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\markboth{\hfil Some integral inequalities for functions of two variables 
\hfil EJDE--2003/10}
{EJDE--2003/10\hfil Sever S. Dragomir \& Young-Ho Kim \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 10, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Some integral inequalities for functions of two variables
 %
\thanks{ {\em Mathematics Subject Classifications:} 26D10, 26D15.
\hfil\break\indent
{\em Key words:} Integral inequality, partial differential equations.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted April 19, 2002. Published February 4, 2003.} }
\date{}
%
\author{Sever S. Dragomir \& Young-Ho Kim}
\maketitle

\begin{abstract}
 In this article, we establish some integral inequalities
 for function with two independent variables. Also we show
 applications of these inequalities for finding bounds
 of solutions to partial differential equations.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}


\section{Introduction}

Let $u:[\alpha , \alpha +h] \to \mathbb{R}$ be a continuous function
satisfying the inequality
$$
0\leq u(t) \leq \int_{\alpha}^t
 [a + bu(s)] \, ds,\quad    \mbox{for } t\in [\alpha , \alpha +h],
$$
where $a,b$ are nonnegative constants. Then $u(t)\leq ahe^{bh}$
for $t\in [\alpha, \alpha +h]$. This result was proved by
Gronwall in 1919, and is the prototype for the study of many
integral inequalities of Volterra type, and also for obtaining
explicit bounds of the unknown function. Therefore integral
inequalities of this type are usually associated with the name of
Gronwall.
Integral inequalities are a necessary tool in the study of various
classes of equations.
During the past few years many authors (please, see
refeences below and some of the reference cited therein) have established
several Gronwall type integral inequalities in two or more
independent variables.
In \cite{p2},  Pachpatte considered the  finite difference inequality
 in two independent variables.
Many of these are analogues of integral inequalities already known
to us.
Our main objective here, as an integral version of  Pachpatte's finite difference
inequalities in \cite{p2}, is to establish some new integral inequalities involving
functions of two independent variables which can be used in the analysis
of certain classes of partial differential equations.

\section{Results}

Throughout this paper,
all the functions which appear in the inequalities are assumed to
be real valued and all the integrals exist on their
 domains of definitions.
We shall introduce some  notation: $\mathbb{R}$ denotes the set
of real numbers and $\mathbb{R}_+ = [0, \infty )$.
 The first order partial derivatives of a functions $z(x,y) $
defined for $x,y \in \mathbb{R}$ with respect to $x$ and $y$ are denoted by
$z_x(x,y)$ and $z_y (x,y) $ respectively.
We need the inequalities in the following lemma, which appear
in  \cite[p. 356]{m2}.

\begin{lemma} \label{lm2.1}
Let $u(t)$ and $k(t)$  be  continuous  and $a(t)$ and $b(t)$
Riemann integrable functions on $J=[\alpha , \beta ]$ with $b(t)$
and $k(t)$ nonnegative on $J$.
\begin{enumerate}
\item If
$u(t) \leq a(t) + b(t) \int_\alpha^t k( s) u(s) ds$ for $t\in J$,
then
$$
u(t) \leq a(t) + b(t) \int_\alpha^t a(s) k( s)
   \exp \Big(  \int_s^t b(r)k(r)  dr\Big)\, ds, \quad t\in J.
$$
\item If $u(t) \leq a(t) + b(t) \int_t^\beta k( s) u(s) ds$ for $t\in J$,
then
$$
u(t) \leq a(t) + b(t) \int_t^\beta a(s) k( s)
   \exp \Big(  \int_t^s b(r)k(r)  dr\Big)\, ds, \quad t\in J.
$$
\end{enumerate}
\end{lemma}

Also, we need the inequalities in the following lemma
which is given in \cite[p. 110]{b1}.

\begin{lemma} \label{lm2.2}
Let $u(x,y), a(x,y), b(x,y) $ be nonnegative continuous functions defined
for  $x,y \in \mathbb{R}_+$. \begin{enumerate}
\item
Assume that $a(x,y)$ is non-decreasing in $x$  and
non-increasing in $y$ for $x,y  \in \mathbb{R}_+$. If
$u(x,y) \leq a(x,y) + \int_0^x \int_y^\infty  b(s,t)
 u(s,t) \, dt  ds$
for all $x,y \in \mathbb{R}_+$, then
$$
u(x,y) \leq a(x,y)
  \exp  \Big( \int_0^x \int_y^\infty  b(s,t) \, dt  ds \Big).
$$

\item
Assume that $a(x,y)$ is non-increasing in each of the variables
$x,y  \in \mathbb{R}_+$. If
$u(x,y) \leq a(x,y) + \int_x^\infty \int_y^\infty  b(s,t)
 u(s,t) \, dt  ds$
for all $x,y \in \mathbb{R}_+$, then
$$
u(x,y) \leq a(x,y)
  \exp  \Big( \int_x^\infty \int_y^\infty  b(s,t) \, dt  ds \Big).
$$
\end{enumerate}
\end{lemma}

The proofs of these inequalities  can be completed as in
\cite[p. 109-111]{b1}; thus, we omit the proof.

\begin{theorem} \label{thm2.3}
Let $u(x,y),  a(x,y), b(x,y), c(x,y), d(x,y), f(x,y) $ be real-valued
non-negative continuous functions defined for  $x,y \in \mathbb{R}_+$.
Let $W(u(x,y))$ be real-valued, positive, continuous, strictly
non-decreasing, subadditive, and submultiplicative function for
$u(x,y)\geq 0$ and let $H(u(x,y))$ be a real-valued, continuous, positive,
and non-decreasing function defined for $x,y \in \mathbb{R}_+$.
Assume that $a(x,y), f(x,y)$ are nondecreasing in $x$  for $x \in \mathbb{R}_+$.
If
\begin{equation}
\begin{aligned}
u(x,y) \leq& a(x,y)+b(x,y) \int_\alpha^x c(s,y)u(s,y)\, ds\\
&+ f(x,y) H\Big( \int_0^x \int_y^\infty d(s,t)
 W \bigl(u(s,t) \bigr) \, dt  ds\Big)
 \end{aligned} \tag{2.1}
\end{equation}
for $\alpha, x,y \in \mathbb{R}_+$ with $ \alpha \leq x$, then
\begin{equation}
\begin{aligned}
u(x,y) \leq & p(x,y)  \Big\{ a(x,y ) +f(x,y) H
 \Big[ G^{-1}\big( G( A(s,t)\big)    \\
 &+  \int_0^x \int_y^\infty   d(s,t)W \bigl(p(s,t)f(s,t) \bigr)
   \, dt  ds \Big)\Big]\Big\}
\end{aligned} \tag{2.2}
\end{equation}
for $\alpha, x,y \in \mathbb{R}_+$ with $ \alpha \leq x$, where
\begin{gather}
p(x,y)=1 + b(x,y)\int_\alpha^x c(s,y)\exp \Big( \int_s^x b(r,y)c(r,y)\, dr\Big)\,
ds, \tag{2.3}\\
A(s,t)= \int_0^\infty \int_0^\infty d(s,t)   W \bigl(p(s,t)a(s,t) \bigr) \, dt
ds, \tag{2.4}\\
G(r)   =\int_{r_0}^r {ds\over W \bigl((H(s)\bigr)},\quad r\geq r_0
>0. \tag{2.5}
\end{gather}
Here $G^{-1}$ is the inverse function of $G$ and
$$
G \Big(\int_0^\infty \int_0^\infty d(s,t)
W \bigl(p(s,t)a(s,t) \bigr) \, dt  ds\Big)
+\int_0^x \int_y^\infty   d(s,t)W \bigl(p(s,t)f(s,t) \bigr) \, dt  ds
$$ is in  the domain of $G^{-1}$
for   $x,y \in \mathbb{R}_+$.
\end{theorem}

\paragraph{Proof.}
Define a function $z(x,y)$ by
\[
z(x,y)= a(x,y)  + f(x,y) H\Big( \int_0^x \int_y^\infty d(s,t)
 W \bigl(u(s,t) \bigr) \, dt  ds\Big). \tag{2.6}
\]
Then (2.1) can be restated as
\[
u(x,y) \leq z(x,y) + b(x,y)\int_\alpha^x c(s,y)u(s,y)\, ds .\tag{2.7}
\]
Clearly $z(x,y)$ is a nonnegative and continuous   in
 $x, x\in \mathbb{R}_+$. Treating $y,y\in \mathbb{R}_+$ fixed in (2.7)
and using 1 of Lemma \ref{lm2.1} to (2.7), we get
$$
u(x,y)\leq z(x,y) +b(x,y) \int_\alpha^x z(s,y)c(s,y)
 \exp \Big(
 \int_s^x b(r,y)c(r,y) \, dr \Big) \, ds.
$$
Moreover, $z(x,y)$ is   nondecreasing   in  $x, x\in \mathbb{R}_+$, we obtain
\[
u(x,y) \leq z(x,y) p(x,y), \tag{2.8}
\]
where $p(x,y)$ is defined by (2.3).
>From (2.6) we have
\[
u(x,y) \leq p(x,y) \Bigl( a(x,y) +f(x,y) H\bigl(v(x,y)\bigr)\Bigr), \tag{2.9}
\]
where $v(s,y)$ is defined by
$$
v(x,y)= \int_0^x \int_y^\infty d(s,t)W \bigl(u(s,t)\bigr)  \, dt  ds.
$$
>From (2.9), we observe that
\begin{equation}
\begin{aligned}
v(x,y)
&\leq  \int_0^x \int_y^\infty d(s,t) W
 \Bigl(p(s,t) \Bigl[ a(s,t) +f(s,t) H\bigl(v(s,t)\bigr)\Bigr]\Bigr)\, dt  ds\\
&\leq  \int_0^x \int_y^\infty d(s,t)
 W\bigl(p(s,t)a(s,t)\bigr)\, dt  ds\\
&\quad +   \int_0^x \int_y^\infty d(s,t)
  W\bigl( p(s,t)f(s,t)\bigr)W \Bigl(H \bigl( v(s,t)\bigr)\Bigr) \, dt  ds\\
&\leq  \int_0^\infty \int_0^\infty  d(s,t)
 W\bigl(p(s,t)a(s,t)\bigr)\, dt  ds\\
&\quad +   \int_0^x \int_y^\infty d(s,t)
  W\bigl( p(s,t)f(s,t)\bigr)W \Bigl(H \bigl( v(s,t)\bigr)\Bigr) \, dt  ds,
\end{aligned} \tag{2.10}
\end{equation}
since $W$ is subadditive and submultiplicative function.
Define  $r(x,y)$ as the right side of (2.10), then
 $r(0,y)=r(x,\infty ) =\int_0^\infty \int_0^\infty d(s,t)
 W\bigl(p(s,t)a(s,t)\bigr)\, dt  ds$, $v(x,y)\leq r(x,y)$, $r(x,y)$ is
 non-increasing in  $y,y\in \mathbb{R}_+$ and
\begin{equation}
\begin{aligned}
r_x (x,y)
&=  \int_y^\infty d(x,t)
  W\bigl( p(x,t)f(x,t)\bigr)W \Bigl(H \bigl( v(x,t)\bigr)\Bigr) \, dt    \\
&\leq
\int_y^\infty d(x,t)
  W\bigl( p(x,t)f(x,t)\bigr)W \Bigl(H \bigl( r(x,t)\bigr)\Bigr) \, dt \\
&\leq
W \Bigl(H \bigl( r(x,y)\bigr)\Bigr)
 \int_y^\infty d(x,t)
  W\bigl( p(x,t)f(x,t)\bigr) \, dt.      \\
\end{aligned}  \tag{2.11}
\end{equation}
Dividing both sides of (2.11) by $W \big(H (r(x,y)\big)\big)$ we get
\[
{r_x (x,y) \over W \big(H ( r(x,y))\big)}
\leq \int_y^\infty d(x,t) W\bigl( p(x,t)f(x,t)\bigr) \, dt.\tag{2.12}
\]
>From (2.5) and (2.12) we have
\[
G_x \bigl(r(x,y)\bigr)
\leq  \int_y^\infty d(x,t)
  W\bigl( p(x,t)f(x,t)\bigr) \, dt.\tag{2.13}
\]
Now setting $x=s$ in (2.13) and then integrating with respect to
$s$ from $0$ to $x$, we obtain
\[
G  \bigl(r(x,y)\bigr)
\leq G  \bigl(r(0,y)\bigr)+ \int_0^x  \int_y^\infty d(s,t)
  W\bigl( p(s,t)f(s,t)\bigr) \, dt ds.\tag{2.14}
\]
Noting that $r(0,y)=\int_0^\infty  \int_0^\infty d(s,t)
 W\bigl(p(s,t)a(s,t)\bigr)\, dt   ds$, we have
\begin{equation}
\begin{aligned}
 r(x,y) \leq& G^{-1}\Big[ G \Big(\int_0^\infty \int_0^\infty d(s,t)
 W\bigl(p(s,t)a(s,t)\bigr)\, dt \,ds\Big)\\
&+  \int_0^x  \int_y^\infty d(s,t)
  W\bigl( p(s,t)f(s,t)\bigr) \, dt ds\Big].
 \end{aligned} \tag{2.15}
\end{equation}
The required inequality in (2.2) follows from the fact that
$v(x,y)\leq r(x,y)$, (2.7) and (2.15). \hfill$\square$


\begin{theorem} \label{thm2.4}
Let $u(x,y)$,  $a(x,y)$, $b(x,y)$, $c(x,y)$, $d(x,y)$, $f(x,y)$,
$W(u(x,y))$,  and
$H(u(x,y))$ be as defined in Theorem \ref{thm2.3}.
 Assume that $a(x,y), f(x,y)$ are non-increasing in $x$
 for $x \in \mathbb{R}_+$. If
\begin{align*}
u(x,y) \leq& a(x,y)+b(x,y) \int_x^\beta c(s,y)u(s,y)\, ds \\
& + f(x,y) H\Big( \int_x^\infty \int_y^\infty d(s,t)
 W \bigl(u(s,t) \bigr) \, dt  ds\Big)
 \end{align*}
for $\beta, x,y \in \mathbb{R}_+$ with $ x \leq \beta$, then
\begin{align*}
u(x,y)  \leq &\overline{p}(x,y)
 \Big\{a(x,y ) +f(x,y) H
 \Big[ G^{-1}\Big( G\Bigl( \overline{A}(s,t)\Bigr)    \\
 &+  \int_x^\infty \int_y^\infty   d(s,t)
  W \bigl(\overline{p}(s,t)f(s,t) \bigr) \, dt  ds
  \Big)\Big]\Big\}
\end{align*}
for  $\beta, x,y \in \mathbb{R}_+$ with $ x \leq \beta$, where
\begin{gather*}
\overline{p}(x,y)=1 + b(x,y)\int_x^\beta c(s,y)
 \exp \Big( \int_x^s b(r,y)c(r,y)\, dr\Big)\, ds,  \\
\overline{A}(s,t)= \int_0^\infty \int_0^\infty d(s,t)
  W \bigl(\overline{p}(s,t)a(s,t) \bigr) \, dt ds, \\
 G(r)   =\int_{r_0}^r {ds\over W \bigl((H(s)\bigr)},
  \quad r\geq r_0 >0,
\end{gather*}
$G^{-1}$ is the inverse function of $G$ and
$$
G \Big(\int_0^\infty \!\int_0^\infty d(s,t)
W \bigl(\overline{p}(s,t)a(s,t) \bigr) \, dt  ds\Big)
+\int_x^\infty \!\int_y^\infty   d(s,t)W \bigl(\overline{p}(s,t)f(s,t) \bigr)
\, dt  ds
$$
is in the domain of $G^{-1}$  for  $x,y \in \mathbb{R}_+$.
\end{theorem}

The details of the proof of Theorem \ref{thm2.4} follows by an argument
similar to that in the proofs of Theorem \ref{thm2.3} with suitable changes.
We omit the proof.

\begin{theorem} \label{thm2.5}
Let $u(x,y)$, $a(x,y)$, $b(x,y)$, $c(x,y)$, $f(x,y)$ be real-valued
nonnegative continuous functions defined for  $x,y \in \mathbb{R}_+$ and
$L:\mathbb{R}_+^3 \to \mathbb{R}_+$ be a continuous function which satisfies
the condition
\[
0\leq L(x,y,u )-L(x,y,v )\leq M(x,y,v )\phi^{-1}(u -v ) \tag{2.16}
\]
for $u\geq v\geq 0$, where $M(x,y,v)$ is a real-valued nonnegative continuous function defined for
$x,y,v\in \mathbb{R}_+$. Assume that $\phi : \mathbb{R}_+  \to \mathbb{R}_+  $ be a continuous and strictly
increasing function with $\phi (0)=0, \phi^{-1}$ is the inverse
function of $\phi $ and
$$
 \phi^{-1}(uv)\leq  \phi^{-1}(u) \phi^{-1}(v)
$$
 for $u,v \in \mathbb{R}_+$.  Assume that $a(x,y), f(x,y)$ are nondecreasing in $x$
 for $x \in \mathbb{R}_+$. If
\[
\begin{aligned}
u(x,y) \leq & a(x,y)
+  b(x,y)\int_\alpha^x c(s,y)u(s,y)\, ds\\
&+ f(x,y) \phi \Big( \int_0^x \int_y^\infty L(s,t, u(s,t) ) \, dt
ds\Big)
\end{aligned} \tag{2.17}
\]
for $\alpha, x, y \in \mathbb{R}_+ $ with $\alpha \leq x$, then
\begin{multline}
u(x,y) \leq  p(x,y)\Big\{ a(x,y)+f(x,y) \phi \Big[ e(x,y)\\
\times \exp  \Big( \int_0^x \int_y^\infty M \Bigl(s,t, p(s,t)a(s,t) \Bigr)
 \phi^{-1}\bigl( p(s,t)f(s,t) \bigr) \, dt  ds \Big)\Big]\Big\}
 \tag{2.18}
\end{multline}
for   $x,y \in \mathbb{R}_+$, where
\begin{gather}
p(x,y)=1+b(x,y) \int_\alpha^xc(s,y) \exp \Big( \int_s^x b(r,y)c(r,y)
\, dr\Big)\, ds, \tag{2.19}\\
e(x,y) = \int_0^x \int_y^\infty
      L \Bigl(s,t,   p(s,t) a(s,t)\Bigr)   \, dt  ds. \tag{2.20}
\end{gather}
\end{theorem}

\paragraph{Proof}
Define the  function
\[
z(x,y)= a(x,y) + f(x,y) \phi \Big(\int_0^x \int_y^\infty
 L(s,t, u(s,t) )  \, dt ds\Big). \tag{2.21}
\]
Then (2.17) can be restated as
\[
u(x,y) \leq z (x,y) +    b(x,y) \int_\alpha^x c(s,y)u(s,y)\, ds.\tag{2.22}
\]
Clearly $z(x,y)$ is a nonnegative and continuous   in
 $x, x\in \mathbb{R}_+$. Treating $y,y\in \mathbb{R}_+$ fixed in (2.22) and using (i)
 of Lemma \ref{lm2.1} to (2.22), we get
$$
u(x,y)\leq z(x,y) +b(x,y) \int_\alpha^x z(s,y)c(s,y)
 \exp \Big(
 \int_s^x b(r,y)c(r,y) \, dr \Big) \, ds.
$$
Moreover, $z(x,y)$ is   nondecreasing   in  $x, x\in \mathbb{R}_+$, we obtain
\[
u(x,y) \leq z(x,y) p(x,y)    ,\tag{2.23}
\]
where $p(x,y)$ is defined by (2.19).
>From (2.21) and (2.23) we have
\[
u(x,y) \leq p(x,y) \Bigl( a(x,y) +f(x,y) \phi \bigl(v(x,y)\bigr)\Bigr),
\tag{2.24}
\]
where $v(s,y)$ is defined by
$$
v(x,y)=\int_0^x \int_y^\infty  L(s,t, u(s,t) )  \, dt ds.
$$
>From (2.24) and the hypotheses on $L$ and $\phi $, it is to
observe that
\begin{equation}
\begin{aligned}
v(x,y)
&\leq  \int_0^x \int_y^\infty
\Big( L\Bigl((s,t,  p(s,t)\Bigl[a(s,t) +f(s,t) \phi \bigl( v(s,t)\bigr)\Bigr]\Bigr)\\
&\quad
-L\Bigl((s,t, p(s,t) a(s,t) \Bigr)+L\Bigl((s,t,  p(s,t)a(s,t) \Bigr) \Big)\, dt  ds\\
&\leq  \int_0^x \int_y^\infty L\Bigl((s,t,  p(s,t)a(s,t) \Bigr)\, dt
ds  \\
&\quad
 +\int_0^x \int_y^\infty M \Bigl((s,t,  p(s,t)a(s,t) \Bigr)\phi^{-1}
 \Bigl(p(s,t) f(s,t)\phi\bigl(v(s,t)\bigr)\Bigr)\, dt ds\\
&\leq
e(x,y)+\int_0^x \int_y^\infty M \Bigl((s,t,  p(s,t)a(s,t) \Bigr)\phi^{-1}
 \bigl( p(s,t)f(s,t) \bigr) v(s,t) \, dt ds ,
\end{aligned} \tag{2.25}
\end{equation}
where $e(x,y)$ is defined by (2.20).
Clearly, $e(x,y)$ is nonnegative, continuous, nondecreasing in $x,x\in
\mathbb{R}_+$ and non-increasing in $y, y\in \mathbb{R}_+$. Now, by
1
of Lemma \ref{lm2.2}, we obtain
\[
v(x,y)\leq e(x,y) \exp \Big( \int_0^x \int_y^\infty
  M \Bigl((s,t,  p(s,t)a(s,t)\Bigr)
\phi^{-1}  \bigl( p(s,t)f(s,t) \bigr) \, dt   ds,\Big). \tag{2.26}
\]
Using (2.24) in (2.26) we get the required inequality in (2.18).
\hfill$\square$

\begin{theorem} \label{thm2.6}
Let $u(x,y)$,  $a(x,y)$, $b(x,y)$, $c(x,y)$,  $f(x,y)$,
$L$, $M$,  $\phi $,  and $\phi^{-1}$ be as defined in Theorem \ref{thm2.5}.
Assume that $a(x,y), f(x,y)$ are non-increasing in $x$  for
$x \in \mathbb{R}_+$. If
\begin{align*}
u(x,y) \leq& a(x,y) + b(x,y) \int_x^\beta c(s,y)u(s,y)\, ds\\
 &+ f(x,y) \phi\Big( \int_x^\infty \int_y^\infty L(s,t, u(s,t) ) \, dt
 ds\Big)
\end{align*}
for $\beta , x,y \in \mathbb{R}_+ $ with $x\leq \beta $, then
\begin{align*}
u(x,y)
& \leq \overline{p}(x,y)\Big\{ a(x,y)+ f(x,y) \phi \Big[ \overline{e}(x,y)\\
&\quad\times \exp
 \Big( \int_x^\infty \int_y^\infty M \Bigl(s,t,\overline{p}(s,t) a(s,t) \Bigr)
 \phi^{-1}\bigl(\overline{p}(s,t)f(s,t) \bigr) \, dt  ds
 \Big)\Big]\Big\}
\end{align*}
for   $x,y \in \mathbb{R}_+$, where
\begin{equation}
\begin{gathered}
\overline{p}(s,t)=1 + b(x,y)\int_x^\beta c(s,y)
 \exp \Big( \int_x^s b(r,y)c(r,y)\, dr\Big)\, ds,  \\
\overline{e}(x,y) = \int_x^\infty \int_y^\infty
      L \Bigl(s,t,  \overline{ p}(s,t) a(s,t)\Bigr)   \, dt  ds. \\
\end{gathered} \tag{2.27}
\end{equation}
\end{theorem}

The  proof of Theorem \ref{thm2.5} follows by an argument
similar to that in the proofs of Theorem \ref{thm2.4} with suitable changes.
We omit the details.


\section{Further Inequalities}

To establish some of our results in this section, we require the class of
functions $S$ as defined in \cite{b2}. A function $g: [0, \infty) \to [0,\infty )
$ is said to belong to the class $S$ if
\\
(i) $g(u)$ is positive, nondecreasing and continuous for $u\geq 0$, and\\
(ii) $(1/v)g(u)\leq g(u/v), u>0, v\geq 1$.

\begin{theorem} \label{thm3.1}
Let $ u(x,y)$, $a(x,y)$, $ c(x,y)$, $ d(x,y)$, $f(x,y)$ be real-valued
nonnegative continuous functions defined for  $x,y \in \mathbb{R}_+ $
and let $g\in S$.
Also let $W(u(x,y))$ be real-valued, positive, continuous,
strictly nondecreasing, subadditive, and submultiplicative function for
$u(x,y)\geq 0 $ and let $H(u(x,y))$ be a real-valued, continuous,
 positive, and nondecreasing function defined for $x,y \in \mathbb{R}_+$.
Assume that a function   $m(x,y)$ is nondecreasing in $x $ and
$m(x,y)\geq 1$, which is defined by
$$
m(x,y)= a(x,y) +f(x,y)H\Big( \int_0^x \int_y^\infty d(s,t)u(s,t)  \, dt
ds\Big)
$$
for $x,y \in \mathbb{R}_+$. If
\[
u(x,y) \leq m(x,y) +    \int_\alpha^x c(s,y)g\bigl(u(s,y)\bigr)\,ds   \tag{3.1}
\]
for $\alpha ,x,y \in \mathbb{R}_+$ and $\alpha \leq x$, then
\begin{equation}
\begin{aligned}
u(x,y)  \leq&  F(x,y) \Big\{ a(x,y ) +f(x,y) H
 \Big[ G^{-1}\Big( G\Bigl( B(s,t)\Bigr)    \\
 &+  \int_0^x \int_y^\infty   d(s,t)W \bigl(F(s,t)f(s,t) \bigr)
   \, dt  ds \Big)\Big]\Big\}
\end{aligned} \tag{3.2}
\end{equation}
for   $x,y \in \mathbb{R}_+$, where
\begin{gather}
F(x,y) = \Omega^{-1}  \Big(\Omega (1) + \int_\alpha^xb(s,y) \, ds\Big) , \tag{3.3}\\
B(s,t)   =  \int_0^\infty  \int_y^\infty   d(s,t)W\bigl(F(s ,t)a(s,t)\bigr)
          \, dt  ds, \tag{3.4} \\
\Omega (u) =\int_{u_0}^u {ds\over g(s)}, \quad u\geq u_0>0,\tag{3.5}
\end{gather}
where $\Omega^{-1}$ is the inverse function of $\Omega$;
$G, G^{-1}$ are defined in Theorem \ref{thm2.3},
$\Omega (1) + \int_\alpha^xb(s,y) \, ds$ is in the domain of $\Omega^{-1}$,
and
$$
G \Big(\int_0^\infty \!\int_0^\infty d(s,t)
W \bigl(F(s,t)a(s,t) \bigr) \, dt  ds\Big)
+\int_0^x \!\int_y^\infty   d(s,t)W \bigl(F(s,t)f(s,t) \bigr) \, dt  ds
$$
is in the domain of $G^{-1}$  for   $x,y \in \mathbb{R}_+$.
 \end{theorem}

\paragraph{Proof}
Let $m(x,y)$ be a positive, continuous, nondecreasing in $x$  and let
$g\in S$. Then (3.1) can be restated as
\[
{u(x,y)\over m(x,y)}\leq 1+
  \int_\alpha^x b(s,y)g\Big( {u(s,y) \over m(s,y)} \Big)\,ds.  \tag{3.6}
\]
The inequality (3.6) may be treated as a one-dimensional Bihari
inequality \cite{b1}  for any fixed $y,y\in \mathbb{R}_+$, which implies
that
$$
u(x,y) \leq F(x,y)m(x,y),
$$
where $F(x,y)$ is defined by (3.3).
Now, by following the last argument as in the proof of Theorem \ref{thm2.3},
we obtain desired  inequality in (3.2).
\hfill$\square$

\begin{theorem} \label{thm3.2}
Let $u(x,y)$,  $a(x,y)$,  $c(x,y)$, $d(x,y)$, $f(x,y)$,
$W(u(x,y))$, and $H(u(x,y))$ be as defined in Theorem \ref{thm3.1} and $g\in S$.
Assume that a function  $\overline{m}(x,y)$ is non-increasing in $x $
and  $\overline{m}(x,y)\geq 1$, which is defined by
$$
\overline{m}(x,y)= a(x,y) +f(x,y)H\Big( \int_x^\infty \int_y^\infty d(s,t)u(s,t)  \, dt
ds\Big)
$$
 for $x,y \in \mathbb{R}_+$. If
\[
u(x,y) \leq \overline{m}(x,y) +    \int_x^\beta
c(s,y)g\bigl(u(s,y)\bigr)\,ds \tag{3.7}
\]
for $\beta ,x,y \in \mathbb{R}_+$ and $x \leq \beta $, then
\begin{equation}
\begin{aligned}
u(x,y) \leq &\overline{F}(x,y)
 \Big\{a(x,y ) +f(x,y) H
 \Big[ G^{-1}\Big( G\Bigl(\overline{ B}(s,t)\Bigr)    \\
 &+  \int_x^\infty \int_y^\infty   d(s,t)W \bigl(\overline{F}(s,t)f(s,t) \bigr)
   \, dt  ds \Big)\Big]\Big\}
\end{aligned} \tag{3.8}
\end{equation}
for   $x,y \in \mathbb{R}_+$, where
\begin{equation}
\begin{gathered}
\overline{F}(x,y) = \Omega^{-1}  \Big(\Omega (1) + \int_x^\beta b(s,y) \,
ds\Big) ,\\
\overline{B}(s,t)   =
 \int_0^\infty  \int_0^\infty   d(s,t)W\bigl(\overline{F}(s ,t)a(s,t)\bigr)
 \, dt  ds ,
\end{gathered} \tag{3.9}
\end{equation}
$\Omega $ is defined in (3.5),  $\Omega^{-1}$ is the inverse function of
$\Omega $;
$G, G^{-1}$ are defined in Theorem \ref{thm2.3},
$\Omega (1) + \int_x^\beta b(s,y) \, ds$ is in the domain of
$\Omega^{-1}$, and
$$
G \Big(\int_0^\infty \!\!\int_0^\infty d(s,t)
W \bigl(\overline{F}(s,t)a(s,t) \bigr) \, dt  ds\Big)
+\int_x^\infty \!\! \int_y^\infty  d(s,t)W \bigl(\overline{F}(s,t)f(s,t) \bigr)
\, dt  ds
$$
is in the domain of $G^{-1}$, for   $x,y \in \mathbb{R}_+$.
 \end{theorem}

\paragraph{Proof}
Let $\overline{m}(x,y)$ be a positive, continuous, nondecreasing in $x$
and let $g\in S$. Then (3.7) can be restated as
\[
{u(x,y)\over \overline{m}(x,y)}\leq 1+
  \int_\alpha^x b(s,y)g\Big( {u(s,y) \over \overline{m}(s,y)} \Big)\,ds.  \tag{3.10}
\]
The inequality (3.10) may be treated as a one-dimensional Bihari
inequality \cite{b1} for any fixed $y,y\in \mathbb{R}_+$, which implies
that
$$
u(x,y) \leq \overline{F}(x,y)\overline{m}(x,y),
$$
where $\overline{F}(x,y)$ is defined by (3.9).
Now, by following the last argument as in the proof of Theorem \ref{thm2.4},
 we obtain desired  inequality in (3.8).
\hfill$\square$


\begin{theorem} \label{thm3.3}
Let $u(x,y)$,  $a(x,y)$, $b(x,y)$, $c(x,y)$,  $f(x,y)$,
$L$, $M$,  $\phi $,  and $\phi^{-1}$ be as defined in Theorem \ref{thm2.5},
and let $g\in S$. Assume that a function $n(x,y)$ is nondecreasing in $x $
and  $n(x,y)\geq 1$, which is defined by
$$
n(x,y)= a(x,y) + f(x,y) \phi
\Big( \int_0^x \int_y^\infty F(s,t, u(s,t) ) \, dt  ds
\Big)
$$
  for $x,y \in \mathbb{R}_+$. If
\[
u(x,y) \leq n(x,y) +
 \int_\alpha^x  b(s,y)b\bigl( u(s,y)\bigr)  \, ds   \tag{3.11}
\]
for $\alpha ,x,y \in \mathbb{R}_+$ and $\alpha \leq x$, then
\begin{multline}
u(x,y)  \leq F(x,y)\Big\{ a(x,y)+f(x,y) \phi \Big[ e(x,y)\\
 \times \exp
 \Big( \int_0^x \int_y^\infty M \Bigl(s,t, F(s,t)a(s,t) \Bigr)
 \phi^{-1}\bigl( F(s,t)f(s,t) \bigr) \, dt  ds \Big)\Big]\Big\}
 \tag{3.12}
\end{multline}
for   $x,y \in \mathbb{R}_+$, where
$F$ is defined in (3.3), $e(x,y)$ is defined in (2.20),
$\Omega $ is defined in (3.5), $\Omega^{-1}$ is the inverse function of
$\Omega $  and
$\Omega (1) + \int_\alpha^x b(s,y) \, ds$ is in the domain of
$\Omega^{-1}$
for   $x,y \in \mathbb{R}_+$.
\end{theorem}

\paragraph{Proof}
The proof of this theorem follows by an argument similar to that
 of Theorem \ref{thm3.1}.
Let $n(x,y)$ is a positive, continuous, nondecreasing in $x$  and let
$g\in S$. Then (3.11) can be restated as
\[
{u(x,y)\over n(x,y)}\leq 1+
  \int_\alpha^x b(s,y)g\Big( {u(s,y) \over n(s,y)} \Big)\,ds.  \tag{3.13}
\]
The inequality (3.13) may be treated as a one-dimensional Bihari
inequality \cite{b1} for any fixed $y,y\in \mathbb{R}_+$, which implies
that
$$
u(x,y) \leq F(x,y)n(x,y),
$$
where $F(x,y)$ is defined by (3.3).
Now, by following the last argument as in the proof of Theorem \ref{thm2.5},
 we obtain desired  inequality in (3.12).
\hfill$\square$

\begin{theorem} \label{thm3.4}
Let $u(x,y)$,  $a(x,y)$, $b(x,y)$, $c(x,y)$,  $f(x,y)$, $L$, $M$, $\phi $,
  and $\phi^{-1}$ be as defined in Theorem \ref{thm2.5}, and let $g\in S$.
Assume that a function $\overline{n}(x,y)$ is non-increasing in $x $
 and  $\overline{n}(x,y)\geq 1$, which is defined by
$$
\overline{n}(x,y)= a(x,y) +f(x,y)\phi
 \Big( \int_x^\infty \int_y^\infty  F(s,t, u(s,t) )  \, dt
 ds\Big)
$$
for $x,y \in \mathbb{R}_+$. If
$u(x,y) \leq \overline{n}(x,y) +
 \int_x^\beta  b(s,y)b\bigl( u(s,y)\bigr)  \, ds$
for $\beta ,x,y \in \mathbb{R}_+$ and $ x\leq \beta$, then
\begin{align*}
u(x,y) \leq &\overline{F}(x,y)\Big\{ a(x,y)+f(x,y) \phi \Big[ \overline{e}(x,y)\\
& \times \exp
 \Big( \int_x^\infty \int_y^\infty M \Bigl(s,t, \overline{F}(s,t)a(s,t) \Bigr)
 \phi^{-1}\bigl( \overline{F}(s,t)f(s,t) \bigr) \, dt  ds
 \Big)\Big]\Big\}
\end{align*}
for   $x,y \in \mathbb{R}_+$, where
$\overline{F} $ is defined in (3.9), $\overline{e}(x,y)$ is defined in
(2.27), $\Omega $ is defined in (3.5), $\Omega^{-1}$ is the inverse function of $\Omega $
 and
$\Omega (1) + \int_x^\beta b(s,y) \, ds$ is in the domain of
$\Omega^{-1}$
for   $x,y \in \mathbb{R}_+$.
\end{theorem}

The proof of this theorem follows by an argument
similar to that in Theorem \ref{thm3.3} with suitable changes.
We omit the details.


\section{Some Applications}

In this section we present some immediate applications of Theorem \ref{thm2.3}
 to study certain properties of solutions of the following
terminal-value problem for the hyperbolic partial differential
equation
\begin{gather}
u_{xy} (x,y)  =h(x,y,u(x,y)) +r(x,y), \tag{4.1}\\
u(x, \infty )=\sigma_\infty (x),  u(0, y )=\tau (y), u(0,\infty
)=k, \tag{4.2}
\end{gather}
where
$ h: \mathbb{R}_+^2 \times R \to R$, $r:\mathbb{R}_+^2 \to R$,
$\sigma_\infty ,\tau (y):\mathbb{R}_+ \to R$ are continuous functions and
$k$ is a real constant.

The following example deals with the estimate on the solution of
the partial differential equation (4.1) with the conditions (4.2).


\paragraph{Example}
Assume that functions are defined and continuous on their respective
domains of definitions and such that
\[
 |h(x,y,u)|   \leq d(x,y) W\bigl( \mid u \mid\bigr)  \tag{4.3}
\]
and
\begin{multline}
 \Big| \sigma_\infty (x)+  \tau (y)-  k -
  \int_0^x  \int_y^\infty  r(s,t)  \, dt ds \Big|\\
 \leq a(x,y)+b(x,y)\int_\alpha^x c(s,y)g \bigl(|u(s,y)|\bigr)\,ds ,
 \tag{4.4}
\end{multline}
where $a(x,y)$, $b(x,y)$, $c(x,y)$ and $W(u)$  are as defined in Theorem
\ref{thm2.3}.
If $u(x,y)$ is a solution of (4.1) with the conditions (4.2),
then it can be  written as \cite[p. 80]{b1}
\[
u(x,y)= \sigma_\infty (x)+  \tau (y)-k - \int_0^x  \int_y^\infty
 \bigl(h (s , t, u(s, t)) +r(s,t)\bigr) \, dt ds  \tag{4.5}
\]
for $x,y \in \mathbb{R}_+$. From (4.3), (4.4), (4.5) we get
\[
\begin{aligned}
|u(x,y)| \leq &a(x,y)+ b(x,y)\int_\alpha^x c(s,y)g \bigl(|u (s,y)|\bigr)\,ds\\
&+  \int_0^x  \int_y^\infty  d(s,t)W\bigl( |u(s,t) |\bigr) \, dt ds.
\end{aligned}\tag{4.6}
\]
Now, a suitable application of Theorem \ref{thm2.3} with $f(x,y)=1$ and $H(u)=u$
to (4.6) yields the required estimate,  Therefore,
\begin{multline*}
|u(x,y)| \leq  p(x,y)\Big\{ a(x,y)+ G^{-1} \Big[
  G \Big( \int_0^\infty  \int_0^\infty
  d(s,t)W \bigl(p(s,t)a(s,t)\bigr)  \, dt ds\Big)\\
 + \int_0^x  \int_y^\infty    d(s,t)W \bigl(p(s,t) \bigr)  \, dt ds
  \Big]\Big\}
\end{multline*}
for $x,y \in \mathbb{R}_+$, where $p(x,y), G$, and $ G^{-1}$ are define
in Theorem \ref{thm2.3}.

\section{Acknowledgements}

The authors are thankful to Prof. Julio G. Dix and the referee for
useful remarks improving this paper.


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\end{thebibliography}

\noindent\textsc{Sever S. Dragomir}\\
School of Communications and Informatics,
Victoria University of Technology\\
P O Box 14428, MCMC
 Melbourne, Victoria 8001, Australia\\
email: sever.dragomir@vu.edu.au
\medskip

\noindent\textsc{Young-Ho Kim}\\
Department of Applied  Mathematics,
 Changwon National University,\\
Changwon 641-773, Korea\\
 email: yhkim@sarim.changwon.ac.kr
\end{document}