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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 196, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/196\hfil Existence of solutions]
{Existence of solutions to fractional-order impulsive
hyperbolic partial differential inclusions}

\author[S. Abbas, M. Benchohra \hfil EJDE-2014/196\hfilneg]
{Sa\"id Abbas, Mouffak Benchohra}  % in alphabetical order

\address{Sa\"id Abbas \newline
Laboratory of Mathematics, University of Sa\"{\i}da,
PO Box 138, 20000 Sa\"{\i}da, Algeria}
\email{abbasmsaid@yahoo.fr}

\address{Mouffak Benchohra \newline
Laboratory of Mathematics, University of Sidi Bel-Abb\`es,
PO Box 89, 22000, Sidi Bel-Abb\`es, Algeria \newline
Department of Mathematics, Faculty of Science, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{benchohra@univ-sba.dz}

\thanks{Submitted February 3, 2014. Published September 18, 2014.}
\subjclass[2000]{26A33, 34A60}
\keywords{Impulsive hyperbolic differential inclusions;  fractional order; 
\hfill\break\indent upper solution; lower solution;
 left-sided mixed Riemann-Liouville integral;
\hfill\break\indent Caputo fractional-order derivative; fixed point}

\begin{abstract}
 In this article we use the upper and lower solution method combined
 with a fixed point theorem for condensing multivalued maps, due to
 Martelli, to study the existence of solutions to impulsive partial hyperbolic
 differential inclusions at fixed instants of impulse.
\end{abstract}

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

\section{Introduction}

The theory of differential equations and inclusions of fractional
order play a very important role in describing some real world
problems. For example some problems in physics, mechanics,
viscoelasticity, electrochemistry, control, porous media,
electromagnetic, etc. (see \cite{Hi,Pod}). Recently, numerous
research papers and monographs have appeared devoted to fractional
differential equations, for example see the monographs of Abbas 
et al \cite{ABN}, Kilbas et al  \cite{KiSrJuTr},
Lakshmikantham et al \cite{LLV}, and Malinowska and Torres
\cite{MT}, and the papers of Abbas and Benchohra \cite{AbBe1,AbBe4},
Abbas et al \cite{AbAgBe,AbBeGo}, Belarbi et al
\cite{BeBeOu}, Benchohra and Ntouyas \cite{BeNt}, Kilbas et al \cite{KiBoTr},
 Kilbas and Marzan \cite{KiMa}, Semenchuk
\cite{Se},Vityuk and Golushkov \cite{ViGo}, and the references
therein.

 The method of upper and lower solutions has been successfully applied to study
the existence of solutions for fractional order ordinary and partial partial 
differential equations and inclusions. See the monographs by  Benchohra 
et al \cite{BeHeNt}, Heikkila and Lakshmikantham \cite{HeLa}, 
Ladde et al \cite{LaLaVa}, the papers of Abbas and Benchohra 
\cite{AbBe2, AbBe3}, Benchohra and Ntouyas \cite{BeNt} and the
references therein.

 This article deals with the existence of solutions to impulsive
fractional order initial value problems (IVP for short), for the system
\begin{gather}\label{e1}
(^{c}D_{\theta_k}^{r}u)(x,y)\in F(x,y,u(x,y)),\quad \text{if }(x,y)\in
J_k;\ k=0,\ldots,m;\\
\label{e2}
u(x_k^+,y)=u(x_k^-,y)+I_{k}(u(x_k^-,y)),\quad  \text{if }y\in [0,b], \;
k=1,\dots,m; \\
\label{e3}
\left\{\begin{gathered}
u(x,0)=\varphi (x),  \quad x\in [0,a],\\
u(0,y)=\psi (y), \quad y\in[0,b],\\
\varphi(0)=\psi(0),
\end{gathered}\right.
\end{gather}
where $J_0=[0,x_1]\times[0,b]$, $J_k:=(x_k,x_{k+1}]\times [0,b]$,
$k=1,\ldots,m$, $\theta_k=(x_k,0)$, $k=0,\ldots,m$, $a,b>0$,
$\theta=(0,0)$, $^{c}D_{\theta}^{r}$ is the fractional caputo
derivative of order $r=(r_1,r_2)\in (0,1]\times (0,1]$, 
$0=x_0<x_1<\dots<x_m<x_{m+1}=a$, 
$F:J\times \mathbb{R}^n\to \mathcal{P}(\mathbb{R}^n)$ is a compact valued
multivalued map, $\mathcal{P}(\mathbb{R}^n)$ is the family of all subsets
of $\mathbb{R}^n$, 
$I_{k}: \mathbb{R}^n\to\mathbb{R}^n$, $k=1,\ldots,m$ are given
functions, $\varphi:[0,a]\to \mathbb{R}^n$, 
$\psi: [0,b]\to \mathbb{R}^n$ are given absolutely continuous functions. 
Here $u(x_k^+,y)$ and $u(x_k^-,y)$ denote the right and left limits of $u(x,y)$ at
$x=x_{k}$, respectively.

 In this article, we provide sufficient conditions for the
existence of solutions for the problem \eqref{e1}-\eqref{e3}. Our
approach is based on the existence of upper and lower solutions and
on a fixed point theorem for condensing multivalued maps, due to
Martelli \cite{Mar}. The present results extend those considered
with integer order derivative \cite{BeHeNt, DaKu1, Kam, KaKr, LaPa,
Pa} and those with fractional derivative and without impulses
\cite{KiMa}.

\section{Preliminaries}

In this section, we introduce notation and preliminary facts which
are used throughout this paper. By $C(J)$ we denote the Banach space of all continuous
functions from $J$ to $\mathbb{R}^n$ with the norm
$$
\|w\|_{\infty}=\sup_{(x,y)\in J}\|w(x,y)\|,
$$
where $\|\cdot\|$ denotes a suitable norm on $\mathbb{R}^n$.
As usual, by $AC(J)$ we denote the space of absolutely continuous
functions from $J$ into $\mathbb{R}^n$ and $L^{1}(J)$ is the space
of Lebesgue-integrable functions $w:J\rightarrow \mathbb{R}^n$\ with
the norm
$$
\|w\|_1=\int_0^{a}\int_0^{b}\|w(x,y)\|\,dy\,dx.
$$

\begin{definition}[\cite{ViGo}] \label{def2.1} \rm 
Let $r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$, $\theta=(0,0)$
and $u\in L^{1}(J)$. The left-sided mixed
Riemann-Liouville integral of order $r$ of $u$ is defined as
$$
(I_{\theta}^{r}u)(x,y)=\frac{1}{\Gamma (r_1)\Gamma (r_{2})}\int_0^{x}%
\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_{2}-1}u(s,t)\,dt\,ds.
$$
\end{definition}
In particular,
$$
(I_{\theta}^{\theta}u)(x,y)=u(x,y), \ ( I_{\theta}^{\sigma}u)(x,y)
=\int_0^{x}\int_0^{y}u(s,t)\,dt\,ds; \ \text{for almost all} \
(x,y)\in J,
$$
where $\sigma=(1,1)$
For instance, $I_{\theta}^{r}u$ exists for all $r_1,r_2\in(0,\infty)$, when
 $u\in L^{1}(J)$. Note also that when $u\in C(J)$, then $(I_{\theta}^{r}u)\in C(J)$,
moreover
$$
(I_{\theta}^{r}u)(x,0)=(I_{\theta}^{r}u)(0,y)=0; \quad
 x\in [0,a], \; y\in [0,b].
$$

\begin{example} \label{examp2.2} \rm
Let $\lambda,\omega\in(-1,0)\cup(0,\infty)$ and 
$r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$,
then
$$
I_{\theta}^{r}x^{\lambda}y^{\omega}=\frac{\Gamma(1+\lambda)\Gamma(1+\omega)}
{\Gamma(1+\lambda+r_1)\Gamma(1+\omega+r_2)}x^{\lambda+r_1}y^{\omega+r_2}, \quad
\text{for almost all }  (x,y)\in J.
$$
\end{example}

By $1-r$ we mean $(1-r_1, 1-r_2)\in[0,1)\times[0,1)$.
Denote by $D^{2}_{xy}:=\frac{\partial ^{2}}{\partial x\partial y}$,
the mixed second order partial derivative.

\begin{definition}[\cite{ViGo}] \label{def2.3} \rm
Let $r\in (0,1]\times (0,1]$ and $u\in L^{1}(J)$. The Caputo fractional-order
 derivative of order $r$ of $u$ is defined by the expression
\begin{align*}
^{c}D_{\theta}^{r}u(x,y)
&=(I_{\theta}^{1-r}D^{2}_{xy}u)(x,y)\\
&=\frac{1}{\Gamma(1-r_1)\Gamma(1-r_{2})}\int_0^{x}
\int_0^{y}\frac{D^{2}_{st}u(s,t)}{(x-s)^{r_1}(y-t)^{r_{2}}}\,dt\,ds.
\end{align*}
\end{definition}
The case $\sigma=(1,1)$ is included and we have
$$
(D_{\theta}^{\sigma}u)(x,y)=(^{c}D_{\theta}^{\sigma}u)(x,y)=(D ^{2}_{xy}u)(x,y), \quad
\text{for almost all } (x,y)\in J.
$$

\begin{example} \label{examp2.4} \rm
Let $\lambda,\omega\in(-1,0)\cup(0,\infty)$ and $r=(r_1,r_2)\in (0,1]\times (0,1]$,
then
$$
D_{\theta}^{r}x^{\lambda}y^{\omega}=\frac{\Gamma(1+\lambda)\Gamma(1+\omega)}
{\Gamma(1+\lambda-r_1)\Gamma(1+\omega-r_2)}x^{\lambda-r_1}y^{\omega-r_2}, \quad
\text{for almost all }  (x,y)\in J.
$$
\end{example}

 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
$$
(I_{z^+}^{r}u)(x,y)=\frac{1}{\Gamma (r_1)\Gamma (r_{2})}\int_{a_1^+}^{x}
\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_{2}-1}u(s,t)\,dt\,ds,
$$
is called the left-sided mixed Riemann-Liouville integral of order $r$ of $u$.

\begin{definition}[\cite{ViGo}] \label{def.25.} \rm
 For $u\in L^{1}(J_z,\mathbb{R}^n)$ where $D^{2}_{xy}u$
is Lebesque integrable on $[x_{k},x_{k+1}]\times[0,b]$, $k=0,\ldots,m$, the
Caputo fractional-order derivative of order $r$ of $u$ is defined by the
expression $(^{c}D_{z^+}^{r}f)(x,y)=( I_{z^+}^{1-r}D ^{2}_{xy}f)(x,y)$.
The Riemann-Liouville fractional-order derivative of order $r$ of $u$ is defined by
$(D_{z^+}^{r}f)(x,y)=(D ^{2}_{xy}I_{z^+}^{1-r}f)(x,y)$.
\end{definition}

 We need also some properties of set-valued Maps.
Let $(X,\|\cdot\|)$ be a Banach space. Denote 
$\mathcal{P}(X)=\{Y\in X: Y\neq\emptyset\}$, 
$\mathcal{P}_{cl}(X)=\{Y\in \mathcal{P}(X): Y\text{ closed}\}$,
 $\mathcal{P}_{b}(X)=\{Y\in \mathcal{P}(X): Y\text{ bounded}\}$,
$\mathcal{P}_{cp}(X)=\{Y\in \mathcal{P}(X): Y\text{ compact}\}$ and
$\mathcal{P}_{cp,cv}(X)=\{Y\in \mathcal{P}(X): Y\text{ compact and convex}\}$.

\begin{definition} \rm
A multivalued map $T:X\to \mathcal{P}(X)$ is convex (closed) valued if
$ T(x)$ is convex (closed) for all $x\in X. \ T$ is bounded on
bounded sets if $T(B)=\cup_{x\in B}T(x)$ is bounded in $X$ for all
$B\in \mathcal{P}_{b}(X)$ (i.e. $\sup_{x\in B}\sup_{y\in T(x)}\|y\|<\infty$).
$T$ is called upper semi-continuous (u.s.c.) on $X$ if for each
$x_0\in X$, the set $T(x_0)$ is a nonempty closed subset of $X$,
and if for each open set $N$ of $X$ containing $T(x_0)$, there
exists an open neighborhood $N_0$ of $x_0$ such that
$T(N_0)\subseteq N$. $T$ is lower semi-continuous (l.s.c.) if the
set $ \{x\in X: T(x)\cap A\not=\emptyset\}$ is open for any open
subset $A\subseteq X$. $T$ is said to be completely continuous if
$T({\mathcal B})$ is relatively compact for every 
${\mathcal B}\in P_{b}(X)$. $T$ has a fixed point if there is $x\in X$ such that
$x\in T(x)$. The fixed point set of the multivalued operator $T$
will be denoted by $Fix T$. A multivalued map 
$G:X\to \mathcal{P}_{cl}(\mathbb{R}^n)$ is said to be measurable if for every 
$v\in \mathbb{R}^n$,
the function $x\mapsto d(v,G(x))=\inf\{\|v-z\|: z\in G(x)\}$
is measurable.
\end{definition}

\begin{lemma}\cite{HuPa} \label{L0}
Let $G$ be a completely continuous multivalued map with nonempty
compact values, then $G$ is u.s.c. if and only if $G$ has a closed
graph (i.e. $u_{n}\to u$, $w_{n}\to w$, $w_{n}\in G(u_{n})$ imply $w\in G(u)$).
\end{lemma}

\begin{definition} \rm
A multivalued map $F: J\times\mathbb{R}^n\to\mathcal{P}(\mathbb{R}^n)$
is said to be Carath\'eodory if
\begin{itemize}
\item[(i)] $(x,y)\mapsto F(x,y,u)$ is  measurable for each $u\in\mathbb{R}^n$;

\item[(ii)] $u\mapsto F(x,y,u)$ is upper semicontinuous for almost
all $(x,y)\in J$.

\end{itemize}
$F$ is said to be $L^{1}$-Carath\'eodory if (i), (ii) and the
following condition holds;
\begin{itemize}
\item[(iii)] for each $c>0$, there exists $\sigma_c
\in L^{1}(J,\mathbb{R}_{+})$ such that
\begin{align*}
\|F(x,y,u)\|_\mathcal{P}
&= \sup\{\|f\|: f\in F(x,y,u)\}\\
&\leq  \sigma_c(x,y) \quad \text{ or all $\|u\|\leq c$ and for a.e. $(x,y)\in J$}.
\end{align*}
\end{itemize}
\end{definition}

 For each $u\in C(J)$, define the set of selections of $F$ by
$$
S_{F,u}=\{w\in L^{1}(J): w(x,y)\in F(x,y,u(x,y)) \text{ a.e. } (x,y)\in J \}.
$$
 Let $(X,d)$ be a metric space induced from the normed
space $(X, \|\cdot \|)$. Consider 
$H_{d}:\mathcal{P}(X)\times \mathcal{P}(X)\to\mathbb{R}_{+}\cup\{\infty\}$
 given by
$$
H_{d}(A,B)=\max\{\sup_{a\in A}d(a,B),\sup_{b\in B}d(A,b)\},
$$
where $d(A,b)=\inf_{a\in A}d(a,b),\ d(a,B)=\inf_{b\in B}d(a,b)$.
Then $(\mathcal{P}_{b,cl}(X),H_{d})$ is a metric space and 
$(\mathcal{P}_{cl}(X),H_{d})$ is a generalized metric space (see \cite{Kis}). 
For more details on multi-valued
maps we refer the reader to the books of Deimling 
\cite{Dei}, Gorniewicz \cite{Gor}, Graef et al \cite{GHO},
Hu and Papageorgiou \cite{HuPa} and Tolstonogov \cite{Tol}.

\begin{lemma}[\cite{LaOp}] \label{L1}
Let $X$ be a Banach space. Let $F:J\times X\to \mathcal{P}_{cp,cv}(X)$ 
be an $L^{1}$-Carath\'eodory multivalued map and let
$\Lambda$ be a linear continuous mapping from $L^{1}(J,X)$ to $C(J,X)$,
then the operator
\begin{align*}
\Lambda \circ S_{F}:C(J,X) & \to \mathcal{P}_{cp,cv}(C(J,X)),  \\
 u & \mapsto (\Lambda \circ S_{F})(u):=\Lambda(S_{F,u})
\end{align*}
is a closed graph operator in $C(J,X)\times C(J,X)$.
\end{lemma}

\begin{lemma}[\cite{Mar}] \label{L2}(Martelli)
Let $X$ be a Banach space and $N:X\to \mathcal{P}_{cl,cv}(X)$ be an u. s. c. and
condensing map. If the set 
$\Omega:=\{u\in X:\lambda N(u)=N(u)\ for\ some\ \lambda>1\}$
is bounded, then $N$ has a fixed point.
\end{lemma}

\section{Main Result}

To define the solutions of problems \eqref{e1}-\eqref{e3},
we shall consider the Banach space
\begin{align*}
PC=\big\{&u: J\to\mathbb{R}^n: u \in C(J_{k}) ;  k=0, \ldots,m, 
\text{  and there exist } u(x_{k}^-,y)\\
&\text{and } u(x_{k}^+,y);\; y\in[0,b],\;
k=1,\ldots,m, \text{ with } u(x_{k}^-,y)=u(x_{k},y) \big\},
\end{align*}
with the norm
$$
\|u\|_{PC}=\sup_{(x,y)\in J}\|u(x,y)\|.
$$

\begin{definition}\label{d1} \rm
A function  $u\in PC\cap\cup_{k=0}^{m} AC(J_k)$ 
whose $r$-derivative exists on $J_k$
is said to be a solution of \eqref{e1}-\eqref{e3} if there exists a
function $f\in L^{1}(J)$ with $f(x,y)\in F(x,y,u(x,y))$ such that
$u$ satisfies $(^{c}D_{\theta_k}^{r}u)(x,y)= f(x,y)$ on 
$J_k$, $k=0,\dots m$  and conditions \eqref{e2}, \eqref{e3} are
satisfied.
\end{definition}

Let $z, \bar z\in C(J)$ be such that
$$
z(x,y)=(z_1(x,y),z_{2}(x,y),\ldots,z_{n}(x,y)), \quad (x,y)\in J,
$$
and
$$
\bar z(x,y)=(\bar z_1(x,y),\bar z_{2}(x,y),\ldots,\bar z_{n}(x,y)),\quad (x,y)\in J.
$$
The notation $z\leq \bar z$ means that
$$
z_{i}(x,y)\leq \bar z_{i}(x,y)\quad \text{for } i=1,\dots,n.
$$
\begin{definition}

A function $z\in PC\cap\cup_{k=0}^{m}AC(J_k)$ is said to be
a lower solution of \eqref{e1}-\eqref{e3} if there exists a function 
$ f\in L^{1}(J)$ with $f(x,y)\in F(x,y,u(x,y))$ such that $z$ satisfies
\begin{gather*}
(^{c}D_{\theta_k}^{r}z)(x,y)\leq f(x,y,z(x,y)), \quad\text{on } J_k;\\
z(x_k^+,y)\leq z(x_k^-,y)+I_{k}(z(x_k^-,y)),\quad \text{if } y\in[0,b],\;
 k=1,\dots,m;\\
z(x,0)\leq \varphi(x),\; x\in[0,a];\\
z(0,y)\leq \psi(y),\quad  y\in[0,b];\\
 z(0,0)\leq \varphi(0).
\end{gather*}
The function $z$ is said to be an upper solution of
\eqref{e1}-\eqref{e3} if the reversed inequalities hold.
\end{definition}

 Let $h\in C(J_k)$, $k=1,\dots,m$
and set
$$
\mu(x,y):=\varphi(x)+\psi(y)-\varphi(0),\quad (x,y)\in J.
$$
For the existence of solutions for  problem \eqref{e1}-\eqref{e3},
we need the following lemma.

\begin{lemma}[\cite{AbBe3}] \label{L3}
Let $r_1,r_2\in(0,1]$ and let $h:J \to\mathbb{R}^n$ be continuous.
A function $u$ is a solution of the fractional integral equation
$$
u(x,y)=\begin{cases}
\mu(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\int_0^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds;\\
\quad \text{if } (x,y)\in [0,x_1]\times[0,b],
\\[3pt]
\mu(x,y)+\sum_{i=1}^{k}(I_{i}(u(x_{i}^{-},y))-I_{i}(u(x_{i}^{-},0)))\\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{i=1}^{k}\int_{x_{i-1}}^{x_{i}}\int_0^{y}
(x_{i}-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds\\
+ \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds;\\
\quad \text{if  } (x,y)\in (x_{k},x_{k+1}]\times[0,b],\ k=1,\dots,m,
\end{cases}
$$
if and only if $u$ is a solution of the fractional IVP
\begin{gather*}
^{c}D^{r}u(x,y)= h(x,y), \quad   (x,y)\in J_k,\\
u(x_{k}^{+},y)= u(x_{k}^{-},y)+I_{k}(u(x_{k}^{-},y)),\quad  y\in[0,b],\;
  k=1,\dots,m.
\end{gather*}
\end{lemma}

 To study problem \eqref{e1}-\eqref{e3},
we first list the following hypotheses:
\begin{itemize}
\item[(H1)]  $F: J\times \mathbb{R}^n\to \mathcal{P}_{cp,cv}(\mathbb{R}^n)$
is $L^1$-Carath\'eodory;

\item[(H2)]  There exist $v$ and
$w\in PC\cap AC(J_k)$, $k=0,\ldots,m$,
lower and upper solutions for the problem \eqref{e1}-\eqref{e3}
such that $v(x,y)\leq w(x,y)$  for each $(x,y)\in J$;

\item[(H3)] For each $y\in[0,b]$, we have
\[
v(x_k^{+},y)\leq\min_{u\in[v(x_{k}^{-},y),w(x_{k}^{-},y)]}I_k(u)
\leq\max_{u\in[v(x_{k}^{-},y),w(x_{k}^{-},y)]}I_k(u)\leq w(x_k^{+},y),
\]
with $k=1,\ldots,m$.
\end{itemize}

\begin{theorem}\label{T1} 
Assume that hypotheses {\rm (H1)-(H3)} hold. Then  problem \eqref{e1}-\eqref{e3} 
has at least one solution $u$ such that
$$
v(x,y)\leq u(x,y)\leq w(x,y),\quad  \text{for all } (x,y)\in J.
$$
\end{theorem}

\begin{proof} 
We transform  problem \eqref{e1}-\eqref{e3} into a fixed point problem.
Consider the  modified problem
\begin{gather}\label{e8}
(^{c}D_{\theta_k}^{r}u)(x,y)\in F(x,y,g(u(x,y))),\quad \text{if }
(x,y)\in J_k,\; k=0,\ldots,m; \\
\label{e9}
u(x_k^+,y)=u(x_k^-,y)+I_{k}(g(x_k^-,y,u(x_k^-,y))),\quad  \text{if }
y\in [0,b],\; k=1,\dots,m; \\
\label{e10}
u(x,0)=\varphi(x),\quad  x\in[0,a],\; u(0,y)=\psi(y)\,; y\in[0,b],\;
\varphi(0)=\psi(0),
\end{gather}
where $g:PC\to PC$ be the truncation operator defined by
$$
(gu)(x,y)=\begin{cases}
v(x,y),  & u(x,y)<v(x,y),\\
u(x,y),  & v(x,y)\leq u(x,y)\leq w(x,y),\\
w(x,y),  & w(x,y)<u(x,y).
\end{cases}
$$
A solution to \eqref{e8}-\eqref{e10} is a fixed point of
the operator $N:PC\to \mathcal{P}(PC)$ defined by
$$
N(u)=\begin{cases}
h\in PC:h(x,y)=\mu(x,y)\\
+\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u(x_{k}^{-},y)))-I_{k}(g(x_{k}^{-},0,u(x_{k}^{-},0))))\\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{0<x_{k}<x}\int_{x_{k-1}}^{x_{k}}\int_0^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}f(s,t)\,dt\,ds\\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_{k}}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}f(s,t)\,dt\,ds,
\end{cases}
$$
where
\begin{gather*}
\begin{aligned}
&f\in\tilde S^{1}_{F,g(u)}\\
&=\big\{f\in S^{1}_{F,g(u)}:
f(x,y)\geq f_1(x,y)  \text{ on $A_1$ and }
f(x,y)\leq f_{2}(x,y)  \text{ on $A_{2}$}  \big\},
\end{aligned} \\
A_1=\{(x,y)\in J: u(x,y)<v(x,y)\leq w(x,y)\},\\
A_{2}=\{(x,y)\in J: u(x,y)\leq w(x,y)<u(x,y)\},
\\
S^{1}_{F,g(u)}=\{f\in L^{1}(J): f(x,y)\in
F(x,y,g(u(x,y))), \text{ for }  (x,y)\in J\}.
\end{gather*}
\end{proof}

\begin{remark} \rm
(A) For each $u\in PC$, the set $\tilde S_{F, g(u)}$ is nonempty.
In fact, $(H1)$ implies there exists $ f_{3}\in S_{F, g(u)}$, so we set
$$
f=f_1\chi_{A_1}+f_{2} \chi_{A_{2}}+f_{3} \chi_{A_{3}},
$$
where
$\chi_{A_{i}}$ is the characteristic function of ${A_{i}};\ i=1,2,3$
and
$$
A_{3}=\{(x,y)\in J: v(x,y)\leq u(x,y)\leq w(x,y)\}.
$$
Then, by decomposability, $f\in \tilde S_{F,g(u)}$.

(B) By the definition of $g$ it is clear
that $F(.,., g(u)(.,.))$ is an $L^{1}$-Carath\'eodory multi-valued
map with compact convex values and there exists $\phi\in C(J,\mathbb{R}_+)$
such that
$$
\|F(x,y,g(u(x,y)))\|_\mathcal{P}\leq
\phi(x,y);\ \text{for each}\ (x,y)\in J \text{and}\ u\in \mathbb{R}^n.
$$
Set
$$
\phi^{*}:=\sup_{(x,y)\in J}\phi(x,y).
$$

(C) By the definition of $g$ and from (H3) we have
$$
u(x_k^+,y)\leq I_{k}(g(x_k,y,u(x_k,y)))\leq w(x_k^+,y); \ y\in
[0,b];\ k=1,\dots,m.
$$
\end{remark}

From Lemma \ref{L3} and the fact that $g(u)=u$ for all $v\leq u\leq w$,
the problem of finding the solutions of the IVP \eqref{e1}-\eqref{e3}
is reduced to finding the solutions of the operator equation $N(u)=u$.
We shall show that $N$ is a completely continuous multivalued map, u.s.c.
with convex closed values. The proof will be given in several steps.
\smallskip

\noindent\textbf{Step 1:}  $N(u)$ is convex for each $u\in PC$.
If $h_1, h_{2}$ belong to $N(u)$, then there
exist $f_1, f_{2}\in \tilde S^{1}_{F,g(u)}$ such that for each
$(x,y)\in J$ we have
\begin{align*}
(h_{i}u)(x,y)
&=\mu(x,y) +\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u(x_{k}^{-},y)))
 -I_{k}(g(x_{k}^{-},0,u(x_{k}^{-},0))))\\
&\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}f_{i}(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}f_{i}(s,t)\,dt\,ds.
\end{align*}
Let $0\leq \xi\leq 1$. Then, for each $(x,y)\in J$, we have
\begin{align*}
&(\xi h_1+(1-\xi)h_{2})(x,y)\\
&= \mu(x,y)+\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u(x_{k}^{-},y))))
 -\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},0,u(x_{k}^{-},0))))\\
&\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}\\
&\quad\times [\xi f_1(s,t)+(1-\xi)f_{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}
 [\xi f_1(s,t)+(1-\xi)f_{2}(s,t)]\,dt\,ds.
\end{align*}
Since $\tilde S^{1}_{F,g(u)}$ is convex (because $F$ has convex
values), we have
$$
\xi h_1+(1-\xi)h_{2}\in G(u).
$$
\smallskip

\noindent\textbf{Step 2:} $N$ sends bounded sets of $PC$ into bounded sets.
We can prove that $N(PC)$ is bounded. It is sufficient to show that there exists
a positive constant $\ell$ such that for each $h\in N(u)$, $u \in PC$
one has $\|h\|_{\infty}\leq\ell$.
If $h \in N(u)$, then there exists $f\in\tilde S^{1}_{F,g(u)}$
such that for each $(x,y)\in J$ we have
\begin{align*}
(hu)(x,y)
&=\mu(x,y) +\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u(x_{k}^{-},y)))-I_{k}
 (g(x_{k}^{-},0,u(x_{k}^{-},0))))\\
&\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}f(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}f(s,t)\,dt\,ds.
\end{align*}
Then, for each $(x,y)\in J$ we get
\begin{align*}
\|(hu)(x,y)\|
&=\|\mu(x,y)\|
+2\sum_{k=1}^{m}\max_{y\in[0,b]}(\|v(x_k^+,y)\|,\|w(x_k^+,y)\|)\\
&\quad +\frac{\phi^{*}}{\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}\,dt\,ds\\
&\quad +\frac{\phi^{*}}{\Gamma(r_1)\Gamma(r_2)}\int_{x_{k}}^{x}
\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}\,dt\,ds.
\end{align*}
Thus,
$$
\|u\|_{\infty}\leq\|\mu\|_{\infty}+
2\sum_{k=1}^{m}\max_{y\in[0,b]}(\|v(x_k^+,y)\|,\|w(x_k^+,y)\|)
+\frac{2a^{r_1}b^{r_{2}}\phi^{*}}{\Gamma(r_1+1)\Gamma(r_2+1)}:=\ell.
$$
\smallskip

\noindent\textbf{Step 3:}  $N$ sends bounded sets of $PC$ into equicontinuous sets.
Let $(\tau_1,y_1)$, $(\tau_{2},y_{2})\in J$, $\tau_1<\tau_{2}$, $y_1<y_{2}$
and $B_{\rho}=\{u\in PC: \|u\|_{\infty}\leq \rho\}$ be a bonded set of $PC$.
For each $u\in B_{\rho}$ and $h\in N(u)$, there exists $f\in \tilde S^{1}_{F,g(u)}$
such that  for each $(x,y)\in J$ we have
\begin{align*}
&\|(hu)(\tau_{2},y_{2})-h(u)(\tau_1,y_1)\|\\
&\leq\| \mu(\tau_1,y_1) -\mu( \tau_{2},y_{2})\|
+\sum_{k=1}^{m}(\|I_{k}(g(x_{k}^{-},y_1,u(x_{k}^{-},y_1)))\\
&\quad -I_{k}(g(x_{k}^{-},y_2,u(x_{k}^{-},y_{2})))\|)\\
&\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 \|f(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}\|f(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 \|f(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}\|f(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}\|f(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}\|f(s,t)\|\,dt\,ds \\
&\leq\|\mu(\tau_1,y_1) -\mu( \tau_{2},y_{2})\|\\
&\quad +\sum_{k=1}^{m}(\|I_{k}(g(x_{k}^{-},y_1,u(x_{k}^{-},y_1)))
-I_{k}(g(x_{k}^{-},y_2,u(x_{k}^{-},y_{2})))\|)\\
&\quad +\frac{\phi^{*}}{\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{\phi^{*}}{\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{\phi^{*}}{\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{\phi^{*}}{\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{\phi^{*}}{\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{\phi^{*}}{\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.
\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.
As a consequence of Steps 1 to 3 together with the Arzel\'a-Ascoli
theorem, we can conclude that $N$ is completely continuous and therefore
a condensing multivalued map.
\smallskip

\noindent\textbf{Step 4:}  $N$ has a closed graph.
Let $u_{n}\to u_{*}$, $h_{n}\in N(u_{n})$ and  $h_{n} \to h_{*}$. 
We need to show that $h_{*}\in N(u_{*})$.
$h_{n}\in N(u_{n})$ means that there exists $f_{n}\in \tilde S^{1}_{F,g(u_{n})}$
such that, for each $(x,y)\in J$, we have
\begin{align*}
h_{n}(x,y)
&= \mu(x,y)+\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u_{n}(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u_{n}(x_{k}^{-},0))))\\
&\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}f_n(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}f_n(s,t)\,dt\,ds.
\end{align*}
We must show that there exists $f_{*}\in \tilde S^{1}_{F, g(u_{*})}$
such that, for each $(x,y)\in J$,
\begin{align*}
h_{*}(x,y)
&= \mu(x,y)+\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u_{*}(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u_{*}(x_{k}^{-},0))))\\
&\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}f_*(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}f_*(s,t)\,dt\,ds.
\end{align*}
Now, we consider the linear continuous operator
$\Lambda:L^{1}(J) \to C(J)$ defined by $f  \mapsto \Lambda(f)(x,y)$,
\begin{align*}
(\Lambda f)(x,y)
&= \sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u(x_{k}^{-},0))))\\
&\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}f(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}f(s,t)\,dt\,ds.
\end{align*}
From Lemma \ref{L1}, it follows that $\Lambda\circ\tilde S^{1}_{F}$
is a closed graph operator. Clearly we have
\begin{align*}
&\Big\|\Big[h_{n}(x,y)-\mu(x,y)-\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u_{n}
(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u_{n}(x_{k}^{-},0))))\Big]\\
&-\Big[h_{*}(x,y)-\mu(x,y)-\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u_{*}(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u_{*}(x_{k}^{-},0))))\Big]\Big\|\\
&\leq\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{x_1<x_{k}<x}
 \int_{x_{k-1}}^{x_{k}}\int_0^{y}
(x_{k}-s)^{r_1-1}(y-t)^{r_2-1}\|f_{n}(s,t)-f_{*}(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}
\|f_{n}(s,t)-f_{*}(s,t)\|\,dt\,ds\to 0,
\end{align*}
as $n\to\infty$. Moreover, from the definition of $\Lambda$, we have
\begin{align*}
&\Big[h_{n}(x,y)-\mu(x,y)-\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u_{n}(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u_{n}(x_{k}^{-},0))))\Big]\\
&\in\Lambda(\tilde S^{1}_{F,g(u_n)}).
\end{align*}
Since $u_n \to u_*$, it follows from Lemma \ref{L1} that,
for some $f_* \in\Lambda(\tilde S^{1}_{F,g(u_*)})$, we have
\begin{align*}
&h_{*}(x,y)_\mu(x,y)-\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u_{*}(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u_{*}(x_{k}^{-},0))))\\
&= \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}f_*(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}f_*(s,t)\,dt\,ds,\quad
 (x,y)\in J.
\end{align*}
From Lemma \ref{L0}, we can conclude that $N$ is u.s.c.
\smallskip

\noindent\textbf{Step 5:}  
The set $\Omega=\{u\in PC:\lambda u=N(u)\text{ for some }\lambda>1\}$ in bounded.
Let $u\in\Omega$. Then, there exists $f \in\Lambda(\tilde S^{1}_{F,g(u)})$,
such that
\begin{align*}
\lambda u(x,y)
&= \mu(x,y)+\sum_{0<x_{k}<x}(I_{k}(g(x_{k}^{-},y,u(x_{k}^{-},y)))
-I_{k}(g(x_{k}^{-},0,u(x_{k}^{-},0))))\\
&\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}f(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}f(s,t)\,dt\,ds.
\end{align*}
As in Step 2, this implies that for each $(x,y)\in J$, we have
$$
\|u\|_{\infty}\leq\|\mu\|_{\infty}+
2\sum_{k=1}^{m}\max_{y\in[0,b]}(\|v(x_k^+,y)\|,\|w(x_k^+,y)\|)
+\frac{2a^{r_1}b^{r_{2}}\phi^{*}}{\Gamma(r_1+1)\Gamma(r_2+1)}=\ell.
$$
This shows that $\Omega$ is bounded. As a consequence of Lemma \ref{L2},
we deduce that $N$ has a fixed point which is a solution of
 \eqref{e8}-\eqref{e10} on $J$.
 \smallskip

\noindent\textbf{Step 6:} The solution $u$ of \eqref{e8}-\eqref{e10} satisfies
$$
v(x,y)\leq u(x,y)\leq w(x,y),\quad \text{for all } (x,y)\in J.
$$
Let $u$ be the above solution to \eqref{e8}-\eqref{e10}. We prove
that
$$
u(x,y)\leq w(x,y) \quad \text{for all } (x,y)\in J.
$$
Assume that $u-w$ attains a positive maximum on 
$[x_{k}^{+},x_{k+1}^{-}]\times[0,b]$
at $(\overline{x}_{k},\overline y)\in [x_{k}^{+},x_{k+1}^{-}]\times[0,b]$,
for some $k=0,\dots,m$; that is,
$$
(u-w)(\overline{x}_{k},\overline y)=\max\{u(x,y)-w(x,y):  (x,y)\in
[x_{k}^{+},x_{k+1}^{-}]\times[0,b]\} > 0,
$$
for some $k=0,\dots,m$. We distinguish the following cases.


\textbf{Case 1.} If $ (\overline{x}_{k},\overline y)\in
(x_{k}^{+},x_{k+1}^{-})\times[0,b]$ there exists
$(x_{k}^{*},y^{*})\in (x_{k}^{+},x_{k+1}^{-})\times[0,b]$
such that
\begin{equation}\label{e11}
\begin{aligned}
&[u(x,y^{*})-w(x,y^{*})]+[u(x_{k}^{*},y)-w(x_{k}^{*},y)]
-[u(x_{k}^{*},y^{*})-w(x_{k}^{*},y^{*})]\\
&\leq 0, \quad \text{for all }   (x,y)\in([x_{k}^{*},\overline x_{k}]
\times\{y^{*}\})\cup(\{x_{k}^{*}\}\times[y^{*},b]),
\end{aligned}
\end{equation}
and
\begin{equation}\label{e12}
u(x,y)-w(x,y)>0,\quad \text{for all } (x,y)\in(x_{k}^{*},
\overline x_{k}]\times(y^{*},b].
\end{equation}
By the definition of $g$, one has
\begin{equation}\label{e13}
^{c}D_{\theta}^{r}u(x,y)\in F(x,y,w(x,y)),\quad \text{for all }
 (x,y)\in [x_{k}^{*},\overline x_{k}]\times[y^{*},b].
\end{equation}
An integration of \eqref{e13}, on $[x_{k}^{*},x]\times[y^{*},y]$
for each $(x,y)\in[x_{k}^{*},\overline{x}_{k}]\times[y^{*},b]$, yields
\begin{equation}\label{e14}
\begin{aligned}
&u(x,y)+u(x_{k}^{*},y^{*})-u(x,y^{*})-u(x_{k}^{*},y) \\
&=\frac{1}{\Gamma(r_1)\Gamma(r_{2})}
\int_{x_{k}^{*}}^{x}\int_{y^{*}}^{y}(x-s)^{r_1-1}(y-t)^{r_{2}-1}f(s,t)\,dt\,ds,
\end{aligned}
\end{equation}
where $f(x,y)\in F(x,y,w(x,y))$.
From \eqref{e14} and using the fact that $w$ is an upper solution to
\eqref{e1}-\eqref{e3} we get
$$
u(x,y)+u(x_{k}^{*}, y^{*})-u(x,y^{*})-u(x_{k}^{*},y)
\leq w(x,y)+w(x_{k}^{*},y^{*})-w(x,y^{*})-w(x_{k}^{*},y),
$$
which gives
\begin{equation}\label{e15}
\begin{aligned}
&u(x,y)-w(x,y)\\
&\leq [u(x,y^{*})-w(x,y^{*})]+[u(x_{k}^{*},y)-w(x_{k}^{*},y)]
-[u(x_{k}^{*},y^{*})-w(x_{k}^{*},y^{*})].
\end{aligned}
\end{equation}
Thus from \eqref{e11}, \eqref{e12} and \eqref{e15} we obtain the
contradiction
\begin{align*}
0&<[u(x,y)-w(x,y)]\leq[u(x,y^{*})-w(x,y^{*})]
+[u(x_{k}^{*},y)-w(x_{k}^{*},y)]\\
&\quad -[u(x_{k}^{*},y^{*})-w(x_{k}^{*},y^{*})]\leq 0,
\quad \text{for all }  (x,y)\in [x_{k}^{*},\overline x_{k}]\times[y^{*},b].
\end{align*}

\textbf{Case 2.} If $ \overline{x}_{k}=x_{k}^{+},\ k=1,\ldots,m$, then
$$
w(x_{k}^{+},\overline y)
<I_{k}(g(x_{k}^{-},u(x_{k}^{-},\overline y)))
\leq w(x_{k}^{+},\overline y),
$$
which is a contradiction. Thus
$$
u(x,y)\leq w(x,y), \quad \text{for all } (x,y)\in J.
$$
Analogously, we can prove that
$$
u(x,y)\geq v(x,y), \quad \text{for all } (x,y)\in J.
$$
This shows that  problem \eqref{e8}-\eqref{e10} has a solution
$u$ satisfying $v\leq u\leq w$ which is solution of \eqref{e1}-\eqref{e3}.

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\end{thebibliography}

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