,u(s,t))\,dt\,ds\Big],\\
\quad \text{if  } (x,y)\in (x_{k},x_{k+1}]\times[0,b],\ k=1,\dots,m,
\end{cases}
\end{equation}
if and only if $u$ is a solution of the fractional initial-value problem
\begin{gather}\label{e8}
^{c}D^{r}\Big(\frac{u(x,y)}{f(x,y,u(x,y))}\Big)
= g(x,y,u(x,y)), \quad   (x,y)\in J', \\
\label{e9} u(x_{k}^{+},y)= u(x_{k}^{-},y)+I_{k}(u(x_{k}^{-},y)), \
y\in [0,b], \  k=1,\dots,m.
\end{gather}
\end{corollary}


\section{Existence of Solutions}

In this section, we are concerned with the existence of solutions
for the problem \eqref{e1}-\eqref{e3}. The
following hypotheses will be used in the sequel.
\begin{itemize}
\item[(A1)] The function $f$ is continuous on
$J\times \mathbb{R}^{n}$.
\item[(A2)] There exists a  function $\alpha\in
C(J,\mathbb{R}_+)$ such that
$$
\|f(x,y,u)-f(x,y,\overline{u})\|\le \alpha(x,y)\|u-\overline{u}\|
; \quad\text{for all} \ (x,y)\in J\ \text{and} \ u, \overline{u}
\in \mathbb{R}^{n}.$$
\item[(A3)] The function $g$ is Carath\'eodory, and there exists
$h\in L^{\infty}(J,\mathbb{R}_+)$ such that
$$
\|g(x,y,u)\|\le h(x,y);  \quad\text{a.e. } (x,y)\in J,\ \text{for
all} \ u\in \mathbb{R}^{n}.
$$
\item[(A4)] There exists a  function $\beta\in
C(J,\mathbb{R}_+)$ such that
$$
\Big\|\frac{I_k(u)}{f(x,y,u)}\Big\|\le \beta(x,y); \quad\text{for
all} \ (x,y)\in J\ \text{and} \ u\in \mathbb{R}^{n}.$$
\end{itemize}

\begin{theorem}\label{tc}
Assume that hypotheses {\rm (A1)--(A4)} hold. If
\begin{equation}\label{e3'}
\|\alpha\|_{\infty}\Big[\|\mu\|_{\infty}+ 2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}\Big]<1,
\end{equation}
Then the initial-value problem \eqref{e1}-\eqref{e3} has at
least one solution on $J$.
\end{theorem}

\begin{proof}
Let $X:=PC(J,\mathbb{R}^{n})$. Define two
operators $A$ and $B$ on $X$ by
\begin{equation}\label{e33}
Au(x,y)=f(x,y,u(x,y));\quad (x,y)\in J,
\end{equation}
and
\begin{equation} \label{e34}
\begin{aligned}
Bu(x,y)&=\mu(x,y)+\sum_{i=1}^{m}
\Big(\frac{I_{i}(u(x_{i}^{-},y))}{f(x_{i}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{i}(u(x_{i}^{-},0))}{f(x_{i}^{+},0,u(x_{i}^{+},0))}\Big)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}
 \sum_{i=1}^{m}\int_{x_{i-1}}^{x_{i}}\int_{0}^{y}
(x_{i}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds \\
&\quad + \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{m}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}
 g(s,t,u(s,t))\,dt\,ds;
\end{aligned}
\end{equation}
with $(x,y)\in J$.
Solving \eqref{e1}-\eqref{e3} is equivalent to solving \eqref{e32},
which is further equivalent to solving the operator equation
\begin{equation}\label{e35}
Au(x,y)\,Bu(x,y)=u(x,y), \quad (x,y)\in J.
\end{equation}
We  show that  operators $A$  and $B$ satisfy all the assumptions of
Theorem \ref{td}. First we shall show that $A$ is a Lipschitz. Let
$u_1, u_2\in X$. Then by (A2),
\begin{align*}
\|Au_1(x,y)-Au_2(x,y)\|
&=\|f(x,y,u_1(x,y))-f(x,y,u_2(x,y))\|\\
&\le \alpha(x,y)\|u_1(x,y)-u_2(x,y)\|\\
&\le \|\alpha\|_{\infty}\|u_1-u_2\|_{PC}.
\end{align*}
Taking the maximum over $(x,y)$, in the above inequality yields
$$
\|Au_1-Au_2\|_{PC}\le \|\alpha\|_{\infty}\|u_1-u_2\|_{PC},
$$
and so $A$ is a Lipschitz with a Lipschitz constant
$\|\alpha\|_{\infty}$.

Next, we show that $B$  is compact operator on $X$. Let $\{u_n\}$ be
a sequence in $X$.   From (A3) and (A4) it follows that
$$
\|Bu_n\|_{PC} \le \|\mu\|_{\infty}+ 2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}.
$$
As a result $\{Bu_n: n\in {\mathbb N}\}$ is a uniformly bounded set
in $X$.

Let $(\tau_{1},y_{1}), (\tau_{2},y_{2})\in J, \,\,\,
\tau_{1}<\tau_{2}$ and $y_{1}<y_{2}$, then for each $(x,y)\in J$,
\begin{align*}
&\|B(u_n)(\tau_{2},y_{2})-B(u_n)(\tau_{1},y_{1})\|\\
&\leq\| \mu(\tau_{1},y_{1}) -\mu(
\tau_{2},y_{2})\|+\sum_{k=1}^{m}
\big\|\frac{I_{k}(u(x_{k}^{-},y_{1}))}{f(x_{k}^{+},y_{1},u(x_{i}^{+},
 y_{1}))}-\frac{I_{k}(u(x_{k}^{-},y_{2}))}{f(x_{k}^{+},y_{2},
 u(x_{i}^{+},y_{2}))}\big\|\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y_{1}}
(x_{k}-s)^{r_1-1}[(y_{2}-t)^{r_2-1}-(y_{1}-t)^{r_2-1}]\\
&\quad\times g(s,t,u(s,t))\,dt\,ds\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_{k}}\int_{y_{1}}^{y_{2}}
(x_{k}-s)^{r_1-1}(y_{2}-t)^{r_2-1}\|g(s,t,u(s,t))\|\,dt\,ds\\
&\quad +\frac{1}{\Gamma(r_{1})\Gamma(r_{2})}\int_0^{\tau_1}\int_0^{y_1}
[(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}-(\tau_1-s)^{r_{1}-1}(y_1-t)^{r_{2}-1}]\\
&\quad\times g(s,t,u(s,t))\,dt\,ds\\
&\quad+\frac{1}{\Gamma(r_{1})\Gamma(r_{2})}\int_{\tau_1}^{\tau_2}\int_{y_1}^{y_2}(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}\|g(s,t,u(s,t))\|\,dt\,ds\\
&\quad+\frac{1}{\Gamma(r_{1})\Gamma(r_{2})}\int_{0}^{\tau_1}\int_{y_1}^{y_2}(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}\|g(s,t,u(s,t))\|\,dt\,ds\\
&\quad+\frac{1}{\Gamma(r_{1})\Gamma(r_{2})}\int_{\tau_1}^{\tau_2}\int_{0}^{y_1}(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}\|g(s,t,u(s,t))\|\,dt\,ds\\
&\leq\|\mu(\tau_{1},y_{1}) -\mu(
\tau_{2},y_{2})\|+\sum_{k=1}^{m}\Big\|\frac{I_{k}(u(x_{k}^{-},y_{1}))}{f(x_{k}^{+},y_{1},u(x_{i}^{+},y_{1}))}
-\frac{I_{k}(u(x_{k}^{-},y_{2}))}{f(x_{k}^{+},y_{2},u(x_{i}^{+},y_{2}))}\Big\|\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y_{1}}
(x_{k}-s)^{r_1-1}[(y_{2}-t)^{r_2-1}-(y_{1}-t)^{r_2-1}]\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_{k}}\int_{y_{1}}^{y_{2}}
(x_{k}-s)^{r_1-1}(y_{2}-t)^{r_2-1}\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_{1})\Gamma(r_{2})}\int_0^{\tau_1}\int_0^{y_1}
[(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}-(\tau_1-s)^{r_{1}-1}(y_1-t)^{r_{2}-1}]\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_{1})\Gamma(r_{2})}\int_{\tau_1}^{\tau_2}\int_{y_1}^{y_2}(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_{1})\Gamma(r_{2})}\int_{0}^{\tau_1}\int_{y_1}^{y_2}(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_{1})\Gamma(r_{2})}\int_{\tau_1}^{\tau_2}\int_{0}^{y_1}(\tau_2-s)^{r_{1}-1}(y_2-t)^{r_{2}-1}\,dt\,ds\\
&\leq\|\mu(\tau_{1},y_{1}) -\mu(
\tau_{2},y_{2})\|+\sum_{k=1}^{m}\Big\|\frac{I_{k}(u(x_{k}^{-},y_{1}))}{f(x_{k}^{+},y_{1},u(x_{i}^{+},y_{1}))}
-\frac{I_{k}(u(x_{k}^{-},y_{2}))}{f(x_{k}^{+},y_{2},u(x_{i}^{+},y_{2}))}\Big\|\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y_{1}}
(x_{k}-s)^{r_1-1}[(y_{2}-t)^{r_2-1}-(y_{1}-t)^{r_2-1}]\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_{k}}\int_{y_{1}}^{y_{2}}
(x_{k}-s)^{r_1-1}(y_{2}-t)^{r_2-1}\,dt\,ds\\
&\quad+\frac{\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)}[2y_2^{r_2}(\tau_{2}-\tau_{1})^{r_{1}
}+2\tau_2^{r_1}(y_{2}-y_{1})^{r_{2} }\\
&\quad+\tau_{1}^{r_{1} }y_{1}^{r_{2} }-\tau_{2}^{r_{1} }y_{2}^{r_{2}
}-2(\tau_{2}-\tau_{1})^{r_{1} }(y_{2}-y_{1})^{r_{2} }].
\end{align*}
As $\tau_{1}\to \tau_{2}$ and $y_{1}\to y_{2}$, the right-hand side
of the above inequality tends to zero.
 From this we conclude that $\{Bu_n: \ n\in {\mathbb N}\}$ is an
equicontinuous set in $X$. Hence $B:X\to X$ is compact by
Arzel\`a-Ascoli theorem. Moreover,
\[
M= \|B(X)\|
\le\|\mu\|_{\infty}+ 2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)},
\]
and so,
$$
\alpha M\le \|\alpha\|_{\infty}\Big(\|\mu\|_{\infty}+
2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}\Big)<1,
$$
by assumption \eqref{e3'}. To finish, it remain to show that either
the conclusion (i) or the conclusion (ii) of Theorem \ref{td} holds.
We now will show that the conclusion (ii) is not possible. Let $u\in
X$ be any solution to \eqref{e1}-\eqref{e3}, then for any
$\lambda\in (0,1)$ we have

\begin{align*}
&u(x,y)\\
&=\lambda f(x,y,u(x,y))\Big[\mu(x,y)+\sum_{0<x_{k}<x}
 \Big(\frac{I_{k}(u(x_{k}^{-},y))}{f(x_{k}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{k}(u(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}\Big)\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\\
&\quad+ \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\Big].
\end{align*}
for $(x,y)\in J$. Therefore,
\begin{align*}
\|u(x,y)\|
&\le \|f(x,y,u(x,y))\|\Big(
\|\mu(x,y)\|+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)}\Big)\\
&\le \big[\|f(x,y,u(x,y))-f(x,y,0)\|+\|f(x,y,0)\|\big]\\
&\quad\times\Big(\|\mu(x,y)\|+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)}\Big)\\
&\le \big[\|\alpha\|_{\infty}\|u(x,y)\|+f^{*}\big]
\Big(\|\mu(x,y)\|+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}\Big)\\
&\le
[\|\alpha\|_{\infty}\|u\|_{PC}+f^{*}]
\Big(\|\mu\|_{\infty}+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}\Big),
\end{align*}
where $f^{*}=\sup\{\|f(x,y,0)\|: (x,y)\in J\}$, and consequently
$$
\|u\|_{PC}\le
\frac{f^{*}\big[\|\mu\|_{\infty}+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
 \Gamma(r_{2}+1)}\big]}{1-\|\alpha\|_{\infty}
\big[\|\mu\|_{\infty}+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}\big]}:=M.
$$
Thus the conclusion (ii)  of Theorem \ref{td} does not hold.
Therefore  \eqref{e1}-\eqref{e3} has
a solution on $J$.
\end{proof}

\section{Existence of extremal solutions}

We equip the space $PC(J,\mathbb{R}^{n})$ with the order relation
$\leq$ with the help of the cone defined by
$$
K=\{u\in PC(J,\mathbb{R}^{n}):  u(x,y) \geq 0, \; \forall (x,y)\in
J\}.
$$
Thus  $u\leq \bar{u}$ if and only if $u(x,y)\leq \bar{u}(x,y)$
for each $(x,y)\in J$.

It is well-known that the cone $K$ is positive and normal in $PC(J,
\mathbb{R}^{n})$ (\cite{HL}).
If $\underline{u}, \bar{u}\in
C(J,\mathbb{R}^{n})$ and ${\underline u}\leq \bar{u}$, we put
$$
[\underline{u}, \overline{u}]=\{u\in PC(J,\mathbb{R}^{n}):
\underline{u} \leq u\leq \bar{u}\}.
$$

\begin{definition} \label{def5.1} \rm
A function $\gamma :J\times \mathbb{R}^{n}\to \mathbb{R}^{n}$
is called \emph{Chandrabhan}  if
\begin{itemize}
\item [(i)] the function $(x,y)\to
\gamma(x,y,u)$ is measurable for each  $u\in \mathbb{R}^{n}$,
\item[(ii)] the function $u\to \gamma(x,y,u)$ is nondecreasing
for almost each $(x,y) \in J$.
\end{itemize}
\end{definition}

\begin{definition} \label{def5.2} \rm
A function $\underline{u}(\cdot,\cdot)\in PC(J,\mathbb{R}^{n})$ is
said to be a lower solution of \eqref{e1}-\eqref{e3} if
\begin{gather*}
^{c}D_{0}^{r}\Big[\frac{\underline{u}(x,y)}{f(x,y,
\underline{u}(x,y))}\Big]
\le g(x,y, \underline{u}(x,y)), \quad (x,y)\in J, \;
x\neq x_k, \; k=1,\ldots,m, \\
\underline{u}(x_k^+,y)\le\underline{u}(x_k^-,y)+I_{k}(\underline{u}
(x_k^-,y)), \ y\in [0,b]; \; k=1,\dots,m,\\
\underline{u}(x,0)\le\varphi(x), \quad \underline{u}(0,y)\le\psi(y),
\quad  (x,y)\in J.
\end{gather*}
 Similarly a function $\bar{
u}(\cdot,\cdot)\in PC (J,\mathbb{R}^{n})$ is said to be an upper
solution of \eqref{e1}-\eqref{e3} if
\begin{gather*}
^{c}D_{0}^{r}\Big[\frac{\bar{u}(x,y)}{f(x,y,\bar{u}(x,y))}\Big]\ge
g(x,y, \bar{u}(x,y)), \quad (x,y)\in J,\; x\neq x_k, \; k=1,\ldots,m, \\
\bar{u}(x_k^+,y)\ge\bar{u}(x_k^-,y)+I_{k}(\bar{u}(x_k^-,y)),
\quad y\in [0,b]; \; k=1,\dots,m,\\
\bar{u}(x,0)\ge\varphi(x), \quad \bar{u}(0,y)\ge\psi(y), \quad
 (x,y)\in J.
\end{gather*}
\end{definition}

\begin{definition} \label{def5.3} \rm
A solution $u_M$ of the problem \eqref{e1}-\eqref{e3} is said to be
maximal if for any other solution $u$ to the problem
\eqref{e1}-\eqref{e3} one has $u(x,y)\leq u_M(x,y)$, for all
$(x,y)\in J$. Again a solution $u_m$ of the problem
\eqref{e1}-\eqref{e3} is said to be minimal if $u_m(x,y)\leq
u(x,y)$, for all $(x,y)\in J$ where $u$ is any solution of the
problem \eqref{e1}-\eqref{e3} on $J$.
\end{definition}

The following hypotheses will be used in the sequel.
\begin{itemize}
\item[(H1)] $f :J\times \mathbb{R}^{n}_{+}\to
\mathbb{R}^{n}_{+}\setminus\{0\}$, $g :J\times \mathbb{R}^{n}_{+}\to
\mathbb{R}^{n}_{+}$, $\psi(y)\ge 0$ on $[0,b]$
 and
 $$
 \frac{\varphi(x)}{f(x,0,\varphi(x))}
\ge\frac{\varphi(0)}{f(0,0,\varphi(0))} \quad \text{for all }
 x\in [0,a].$$
\item[(H2)] The functions $f$ and $g$ are Chandrabhan.
\item[(H3)] There exists a function
 $\tilde h\in L^{\infty}(J,\mathbb{R}_+)$ such that
$$
\|g(x,y,u)\|\le \tilde h(x,y) , \quad\text{a.e. } (x,y)\in J,
\text{ for all }  u\in \mathbb{R}^{n}.
$$
\item[(H4)] There exists a function $\tilde \beta\in C(J,\mathbb{R}_+)$
such that
$$
\big\|\frac{I_k(u)}{f(x,y,u)}\big\|\le \tilde \beta(x,y) ,
\quad\text{for all }  (x,y)\in J, \text{ for all }
 u\in \mathbb{R}^{n}.
$$
\item[(H5)] The problem \eqref{e1}-\eqref{e3} has a lower solution
$\underline{u}$
and an upper solution $\overline{u}$ with $\underline{u}\le
\overline{u}$.
\end{itemize}

\begin{theorem}\label{t31}
Assume that hypotheses {\rm (A2), (H1)--(H5)} hold. If
$$
\|\alpha\|_{\infty}\big[\|\mu\|_{\infty}+2m\|\tilde
\beta\|+\frac{2a^{r_{1}}b^{r_{2}}\|\tilde
h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)}\big]<1,
$$
then  \eqref{e1}-\eqref{e3} has a minimal and a maximal
positive solution on $J$.
\end{theorem}

 \begin{proof}
Let $X=PC(J,\mathbb{R}^{n})$ and consider a
closed interval $[\underline{u}, \overline{u}]$ in $X$ which is
well defined in view of hypothesis (H5). Define two operators $A,B :
[\underline{u}, \overline{u}]\to X$   by \eqref{e33} and
\eqref{e34}, respectively. Clearly $A$ and $B$ define the operators
$A, B : [\underline{u}, \overline{u}]\to K$.

Now solving \eqref{e1}-\eqref{e3} is equivalent to solving
\eqref{e32}, which is further equivalent to solving the operator
equation
\begin{equation}\label{e41}
Au(x,y)\,Bu(x,y)=u(x,y), \quad (x,y)\in J.
\end{equation}
We  show that  operators $A$  and $B$ satisfy all the assumptions of
Theorem \ref{t22}. As in Theorem \ref{tc} we can prove that $A$ is
Lipschitz with a Lipschitz constant $\|\alpha\|_{\infty}$ and $B$ is
completely continuous operator on $[\underline{u}, \overline{u}]$.
Now hypothesis (H2) implies that $A$ and $B$ are nondecreasing on
$[{\underline u},\overline{u}]$. To see this, let $u_1,u_2\in
[{\underline u},\overline{u}]$ be such that $u_1\le u_2$. Then by
(H2), we obtain
$$
Au_1(x,y)=f(x,y,u_1(x,y))\le f(x,y,u_2(x,y))=Au_2(x,y),\quad \forall
(x,y) \in J,
$$
and
\begin{align*}
&Bu_1(x,y)\\
&=\mu(x,y)+\sum_{0<x_{k}<x}\Big(\frac{I_{k}(u_1(x_{k}^{-},y))}{f(x_{k}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{k}(u_1(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}\Big)\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u_1(s,t))\,dt\,ds\\
&\quad + \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u_1(s,t))\,dt\,ds\\
&\le\mu(x,y)+\sum_{0<x_{k}<x}\Big(\frac{I_{k}(u_2(x_{k}^{-},y))}{f(x_{k}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{k}(u_2(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}\Big)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u_2(s,t))\,dt\,ds\\
&\quad + \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u_2(s,t))\,dt\,ds\\
&= Bu_2(x,y), \quad \forall (x,y)\in J.
\end{align*}
So $A$ and $B$ are nondecreasing operators on $[{\underline
u},\overline{u}]$. Again hypothesis (H5) implies
\begin{align*}
&\underline{u}(x,y)\\
&= [f(x,y,\underline{u}(x,y))]\Big(\mu(x,y)+\sum_{0<x_{k}<x}
\Big(\frac{I_{k}(\underline{u}(x_{k}^{-},y))}{f(x_{k}^{+},
y,u(x_{i}^{+},y))}
-\frac{I_{k}(\underline{u}(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}
\Big)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}
\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,\underline{u}(s,t))\,dt\,ds \\
&\quad + \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}
g(s,t,\underline{u}(s,t))\,dt\,ds\Big)\\
&\le [f(x,y,u(x,y))]\Big(\mu(x,y)+\sum_{0<x_{k}<x}
\Big(\frac{I_{k}(u(x_{k}^{-},y))}{f(x_{k}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{k}(u(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}\Big)\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}
\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\\
&\quad + \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\Big)\\
&\le
[f(x,y,\overline{u}(x,y))]\Big(\mu(x,y)+\sum_{0<x_{k}<x}
\Big(\frac{I_{k}(\overline{u}(x_{k}^{-},y))}{f(x_{k}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{k}(\overline{u}(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}
\Big)\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}
\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,\overline{u}(s,t))\,dt\,ds \\
&\quad+ \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}
g(s,t,\overline{u}(s,t))\,dt\,ds\Big)\\
&\le \overline{u}(x,y),
\end{align*}
for all $(x,y)\in J$ and $u\in [{\underline u},\overline{u}]$. As a
result
$$
\underline{u}(x,y)\le Au(x,y)Bu(x,y)\le \overline{u}(x,y), \quad
\forall (x,y)\in J  \text{ and } u\in [{\underline u},\overline{u}].
$$
Hence $Au \,Bu\in [{\underline u},\overline{u}]$, for all $u\in
[{\underline u},\overline{u}]$.

Notice for any $u\in [{\underline u},\overline{u}]$,
\begin{align*}
M&= \|B([{\underline u},\overline{u}])\| \\
&\le \|\mu(x,y)\|
+\Big\|\sum_{0<x_{k}<x}\Big(\frac{I_{k}(u(x_{k}^{-},y))}{f(x_{k}^{+},y,u(x_{i}^{+},y))}
-\frac{I_{k}(u(x_{k}^{-},0))}{f(x_{k}^{+},0,u(x_{i}^{+},0))}\Big)\\
&\quad+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}\int_{x_{k-1}}^{x_{k}}\int_{0}^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\\
&\quad + \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_{0}^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t,u(s,t))\,dt\,ds\Big\|\\
&\le \|\mu\|_{\infty}+2m\|\tilde \beta\|+
\frac{2a^{r_{1}}b^{r_{2}}\|\tilde
h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)}.
\end{align*}
and so,
$$
\alpha M\le \|\alpha\|_{\infty}\Big( \|\mu\|_{\infty}+2m\|\tilde
\beta\|+ \frac{2a^{r_{1}}b^{r_{2}}\|\tilde
h\|_{L^{\infty}}}{\Gamma(r_{1}+1)\Gamma(r_{2}+1)}\Big)<1.
$$
Thus the operators $A$ and $B$ satisfy all the conditions of Theorem
\ref{t22} and so the operator equation \eqref{e34} has a least and a
greatest solution in $[\underline{u}, \overline{u}]$. This further
implies that the problem \eqref{e1}-\eqref{e3} has a minimal and a
maximal positive solution on $J$.
\end{proof}

\begin{theorem}\label{t32}
Assume that hypotheses {\rm (A1), (H1)--(H5)} hold. Then
\eqref{e1}-\eqref{e3} has a minimal and a maximal positive solution
on $J$.
\end{theorem}

 \begin{proof}
Let $X=PC(J,\mathbb{R}^{n})$. Consider the
order interval $[{\underline u},\overline{u}]$ in $X$ and define
two operators $A$ and $B$ on $[{\underline u},\overline{u}]$ by
\eqref{e33} and \eqref{e34} respectively. Then the problem
\eqref{e1}-\eqref{e3} is transformed into an operator equation
$Au(x,y)\,Bu(x,y)=u(x,y)$, $(x,y)\in J$ in a Banach algebra $X$.
Notice that(H1) implies $A,B: [{\underline u},\overline{u}]\to K$.
Since the cone $K$  in $X$ is normal, $[{\underline u},\overline{
u}]$ is a norm bounded set in $X$.

Next we show that $A$ is completely continuous on $[{\underline
u},\overline{u}]$. Now the cone $K$ in $X$ is normal, so the order
interval $[{\underline u}, \overline{u}]$ is norm-bounded. Hence
there exists a constant $r> 0$ such that $\|u\|\le r$ for all $u\in
[{\underline u},\overline{u}]$. As $f$ is continuous on compact set
$J\times[-r,r]$, it attains its maximum, say $M$. Therefore, for any
subset $S$ of $[{\underline u},\overline{u}]$ we have
\begin{align*}
\|A(S)\|&= \sup\{ \|Au\| : u\in S\}\\
&=\sup\Big\{ \sup_{(x,y)\in J} \|f(x,y,
u(x,y))\|: u\in S\Big\}\\
& \le \sup\Big\{\sup_{(x,y)\in J}\|f(x,y,
u)\| :u\in [-r,r]\Big\}
\le M.
\end{align*}
This shows that $A(S)$ is a uniformly bounded subset of $X$.

We note that the function $f(x,y,u)$ is uniformly continuous on
$J\times[-r,r]$. Therefore, for any $(\tau_1,y_1),(\tau_2,y_2)\in J$
we have
$$
\|f(\tau_1,y_1, u)-f(\tau_2,y_2, u)\|\to 0\quad \mbox{as }
 (\tau_1,y_1)\to (\tau_2,y_2),
$$
for all $u\in [-r,r]$. Similarly for any $u_1,u_2\in [-r,r]$
$$
\|f(x,y,u_1)-f(x,y,u_2)\|\to 0\quad \mbox{as } u_1\to u_2,
$$
for all $(x,y)\in J$. Hence for any $(\tau_1,y_1),(\tau_2,y_2)\in J$
and for any $u\in S$ one has
\begin{align*}
\|Au(\tau_1,y_1)-Au(\tau_2,y_2)\|&=\|f(\tau_1,y_1,
u(\tau_1,y_1))-f(\tau_2,y_2, u(\tau_2,y_2))\|\\
&\le \|f(\tau_1,y_1, u(\tau_1,y_1))-f(\tau_2,y_2, u(\tau_1,y_1)\|\\
&\quad +\|f(\tau_2,y_2, u(\tau_1,y_1))-f(\tau_2,y_2,
u(\tau_2,y_2))\|\\
&\to 0\quad \mbox{as } (\tau_1,y_1)\to (\tau_2,y_2).
\end{align*}
This shows that $A(S)$ is an equicontinuous set in $K$. Now an
application of Arzel\`a-Ascoli theorem yields that $A$ is a
completely continuous operator on $[{\underline u},\overline{u}]$.
\end{proof}

Next it can be shown as in the proof of Theorem \ref{t31} that $B$
is a compact operator on $[{\underline u},\overline{u}]$. Now an
application of Theorem \ref{t21} yields that the problem
\eqref{e1}-\eqref{e3} has a minimal and maximal positive solution on
$J$.

\section{An Example}

As an application of our results we consider the following partial
hyperbolic functional differential equations of the form
\begin{gather}\label{ex1}
^{c}D_{0}^{r}\Big(\frac{u(x,y)}{f(x,y,u(x,y))}\Big)
=g(x,y,u(x,y)),\quad (x,y)\in [0,1]\times [0,1], \\
\label{ex2}
u\Big(\frac{1}{2}^+,y\Big)=u\Big(\frac{1}{2}^-,y\Big)
+I_{1}\Big(u\Big(\frac{1}{2}^-,y\Big)\Big),
\quad   y\in [0,1], \\
\label{ex3}
u(x,0)=\varphi (x), \quad x\in [0,1], \quad u(0,y)=\psi (y),
\quad y\in [0,1],
\end{gather}
where $f,g : [0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$ and
$I_{1}:\mathbb{R}\to\mathbb{R}$ are  defined by
\begin{gather*}
f(x,y,u)= \frac{1}{e^{x+y+10}(1+|u|)},\\
g(x,y,u)=\frac{1}{e^{x+y+8}(1+u^{2})},\\
I_{1}(u)=\frac{(8+e^{-10})^{2}}{512e^{10}(1+|u|)^{2}}.
\end{gather*}
The functions $\varphi, \psi:[0,1]\to \mathbb{R}$ are defined by
$$
\varphi(x)=\begin{cases}
\frac{x^2}{2}e^{-10}; & \text{if }  x\in [0,\frac{1}{2}],\\
x^2e^{-10}; & \text{if }   x\in (\frac{1}{2},1],
\end{cases}
$$
and
$$
\psi(y)=ye^{-10}, \quad \text{for all }  y\in [0,1].
$$
We show that the functions $\varphi, \psi, f, g$ and $I_1$ satisfy
all the hypotheses of Theorem \ref{tc}.  Clearly, the function $f$
satisfies (A1) and (A2) with $\alpha(x,y)=\frac{1}{e^{x+y+10}}$ and
$$
\|\alpha\|_{\infty}=1/e^{10}.
$$
Also, the function $g$
satisfies (A3) with $h(x,y)=\frac{1}{e^{x+y+8}}$ and
$$
\|h\|_{L^{\infty}}=1/e^{8}.
$$
Finally, condition (A4) holds
with $\beta(x,y)=\frac{81e^{x+y}}{512}$ and
$\|\beta\|_{\infty}=\frac{81e^{2}}{512}$.
A simple computation gives
$\mu(x,y)<4e$.
 Condition \eqref{e3'} holds. Indeed
\begin{align*}
&\|\alpha\|_{\infty}\Big[\|\mu\|_{\infty}+2m\|\beta\|_{\infty}+
\frac{2a^{r_{1}}b^{r_{2}}\|h\|_{L^{\infty}}}{\Gamma(r_{1}+1)
\Gamma(r_{2}+1)}\Big]\\
&<\frac{1}{e^{10}}
\Big[4e+\frac{81e^{2}}{256}+\frac{2}{e^{8}\Gamma(r_{1}+1)\Gamma(r_{2}+1)}
\Big]<1,
\end{align*}
for each $(r_1,r_2)\in (0,1]\times (0,1]$.
Hence by Theorem \ref{tc}, problem \eqref{ex1}-\eqref{ex3} has a
solution defined on $[0,1]\times [0,1]$.

\begin{thebibliography}{00}

\bibitem{AbBe1} S. Abbas and M. Benchohra;
 Partial hyperbolic differential
equations with finite delay involving the Caputo fractional
derivative, {\em Commun. Math. Anal.} {\bf 7} (2009), 62-72.

\bibitem{AbBe2} S. Abbas and M. Benchohra;
 Darboux problem for perturbed
partial differential equations of fractional order with finite
delay, {\em Nonlinear Anal. Hybrid Syst.} {\bf 3} (2009), 597-604.

\bibitem{AbBe3} S. Abbas and M. Benchohra;
 Upper and lower solutions method for impulsive
partial hyperbolic differential equations with fractional order,
{\em Nonlinear Anal. Hybrid Syst.} {\bf 4} (2010), 604-613.

\bibitem{AbBe4} S. Abbas and M. Benchohra;
 Darboux problem for fractional order discontinuous hyperbolic
partial differential equations in Banach algebras, {\em Math.
Bohemica} (to appear).

\bibitem{ABH1} R.P Agarwal, M. Benchohra and  S. Hamani;
A survey on existence result for boundary value problems of
nonlinear fractional differential equations and inclusions, {\em
Acta. Appl. Math.} {\bf 109} (3) (2010), 973-1033.

\bibitem{ABD} A. Arara, B. D. Karande, M. Benchohra and B. C. Dhage;
Existence theorems for a class of hyperbolic differential equations
in Banach algebras, {\em J. Nat. Physic. Sci.} {\bf 20} (1-2)
(2006), 51-63.

\bibitem{BBO} A. Belarbi, M. Benchohra and A. Ouahab;
 Uniqueness results for fractional functional differential equations with
infinite delay in Fr\'echet spaces, {\em Appl. Anal.} {\bf 85}
(2006), 1459-1470.

\bibitem{BeHaGr} M. Benchohra,  J. R. Graef and S. Hamani;
Existence results for boundary value problems of nonlinear
fractional differential equations with integral conditions, {\em
Appl. Anal.} {\bf 87} (7) (2008), 851-863.

\bibitem{BeHaNt} M. Benchohra, S. Hamani and S. K. Ntouyas;
Boundary value problems for differential equations with fractional order, {\em
Surv. Math. Appl.} {\bf 3} (2008), 1-12.

\bibitem{BeHeNtOu} M. Benchohra, J. Henderson, S. K. Ntouyas
and A. Ouahab;
Existence results for functional differential equations of
fractional order, {\it J. Math. Anal. Appl.} {\bf 338} (2008),
1340-1350.

\bibitem{BHN} M. Benchohra, J. Henderson and S. K. Ntouyas;
{\em Impulsive Differential Equations and Inclusions},
Hindawi Publishing Corporation,  Vol 2, New York, 2006.

\bibitem{DaKu1} M. Dawidowski and I. Kubiaczyk;
 An existence theorem for the generalized hyperbolic
equation $z''_{xy}\in F(x,y,z)$ in Banach space,
{\em Ann. Soc. Math. Pol. Ser. I,
Comment. Math.}, {\bf 30} (1) (1990), 41-49.

\bibitem {D1} B. C. Dhage;
 A nonlinear alternative in Banach algebras with
applications to functional differential equations, {\em Nonlinear
Funct. Anal. \&  Appl.} {\bf 8} (2004), 563--575.

\bibitem {D2} B. C. Dhage;
 Some algebraic  fixed
point theorems for multi-valued mappings with applications, {\em
Diss. Math. Differential Inclusions, Control \& Optim.} {\bf 26}
(2006), 5-55.

\bibitem{DhNt} B. C. Dhage, S. K. Ntouyas;
 Existence of solutions for discontinuous hyperbolic partial differential
equations in Banach algebras, {\em Electron. J. Differential
Equations} Vol. 2006 (2006), No. 28, pp. 1-10.

\bibitem{DiFo} K. Diethelm and N. J. Ford;
Analysis of fractional differential equations,
{\it J. Math. Anal. Appl.} {\bf 265} (2002), 229-248.

\bibitem{HL} S. Heikkila and V. Lakshmikantham;
 \emph{ Monotone Iterative Techniques for Discontinuous Nonlinear
Differential Equations}, Pure and Applied Mathematics, Marcel
Dekker, New York, 1994.

\bibitem{Hi} R. Hilfer;
 {\it Applications of Fractional Calculus in Physics},
World Scientific, Singapore, 2000.


\bibitem{Kam} Z. Kamont;
{\em Hyperbolic Functional Differential
Inequalities and Applications}. Kluwer
Academic Publishers, Dordrecht, 1999.


\bibitem{KiSrJuTr} A. A. Kilbas, Hari M. Srivastava,
and Juan J. Trujillo;
{\it Theory and Applications of Fractional Differential Equations}.
Elsevier Science B.V., Amsterdam, 2006.

\bibitem{LaBaSi} V. Lakshmikantham, D. D. Bainov and P. S. Simeonov;
\textit{Theory of Impulsive Differential Equations}, World Scientific,
 Singapore, 1989.

\bibitem{LLV} V. Lakshmikantham, S. Leela and J. Vasundhara;
 {\em Theory of Fractional Dynamic Systems}, Cambridge Academic
Publishers, Cambridge, 2009.


\bibitem{MeScKiNo} F. Metzler, W. Schick, H. G. Kilian and T. F.
Nonnenmacher;
Relaxation in filled polymers: A fractional calculus
approach, {\it J. Chem. Phys.} {\bf 103} (1995), 7180-7186.

\bibitem{Pod} I. Podlubny;
 {\em Fractional Differential Equation}, Academic  Press, San Diego, 1999.

\bibitem{SaKiMa1} S. G. Samko, A. A. Kilbas and O. I. Marichev;
 {\it Fractional Integrals and Derivatives. Theory and Applications},
Gordon and Breach, Yverdon, 1993.

\bibitem{SaPe} A. M. Samoilenko and N. A. Perestyuk;
 \textit{Impulsive Differential Equations, World Scientific},
Singapore, 1995.

\bibitem{ViGo} A. N. Vityuk and A. V. Golushkov;
 Existence of solutions of
systems of partial differential equations of fractional order, {\it
Nonlinear Oscil.} {\bf 7} (3) (2004), 318-325.

\end{thebibliography}

\end{document}