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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 28, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/28\hfil Hyperbolic differential equations in Banach algebras]
{Existence of solutions for discontinuous hyperbolic partial  differential
equations in Banach algebras}
\author[B. C. Dhage and S. K. Ntouyas\hfil EJDE-2006/28\hfilneg]
{Bapurao C. Dhage, Sotiris  K. Ntouyas}  % in alphabetical order

\address{Bapurao C. Dhage \newline
Kasubai, Gurukul Colony,
Ahmedpur-413 515,  Dist: Latur
Maharashtra, India}
\email{bcd20012001@yahoo.co.in}

\address{Sotiris K. Ntouyas \newline
Department of Mathematics,
University of Ioannina, 451 10 Ioannina, Greece}
\email{sntouyas@cc.uoi.gr}

\date{}
\thanks{Submitted December 22, 2005. Published March 9, 2006.}
\subjclass[2000]{35L70, 35L15}
\keywords{Hyperbolic differential equation; Banach algebras}

\begin{abstract}
 In this paper, we prove an existence theorem  for
 hyperbolic differential equations in Banach algebras
 under Lipschitz and Carath\'eodory conditions.
 The existence  of extremal solutions is also proved  under
 certain monotonicity conditions.
\end{abstract}

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

\section{Introduction}

Let $\mathbb{R}$ denote the real line. Given two closed and
bounded intervals $J_a =[0,a]$ and $J_b=[0,b]$ in $\mathbb{R}$, we
consider the second order hyperbolic differential
equation (HDE)
\begin{equation}\label{e11}
\begin{gathered}
\frac{\partial^2}{{\partial x}{\partial
y}}\left[\frac{u(x,y)}{f(x,y,u(x,y))}\right] = g(x,y,
u(x,y)), \quad (x,y)\in J_a\times J_b,\\
 u(x,0)=\phi(x),\quad u(0,y)=\psi(y),
\end{gathered}
\end{equation}
where $ f: J_a\times J_b\times \mathbb{R}\to \mathbb{R}\setminus\{0\}$,
$g: J_a\times J_b\times \mathbb{R} \to \mathbb{R}$,
and $\phi: J_a\to \mathbb{R}$,  $\psi:J_b\to \mathbb{R}$ are
continuous
functions with $\phi(0)=\psi(0)$.

By a \emph{solution} of the HDE \eqref{e11} we mean a
function $u\in AC(J_a\times J_b, \mathbb{R})$ satisfying
\begin{itemize}
\item[(i)] the function $(x,y)\mapsto
\big(\frac{u(x,y)}{f(x,y,u(x,y))}\big)$ is absolutely
continuous, and

\item[(ii)] $u$ satisfies the equations in \eqref{e11},
\end{itemize}
where $AC(J_a\times J_b, \mathbb{R})$ is the space of
absolutely continuous
real-valued functions on $J_a\times J_b$.


The existence of solutions and the topological properties of the
solutions set of hyperbolic differential equations have received
much attention during the last two decades, see for example, De
Blasi and Myjak \cite{DeMy1} and the references cited therein.
Lakshmikantham and Pandit \cite{LaPa,Pa} coupled the method of
upper and lower solutions with the monotone method to obtain
existence of extremal solutions for hyperbolic differential
equations. The method of upper and lower solutions has been
successfully applied to study the existence of multiple solutions
for initial and boundary value problems of the first and second
order partial differential equations. We refer to the books by
Carl and Heikkila \cite {CaHe}, Heikkila and Lakshmikantham
\cite{HL},  Lakshmikantham and Pandit \cite{LaPa}, Pandit
\cite{Pa} and the references cited therein. The hyperbolic
differential equation \eqref{e11} is new to the literature and the
physical situations in which HDE \eqref{e11} occurs are yet to be
investigated. Existence results for the hyperbolic differential
equations \eqref{e11} are proved in Arara et.\ al,\ \cite{ABD}
under Carath\'eodory conditions via nonlinear alternative of
Leray-Schauder type. In this paper, we prove   existence of
extremal solutions under discontinuous nonlinearity involved in
the equations. The rest of the paper is organized as follows. In
the following section we present notations, definitions and
preliminary results needed in the following sections. In Section 3
we prove the main existence result. Section 4 deals with existence
theorems for extremal solutions of the HDE \eqref{e11} under
certain Lipschitz and monotonicity conditions. Finally, an example
illustrating the abstract results is presented in Section 5.

\section{Auxiliary Results}

Let $X$  be a Banach algebra with norm $\|\cdot \|$. A mapping $A:
X\to X $ is called $\mathcal D$-Lipschitz if there exists a
continuous nondecreasing
 function $\psi : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying
\begin{equation}\label{e21}
\|Ax - Ay\| \le \psi( \| x-y \|)
\end{equation}
for all $x,y \in X$ with $\psi(0)=0$. In the special case when
$\psi (r) =\alpha r$ ($\alpha > 0$), $A$ is called a Lipschitz
with a Lipschitz constant $ \alpha $. In particular, if $\alpha
<1$, $A$ is called a contraction with a contraction constant
$\alpha$. Further, if $\psi(r)< r$ for all $r>0$, then $A$ is
called a nonlinear contraction on $X$. Sometimes we call the
function $\psi$ a $D$-function for convenience.

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.

The nonlinear alternative of Schaefer type recently
proved by Dhage \cite{D4}
 is embodied in the following theorem.

\begin{theorem}[Dhage \cite{D4}] \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)] $\alpha < 1$,   where $M= \|B(X)\|:=\sup\{\|Bx\| : x\in
X\}$.
\end{itemize}
Then either
\begin{itemize}
\item[(i)] the equation $\lambda [A x \, Bx ]=x $
 has a solution for $\lambda = 1$, or
\item[(ii)] the set ${\mathcal  E }=\{u \in X
\mid\lambda
   [Au\,Bu] =u,\, 0<\lambda <1 \}$ is unbounded.
\end{itemize}
\end{theorem}

A non-empty closed set $K$ in a Banach algebra $X$ is called a
\emph{cone} if (i) $K+K\subseteq K$, (ii) $\lambda K\subseteq K$
for $\lambda\in \mathbb{R}, \lambda\ge 0$ and (iii) $\{-K\}\cap
K=0$, where $0$ is the zero element of $X$. A cone $K$ is called
to be \emph{positive} if (iv) $K\circ K\subseteq K$, where
"$\circ$" is a multiplication composition in $X$. We introduce an
order relation $\le$ in $X$ as follows. Let $x,y\in X$. Then $x\le
y$ if and only if $y-x\in K$. 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. The details of cones and
their properties appear in Heikkila and Lakshmikantham \cite{HL}.

\begin{lemma}[Dhage \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 $a,b\in X, a\le b$, the order interval $[a,b]$ is a set in
$X$ given by
$$[a,b]=\{x\in X: a\le x\le b\}.$$
We use the following fixed point theorem of Dhage
\cite{D2} for proving the
existence of extremal solutions for the HDE
\eqref{e11}
under certain monotonicity conditions.

\begin{theorem}[Dhage \cite{D2}]\label{t21}
Let $K$ be a cone in a Banach algebra $X$ and let $a,b\in X$.
Suppose that $A,B: [a,b]\to K$ are two operators such that
\begin{itemize}
\item[(a)] $A$ is completely continuous,
\item[(b)] $B$ is totally bounded,
\item[(c)] $Ax\,By\in [a,b]$ for all $x,y\in [a,b]$, and
\item[(d)] $A$ and $B$ are nondecreasing.
\end{itemize}
Further if the cone $K$ is positive and normal, then
the operator equation $Ax\,Bx=x$ has a least and a greatest
positive solution
in $[a,b]$.
\end{theorem}

\begin{theorem}[Dhage \cite{D2}] \label{t22}
Let $K$ be a cone in a Banach algebra $X$ and let $a,b\in X$.
Suppose that
$A,B: [a,b]\to K$ 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)]
$Ax\,By\in [a,b]$
for all $x,y\in [a,b]$, and \item[(d)]  $A$ and $B$
are nondecreasing.
\end{itemize}
Further if the cone $K$ is positive and normal, then
the operator equation $Ax\,Bx=x$ has least and a greatest positive
solution in $[a,b]$, whenever $\alpha M<1$, where
$M=\|B([a,b])\|:=\sup\{\|Bx\|: x\in [a,b]\}$.
\end{theorem}

\begin{remark} \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 $a$ and $b$ in $X$
such that $a\le Aa\,Ba$ and $Ab\,Bb\le b$.
\end{remark}

\section{Existence Results} \label{S3}

Let $B(J_a\times J_b,\mathbb{R})$ denote the space of
real-valued bounded functions 0n $J_a\times J_b$ and
let $C(J_a\times J_b,\mathbb{R})$ be the Banach
space of all continuous functions from $J_a\times J_b$
into $\mathbb{R}$
with the norm
\begin{equation}\label{e31}
\|u\|_{\infty}=\sup\{|u(x,y)|:(x,y)\in J_a\times
J_b\}. \end{equation}
Define a multiplication
$``\,\cdot\,"$  by
$$
(u\cdot v)(x,y)=u(x,y)v(x,y) $$  for each  $(x,y)\in
J_a\times
J_b$. Then $C(J_a\times J_b,\mathbb{R})$ is a Banach algebra
with above norm and
multiplication. Let
$L^{1}(J_a\times J_b,\mathbb{R})$ denotes the Banach space of
measurable
functions $u: J_a\times J_b \longrightarrow \mathbb{R}$ which
are Lebesgue
integrable normed by
$$
\|u\|_{L^{1}}=\int_{0}^{a}\int_{0}^{b}|u(x,y)|dx\,dy.
$$
The HDE \eqref{e11} is equivalent to the
functional integral equation (in short FIE).
\begin{equation} \label{e32}
u(x,y) = \big[f(x,y,u(x,y))\big]\Big(z_0(x,y)+
\int_{0}^{x}\int_{0}^{y}g(t,s,u(t,s))\,ds\,dt\Big)
\end{equation}
for $(x,y) \in J_a\times J_b$, where
$z_0(x,y)=\frac{\psi(y)}{f(0,y,\psi(y))}
+\frac{\phi(x)}{f(x,0,\phi(x))}-\frac{\phi(0)}{f(0,0,\phi(0))}$.

Note that if the function $f$ is continuous on
$J_a\times J_b\times \mathbb{R}$,
then from
the continuity of $\phi$ and $\psi$ it follows that
$z_0\in C(J_a\times
J_b,\mathbb{R})$.

We need the following definition in the sequel.

\begin{definition}\label{Car}
A function $\beta :J_a\times J_b \times \mathbb{R} \to
\mathbb{R}$ is called
\emph{Carath\'eodory}'s  if
\begin{enumerate}
\item [(i)] the function $(x,y)\to \beta(x,y,z)$ is measurable
for each   $z\in \mathbb{R}$,
\item [(ii)] the function $z\to \beta(x,y,z)$ is continuous for
almost each $(x,y) \in J_a\times J_b$.
\end{enumerate}
Further a \emph{Carath\'eodory} function $\beta(x,y,z)$
is called $L^1$-\emph{Carath\'eodory} if
\begin{enumerate}
\item[(iii)] for each number $r>0$, there exists a
function $h_r\in
L^1 (J_a\times J_b, \mathbb{R})$  such that
$$
|\beta(x,y,z)|\leq h_r(x,y) \quad\text{a.e. }(x,y)\in J_a\times
J_b
$$
for all $z\in \mathbb{R}$ with $|z|\leq r$.
\end{enumerate}
Finally, a \emph{Carath\'eodory} function
$\beta(x,y,z)$ is called $L^1_X$-\emph{Carath\'eodory} if
\begin{enumerate}
\item[(iv)] there exists a function $h\in
L^1 (J_a\times J_b, \mathbb{R})$  such that
$$
|\beta(x,y,z)|\leq h(x,y) \quad \text{a.e. }
(x,y)\in J_a\times J_b
$$
for all $z\in \mathbb{R}$.
\end{enumerate}
\end{definition}

The following hypotheses will be used in the sequel.
\begin{itemize}
\item[(A1)] The function $f$ is continuous on
$J_a\times J_b\times \mathbb{R}$.
\item[(A2)] There exists a   function $\alpha\in
B(J_a\times J_b, \mathbb{R}^+)$
such that
$$
|f(x,y,z)-f(x,y,{\overline z})|\le \alpha(x,y)|z-{\overline z}| ,
\quad\text{a.e. } (x,y)\in J_a\times J_b,
$$
 for all $z, {\overline z} \in \mathbb{R}$.
\item[(A3)] The function $g$ is $L^1_X$-Carath\'eodory.
\end{itemize}

\begin{theorem}\label{tc}
Assume that hypotheses    (A1)-(A4) hold. If
$$
\|\alpha\|_{\infty}\big[\|z_0\|_{\infty}+ \|h\|_{L^1}\big]<1,
$$
then the hyperbolic equation \eqref{e11} has a  solution
on $J_a\times J_b$.
\end{theorem}

\begin{proof}
Let $X=C(J_a\times J_b, \mathbb{R})$. Define
two operators $A$ and $B$ on $X$   by
\begin{gather}\label{e33}
Au(x,y)=f(x,y,u(x,y)),\quad (x,y)\in J_a\times J_b,
\\ \label{e34}
 Bu(x,y)=
z_0(x,y)+\int_{0}^{x}\int_{0}^{y}g(t,s,u(t,s))\,
ds\,dt,\ \ \ (x,y)\in
J_a\times J_b.
\end{gather}
Clearly $A$ and $B$ define the operators $A, B : X\to X$.
Now solving \eqref{e11} is equivalent to solving
FIE (\ref{e31}), 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_a\times J_b.
\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\|_{\infty}.
\end{align*}
Taking the maximum over $(x,y)$, in the above
inequality yields
$$
\|Au_1-Au_2\|_{\infty}\le
\|\alpha\|_{\infty}\|u_1-u_2\|_{\infty},
$$
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 $(A3)$ it
follows that
$$
\|Bu_n\|_{\infty} \le \|z_0\|_{\infty}+ \|h\|_{L^1},
$$
where $h$ is given in Definition \ref{Car} (iv). As a
result $\{Bu_n: n\in {\mathbb N}\}$ is a uniformly bounded set in $X$.
Let $(x_1,y_1),  (x_2,y_2)\in J_a\times J_b$. Then
\begin{align*}
&|Bu_n(x_1,y_1)&-Bu_n(x_2,y_2)|\\
& \le |z_0(x_1,y_1)-z_0(x_2,y_2)|
+ \int_{x_1}^{x_2}\int_{y_1}^{y_2}|g(t,s,u_n(t,s))|ds\,dt\\
&\le |z_0(x_1,y_1)-z_0(x_2,y_2)|+\int_{x_1}^{x_2}
\int_{y_1}^{y_2}h(t,s)ds\,dt\\
& \to   0,  \quad \hbox {as }  (x_1,y_1)\to (x_2,y_2).
\end{align*}
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,
\begin{align*}
M&= \|B(X)\| \\
&\le |z_0(x,y)|+\sup_{(x,y) \in J_a\times J_b
}\int_{0}^{x}\int_{0}^{y}|g(t,s,u(t,s))|\,ds\,dt\\
&\le \|z_0\|_{\infty}+ \|h\|_{L^1},
\end{align*}
and so,
$$
\alpha M\le \|\alpha\|_{\infty}(\|z_0\|_{\infty}+ \|h\|_{L^1})<1,
$$
by assumption. 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{e11}. Then, for
any $\lambda\in (0,1)$
we have
$$
u(x,y)=\lambda[f(x,y,u(x,y))]\Big(z_0(x,y)+\int_{0}^{x}
\int_{0}^{y}g(t,s,u(t, s))\,
ds\,dt\Big),
$$
for $(x,y)\in J_a\times J_b$.
Therefore,
\begin{align*}
|u(x,y)|&\le \big[f(x,y,u(x,y))\big]\Big(
|z_0(x,y)|+\int_{0}^{x}\int_{0}^{y}|g(t,s,u(t,s))|\,
dt\, ds\Big)\\
&\le \big[|f(x,y,u(x,y))-f(x,y,0)|+|f(x,y,0)|\big]\times\\
&\quad\times\Big(|z_0(x,y)|+\int_{0}^{x}\int_{0}^{y}h(s,t)\, dt\,
ds\Big)\\
 &\le \big[\|\alpha\|_{\infty}|u(x,y)|+F\big]\left(
|z_0(x,y)|+\|h\|_{L^1}\right)\\
 &\le
\big[\|\alpha\|_{\infty}\|u\|_{\infty}+F\big][\|z_0\|_{\infty}+\|h\|_{L^1}],
\end{align*}
where $F=|f(x,y,0)|$, and consequently
$$
\|u\|_{\infty}\le
\frac{F(\|z_0\|_{\infty}+\|h\|_{L^1})}{1-\|\alpha\|_{\infty}
(\|z_0\|_{\infty}+\|h\|_{L^1})}:=M.
$$
 Thus the conclusion (ii)  of Theorem \ref{td} does
not hold.
  Therefore the hyperbolic differential equation
\eqref{e11} has a
  solution on $J_a\times J_b$. This completes the
proof.
\end{proof}

\section{Existence Results for Extremal Solutions}
\label{S4}


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

It is well-known that the cone $K$ is positive
and normal in $C(J_a\times J_b, \mathbb{R})$.
If $\underline{u}, {\bar u}\in C(J_a\times J_b,\mathbb{R})$
and
${\underline u}\leq {\bar u}$, we put
$$
[\underline{u}, {\overline
u}]=\{u\in C(J_a\times J_b,\mathbb{R}):\underline{u} \leq
u\leq {\bar u}\}.
$$
\begin{definition}
A function $\beta :J_a\times J_b \times \mathbb{R} \to
\mathbb{R}$ is called
\emph{Chandrabhan}  if
\begin{itemize}
\item [(i)] the function $(x,y)\to
\beta(x,y,z)$ is measurable
for each   $z\in \mathbb{R}$, \item [(ii)] the function
$z\to \beta(x,y,z)$ is nondecreasing for almost each
$(x,y) \in J_a\times J_b$.
\end{itemize}

Further a \emph{Chandrabhan} function $\beta(x,y,z)$ is
called $L^1$-\emph{Chandrabhan} if
\begin{itemize}
\item[(iii)] for each number $r>0$, there exists a
function $h_r\in
L^1 (J_a\times J_b, \mathbb{R})$  such that
$$
|\beta(x,y,z)|\leq h_r(x,y) \quad\text{a.e. }(x,y)\in J_a\times
J_b,
$$
for all $z\in \mathbb{R}$ with $|z|\leq r$.
\end{itemize}
\end{definition}

\begin{definition}
A function $\underline{u}(\cdot,\cdot)\in C(J_a\times J_b,\mathbb{R})$ is
said to be a lower solution of \eqref{e11} if we have
\begin{gather*}
\frac{\partial^2}{{\partial x}{\partial
y}}\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_a\times J_b,\\
\underline{u}(x,0)\le\varphi(x), \quad
\underline{u}(0,y)\le\psi(y),
\end{gather*}
 for each $(x,y)\in J_a\times J_b$.  Similarly a
function
${\bar u}(\cdot,\cdot)\in C (J_a\times J_b,\mathbb{R})$ is
said to be an
upper solution of \eqref{e11} if we have
\begin{gather*}
\frac{\partial^2}{{\partial x}{\partial
y}}\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_a\times J_b,\\
{\bar u}(x,0)\ge\varphi(x), \quad {\bar u}(0,y)\ge\psi(y),
\end{gather*}
for each $(x,y)\in J_a\times J_b $.
\end{definition}
\begin{definition}

A solution $u_M$ of the problem \eqref{e11} is said to
be maximal if for any other solution $u$ to the problem \eqref{e11}
one has
$u(x,y)\leq u_M(x,y)$, for all $(x,y)\in J_a\times J_b$. Again a
solution $u_m$ of the problem \eqref{e11} is said to
be minimal if
$u_m(x,y)\leq u(x,y)$, for all $(x,y)\in J_a\times J_b$ where
$u$ is any solution of the problem \eqref{e11} on
$J_a\times J_b$.
\end{definition}

The following hypotheses will be used in the sequel.
\begin{itemize}
\item[(H1)] $f :J_a\times J_b\times \mathbb{R}_{+}\to
\mathbb{R}_{+}\setminus\{0\}$,
$g :J_a\times J_b\times \mathbb{R}_{+}\to \mathbb{R}_{+}$,
$\psi(y)\ge 0$ on $J_b$
 and $\frac{\phi(x)}{f(x,0,\phi(x))}
\ge\frac{\phi(0)}{f(0,0,\phi(0))}$ for all $x\in J_a$.

\item[(H2)] The functions $f(x,y,u)$ and $g(x,y,u)$
are nondecreasing in $u$ almost everywhere for
$(x,y)\in J_a\times J_b$.
\item[(H3)] The function $g$ is $L^1$-Chandrabhan.
\item[(H4)] The hyperbolic differential equation
\eqref{e11}
has a lower solution $\underline{u}$ and an upper
solution $\overline{u}$
with $\underline{u}\le \overline{u}$.
\end{itemize}

\begin{remark}\label{r31} \rm
Assume that (H2)-(H4) hold. Define a function
$h:J_a\times J_b\to \mathbb{R}^{+}$ by
$$
h(x,y)=|g(x,y,{\overline u}(x,y))|= g(x,y,{\overline
u}(x,y))\quad
\forall \,(x,y) \in J_a\times J_b.
$$
Then $h$ is Lebesgue integrable and
$$
|g(x,y,z)|\le h(x,y), \quad\text{a.e. } (x,y)\in J_a\times J_b,
\quad \forall z\in [\underline{u},\overline{u}].
$$
\end{remark}

\begin{theorem}\label{t31}
Assume that hypotheses $(A2),  (H1)-(H4)$ hold. If
$$
\|\alpha\|_{\infty}\big[\|z_0\|_{\infty}+\|h\|_{L^1}\big]<1,
$$
then the hyperbolic equation \eqref{e11} has a minimal and a
maximal positive solution
on $J_a\times J_b$.
\end{theorem}

\begin{proof}
Let $X=C(J_a\times J_b, \mathbb{R})$ and consider
a closed interval
$[\underline{u}, {\overline u}]$ in $X$ which is well defined
in view of hypothesis (H4). Define two
operators $A,B : [\underline{u}, {\overline u}]\to X$   by
(\ref{e33}) and (\ref{e34}) respectively. Clearly $A$
and $B$ define the operators $A,
B : [\underline{u}, {\overline u}]\to K$.

Now solving  \eqref{e11} is equivalent to solving
(\ref{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_a\times J_b.
\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 the 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),
$$
Au_1(x,y)=f(x,y,u_1(x,y))\le
f(x,y,u_2(x,y))=Au_2(x,y),\quad \forall (x,y) \in J_a\times J_b,
$$
and
\begin{align*}
Bu_1(x,y)&= z_0(x,y) +\int_{0}^{x}\int_{0}^{y}g(t,s,u_1(t,s))\,
ds\,dt \\
&\le z_0(x,y)
+\int_{0}^{x}\int_{0}^{y}g(t,s,u_2(t,s))\,
ds\,dt\\
&= Bu_2(x,y), \quad \forall (x,y)\in J_a\times J_b.
\end{align*}
So $A$ and $B$ are nondecreasing operators on
$[{\underline u},{\overline u}]$. Again
  hypothesis (H4) imply
 \begin{align*}
 \underline{u}(x,y)&=
[f(x,y,\underline{u}(x,y))]\Big(z_0(x,y)+\int_{0}^{x}
\int_{0}^{y}g(t,s,\underline{u}(t,s))\
,ds\,dt\Big)\\
&\le
[f(x,y,z(x,y))]\Big(z_0(x,y)+\int_{0}^{x}\int_{0}^{y}g(t,s,z(t,s))dsdt\Big)\\
&\le [f(x,y,\overline{u}(x,y))]\Big(z_0(x,y)
+\int_{0}^{x}\int_{0}^{y}g(t,s,\overline{u}(t,s))\,ds\,dt\Big)\\
&\le \overline{u}(x,y),
\end{align*}
for all $(x,y)\in J_a\times J_b$ and $z\in
[{\underline u},{\overline u}]$.
As a result
 $$
 \underline{u}(x,y)\le Az(x,y)Bz(x,y)\le
\overline{u}(x,y), \quad \forall (x,y)\in
J_a\times J_b  \hbox{ and }
z\in [{\underline u},{\overline u}].
$$
Hence $Az \,Bz\in [{\underline u},{\overline u}]$, for all
$z\in [{\underline u},{\overline u}]$.

Notice for any $u\in [{\underline u},{\overline u}]$,
\begin{align*}
M&= \|B([{\underline u},{\overline u}])\| \\
&\le |z_0(x,y)|+\sup_{(x,y) \in J_a\times J_b
}\int_{0}^{x}\int_{0}^{y}|g(t,s,u(t,s))|\,ds\,dt\\
&\le \|z_0\|_{\infty}+ \|h_r\|_{L^1},
\end{align*}
and so,
$$
\alpha M\le \|\alpha\|_{\infty}(\|z_0\|_{\infty}+
\|h_r\|_{L^1})<1.
$$
Thus the operators $A$ and $B$ satisfy all the
conditions of Theorem
\ref{t22} and so the operator equation (\ref{e34}) has
a least and a
greatest solution in $[\underline{u}, {\overline u}]$.
This further
implies that the hyperbolic differential equation
\eqref{e11} has a
minimal and a maximal positive solution on $J_a\times J_b$.
This completes the proof.
\end{proof}

\begin{theorem}\label{t32}
Assume that hypotheses {\rm (A1), (A2), (H1)-(H4)} hold.
Then the
hyperbolic equation \eqref{e11} has a minimal and a
maximal positive solution
on $J_a\times J_b$.
\end{theorem}

\begin{proof} Let $X=C(J_a\times J_b,\mathbb{R})$. Consider
the order interval
$[{\underline u},{\overline u}]$ in $X$ and define two operators
$A$ and $B$ on $[{\underline u},{\overline u}]$ by
(\ref{e33}) and (\ref{e34}) respectively.
Then  HDE \eqref{e11} is transformed into an operator
equation
$Au(x,y)\,Bu(x,y)=u(x,y)$, $(x,y)\in J_a\times J_b$
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 $J_a\times J_b\times [-r, r]$, 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_a\times J_b} |f(x,y,
u(x,y))|: u\in S\Big\}\\
& \le \sup\Big\{\sup_{(x,y)\in J_a\times J_b}|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
$J_a\times J_b\times [-r,r]$. Therefore, for any
$(x_1,y_1),(x_2,y_2)\in J_a\times J_b$ we have
\[
|f(x_1,y_1, u)-f(x_2,y_2, u)|\to 0\quad \mbox{as }
 (x_1,y_1)\to (x_2,y_2),
\]
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 $(x,y)\in J_a\times J_b$. Hence any
$(x_1,y_1),(x_2,y_2)\in J_a\times J_b$ and for any
$u\in S$ one has
\begin{align*}
|Au(x_1,y_1)-Au(x_2,y_2)|&=|f(x_1,y_1,
u(x_1,y_1))-f(x_2,y_2, u(x_2,y_2))|\\
&\le |f(x_1,y_1, u(x_1,y_1))-f(x_2,y_2, u(x_1,y_1)|\\
&\quad +|f(x_2,y_2, x(x_1,y_1))-f(x_2,y_2,
x(x_2,y_2))|\\
&\to 0\quad \mbox{as } (x_1,y_1)\to (x_2,y_2).
\end{align*}
This shows that $A(S)$ is an equi-continuous set in
$K$. Now an
application of Arzel\`a-Ascoli theorem yields that A
is a completely
continuous operator on $[{\underline u},{\overline u}]$.

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
hyperbolic
differential equation \eqref{e11} has a minimal and
maximal positive solution
on $J$. This completes the proof.
\end{proof}

\section{An Example}

Let $J_a=[0,1]=J_b$  and let $\phi, \psi:[0,1]\to \mathbb{R}$
be two functions defined by
\begin{equation}\label{e51}
\phi(x)=x^2 \quad\mbox{and}\quad \psi(x) =x.
\end{equation}
Define two functions $f,g : [0,1]\times [0,1]\times
\mathbb{R}\to \mathbb{R}$ by
\begin{equation}\label{e52}
f(x,y,u)=\begin{cases}
1, & \mbox{if }u<0\\
1+\frac{u}{9}, & \mbox{if }u \ge 0,
\end{cases}
\end{equation}
and
\begin{equation}\label{e53}
g(x,y,u)=\begin{cases}
0, & \mbox{if }u<0\\
\frac{[u]}{15+[u]}, & \mbox{if } u \ge 0
\end{cases}
\end{equation}
for all $x,y\in [0,1]$, where $[u]$ is the greatest
integer less than or equal to $u$.

Now consider the hyperbolic differential equation
\eqref{e11} with the
functions $\phi$, $\psi$ $f$ and $g$ defined by
(\ref{e51}),  (\ref{e52}) and (\ref{e53})
respectively.

We show that the functions $\phi, \psi, f$ and $g$
satisfy all the hypotheses of Theorem 4.2.
Clearly $\phi(0)=0=\psi(0)$. Again, here we have
$f :[0,1]\times [0,1]
\times \mathbb{R} \to \mathbb{R}_{+}\setminus\{0\}$,
$g :[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}_{+}$,
$\psi(x)\ge 0$ on $[0,1]$
 and $\frac{\phi(x)}{f(x,0,\phi(x))}
\ge\frac{\phi(0)}{f(0,0,\phi(0))}$ for all
$x\in [0,1]$.

Also the maps $u\mapsto f(x,y,u)$ and $u\mapsto g(x,y,u)$
are nondecreasing in $\mathbb{R}$ for all $x,y\in [0,1]$. Note
that $f$ is continuous
and $g$ is $L^1$-Chandrabhan on $[0,1]\times [0,1]\times \mathbb{R}$.
Further, it is easy to verify that identically zero
function $\underline{u}\equiv 0$
and the constant function $\overline{u}\equiv 3$ are
the lower and
upper solutions of the \eqref{e11} respectively.
Hence by
Theorem 4.2, the HDE \eqref{e11} has a maximal and a
minimal positive
solution in the order interval $[0,3]$ in the space
$C([0,1]\times [0,1],\mathbb{R})$ defined
on $[0,1]\times [0,1]$.

\begin{remark} \rm
Note that the function $g$ in the above
example is not continuous, but   Lebesgue integrable
on $[0,1]\times [0,1]$.
 \end{remark}

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\end{document}