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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 65, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/65\hfil Integral equations of fractional order]
{Integral equations of fractional order with multiple time delays
 in Banach spaces}

\author[M. Benchohra, D. Seba\hfil EJDE-2012/65\hfilneg]
{Mouffak Benchohra, Djamila Seba}

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

\address{Djamila Seba \newline
D\'epartement de Math\'ematiques, Universit\'e de Boumerd\`es\\
Avenue de l'ind\'ependance, 35000 Boumerd\`es, Alg\'erie}
\email{djam\_seba@yahoo.fr}

\thanks{Submitted  March 7, 2012. Published April 27, 2012.}
\subjclass[2000]{26A33, 45N05}
\keywords{Integral equation; left sided mixed Riemann Liouville integral;
\hfill\break\indent
 measure of noncompactness; fixed point; Banach space}

\begin{abstract}
 In this article, we give sufficient conditions for the existence of
 solutions for an integral equation of fractional order with multiple
 time delays in Banach spaces.
 Our main tool is a fixed point theorem of M\"onch type associated with
 measures of noncompactness. Our results are illustrated by an example.
\end{abstract}

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


\section{Introduction}

 Fractional differential and integral equations play an important role in 
characterizing many chemical, physical, viscoelasticity, control and 
engineering problems. For more details,  see \cite{BDST, Die, Hi, Ma, Tar}, 
and references therein. In consequence, the subject  of fractional differential 
and integral equations is gaining much importance and attention; see, for instance,
the monograph of Abbas \emph{et al.} \cite{ABN}, Kilbas \emph{et al.} \cite{KST},
and the papers of Abbas and Benchohra \cite{AbBe1},
Agarwal \emph{et al.} \cite{ABH1}, Bana\`{s} and Zaj\c{a}c \cite{BaZa},
Benchohra and Seba \cite{BeSe, BeSe1},
Vityuk  and Golushkov \cite{ViGo}  and the references therein.

 Ibrahim and Jalab \cite{IbJa} studied the existence of solutions of the 
fractional integral inclusion
$$
u(t)-\sum_{i=1}^{m}b_i(t)u(t-\tau_i)\in I^{\alpha}F(t,u(t)),\quad
t\in [0,T],
$$
where $\tau_i<t\in[0,T]$, $b_i:[0,T]\to \mathbb{R}$,
$i=1,\dots,n$ are continuous functions, and
$F:[0,T]\times\mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a given multivalued map.
Motivated by their work, we study the fractional integral equation
\begin{gather}\label{e1}
\begin{gathered}
u(x,y)=\sum_{i=1}^{m}g_i(x,y)u(x-\xi_i,y-\mu_i)+
I_{\theta}^{r}f(x,y,u(x,y)), \\
 (x,y)\in J:=[0,a]\times [0,b];
\end{gathered}\\
\label{e1'}
u(x,y)=\Phi(x,y), \quad (x,y)\in \tilde
J:=[-\xi,a]\times[-\mu,b]\backslash(0,a]\times (0,b],
\end{gather}
where $a,b>0$, $\theta=(0,0)$, $\xi_i,\mu_i\geq 0$; $i=1,\dots,m$,
$\xi:=\max_{i=1,\dots,m}\{\xi_i\}$,
$\mu:=\max_{i=1,\dots,m}\{\mu_i\}$, $f:J\times {E} \to {E}$ is a
 function satisfying some assumptions  specified
later, $I_{\theta}^{r}$ is the left-sided mixed Riemann-Liouville
integral of order $r=(r_1,r_2)\in(0,\infty)\times(0,\infty)$, 
$g_i:J\to{E}$; $i=1,\dots m$, are  continuous functions,
$\Phi:\tilde J\to E$ is a  continuous function such that
\begin{gather*}
\Phi(x,0)=\sum_{i=1}^{m}g_i(x,0)\Phi(x-\xi_i,-\mu_i),\quad x\in [0,a], \\
\Phi(0,y)=\sum_{i=1}^{m}g_i(0,y)\Phi(-\xi_i,y-\mu_i), \quad y\in[0,b],
\end{gather*}
and $E$ is a real Banach space with norm $\|\cdot\|$.

Using properties of the Kuratowski measure of
noncompactness and a fixed point theorem of M\"onch type, we prove the
existence  of  solutions to  \eqref{e1}-\eqref{e1'}.
Let us note here that the technique of measures of noncompactness
is a very important tool for finding solutions for
differential and integral equations; for more details see
\cite{AgBeSe, BeSe, BeSe1} and references therein.

\section{Preliminaries}

In this section, we collect  a few auxiliary results
which will be needed in the sequel.
By $C(J, E)$ we denote the Banach space of continuous functions $u:J\to E$,
with the norm
$$
\|u\|_{\infty}=\sup_{(x,y)\in J}\|u(x,y)\|.
$$
Let $L^1(J,E)$ be the space of Lebesgue integrable functions $u: J \to E$
with the norm
$$
\| u\|_{L^1}=\int_{0}^{a}\int_{0}^{b}\|u(x,y)\|dxdy.
$$
Let $C([-\xi,a]\times[-\mu,b],E)$ be a Banach space endowed with the norm
$$
\|u\|_{C}=\sup_{(x,y)\in [-\xi,a]\times[-\mu,b]}\|u(x,y)\|.
$$

\begin{definition}[\cite{ViGo}] \rm
 Let $r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$,
 $\theta=(0,0)$ and $u\in L^{1}(J, E).$ The left-sided mixed
Riemann-Liouville integral of order $r$ of $u$ is defined by
$$
(I_{\theta}^{r}u)(x,y)=\frac{1}{\Gamma (r_{1})\Gamma (r_{2})}\int_{0}^{x}%
\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}u(s,t)dtds.
$$
\end{definition}

In particular,
$$
( I_{\theta}^{\theta}u)(x,y)=u(x,y), \ ( I_{\theta}^{\sigma}u)(x,y)
=\int_{0}^{x}\int_{0}^{y}u(s,t)dtds;
$$
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 
$u\in L^{1}(J,E).$ Note also that when $u\in C(J,E)$, then 
$(I_{\theta}^{r}u)\in C(J,E)$, moreover
$$
(I_{\theta}^{r}u)(x,0)=(I_{\theta}^{r}u)(0,y)=0, \quad x\in [0,a], \; y\in [0,b].
$$

Now we recall some fundamental facts of the notion of
Kuratowski measure of noncompactness.

\begin{definition}[\cite{AkKaPaRoSa, BaGo}] \rm
Let $F$ be a Banach space and let $\Omega_{F}$ be the family of
bounded subsets of $F$. The Kuratowski measure of noncompactness is
the map $\alpha:\Omega_{F}\to [0,\infty]$ defined by
$$
\alpha(B)=\inf\{\epsilon>0: B\subseteq\cup_{i=1}^{n}B_{i} \text{ and }
\operatorname{diam}(B_{i})\leq\epsilon\}, 
\quad \text{here } B\in \Omega_{E}.
$$
\end{definition}

The Kuratowski measure of noncompactness satisfies the following
 properties (For more details see \cite{AkKaPaRoSa,BaGo}).
\begin{itemize}
\item[(a)] $\alpha(B)= 0 \Leftrightarrow \overline{B} \;$ is
compact ($B$ is relatively compact).
\item[(b)]$\alpha(B)=\alpha(\overline{B})$.
\item[(c)]$A \subset B
\Rightarrow\alpha(A)\leq\alpha(B)$.
\item[(d)]$\alpha(A+B)\leq\alpha(A)+\alpha(B)$
\item[(e)]$\alpha(cB)=|c|\alpha(B);\,c\in\mathbb{R}$.
\item[(f)]$\alpha(\operatorname{conv} B)=\alpha(B)$.
\end{itemize}

For our purpose we will need the following auxiliary results.

\begin{theorem}[\cite{Mon}] \label{thm1}
Let $D$ be a bounded, closed and convex subset of a Banach space
such that $0\in{D}$, and let $N$ be a continuous mapping of $D$ into
itself. If the implication
$$
V=\overline{\operatorname{conv}} N(V) \quad \text{or} \quad
V=N(V)\cup\{0\}\Rightarrow\alpha(V)=0
$$
holds for every subset $V$ of $D$, then $N$ has a fixed point.
\end{theorem}

\begin{lemma}[\cite{GuLaLi}] \label{lem3}
Let $V\subset C(J,E)$ be bounded and equicontinuous on $J$. Then
the map $(s,t)\mapsto \alpha(V(s,t))$ is continuous  on $J$ and
$$
\alpha\Big(\int_{J}V(s,t)\,ds\,dt\Big)\leq
\int_{J}\alpha(V(s,t))\,ds\,dt,
$$ 
where $V(s,t)=\{u(s,t): u\in V\}$.
\end{lemma}

\section{Main Results}

\begin{definition} \rm
A function $u\in C(J, E)$ is said to be a solution of
\eqref{e1}-\eqref{e1'} if $u$ satisfies equation \eqref{e1} on $J$
and condition \eqref{e1'}.
\end{definition}
Set
$$
B=\max_{i=1,\dots m}\big\{\sup_{(x,y)\in J}\|g_i(x,y)\|\big\}.
$$
Let us impose two conditions for convenience.
\begin{itemize}
\item[(H1)] $f: J\times E\to E$ is a continuous map.
\item[(H2)] There exists $p\in C(J, \mathbb{R}_+)$, such that
$$
\|f(x,y, u)\|\leq p(x,y)\|u\|, \quad 
\text{for $(x,y)\in {J}$  and each } u\in{E}.
$$
\end{itemize}
Let $p^* = \|p\|_{\infty}$. The main result in this paper reads
as follows.

\begin{theorem} \label{thm2}
Assume that assumptions {\rm (H1)} and {\rm (H2)} hold. If
\begin{equation}\label{e5}
m B+\frac{p^*a^{r_1}b^{r_2}}{\Gamma (r_1+1)\Gamma (r_2+1)}<1
\end{equation}
then the problem \eqref{e1}-\eqref{e1'} has at least
one solution.
\end{theorem}

\begin{proof}
 Transform the problem \eqref{e1}-\eqref{e1'}
into a fixed point problem. Consider the operator
 $N:C(J,E)\to C(J, E)$ defined by
\begin{equation}\label{e2}
N(u)(x,y)=\sum_{i=1}^{m}g_i(x,y)u(x-\xi_i,y-\mu_i)+I_{\theta}^{r}f(x,y,u(x,y)).
\end{equation}
Since $f$ is continuous, the operator $N$ is well defined;
 i.e., $N$ maps $C(J,E)$ into itself. The problem of finding the 
solutions of equation \eqref{e1}-\eqref{e1'} is reduced to finding the 
solutions of the
operator equation $N(u)=u.$ Let $R>0$ and consider the set 
$$
 D_R=\{u\in C(J,E):\|u\|_{\infty}\leq R\}.
$$ 
It is clear that $D_R$ is a closed
bounded and convex subset of $C(J, E).$  We shall show that $N$
satisfies the assumptions of Theorem \ref{thm1}. The proof will be
given in three steps. 
\end{proof}

\noindent\textbf{Step 1:} $N$ is continuous. 
Let $\{u_n\}$ be a sequence such that $u_n\to u$ in ${C(J, E)}$, then for each
$(x,y)\in J$,
\begin{align*}
&\|N(u_{n})(x,y)-N(u)(x,y)\|\\
&\leq \frac{1}{\Gamma (r_{1})\Gamma (r_{2})}\int_{0}^{x}
\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\|f(s,t,u_n)-f(s,t,u)\|\,ds\,dt.
\end{align*}
Let $\rho>0$ be such that
$$
\|u_{n}\|_{\infty}\leq \rho, \ \|u\|_{\infty}\leq \rho.$$ By (H2)
we have
$$
(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\|f(s,t,u_n)-f(s,t,u)\|\leq
2\rho p^{*}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}
$$
which belongs to $L^{1}(J,\mathbb{R}_+)$.
Since $f$ is continuous, then by the Lebesgue dominated
convergence theorem we have
$$
\| N(u_{n})-N(u)\|_{\infty }\to{0} \text{ as } n\to\infty.
$$

\noindent\textbf{Step 2:} $N$ maps $D_R$ into itself. 
 For each $u\in D_R$, by (H2) and \eqref{e5}  we have for each $(x,y)\in J$,
\begin{align*}
&\|N(u)(x,y)\|\\
&\leq  \sum_{i=1}^{m}\|g_i(x,y)\|\|u(x-\xi_i,y-\mu_i)\|\\
&\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}\|f(s,t,u(s,t))\|\,ds\,dt\\
&\leq  mB\|u\|_{\infty}+\frac{1}{\Gamma (r_{1})\Gamma (r_{2})}
\int_{0}^{x}\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}p(s,t)\|u\|_{\infty}\,ds\,dt\\
&\leq  m B R+\frac{p^*R}{\Gamma (r_{1})\Gamma (r_{2})}
\int_{0}^{x}\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\,ds\,dt\\
&\leq  m B R+\frac{p^*R\, a^{r_{1}}b^{r_{2}}}
{\Gamma (r_{1}+1)\Gamma (r_{2}+1)}
< R.
\end{align*}

\noindent\textbf{Step 3:} $N(D_R)$ is bounded and equicontinuous.
By Step 2 we have $N(D_R)=\{ N(u): u\in D_R\}\subset D_R$. Thus, for
each $u\in D_R$ we have $\|N(u)\|_{\infty}\leq R$ which means that
$N(D_R)$ is bounded.

For the equicontinuity of $N(D_R)$, let
$(x_1,y_1),(x_2,y_2)\in J$, $x_1<x_2, y_1<y_2$,
and $u\in D_R$. Then
\begin{align*}
&\|N(u)(x_2,y_2)-N(u)(x_1,y_1)\|\\
&=\Big\|\sum_{i=1}^{m}\big[g_i(x_2,y_2)u(x_2-\xi_i,y_2-\mu_i)-g_i(x_1,y_1)u(x_1-\xi_i,y_1-\mu_i)\big]\\
&\quad +\frac{1}{\Gamma (r_1)\Gamma (r_2)}
\int_{0}^{x_1}\int_{0}^{y_1}[(x_2-s)^{r_1-1}(y_2-t)^{r_2-1}-(x_1-s)^{r_1-1}(y_1-t)^{r_2-1}]\\
&\times f(s,t,u(s,t))\,ds\,dt\\
&\quad +\frac{1}{\Gamma (r_1)\Gamma (r_2)}
\int_{x_1}^{x_2}\int_{0}^{y_2}(x_2-s)^{r_1-1}(y_2-t)^{r_2-1}f(s,t,u)\,ds\,dt\\
&\quad +\frac{1}{\Gamma (r_1)\Gamma (r_2)}
\int_{0}^{x_1}\int_{y_1}^{y_2}(x_2-s)^{r_1-1}(y_2-t)^{r_2-1}f(s,t,u)\,ds\,dt\Big\|\\
&\leq \sum_{i=1}^{m}\|g_i(x_2,y_2)u(x_2-\xi_i,y_2-\mu_i)-g_i(x_1,y_1)u(x_1-\xi_i,y_1-\mu_i)\|\\
&\quad +\frac{p^*R}{\Gamma (r_1)\Gamma (r_2)}\int_{0}^{x_1}\int_{0}^{y_1}[(x_2-s)^{r_1-1}(y_2-t)^{r_2-1}-(x_1-s)^{r_1-1}(y_1-t)^{r_2-1}]\,ds\,dt\\
&\quad +\frac{p^*R}{\Gamma (r_1)\Gamma (r_2)}\int_{x_1}^{x_2}
\int_{0}^{y_2}(x_2-s)^{r_1-1}(y_2-t)^{r_2-1}\,ds\,dt\\
&\quad +\frac{p^*R}{\Gamma (r_1)\Gamma (r_2)}\int_{0}^{x_1}
\int_{y_1}^{y_2}(x_2-s)^{r_1-1}(y_2-t)^{r_2-1}\,ds\,dt\\
&\leq \sum_{i=1}^{m}\|g_i(x_2,y_2)u(x_2-\xi_i,y_2-\mu_i)-g_i(x_1,y_1)u(x_1-\xi_i,y_1-\mu_i)\|\\
&\quad +\frac{p^*R}{\Gamma (r_1+1)\Gamma (r_2+1)}
[(x_2-x_1)^{r_1}(y_2-y_1)^{r_2}+{x_1}^{r_1}{y_1}^{r_2}-{x_2}^{r_1}{y_2}^{r_2}]\\
&\quad +\frac{p^*R}{\Gamma (r_1+1)\Gamma (r_2+1)}
[{y_2}^{r_2}(x_2-x_1)^{r_1}-(x_2-x_1)^{r_1}(y_2-y_1)^{r_2}]\\
&\leq \sum_{i=1}^{m}\|g_i(x_2,y_2)u(x_2-\xi_i,y_2-\mu_i)-g_i(x_1,y_1)u(x_1-\xi_i,y_1-\mu_i)\|\\
&\quad +\frac{p^*R}{\Gamma (r_1+1)\Gamma (r_2+1)}
[{y_2}^{r_2}(x_2-x_1)^{r_1}+{x_1}^{r_1}{y_1}^{r_2}-{x_2}^{r_1}{y_2}^{r_2}].
\end{align*}
As $x_{1}\to x_{2}$, $y_{1}\to y_{2}$ the right-hand side of the
above inequality tends to zero. 

Now let $V$ be a subset of $D_R$ such that 
$V\subset\overline{\operatorname{conv}}(N(V)\cup\{0\})$. $V$ is bounded and
equicontinuous and therefore the function
$(x,y)\to{v(x,y)}=\alpha(V(x,y))$ is continuous on ${J}$. Using
Lemma \ref{lem3} and the properties of the measure $\alpha$ we have
for each $(x,y)\in J$,
\begin{align*}
v(t)&\leq \alpha(N(V)(x,y)\cup\{0\})\\
&\leq \alpha(N(V)((x,y))\\
&\leq  m B \alpha(V(x-\xi_i,y-\mu_i))
 +\frac{1}{\Gamma (r_{1})\Gamma (r_2)}\int_{0}^{x}\int_{0}^{y}p(s,t)\alpha(V(s,t))\,ds\,dt\\
&\leq  m B v(x-\xi_i,y-\mu_i)+\frac{1}{\Gamma (r_{1})\Gamma (r_2)}
 \int_{0}^{x}\int_{0}^{y}p(s,t)v(s,t)\,ds\,dt\\
&\leq  m B \|v\|_{\infty}+\|v\|_{\infty}\frac{1}{\Gamma (r_{1})\Gamma (r_2)}
 \int_{0}^{x}\int_{0}^{y}p(s,t)\,ds\,dt\\
&\leq \|v\|_{\infty}\Big(m B +\frac{p^*a^{r_1}b^{r_2}}{\Gamma
(r_1+1)\Gamma (r_2+1)}\Big).
\end{align*}
This implies
$$
\|v\|_{\infty}\leq\|v\|_{\infty}\Big(m B
+\frac{p^*a^{r_1}b^{r_2}}{\Gamma (r_1+1)\Gamma (r_2+1)}\Big).
$$ 
By \eqref{e5}  it follows that $\|v\|_{\infty}=0$; that is,
 $v(x,y)=0$ for each $(x,y) \in J$, and then $V(x,y)$ is relatively
compact in $E$. In view of the Ascoli-Arzel$\grave{a}$ theorem, $V$
is relatively compact in $D_R$. Applying now Theorem \ref{thm1} we
conclude that $N$ has a fixed point which is a solution of
problem \eqref{e1}-\eqref{e1'}. \hfill$\Box$\smallskip

\section{An Example}

As an application, we consider the infinite system of partial hyperbolic 
fractional differential equations
\begin{gather}\label{ex1}
\begin{aligned}
u_n(x,y)
&=\frac{x^{4}y}{7}u_n\big(x-\frac{1}{2},y-\frac{3}{5}\big)
+\frac{x^{5}y^{2}}{12}u_n\big(x-\frac{2}{3},y-\frac{1}{4}\big)\\
&\quad +\frac{1}{9}u_n\big(x-\frac{2}{5},y-\frac{1}{3}\big)
 +I_{\theta}^{r}\big(\frac{1}{3e^{x+y+4}}u_{n}(x,y)\big),\\
&\quad  (x,y)\in J:=[0,1]\times [0,1];
\end{aligned} \\
\label{ex2}
u_n(x,y)=\Phi(x,y), \quad (x,y)\in \tilde
J:=[-\frac{2}{3},1]\times[-\frac{3}{5},1]\backslash(0,1]\times (0,1],
\end{gather}
where
$n=1,2,\dots,n,\dots$, $r=(\frac{1}{2},\frac{1}{5})$,
 and $\Phi:\tilde J \to E $ is continuous with
\begin{equation}\label{ex3}
\Phi(x,0)=\frac{1}{9}\Phi\big(x-\frac{2}{3},-\frac{3}{5}\big),\quad
\Phi(0,y)=\frac{1}{9}\Phi\big(-\frac{2}{3},y-\frac{3}{5}\big),\quad
x,y\in(0,1]
\end{equation}
Let
$$
E=l^{1}=\big\{u=(u_{1},u_{2},\dots,u_{n},\dots): 
\sum_{n=1}^{\infty}|u_{n}|<\infty\big\}
$$
with the norm
$$
\|u\|_{E}=\sum_{n=1}^{\infty}|u_{n}|.
$$
Set
$u=(u_{1},u_{2},\dots,u_{n},\dots)$ and 
$f=(f_{1},f_{2},\dots,f_{n},\dots)$,
with
\begin{gather*}
f_{n}(x,y,u_{n})=\frac{1}{3e^{x+y+4}}u_{n}, \quad (x,y)\in [0,1]\times [0,1],\\
g_1(x,y)=\frac{x^{4}y}{7}, \quad g_2(x,y)=\frac{x^{5}y^{2}}{12}, \quad
g_3(x,y)=\frac{1}{9}.
\end{gather*}
Then  problem \eqref{ex1}--\eqref{ex2} can be written as \eqref{e1}--\eqref{e1'}.
In which case, we have
\begin{equation}\label{c1}
|f_{n}(x,y,u_{n})|\leq \frac{1}{3e^{x+y+4}}|u_{n}|, \quad 
\text{for } (x,y)\in [0,1]\times [0,1],  \text{ and }  u_n\in \mathbb{R}.
\end{equation}
Hence conditions (H1) and (H2) are satisfied with $p(x,y)=\frac{1}{3e^{x+y+4}}$.
Condition \eqref{e5} holds with $a=b=1$. Indeed
$$
m B +\frac{p^*a^{r_1}b^{r_2}}{\Gamma (r_1+1)\Gamma (r_2+1)}
=\frac{3}{7}+\frac{1}{3e^{4}\Gamma (r_1+1)\Gamma (r_2+1)}<1
$$
which is satisfied for each $(r_1,r_2)\in (0,1]\times (0,1]$.
Consequently, Theorem \ref{thm2} implies that
\eqref{ex1}--\eqref{ex2} has a solution defined on
$[-\frac{2}{3},1]\times [-\frac{3}{5},1]$.

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