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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 150, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/150\hfil Implicit impulsive differential equations]
{Darboux problem for implicit impulsive partial hyperbolic fractional
order differential equations}

\author[S. Abbas, M. Benchohra \hfil EJDE-2011/150\hfilneg]
{Sa\"id Abbas, Mouffak Benchohra}

\address{Sa\"id Abbas \newline
Laboratoire de Math\'ematiques, Universit\'e de Sa\"{\i}da,
B. P. 138, 20000, Sa\"{\i}da, Alg\'erie}
\email{abbasmsaid@yahoo.fr}

\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 August 19, 2011. Published November 8, 2011.}
\subjclass[2000]{26A33, 34A08}
\keywords{Hyperbolic differential equation; fractional order;
\hfill\break\indent Riemann-Liouville integral;
mixed regularized derivative; impulse; fixed point}

\begin{abstract}
 In this article we investigate the existence and uniqueness of
 solutions for the initial value problems, for a
 class of hyperbolic impulsive fractional order differential
 equations by using some fixed point theorems.
\end{abstract}

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

\section{Introduction}

Fractional calculus is a generalization of the ordinary
differentiation and integration to arbitrary non-integer order. The
subject is as old as the differential calculus since, starting from
some speculations of  Leibniz (1697) and  Euler (1730), it has
been developed up to nowadays. The idea of fractional calculus and
fractional order differential equations and inclusions has been a
subject of interest not only among mathematicians, but also among
physicists and engineers. Indeed, we can find numerous applications
in rheology, control, porous media, viscoelasticity,
electrochemistry, electromagnetism, etc.
\cite{GlNo,Hi,Ma,MeScKiNo,OlSp}. There has been a significant
development in ordinary and partial fractional differential
equations in recent years; see the monographs of Kilbas \emph{et al.}
\cite{KST}, Miller and Ross \cite{MiRo}, Podlubny \cite{Pod},
Samko \emph{et al.} \cite{SaKiMa}, the papers of Abbas and
Benchohra \cite{AbBe1,AbBe2,AbBe3}, Abbas \emph{et al.}
\cite{AbAgBe,AbBeGo,AbBeNi}, Belarbi \emph{et al.} \cite{BBO},
Benchohra \emph{et al.} \cite{BeHaGr, BeHaNt, BeHeNtOu}, Diethelm
\cite{DiFo}, Kilbas and Marzan \cite{KiMa}, Mainardi
\cite{Ma},  Podlubny \emph{et al.} \cite{PoPeViOlDo}, Vityuk and
Golushkov \cite{ViGo}, Yu and Gao \cite{YuGa}, Zhang \cite{Zh}
and the references therein.

 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
\emph{et al.} \cite{BHN}, Lakshmikantham \emph{et al.} \cite{LaBaSi},
the papers of Abbas and Benchohra \cite{AbBe2,AbBe3},
Abbas \emph{et al.} \cite{AbAgBe, AbBeGo} and the references therein.

 The Darboux problem for partial hyperbolic differential
equations was studied in the papers of Abbas and Benchohra
\cite{AbBe1,AbBe2}, Abbas \emph{et al.} \cite{AbBeVi}, Vityuk
\cite{Vi}, Vityuk and Golushkov \cite{ViGo}, Vityuk and Mykhailenko
\cite{ViMy0,ViMy} and by other authors.

  In the present article we are concerned with the existence
and uniqueness of solutions to fractional order initial-value problem
(IVP) for the system
\begin{gather}\label{e1}
\overline{D}_{\theta}^{r}u(x,y)=f(x,y,u(x,y),
 \overline{D}_{\theta}^{r}u(x,y));\quad \text{for }(x,y)\in J,\ x\neq x_k,\ k=1,\dots,m,
\\ \label{e2}
u(x_k^+,y)=u(x_k^-,y)+I_k(u(x_k^-,y)); \quad  \text{for }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 $J:=[0,a]\times [0,b]$, $a,b>0$,
$\theta=(0,0)$, $\overline{D}_{\theta}^{r}$ is the mixed regularized
derivative of order $r=(r_1,r_2)\in(0,1]\times (0,1]$,
$0=x_0<x_1<\dots<x_{m}<x_{m+1}=a$,
$f:J\times \mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}^n$,
$I_k: \mathbb{R}^n\to\mathbb{R}^n$, $k=1,\dots,m$,
$\varphi:[0,a]\to \mathbb{R}^n$ and $\psi:[0,b]\to \mathbb{R}^n$
are given absolutely continuous functions.

We present two results for the problem \eqref{e1}-\eqref{e3}, the
first one is based on Banach's contraction principle and the second
one on the nonlinear alternative of Leray-Schauder type \cite{GrDu}.

\section{Preliminaries}

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

\begin{definition}[\cite{KST,SaKiMa}] \label{def2.1}
Let $\alpha\in (0,\infty)$ and $u\in L^1(J)$.
The partial Riemann-Liouville integral of order $\alpha$ of
$u(x,y)$ with respect to $x$ is defined by the expression
$$
I_{0,x}^{\alpha}u(x,y)=\frac{1}{\Gamma (\alpha)}\int_0^{x}
(x-s)^{\alpha-1}u(s,y)ds,
$$
for almost all $x\in [0,a]$  and all $y\in [0,b]$,
where $\Gamma (.)$ is the (Euler's) Gamma function defined by
$\Gamma(\varsigma)=\int_0^{\infty}t^{\varsigma-1}e^{-t}\,dt$;
$\varsigma>0$.
\end{definition}

Analogously, we define the integral
$$
I_{0,y}^{\alpha}u(x,y)=\frac{1}{\Gamma (\alpha)}\int_0^{y}
(y-s)^{\alpha-1}u(x,s)ds,
$$
for almost all $x\in [0,a]$  and almost all $y\in [0,b]$.

\begin{definition}[\cite{KST,SaKiMa}] \label{def2.2} \rm
Let $\alpha\in (0,1]$
and $u\in L^1(J)$. The Riemann-Liouville
fractional derivative of order $\alpha$ of $u(x,y)$ with respect to
$x$ is defined by
$$
(D_{0,x}^{\alpha}u)(x,y)=\frac{\partial}{\partial
x}I_{0,x}^{1-\alpha}u(x,y),
$$
for almost all $x\in [0,a]$  and all $y\in [0,b]$.
\end{definition}

Analogously, we define the derivative
$$
(D_{0,y}^{\alpha}u)(x,y)=\frac{\partial}{\partial
y}I_{0,y}^{1-\alpha}u(x,y),
$$
for almost all $x\in [0,a]$  and almost all $y\in [0,b]$.

\begin{definition}[\cite{KST,SaKiMa}] \label{def2.3} \rm
 Let $\alpha \in (0,1]$ and $u\in L^1(J)$.
The Caputo fractional derivative of order $\alpha$ of $u(x,y)$
with respect to $x$ is defined by the expression
$$
^{c}D_{0,x}^{\alpha}u(x,y)=I_{0,x}^{1-\alpha}\frac{\partial}{\partial
x}u(x,y),
$$
for almost all $x\in [0,a]$  and all $y\in [0,b]$.
\end{definition}

Analogously, we define the derivative
$$
^{c}D_{0,y}^{\alpha}u(x,y)=I_{0,y}^{1-\alpha}\frac{\partial}{\partial
y}u(x,y),
$$
for almost all $x\in [0,a]$ and almost all $y\in [0,b]$.

\begin{definition}[\cite{ViGo}] \label{def2.4}
 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 by
$$
(I_{\theta}^{r}u)(x,y)
=\frac{1}{\Gamma (r_1)\Gamma (r_2)}\int_0^{x}
\int_0^y (x-s)^{r_1-1}(y-t)^{r_2-1}u(s,t)\,dt\,ds.
$$
\end{definition}

In particular,
$$
( I_{\theta}^{\theta}u)(x,y)=u(x,y), \ ( I_{\theta}^{\sigma}u)(x,y)
=\int_0^{x}\int_0^{y}u(s,t)\,dt\,ds;
$$
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{examp2.5} \rm
Let $\lambda,\omega\in(-1,\infty)$ and
$r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$, then
$$
I_{\theta}^{r}x^{\lambda}y^{\omega}
=\frac{\Gamma(1+\lambda)\Gamma(1+\omega)}
{\Gamma(1+\lambda+r_1)\Gamma(1+\omega+r_2)}x^{\lambda+r_1}
y^{\omega+r_2},
$$
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{def2.6} \rm
Let $r\in (0,1]\times (0,1]$ and $u\in L^1(J)$. The mixed
fractional Riemann-Liouville derivative of order $r$ of $u$ is
defined by the expression
$D_{\theta}^{r}u(x,y)=(D^2_{xy}I_{\theta}^{1-r}u)(x,y)$
and the Caputo fractional-order
derivative of order $r$ of $u$ is defined by the expression
$^{c}D_{\theta}^{r}u(x,y)=(I_{\theta}^{1-r}D^2_{xy}u)(x,y)$.
\end{definition}

The case $\sigma=(1,1)$ is included and we have
$$
(D_{\theta}^{\sigma}u)(x,y)=(^{c}D_{\theta}^{\sigma}u)(x,y)
=(D ^2_{xy}u)(x,y),
$$
for almost all $(x,y)\in J$.

\begin{example}\label{examp2.7} \rm
 Let $\lambda,\omega\in(-1,\infty)$ and
$r=(r_1,r_2)\in (0,1]\times (0,1]$, then
$$
D_{\theta}^{r}x^{\lambda}y^{\omega}
=\frac{\Gamma(1+\lambda)\Gamma(1+\omega)}
{\Gamma(1+\lambda-r_1)\Gamma(1+\omega-r_2)}x^{\lambda-r_1}
y^{\omega-r_2},
$$
for almost all $(x,y)\in J$.
\end{example}

\begin{definition}[\cite{ViMy}] \label{def2.8}\rm
 For a function $u:J\to \mathbb{R}^n$, we set
$$
q(x,y)=u(x,y)-u(x,0)-u(0,y)+u(0,0).
$$
By the mixed regularized derivative of order
$r=(r_1,r_2)\in (0,1]\times (0,1]$ of a function $u(x,y)$,
we name the function
$$
\overline{D}_{\theta}^{r}u(x,y)=D_{\theta}^{r}q(x,y).
$$
\end{definition}

The function
$$
\overline{D}_{0,x}^{r_1}u(x,y)=D_{0,x}^{r_1}[u(x,y)-u(0,y)],
$$
is called the partial $r_1-$order regularized derivative of
the function $u(x,y):J\to \mathbb{R}^n$ with respect to the
variable $x$. Analogously,
we define the derivative
$$
\overline{D}_{0,y}^{r_2}u(x,y)=D_{0,y}^{r_2}[u(x,y)-u(x,0)].
$$

 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{def2.9} \rm
 For $u\in L^1(J_z,\mathbb{R}^n)$ where $D^2_{xy}u$ is Lebesque
integrable on $[x_k,x_{k+1}]\times[0,b], \ k=0,\dots,m$, 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}

Analogously, we define the derivatives
\begin{gather*}
\overline{D}_{z^+}^{r}u(x,y)=D_{z^+}^{r}q(x,y), \\
\overline{D}_{a_1,x}^{r_1}u(x,y)=D_{a_1,x}^{r_1}[u(x,y)-u(0,y)],\\
\overline{D}_{a_1,y}^{r_2}u(x,y)=D_{a_1,y}^{r_2}[u(x,y)-u(x,0)].
\end{gather*}

\section{Existence of solutions}

In what follows set
$$
J_k:=(x_k,x_{k+1}]\times [0,b].
$$
To define the solutions of  \eqref{e1}-\eqref{e3}, we shall consider
the  space
\begin{align*}
PC(J)=&\big\{u: J\to\mathbb{R}^n: u \in
C(J_k, \mathbb{R}^n) ; \ k=0,1, \dots,m, \text{  and} \\
&\text{ there exist }u(x_k^-,y) \text{ and } u(x_k^+,y); \
 k=1,\dots,m, \\
&\text{ with } u(x_k^-,y)=u(x_k,y)  \text{ for each }  y\in [0,b]\big\}.
\end{align*}
This set is a Banach space with the norm
$$
\|u\|_{PC}=\sup_{(x,y)\in J}\|u(x,y)\|.
$$
Set
$$
J':=J\backslash\{(x_1,y),\dots,(x_{m},y),\  y\in [0,b]\}.
$$

\begin{definition} \label{def3.1} \rm
A function $u\in PC(J)$ such that $u, \overline{D}_{x_k,x}^{r_1}u,
\overline{D}_{x_k,y}^{r_2}u, \overline{D}_{z_k^+}^{r}u;\ k=0,\dots,m$,
are continuous on $J'$ and $I_{z^+}^{1-r}u\in AC(J')$ is said to be
a solution of \eqref{e1}-\eqref{e3} if $u$ satisfies  \eqref{e1}
on $J'$ and conditions \eqref{e2}, \eqref{e3} are satisfied.
\end{definition}

For the existence of solutions for \eqref{e1}-\eqref{e3} we need
the following lemmas.

\begin{lemma}[\cite{ViMy}] \label{lem1}
Let the function $f:J\times\mathbb{R}^n\times\mathbb{R}^n\to
\mathbb{R}^n$ be continuous on its variables. Then the problem
\begin{gather}\label{e1'}
\overline{D}_{\theta}^{r}u(x,y)=f(x,y,u(x,y),
\overline{D}_{\theta}^{r}u(x,y));\quad \text{if }(x,y)\in
J:=[0,a]\times [0,b], \\
\label{e2'}
\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}
\end{gather}
is equivalent to the problem
$$
g(x,y)=f(x,y,\mu(x,y)+I_{\theta}^{r}g(x,y),g(x,y)),
$$
and if $g\in C(J)$ is the solution of this equation,
then $u(x,y)=\mu(x,y)+I_{\theta}^{r}g(x,y)$,
where
$$
\mu(x,y)=\varphi(x)+\psi(y)-\varphi(0).
$$
\end{lemma}

\begin{lemma}[\cite{AbBeGo}] \label{lem2}
Let $0< r_1,r_2\leq 1$ and let $ h: 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}
\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;\\
\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;\\
\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_{z_k^{+}}^{r}u(x,y)= h(x,y), \quad   (x,y)\in J',
\; k=1,\dots,m,\\ \label{e6'}
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}

By Lemmas \ref{lem1} and \ref{lem2}, we conclude the
following statement.

\begin{lemma} \label{lem3}
Let the function
$f:J\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$
be continuous. Then problem \eqref{e1}-\eqref{e3} is equivalent
to the problem
\begin{equation}\label{eq1}
g(x,y)=f(x,y,\xi(x,y),g(x,y)),
\end{equation}
where
$$
\xi(x,y)=\begin{cases}
\mu(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\int_0^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds;\\
\text{  if } (x,y)\in [0,x_1]\times[0,b],
\\[3pt]
\mu(x,y)+\sum_{i=1}^{k}(I_{i}(u(x_{i}^{-},y))-I_{i}(u(x_{i}^{-},0)))\\
+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{i=1}^{k}\int_{x_{i-1}}^{x_{i}}\int_0^{y}
(x_{i}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\\
+ \frac{1}{\Gamma(r_1)\Gamma(r_2)}
\int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds;\\
\text{ if  } (x,y)\in (x_k,x_{k+1}]\times[0,b],\ k=1,\dots,m,
\end{cases}
$$
$$
\mu(x,y)=\varphi(x)+\psi(y)-\varphi(0).
$$
And if $g\in C(J)$ is the solution of \eqref{eq1}, then
$u(x,y)=\xi(x,y)$.
\end{lemma}

 Further, we present conditions for the existence and
uniqueness of a solution of problem \eqref{e1}-\eqref{e3}.
We will us the following hypotheses.
\begin{itemize}
\item[(H1)] The function
$f:J\times \mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$
is continuous;
\item[(H2)] For any $u,v,w,z\in \mathbb{R}^n$ and $(x,y) \in J$,
there exist constants $l>0$ and $0<l_*<1$ such that
$$
\|f(x,y,u,z)-f(x,y,v,w)\|\leq l\|u-v\|+l_*\|z-w\|,
$$
\item[(H3)] There exists a constant $l^*>0$ such that
$$
\|I_k(u)-I_k(\overline u)\|\leq l^*\|u-\overline u\|, \quad
\text{for }  u, \overline u \in \mathbb{R}^n, \; k=1,\dots,m.
$$
\end{itemize}

\begin{theorem}\label{thm3}
Assume {\rm (H1)--(H3)} are satisfied. If
\begin{equation} \label{e6}
2ml^*+\frac{2la^{r_1}b^{r_2}}{(1-l_{*})\Gamma(r_1+1)\Gamma(r_2+1)}<1,
\end{equation}
then there exists a unique solution for $IVP$ \eqref{e1}-\eqref{e3}
on $J$.
\end{theorem}

\begin{proof}
 Transform the problem \eqref{e1}-\eqref{e3} into a
fixed point problem. Consider the operator
$N:PC(J)\to PC(J)$ defined by
\begin{equation} \label{e7}
\begin{split}
N(u)(x,y)
&= \mu(x,y)+\sum_{0<x_k<x}(I_k(u(x_k^{-},y))-I_k(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}g(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)\,dt\,ds,
\end{split}
\end{equation}
where $g\in C(J)$ such that
$$
g(x,y)=f(x,y,u(x,y),g(x,y)),
$$
By Lemma \ref{lem3}, the problem of finding the solutions of
\eqref{e1}-\eqref{e3} is reduced to finding the solutions of the
operator equation $N(u)=u$.
 Let $v,w\in PC(J)$. Then, for $(x,y)\in J$, we have
\begin{equation}\label{e07}
\begin{split}
&\|N(v)(x,y)-N(w)(x,y)\|\\
&\leq \sum_{k=1}^{m}(\|I_k(v(x_k^{-},y))-I_k(w(x_k^{-},y))\|
 +\|I_k(v(x_k^{-},0))-I_k(w(x_k^{-},0))\|)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}
 \int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)-h(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)-h(s,t)\|\,dt\,ds,
\end{split}
\end{equation}
where $g,h\in C(J)$ such that
\begin{gather*}
g(x,y)=f(x,y,v(x,y),g(x,y)),\\
h(x,y)=f(x,y,w(x,y),h(x,y)).
\end{gather*}
By (H2), we obtain
\[
\|g(x,y)-h(x,y)\|\leq l\|v(x,y)-w(x,y)\|+l_*\|g(x,y)-h(x,y)\|.
\]
Then
\[
\|g(x,y)-h(x,y)\| \leq \frac{l}{1-l_*}\|v(x,y)-w(x,y)\|
\leq \frac{l}{1-l_*}\|v-w\|_{PC}.
\]
Thus, (H3) and \eqref{e07} imply
\begin{align*}
&\|N(v)-N(w)\|_{PC}\\
&\leq \sum_{k=1}^{m}l^*(\|v(x_k^{-},y)-w(x_k^{-},y)\|
 +\|v(x_k^{-},0)-w(x_k^{-},0)\|)\\
&\quad +\frac{l}{(1-l_*)\Gamma(r_1)\Gamma(r_2)}
 \sum_{k=1}^{m}\int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-s)^{r_1-1}(y-t)^{r_2-1}\|v-w\|_{PC}\,dt\,ds\\
&\quad +\frac{l}{(1-l_{*})\Gamma(r_1)\Gamma(r_2)}
\int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}\|v-w\|_{PC}\,dt\,ds.
\end{align*}
However,
\begin{align*}
&\frac{l}{(1-l_*)\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}
 \int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-s)^{r_1-1}(y-t)^{r_2-1}\|v-w\|_{PC}\,dt\,ds\\
&\leq  \frac{l}{(1-l_*)\Gamma(r_1)\Gamma(r_2)}\frac{b^{r_2}}{r_2}
 \|v-w\|_{PC}
\sum_{k=1}^{m}\int_{x_{k-1}}^{x_k}(x_k-s)^{r_1-1}ds\\
&\leq  \frac{l}{(1-l_*)\Gamma(r_1)\Gamma(r_2)}\frac{b^{r_2}}{r_2}
 \|v-w\|_{PC}
\sum_{k=1}^{m}\frac{x_{k-1}^{r_1}}{r_1}\\
&=  \frac{l}{(1-l_*)\Gamma(r_1)\Gamma(r_2)}\frac{b^{r_2}}{r_2}
 \|v-w\|_{PC} \frac{x_{m-1}^{r_1}-x_0^{r_1}}{r_1}\\
&\leq  \frac{l}{(1-l_*)\Gamma(r_1)\Gamma(r_2)}
 \frac{b^{r_2}}{r_2}\|v-w\|_{PC} \frac{a^{r_1}}{r_1}\\
&=  \frac{la^{r_1}b^{r_2}}{(1-l_*)\Gamma(1+r_1)\Gamma(1+r_2)}
 \|v-w\|_{PC}.
\end{align*}
Then
\begin{align*}
&\|N(v)-N(w)\|_{PC}\\
&\leq \Big(2ml^*+\frac{la^{r_1}b^{r_2}}{(1-l_{*})
 \Gamma(r_1+1)\Gamma(r_2+1)}+\frac{la^{r_1}b^{r_2}}
{(1-l_{*})\Gamma(r_1+1)\Gamma(r_2+1)}\Big)\|v-w\|_{PC}\\
&\leq \Big(2ml^*+\frac{2la^{r_1}b^{r_2}}{(1-l_{*})
 \Gamma(r_1+1)\Gamma(r_2+1)}\Big)\|v-w\|_{PC}.
\end{align*}
Hence
\[
\|N(v)-N(w)\|_{PC}\leq\Big(2ml^*
+\frac{2la^{r_1}b^{r_2}}{(1-l_{*})\Gamma(r_1+1)\Gamma(r_2+1)}\Big)
 \|v-w\|_{PC}.
\]
By \eqref{e6}, $N$ is a contraction, and hence $N$ has a unique
fixed point by Banach's contraction principle.
\end{proof}

\begin{theorem}[Nonlinear alternative of Leray-Schauder type
\cite{GrDu}] \label{thm0}
Let $X$ be a \\ Banach space and $C$ a nonempty
convex subset of $X$. Let $U$ a nonempty open subset of $C$ with
$0\in U$ and $T:\overline{U}\to C $ continuous and compact
operator.
Then either
\begin{itemize}
\item[(a)] $T$ has fixed points. Or
\item[(b)] There exist $u\in \partial U$ and $\lambda\in [0,1]$
with $u=\lambda T(u)$.
\end{itemize}
\end{theorem}

For the next theorem, we use the following assumptions:
\begin{itemize}
\item [(H4)] There exist $p, q, d\in C(J,\mathbb{R}_{+})$ such that
$$
\|f(x,y,u,z)\|\leq p(x,y)+q(x,y)\|u\|+d(x,y)\|z\|
$$
for $(x,y)\in J$ and each $u,z\in \mathbb{R}^n$,

\item[(H5)] There exists $\psi^*:[0,\infty)\to (0,\infty)$ continuous
and nondecreasing such that
$$
\|I_k(u)\|\leq \psi^*(\|u\|);\ k=1,\dots,m, \quad
\text{for all }  u\in \mathbb{R}^n,
$$

\item[(H6)] There exists a number $\overline M>0$ such that
$$
\|\mu\|_{\infty}+2m\psi^{*}(\overline M)
+\frac{2a^{r_1}b^{r_2}\big(p^{*}+q^{*}\|\mu\|_{\infty}+
2mq^{*}\psi^{*}(\overline M)\big)}{\big(1-d^*
-\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\big)
\Gamma(1+r_1)\Gamma(1+r_2)}
< \overline{M},
$$
where $p^{*}=\sup_{(x,y)\in J}p(x,y)$,
$ q^{*}=\sup_{(x,y)\in J}q(x,y)$ and\\
$d^{*}=\sup_{(x,y)\in J}d(x,y)$.
\end{itemize}

\begin{theorem} \label{thm3.7}
Assume {\rm (H1), (H4), (H5), (H6)} hold.
If
\begin{equation}\label{e''}
d^*+\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}<1,
\end{equation}
then \eqref{e1}-\eqref{e3} has at least one solution on $J$.
\end{theorem}

\begin{proof} Transform  problem \eqref{e1}-\eqref{e3}
into a fixed point problem. Consider the operator $N$ defined in
\eqref{e7}. We shall show that the operator $N$ is continuous and
compact.

\textbf{Step 1:}  $N$ is continuous.
 Let $\{u_{n}\}_{n\in \mathbb{N}}$ be a sequence such that $u_{n}\to u$
in $PC(J)$. Let $\eta>0$ be such that $\|u_{n}\|_{PC} \leq \eta$.
Then for each $(x,y)\in J$ we have
\begin{equation}\label{e007}
\begin{split}
&\|N(u_n)(x,y)-N(u)(x,y)\|\\
&\leq \sum_{k=1}^{m}(\|I_k(u_n(x_k^{-},y))-I_k(u(x_k^{-},y))\|
+\|I_k(u_n(x_k^{-},0))-I_k(u(x_k^{-},0))\|)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}
\int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-s)^{r_1-1}(y-t)^{r_2-1}\|g_n(s,t)-g(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_n(s,t)-g(s,t)\|\,dt\,ds,
\end{split}
\end{equation}
where $g_n,g\in C(J)$ such that
\begin{gather*}
g_n(x,y)=f(x,y,u_n(x,y),g_n(x,y)), \\
g(x,y)=f(x,y,u(x,y),g(x,y)).
\end{gather*}
Since
$u_n\to u$ as $n\to \infty$ and $f$ is a continuous function,
we obtain
$$
g_n(x,y)\to g(x,y) \quad\text{as $n\to \infty$,  for each $(x,y)\in J$}.
$$
Hence, \eqref{e007} gives
\[
\|N(u_{n})-N(u)\|_{PC} \leq 2ml^*\|u_n-u\|_{PC}
+\frac{2a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}
\|g_n-g\|_{\infty}\to 0
\]
as $n\to \infty$.


\textbf{Step 2:} $N$ maps bounded sets into bounded sets in $PC(J)$.
Indeed, it is sufficinet show that for any
$\eta^{*}>0$, there exists a positive constant $M^{*}$ such that,
for each $u\in B_{\eta^{*}}=\{u\in PC(J):\|u\|_{PC}\leq \eta^{*}\}$,
we have $\|N(u)\|_{PC}\leq M^{*}$. For $(x,y)\in J$, we have
\begin{equation}\label {e0007}
\begin{split}
&\|N(u)(x,y)\|\\
&\leq \|\mu(x,y)\|+\sum_{k=1}^{m}(\|I_k(u(x_k^{-},y))\|
 +\|I_k(u(x_k^{-},0))\|)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}
 \int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-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_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds,
\end{split}
\end{equation}
where $g\in C(J)$ such that
$g(x,y)=f(x,y,u(x,y),g(x,y))$.
By (H4), for each $(x,y)\in J$, we have
$$
\|g(x,y)\|\leq p(x,y)+q(x,y)\|\xi(x,y)\|+d(x,y)\|g(x,y)\|.
$$
On the other hand, for each $(x,y)\in J$,
\begin{align*}
\|\xi(x,y)\|
&\leq \|\mu(x,y)\|+\sum_{k=1}^{m}(\|I_k(u(x_k^{-},y))\|
 +\|I_k(u(x_k^{-},0))\|)\\
&\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}
 \int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-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_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds\\
&\leq \|\mu\|_{\infty}+ 2m\psi^{*}(\eta^{*})
 +\frac{2a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\|g\|_{\infty}.
\end{align*}
Hence, for each $(x,y)\in J$, we have
\begin{align*}
\|g\|_{\infty}&\leq p^{*}+q^{*}\Big(\|\mu\|_{\infty}+
2m\psi^{*}(\eta^{*})+\frac{2a^{r_1}b^{r_2}}
{\Gamma(1+r_1)\Gamma(1+r_2)}\|g\|_{\infty}\Big)+d^{*}\|g\|_{\infty}.
\end{align*}
Then, by \eqref{e''}, we have
\begin{align*}
\|g\|_{\infty}\leq\frac{p^{*}+q^{*}\big(\|\mu\|_{\infty}+
2m\psi^{*}(\eta^{*})\big)}{1-d^*
-\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}}:=M.
\end{align*}
Thus, \eqref{e0007} implies
\begin{align*}
\|N(u)\|_{PC}\leq\|\mu\|_{\infty}+2m\psi^{*}(\eta^{*})
+\frac{2Ma^{r_1}b^{r_2}}
{\Gamma(1+r_1)\Gamma(1+r_2)}:= M^{*}.
\end{align*}

\textbf{Step 3:}  $N$ maps bounded sets into equicontinuous sets in
$PC(J)$.
Let \\
$(\tau_1,y_1), (\tau_2,y_2)\in J$, $\tau_1<\tau_2 $ and
$y_1<y_2 $, $B_{\eta^*}$ be a
bounded set of $PC(J)$ as in Step 2, and let $u\in B_{\eta^*}$. Then
for each $(x,y)\in J$, we have
\begin{align*}
&\|N(u)(\tau_2,y_2)-N(u)(\tau_1,y_1)\|\\
&\leq\| \mu(\tau_1,y_1) -\mu( \tau_2,y_2)\|
 +\sum_{k=1}^{m}(\|I_k(u(x_k^{-},y_1))-I_k(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 g(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)\|\,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)\,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)\|\,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)\|\,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)\|\,dt\,ds,
\end{align*}
where $g\in C(J)$ such that
$g(x,y)=f(x,y,u(x,y),g(x,y))$.
However, $\|g\|_{\infty}\leq M$. Thus
\begin{align*}
&\|N(u)(x_2,y_2)-N(u)(x_1,y_1)\|\\
&\leq\|\mu(\tau_1,y_1) -\mu( \tau_2,y_2)\|
 +\sum_{k=1}^{m}(\|I_k(u(x_k^{-},y_1))-I_k(u(x_k^{-},y_2))\|)\\
&\quad +\frac{M}{\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{M}{\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{M}{\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{M}{\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{M}{\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{M}{\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
Arzela-Ascoli theorem, we can conclude that $N$ is continuous and
completely continuous.

\textbf{Step 4}: A priori bounds.
 We now show there exists an open set
$U\subseteq PC(J)$ with $u\neq \lambda N(u)$, for $\lambda \in
(0,1)$ and $u\in \partial U$.
Let $u\in PC(J)$ and $u=\lambda N(u)$ for
some $0<\lambda <1$. Thus for each $(x,y)\in J$, we have
\begin{align*}
\|u(x,y)\|&\leq \|\lambda\mu(x,y)\|+\sum_{k=1}^{m}\lambda(\|I_k(u(x_k^{-},y))\|+\|I_k(u(x_k^{-},0))\|)\\
&\quad +\frac{\lambda}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m}\int_{x_{k-1}}^{x_k}\int_0^{y}
(x_k-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds\\
&\quad +\frac{\lambda}{\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\\
&\leq \|\mu\|_{\infty}+
2m\psi^{*}(\|u(x,y\|)+\frac{2a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\|g\|_{\infty}.
\end{align*}
However,
\begin{align*}
\|g\|_{\infty}\leq\frac{p^{*}+q^{*}\big(\|\mu\|_{\infty}+
2m\psi^{*}(\|u\|_{PC})\big)}{1-d^*
-\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}}.
\end{align*}
Thus, for each $(x,y)\in J$, we have
\[
\|u\|_{PC}\leq \|\mu\|_{\infty}+
2m\psi^{*}(\|u\|_{PC})+\frac{2a^{r_1}b^{r_2}
\big(p^{*}+q^{*}\|\mu\|_{\infty}+ 2mq^{*}\psi^{*}(\|u\|_{PC}) \big)}
{\big(1-d^*-\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\big)
\Gamma(1+r_1)\Gamma(1+r_2)}.
\]
Hence
\[
\|u\|_{PC}\leq
\|\mu\|_{\infty}+ 2mq^{*}\psi^{*}(\|u\|_{PC})
 +\frac{2a^{r_1}b^{r_2}\big(p^{*}+q^{*}\|\mu\|_{\infty}+
2m\psi^{*}(\|u\|_{PC}) \big)}{\big(1-d^*
-\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\big)
\Gamma(1+r_1)\Gamma(1+r_2)}.
\]
By (H6), there exists  $\overline M$ such that
 $\|u\|_{PC} \neq \overline M$.
Let
$$
U=\{u\in PC(J):\|u\|_{PC}<\overline M+1\}.
$$
By our choice of $U$, there is no $u\in \partial U$ such that
$u=\lambda N(u)$, for $\lambda \in (0,1)$. As a consequence
of Theorem \ref{thm0}, we deduce that $N$ has a
fixed point $u$ in $\overline{U}$ which is a solution to
 \eqref{e1}-\eqref{e3}.
\end{proof}

\section{An Example}

As an application of our results we consider the following impulsive
implicit partial hyperbolic differential equations
\begin{equation} \label{ex1}
\begin{split}
\overline{D}_{\theta}^{r}u(x,y)
=\frac{1}{10e^{x+y+2} (1+|u(x,y)|+|\overline{D}_{\theta}^{r}u(x,y)|)},\\
\text{for }(x,y)\in [0,1]\times [0,1], \; x\neq x_k, \; k=1,\dots,m;
\end{split}
\end{equation}
\begin{equation}\label{ex2}
u(x_k^+,y)=u(x_k^-,y)+\frac{1}{6e^{x+y+4}(1+|u(x_k^-,y)|)}; \
\text{for } y\in [0,1], \; k=1,\dots,m;
\end{equation}
\begin{equation}\label{ex3}
 u(x,0)=x, \quad u(0,y)=y^2; \text{for }  x,y\in [0,1].
\end{equation}
Set
\begin{gather*}
f(x,y,u,v)=\frac{1}{10e^{x+y+2}(1+|u|+|v|)}, \quad
(x,y)\in [0,1]\times [0,1], \\
I_k(u(x_k^-,y))=\frac{1}{6e^{x+y+4}(1+|u(x_k^-,y)|)}, \quad
 y\in [0,1].
\end{gather*}
Clearly, the function $f$ is continuous.
For each $u,v, \overline u, \overline v\in\mathbb{R}$ and
$(x,y)\in [0,1]\times[0,1]$, we have
\begin{gather*}
|f(x,y,u,v)-f(x,y,\overline u,\overline v)|\leq \frac{1}{10e^2}
(|u-\overline u|+|v-\overline v|), \\
|I_k(u)-I_k(\overline u)|\leq\frac{1}{6e^{4}}|u-\overline u|.
\end{gather*}
 Hence condition (H2) and (H3) are satisfied with
$l=l_*=\frac{1}{10e^2}$ and $l^*=\frac{1}{6e^{4}}$.
We shall show that  \eqref{e6} holds with $a=b=1$. Indeed,
if we assume, for instance, that the number of impulses $m=3$, then
we have
$$
2ml^*+\frac{2la^{r_1}b^{r_2}}{(1-l_*)\Gamma(r_1+1)\Gamma(r_2+1)}
=\frac{1}{e^{4}} +\frac{2}{(10e^2-1)\Gamma(r_1+1)\Gamma(r_2+1)}<1,
$$
which is satisfied for each $(r_1,r_2)\in (0,1]\times (0,1]$.
Consequently Theorem \ref{thm3} implies that
 \eqref{ex1}-\eqref{ex3} has a unique
solution defined on $[0,1]\times [0,1]$.

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