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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 117, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/117\hfil Integral inequalities with time delay]
{Integral inequalities with time delay in two independent variables}

\author[H. Khellaf, M. Smakdji, M. Denche \hfil EJDE-2014/117\hfilneg]
{Hassene Khellaf, Mouhamed Smakdji, Mouhamed Denche}  % in alphabetical order

\address{Hassene Khellaf \newline
Laboratory of Differential Equations,
Department of Mathematics, University of Constantine 1,
Constantine 25000, Algeria}
\email{khellafhassane@umc.edu.dz}

\address{Mouhamed El-Hadi Smakdji \newline
Laboratory of Differential Equations,
Department of Mathematics, University of Constantine 1,
Constantine 25000, Algeria}
\email{smakelhadi71@gmail.com}

\address{Mouhamed Denche\newline
Laboratory of Differential Equations,
Department of Mathematics, University of Constantine 1,
Constantine 25000, Algeria}
\email{m\_denche@hotmail.com}

\thanks{Submitted August 2, 2013. Published April 22, 2014.}
\subjclass[2000]{26D10, 26D15}
\keywords{Retarded integral inequality; Gronwall-like type inequality;
\hfill\break\indent submultiplicative function;  differential equations with
time delay}

\begin{abstract}
 In this article, we generalize some retarded integral inequalities in two
 independent variables to more general situations. These integral
 inequalities can be applied as tools to the study of certain class of
 integral and differential equations with time delay.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 Integral inequalities play an important role in the qualitative
analysis of the solutions to differential and integral equations. Over the
years many retarded inequalities have been discovered (see \cite{d1,p2,p4}).
The literature on such inequalities and their applications is vast; see 
\cite{k2,k4,p2} and the references given therein.

  In his study of boundedness of solutions to linear second
order differential equations, Pachpatte \cite{p3} established and applied the
following useful nonlinear integral inequality.
\begin{equation}
u(t)\leq a+\int_{t_0}^tf(s)w(u(s))ds  \label{1.1'}
\end{equation}

where $a>0$ is a constant. Replacing $t$ by a function $b(t)$ in \eqref{1.1'},
Lipovan \cite{l1} investigates the retarded Gronwall-like inequalities
\begin{equation}
u(t)\leq a+\int_{t_0}^tf(s)w(u(s))ds+\int_{b(t)}^{b(t)}g(s)w(u(s))ds
\label{1.1"}
\end{equation}

 In recent years, Pachpatte \cite{p4} discovered some new integral
inequalities involving functions in two independent variables. These
inequalities are applied to study the boundedness and uniqueness of the
solutions of the following terminal value problem for the hyperbolic partial
differential equation \eqref{1.3} with conditions \eqref{1.4},
\begin{gather}
D_1D_2u(x,y) = h(x,y,u(x,y))+r(x,y),  \label{1.3} \\
u(x,\infty ) = \sigma _{\infty }(x),\quad u(\infty ,y)=\tau _{\infty }(y),\quad
\phi (\infty ,\infty )=k,  \label{1.4}
\end{gather}
These inequalities have been generalized to more than one variable. Many
authors have established Gronwall-like type integral inequalities in two
independent variables; see for example \cite{k2,y1,y2}. Recently,  Khan in
\cite{k1} obtained the  inequality
\begin{equation} \label{1.1}
\begin{aligned}
\phi (x,y) &= c+\int_0^{x}A(s,y)\phi (s,y)ds+\int_0^{y}B(x,t)\phi
^p(x,t)dt   \\
&\quad +\int_0^{x}\int_0^{y}H(s,t)\phi ^p(x,y)dtds,\ \
\end{aligned}
\end{equation}
and its variants, where $1>p>0$ and $c>0$ are constants and
$\phi (x,y)\in C(\mathbb{R}_{+}^{2},\mathbb{R}_{+})$.

However, sometimes we need to study such inequalities with a
function $c(x,y)$ in place of the constant term $c$. Our main aim here,
motivated by the works of \cite{k1,l1,p4}, is to
establish some new and more general retarded Gronwall-like integral
inequalities with two independent variables which are useful in the analysis
of certain classes of partial differential equations.

In this article we discuss more general forms of  integral inequality
\begin{equation} \label{1.2}
\begin{gathered}
\phi ^{q}(x,y) \leq c(x,y)+\sum_{i=1}^{n_1}\int_{x_0}^{x}a_{i}(s,y)\phi
^p(s,y)ds+\sum_{j=1}^{n_2}\int_{y_0}^{y}b_{j}(x,t)\phi
^p(x,t)dt   \\
+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{x_0}^{x}
\int_{y_0}^{y}d_{k}(x,y,s,t)g(\phi (s,t))\,ds\,dt,
\end{gathered}
\end{equation}
where $c(x,y)\geq 0$ is a function and $q\geq p>0$ are constants for all
$(x,y)\in \Delta $. Our results remain valid if we replace $\phi ^p(x,t)$
by $w(u(s,y))$ in \eqref{1.2} where
$w\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ be nondecreasing  function with
$w(\phi )>0$ for $\phi >0$.
Furthermore, we show that the results of \cite{k1,l1} can be deduced from
our results in some special cases.

Motivated by the hyperbolic partial differential equation 
\eqref{1.3}-\eqref{1.4} in \cite[Pachpatte]{p4}, we give the boundedness of the
solutions of the initial boundary value problem for hyperbolic partial delay
differential equations.

\section{Main results}

In what follows, we define $I=[x_0,X)$ and $J=[x_0,Y)$ are the given
subsets of $\mathbb{R}_{+}$, and $\Delta =I\times J$, 
$E=\{(x,y,s,t)\in \Delta ^{2}:x_0\leq s\leq x\leq X; y_0\leq t\leq y\leq Y\} $. 
We also assume that all improper integrals appeared in the
sequel are always convergent, and suppose that
\begin{itemize}
\item[(H1)] All $a_{i}(x,y)$ $(i=1,2,\dots,n_1); b_{j}(x,y)$
$(j=1,\dots,n_2);$ $c(x,y)$ and $\phi (x,y)$ are nonnegative, continuous
functions and nondecreasing in each variables on $\Delta$.

\item[(H2)] All $\alpha :I\to I$, $\beta :J\to J$ are
continuously differentiable and nondecreasing such that $\alpha (x)\leq x$
on $I,\beta (y)\leq y$ on $J$.

\item[(H3)] All $v_{k}(x,y)$ $(k=1,2,\dots,n_3)$ are nonnegative,
continuous functions and nondecreasing in each variables on $\Delta$.

\item[(H4)] All $d_{k}(x,y,s,t):E\to \mathbb{R}_{+}$
$(k=1,2,\dots,n_3)$ are nonnegative, continuous functions and nondecreasing
in $x$ and $y$ for each variables $(s,t)$ on $\Delta $. 

\item[(H5)]  $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$ be
nonnegative, continuous, nondecreasing  and submultiplicative function with
$w(\phi )>0$ for $\phi >0$. 

\end{itemize}
The following lemma is useful in our main results.

\begin{lemma} \label{lm1}
Let $c, \phi  $ and $a_{i}\in C(I,\mathbb{R}_{+})$ be
nonnegative continuous functions for any $x\in I$and $i=1,2,\dots,n$ with 
$c(x)$ is nondecreasing function for $x\in I$ and assume that 
$\alpha \in C^1(I,J)$, $\beta \in C^1(I,J)$ be nondecreasing with 
$\alpha (x)\leq x$ on $I$, $\beta (y)\leq y$ on $J$.  Suppose that $q\geq p>0$ 
are constants. If $\phi (x)$ satisfies the inequality 
\begin{equation}
\phi ^{q}(x)\leq c(x)+\sum_{i=1}^{n}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s)\phi ^p(s)ds,  \label{2.1}
\end{equation}
for $x_0\leq s\leq x$, then the following inequalities hold
\begin{equation}
\phi (x)\leq \begin{cases}
c^{1/p}(x)\exp \Big( \frac{1}{p}\sum_{i=1}^{n}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s)ds\Big) , &\text{if }p=q, \\
c^{1/q}(x)+\Big( 1+\frac{q-p}{q}\sum_{i=1}^{n}\int_{\alpha
(x_0)}^{\alpha (x)}c^{(p-q)/q}(s)a_{i}(s).ds\Big)^{\frac{1}{q-p}}
&\text{if } p<q,
\end{cases}  \label{2.2}
\end{equation}
for $x\in I$.
\end{lemma}

 Now, let us list our main results. 

\begin{theorem} \label{th1}
 Suppose {\rm (H1)--(H2)} hold and the constant $p$ satisfies $1>p>0$.

(1) If $\phi (x,y)$ satisfies
\begin{equation}
\phi (x,y)\leq c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)\phi
(s,y)ds+\sum_{i=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)\phi ^p(x,t)dt,  \label{2.4}
\end{equation}
for all $(x,y)\in \Delta $, then
\begin{equation}
\phi (x,y)\leq c(x,y)E_1(x,y)Q_1(x,y),  \label{2.5}
\end{equation}
for all $(x,y)\in \Delta $.  Where
\begin{gather}
E_1(x,y) =\exp \Big( \sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)ds\Big)  \label{2.6} \\
Q_1(x,y) =\Big( 1+(1-p)\sum_{i=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}b_{j}(x,t)c^{(p-1)}(x,t)E_1^p(x,t)dt\Big) ^{\frac{1}{1-p}}.  \label{2.7}
\end{gather}

(2) If $\phi (x,y)$ satisfies
\begin{equation}
\phi (x,y)\leq c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a(s,y)\phi
^p(s,y)ds+\sum_{i=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b(x,t)\phi (x,t)dt,  \label{2.8}
\end{equation}
for all $(x,y)\in \Delta $, then
\begin{equation}
\phi (x,y)\leq c(x,y)E_2(x,y)Q_2(x,y),  \label{2.9}
\end{equation}
for all $(x,y)\in \Delta $.  Where
\begin{gather}
E_1(x,y) =\exp \Big( \sum_{j=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}b(x,t)dt\Big)  \label{2.10} \\
Q_1(x,y) = \Big( 1+(1-p)\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)c^{(p-1)}(s,y)E_2^p(s,y)dt\Big)^{\frac{1}{1-p}}.  
\label{2.11}
\end{gather}
\end{theorem}

The proof of the theorem will be given in the next section.

\begin{theorem} \label{th2} 
Suppose {\rm (H1)--(H2)} hold and $q\geq p>0$ are constants. 
If $\phi (x,y)$ satisfies the inequality
\begin{equation}
\phi ^{q}(x,y)\leq c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s,y)\phi ^p(s,y)ds+\sum_{j=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}b_{j}(x,t)\phi ^p(x,t)dt,  \label{2.12}
\end{equation}
for all $(x,y)\in \Delta $, then we have:

(1) If $p=q$, then
\begin{equation}
\phi (x,y)\leq c^{1/p}(x,y)E_1^{1/p}(x,y)Q_3^{1/p} (x,y),  \label{2.13}
\end{equation}
for all $(x,y)\in \Delta $, where
\begin{equation}
Q_3(x,y)=\exp \Big( \sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)E_1(x,t)dt\Big) ,  \label{2.14}
\end{equation}
and $E_1$ is defined in \ref{2.6}.

(2) If $p<q$, then
\begin{equation}
\phi (x,y)\leq c^{1/q}(x,y)E_4(x,y)Q_4(x,y),  \label{2.15}
\end{equation}
for all $(x,y)\in \Delta $, where
\begin{gather}
Q_4(x,y) =\Big[ 1+\frac{q-p}{q}\sum_{j=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}z^{\frac{p-q}{q}}(x,t)b_{j}(x,t)dt\Big] ^{\frac{1}{q-p}
},  \label{2.16} \\
E_4(x,y) =\Big[ 1+\frac{q-p}{q}\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}c^{\frac{p-q}{q}}(s,y)a_{i}(s,y)Q_4^p(s,y)dt\Big]
^{\frac{1}{q-p}}.  \label{2.17}
\end{gather}
Where $z(x,y)\leq c(x,y)E_4^{q}(x,y)$, for all all $(x,y)\in \Delta $.
\end{theorem}

\begin{remark} \rm
If we take $b(x,y)=0$ and keep $y$ fixed, then Theorem \ref{th2} reduce
exactly to Lemma \ref{lm1}.
\end{remark}

\begin{remark}\rm
 Using similar methods to those in the proof our main result above, if we
replace $\phi ^p(x,t)$ by $w(u(s,y))$ where
$w\in C(\mathbb{R}_{+},\mathbb{R}_{+\ })$ is a nondecreasing  function 
with $w(\phi )>0$ for $\phi >0$, an
estimate of the inequality \eqref{2.12} can be easily obtained; in this
case our result above reduces to the main results in \cite{k1}.
\end{remark}

Using Theorems \ref{th1} and \ref{th2}, we can get some more generalized
results as follow.

\begin{theorem} \label{th3} 
Suppose {\rm (H1)--(H5)} hold and $1\geq p>0$ is  constant.

(1) If  $\phi (x,y)$ satisfies
\begin{equation} \label{2.18}
\begin{aligned}
\phi (x,y)
&\leq c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha (x)}a_{i}(s,y)
\phi (s,y)ds+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)\phi ^p(x,t)dt   \\
&\quad +\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta
(y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(\phi (s,t))\,ds\,dt
\end{aligned}
\end{equation}
for all $(x,y)\in \Delta $, then
\begin{equation}
\phi (x,y)\leq M_1(x,y)E_1(x,y)\tilde{Q}_1(x,y),  \label{2.19}
\end{equation}
for all $x_0\leq x\leq x_1,\ y_0\leq y\leq y_1$.  Where
\begin{equation}
\begin{aligned}
M_1(x,y)&\leq G^{-1}
\Big[ G(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}
\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)\\
&\quad\times g(E_1(s,t))g(\tilde{Q}_1(s,t))\,ds\,dt\Big] ,
\end{aligned} \label{2.20}
\end{equation}
for all $x_0\leq x\leq x_1$, $y_0\leq y\leq y_1$ and
\begin{equation}
G(\phi )=\int_{\phi _0}^{\phi }\frac{\delta t}{g(t)},\quad
\phi \geq \phi _0>0.  \label{2.21}
\end{equation}
Where $E_1(x,y)$ is defined in \ref{2.10} and
\begin{equation}
\tilde{Q}_1(x,y)=\Big[ 1+(1-p)\sum_{j=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}b_{j}(x,t)M_1^{p-1}(x,t)E_1^p(x,t)dt\Big]
^{\frac{1}{1-p}}.  \label{2.22}
\end{equation}
Where $G^{-1}$ is the inverse function of $G$ and the real numbers
$x_1,y_1\in \mathbb{R}_{+}$ are chosen so that
$G(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}
\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(E_1(s,t))g(\tilde{Q}_1(s,t))\,ds\,dt
$ is in $\operatorname{Dom}(G^{-1})$.

(2) If  $\phi (x,y)$ satisfies
\begin{equation}
\begin{aligned}
\phi (x,y)
&\leq c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s,y)\phi ^p(s,y)ds+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)\phi (x,t)dt   \\
&\quad +\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta
(y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(\phi (s,t))\,ds\,dt,
\end{aligned} \label{2.23}
\end{equation}
for all $(x,y)\in \Delta $, then
\begin{equation}
\phi (x,y)\leq M_2(x,y)E_2(x,y)\tilde{Q}_2(x,y),  \label{2.24}
\end{equation}
for all $x_0\leq x\leq x_2,\ y_0\leq y\leq y_2\ $.  Where
\begin{equation}
\begin{aligned}
M_2(x,y)
&\leq G^{-1}\Big[ G(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{
\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)\\
&\quad\times  g(E_2(s,t))g(\tilde{Q}_2(s,t))\,ds\,dt\Big] ,
\end{aligned} \label{2.25}
\end{equation}
for all $x_0\leq x\leq x_1,\ y_0\leq y\leq y_1$,  $G$ and $E_1$
are defined in \ref{2.21} and \ref{2.10}, with
\begin{equation}
\tilde{Q}_2(x,y)=\Big[ 1+(1-p)\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)M_2^{p-1}(s,y)E_2^p(s,y)ds\Big]
^{\frac{1}{1-p}}.  \label{2.26}
\end{equation}
Where $G^{-1}$ is the inverse function of $G$ and the real numbers
$x_2,y_2\in \mathbb{R}_{+}$ are chosen so that
$G(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}
\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(E_2(s,t))
g(\tilde{Q}_2(s,t))\,ds\,dt$ is in $\operatorname{Dom}(G^{-1})$.
\end{theorem}

\begin{remark} \rm
If we take $d_{k}(x,y,s,t)=0$ for any $k=1,2,\dots,n_3$ in the previous Theorem,
then Theorem \ref{th3} reduce to Theorem \ref{th1}. 
\end{remark}

By choosing suitable functions for $g$, some interesting new Gronwall-like
type inequalities of two variables can be obtained from Theorem \ref{th3}.
For example if we take $g(s)=s^{r}$, the following interesting inequalities
are easily obtained.

\begin{corollary}\label{c1}
Suppose {\rm (H1), (H2), (H4)} hold. 
Suppose $1\geq p>0$ and  $0<r<1$ are constants and if $\phi (x,y)$ 
satisfies the inequality
\begin{equation}
\begin{aligned}
\phi (x,y)
&\leq  c(x,y)+\int_{\alpha (x_0)}^{\alpha (x)}a_1(s,y)\phi (s,y)ds
+\int_{\beta (y_0)}^{\beta (y)}b_1(x,t)\phi ^p(x,t)dt   \\
&\quad +\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_1(x,y,s,t)\phi ^{r}(s,t)\,ds\,dt
\end{aligned}  \label{2.27}
\end{equation}
for all $(x,y)\in \Delta $, then
\begin{equation}
\phi (x,y)\leq m_1(x,y)e_1(x,y)\tilde{q}_1(x,y),  \label{2.28}
\end{equation}
for all $(x,y)\in \Delta$.  Where
\begin{equation}
m_1(x,y)\leq \Big[ c^{1-r}(x,y)+(1-r)\int_{\alpha (x_0)}^{\alpha
(x)}\int_{\beta (y_0)}^{\beta (y)}d_1(x,y,s,t)e_1^{r}(s,t)\tilde{q}
_1^{r}(s,t)\,ds\,dt\Big] ^{\frac{1}{1-r}},  \label{2.29}
\end{equation}
for all $(x,y)\in \Delta$, and
\begin{gather}
\tilde{q}_1(x,y) = \Big[ 1+(1-p)\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)m_1^{p-1}(x,t)e_1^p(x,t)dt\Big] ^{\frac{1}{1-p}},
\label{2.30} \\
e_1(x,y) = \exp \Big( \int_{\alpha (x_0)}^{\alpha
(x)}a_1(s,y)ds\Big) .  \label{2.31}
\end{gather}
\end{corollary}

\begin{remark} \rm
(i) If $r=0$, by using Theorem \ref{th3}, an estimation of \eqref{2.27}
 can be easily obtained.

(ii) also when $r=1$, an estimation of the inequality \eqref{2.27} can be
easily obtained;  for space-saving, the details are omitted here.
\end{remark}

\begin{remark} \rm
Corollary \ref{c1} reduces to the main results in \cite[Theorem 2.3]{k1}, 
when $c(x,y)=c$ (constant), $\alpha (x)=x$, $\beta (y)=y$,
$d_1(x,y,s,t)=d(s,t)$, $x_0=y_0=0$ and $r=p$.
\end{remark}

We can also get an interesting result as follows.

\begin{corollary}  \label{c2} 
Suppose {\rm (H1), (H2), (H4)} hold. Suppose $1\geq p>0$,
$0<r<1$ are a constants and if $\phi (x,y)$  satisfies the  inequality
\begin{align*}
\phi (x,y) 
&\leq c(x,y)+\int_{\alpha (x_0)}^{\alpha (x)}a_1(s,y)\phi
^p(s,y)ds+\int_{\beta (y_0)}^{\beta (y)}b_1(x,t)\phi (x,t)dt \\
&\quad +\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_1(x,y,s,t)\phi ^{r}(s,t)\,ds\,dt
\end{align*}
for all $(x,y)\in \Delta $, then
\begin{equation*}
\phi (x,y)\leq m_2(x,y)e_2(x,y)\tilde{q}_2(x,y),
\end{equation*}
for all $(x,y)\in \Delta$.  Where
\begin{equation*}
m_2(x,y)\leq \Big[ c^{1-r}(x,y)+(1-r)\int_{\alpha (x_0)}^{\alpha
(x)}\int_{\beta (y_0)}^{\beta (y)}d_1(x,y,s,t)e_2^{r}(s,t)\tilde{q}
_2^{r}(s,t)\,ds\,dt\Big] ^{\frac{1}{1-r}},
\end{equation*}
for all $(x,y)\in \Delta$, and
\begin{gather*}
\tilde{q}_2(x,y) =\Big[ 1+(1-p)\int_{\alpha (x_0)}^{\alpha
(x)}a_1(s,y)m_2^{p-1}(s,y)e_2^p(s,y)ds\Big] ^{\frac{1}{1-p}}, \\
e_2(x,y) =\exp \Big( \int_{\beta (y_0)}^{\beta
(y)}b_1(x,t)dt\Big) .
\end{gather*}
\end{corollary}

\begin{remark} \rm
Under some suitable conditions, Corollary \ref{c2} is also a generalization
of the main result in \cite[Theorem 2.4]{k1}.
\end{remark}

Using Theorem \ref{th2}, we can get some more generalized results as follows.

\begin{theorem} \label{th4}
Suppose {\rm (H1)--(H5)} hold. Suppose that $q\geq p>0$ are constants.
If  $\phi (x,y)$ satisfies
\begin{equation}
\begin{aligned}
\phi ^{q}(x,y)
&\leq  c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s,y)\phi ^p(s,y)ds
+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta (y)}b_{j}(x,t)\phi ^p(x,t)dt   \\
&\quad +\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta
(y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(\phi (s,t))\,ds\,dt
\end{aligned} \label{2.32'}
\end{equation}
for all $(x,y)\in \Delta$, then the following conclusions are true:

(1) If $p=q$, then
\begin{equation}
\phi (x,y)\leq N_1^{1/p}(x,y)E_1^{1/p}(x,y)Q_3^{1/p}(x,y),
\label{2.32}
\end{equation}
for all $x_0\leq x\leq x_3$, $y_0\leq y\leq y_3$.  Where
\begin{equation}
\begin{aligned}
N_1(x,y)
&\leq H^{-1}\Big[ H(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{
\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)\\
&\quad\times g(E_1^{1/p}(s,t))g(Q_3^{1/p}(s,t))\,ds\,dt\Big] ,
\end{aligned}\label{2.33}
\end{equation}
for all $x_0\leq x\leq x_3$, $y_0\leq y\leq y_3$and
\begin{equation}
H(\phi )=\int_{\phi _0}^{\phi }\frac{dt}{g(t^{1/q})},\quad 
\phi \geq \phi _0>0.  \label{2.34}
\end{equation}
Where $E_1(x,y)$ and $Q_3(x,y)$ are defined in in \ref{2.6} and 
\ref{2.14}. Where $H^{-1}$ is the inverse function of $H$ and the real numbers 
$x_3,y_3$ are chosen so that 
$H(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha (x_0)}^{\alpha (x)}
\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(E_1^{1/p}(s,t))
g(Q_3^{1/p}(s,t))\,ds\,dt\in \operatorname{Dom}(H^{-1})$.


(2) If  $p<q$, then
\begin{equation}
\phi (x,y)\leq N_2^{1/q}(x,y)\tilde{E}_4(x,y)\tilde{Q}_4(x,y),
\label{2.35}
\end{equation}
for all $x_0\leq x\leq x_4,\ y_0\leq y\leq y_4\ $.  Where
\begin{equation}
\begin{aligned}
N_2(x,y)
&\leq H^{-1}\Big[ H(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{
\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)\\
&\quad\times g(\tilde{E}_4(s,t)\tilde{Q}_4(s,t))\,ds\,dt\Big] ,
\end{aligned}  \label{2.36}
\end{equation}
for all $x_0\leq x\leq x_4$, $y_0\leq y\leq y_4$, $H$ is defined in
\ref{2.34}, with
\begin{gather}
\tilde{Q}_4(x,y) =\Big[ 1+\frac{(q-p)}{q}\sum_{j=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}b_{j}(s,y)\tilde{z}^{(p-q\grave{a}/q}(x,t)dt\Big] ^{\frac{1}{q-p}},
  \label{2.37} \\
\tilde{E}_4(x,y) =\Big[ 1+\frac{(q-p)}{q}\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)N_2^{(p-q\grave{a}/q}(s,y)\tilde{Q}
_4^p(s,y)ds\Big] ^{\frac{1}{q-p}}  \label{2.38}
\end{gather}
for all $x_0\leq x\leq x_4$, $y_0\leq y\leq y_4$, where 
$\tilde{z} (x,y)\leq N_2(x,y)\tilde{E}_4^{q}(x,y)$. 
Here $H^{-1}$ is the inverse function of $H$ and the real numbers 
$x_4,y_4$ are chosen so that 
$H(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)
\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)g(\tilde{E}_4(s,t)\tilde{Q}_4(s,t))\,ds\,dt\in
\operatorname{Dom}(H^{-1})$.
\end{theorem}

\begin{remark} \rm
Various choices of $c,g,d_{k},v_{k}$ and $p,q,\alpha ,\beta $ 
can give many different inequalities. Obviously, our results generalize 
many results obtained before, for example, let $p=1$, $\alpha (x)=x$,
$\beta (y)=y $, $g(s)=s$, $c(x,y)=c>0$ (constant), $n_1=n_2=n_3=1$,
$x_0=y_0=0$ and $d_1(x,y,s,t)=d_1(s,t)$, then our Theorem \ref{th4}(1)
reduces to \cite[Theorem 2.1]{k1}.
Considering $q=1$, $\alpha (x)=x$, $\beta (y)=y$, $w(s)=s^p$,
$n_1=n_2=n_3=1$, $x_0=y_0=0$, $d_1(x,y,s,t)=d_1(s,t)$ and 
$c(x,y)=c\geq 0$ (constant) in Theorem \ref{th4}(2), we obtain 
\cite[Theorem 2.5]{k1}.
If we take $q=p=1$, $g(s)=s^{r}$, $1>r>0$, $\alpha (x)=x$, 
$\beta (y)=y$, $c(s,y)=c\geq 0$ (constant), $n_1=n_2=n_3=1$, 
$x_0=y_0=0$, $v_1(x,y)=1$ and $d_1(x,y,s,t)=d_1(s,t)$, then the inequality
established in Theorem \ref{th4}(1) reduces to the 
\cite[Theorem 2.2]{k1}.
\end{remark}

\begin{remark} \rm
By replacing $\phi ^p(x,t)$ by $w(u(s,y))$ where
$w\in C(\mathbb{R}_{+},\mathbb{R}_{+\ })$ be nondecreasing 
function with $w(\phi )>0$ for $\phi >0$ in the line above \eqref{2.32'} 
and using the same arguments in the proof of our theorem \ref{th4}, 
an estimation of the inequality \eqref{2.32'} can
be easily obtained. In particular, when $a$ (a nonnegative constant) and 
$d_{k}(x,y,s,t)=$0 for all $k=0,\dots,n_3$, the inequality \eqref{2.32'}
becomes
\begin{equation}
\phi ^{q}(x,y)\leq a+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s,y)w(\phi (s,y))ds+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)w(\phi (x,t))dt
\end{equation}
the general form of \eqref{2.32'} in the case of two independent variables.
Thus our result implies to the main result in \cite{k1}.
\end{remark}

\begin{remark} \rm
By choosing suitable functions for $g$ for example $g(s)=s^{r}$ with 
 $ q\geq r\geq 0$ or when we take $q=r>0$ (with $p=q$ or $q>p$), using similar
arguments in the proof of Theorem \ref{th4}, we can obtain many interesting
new retarded integral inequalities, but, for space-saving, the details are
omitted here.
\end{remark}

\begin{remark} \rm
Using similar method of those in the proof of  our main results above, with
a suitable conditions, we can obtain some new reversed inequalities of our
results.
\end{remark}


\section{Proof of theorems}

Since the proofs resemble each other, we give the details only for
Theorem \ref{th1}(1), Theorem \ref{th2}(2), Throem\ref{th3}(1) and
Theorem \ref{th4}(2). The proofs of the remaining
inequalities can be completed by following the proofs of the above-mentioned
inequalities.
To the best our knowledge, Lemma \ref{lm1} is not found in the literature
(in this form). Therefore, we give a proof here.

\begin{proof} [Proof of Lemma \ref{lm1}]
If $c(x)>0$
(i) If $p=q$ holds, letting $z(x)=[ \frac{\phi (x)}{c^{1/p}(x)}] ^p$,
 from \eqref{2.1} derive that 
\begin{equation}
z(x)\leq 1+\sum_{i=1}^{n}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s)z(s)ds,  \label{4.1}
\end{equation}
for $x\in I$.  define a positive, continuous and nondecreasing function $v(x)$
by the right hand of \eqref{4.1}, then $z(x)\leq v(x)$ and $v(x_0)=1$
hold. Since $v(x)$ is positive and by differentiation we obtain
\begin{gather}
v(x)=\sum_{i=1}^{n}\alpha '(x)a_{i}(\alpha (x))z(\alpha (x)), \notag \\
\frac{v'(x)}{v(x)}\leq \sum_{i=1}^{n}\alpha '(x)a_{i}(\alpha (x)),\quad
 x\in I.  \label{4.2}
\end{gather}
By integration of \eqref{4.2} from $x_0$ to $x$,  we have
\begin{equation*}
v(x)\leq \exp \Big( \sum_{i=1}^{n}\int_{\alpha (x_0)}^{\alpha(x)}a_{i}(s)ds\Big) ,
\end{equation*}
hence we obtain
\[
[ \frac{\phi (x)}{c^{1/p}(x)}] ^p=z(x)
\leq \exp [ \sum_{i=1}^{n}\int_{\alpha (x_0)}^{\alpha (x)}a_{i}(s)ds].
\]
This inequality implies the desired inequality \eqref{2.2} immediately.

(ii) If $p<q$ holds, letting $y(x)=\frac{\phi (x)}{c^{1/q}(x)}$,
from \eqref{2.1} we obtain
\begin{equation}
y^{q}(x)\leq 1+\sum_{i=1}^{n}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s)c^{(p-q)/q}(s)y^p(s)ds,  \label{4.3}
\end{equation}
Define a positive, continuous and nondecreasing function $h(x)$ by the right
hand of \eqref{4.3}, then $y(x)\leq h^{1/q}(x)$ and $h(x_0)=1$ hold. we
carry out the above procedure, we obtain
\begin{equation*}
\frac{\phi (x)}{c^{1/q}(x)}=y(x)
\leq \Big[ 1+\frac{q-p}{q}
\sum_{i=1}^{n}\int_{\alpha (x_0)}^{\alpha
(x)}c^{(p-q)/q}(s)a_{i}(s)ds\Big] ^{\frac{1}{q-p}},
\end{equation*}
This inequality implies the desired inequality \eqref{2.2} immediately.

If $c(x)\geq 0$ is nonnegative, we carry out the above procedure in 
(i) and (ii) with $c(x)+\varepsilon $ instead of $c(x)$, where 
$\varepsilon >0$ is an arbitrary small constant, and subsequently pass to the
limit as $\varepsilon \to 0$ to obtain \eqref{2.2}. This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th1}]
(1) We define a function
\begin{equation}
z(x,y)=c(x,y)+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)\phi ^p(x,t)dt,  \label{3.5}
\end{equation}
by substituting \eqref{3.5} in \eqref{2.4}, we obtain
\begin{equation}
\phi (x,y)\leq z(x,y)+\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)\phi (s,y)ds,\quad  (x,y)\in \Delta .
\label{3.6}
\end{equation}
Clearly, $z(x,y)$ is a nonnegative, continuous and nondecreasing function in
$x$. Treating $y$, $y\in I_2$ fixed in \eqref{3.6}, a suitable application
of Lemma \ref{lm1} to \eqref{3.6} we obtain
\begin{equation}
\phi (x,y)\leq z(x,y)E_1(x,y),  \label{3.7}
\end{equation}
for $(x,y)\in \Delta$, where $E_1(x,y)$ is defined as in \eqref{2.6}.

By \eqref{3.5} and \eqref{3.7}, we obtain
\begin{equation}
z(x,y)\leq c(x,y)+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)E_1^p(x,t)z^p(x,t)dt.  \label{3.8}
\end{equation}
Keeping $x$ fixed in \eqref{3.8}, an estimation of $z(x,y)$ can be obtained
by a suitable application of Lemma \ref{lm1}  to \eqref{3.8}, after that,
we obtain
\begin{equation}
z(x,y)\leq c(x,y)Q_1(x,y),  \label{3.9}
\end{equation}
for $(x,y)\in \Delta$, where $Q_1(x,y)$ is defined as in \eqref{2.7}.

Finally, substituting the last inequality into \eqref{3.7}, the desired
inequality \eqref{2.5} follows immediately.
\end{proof}

\begin{proof} [Proof of Theorem \ref{th2}(2)]
We define a function 
\begin{equation}
z(x,y)=c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s,y)\phi ^p(s,y)ds,  \label{3.10}
\end{equation}
by substituting \eqref{3.10} in \eqref{2.12}, we obtain
\begin{equation}
\phi ^{q}(x,y)\leq z(x,y)+\sum_{j=1}^{n_2}\int_{\beta
(y_0)}^{\beta (y)}b_{j}(x,t)\phi ^p(x,t)dt.  \label{3.11}
\end{equation}
Clearly, $z(x,y)$ is a nonnegative, continuous and nondecreasing function in
$y$. Treating $x$ fixed in \eqref{3.11}, a suitable application of 
Lemma \ref{lm1} to \eqref{3.11} we obtain
\begin{equation}
\phi (x,y)\leq z(x,y)^{1/q}Q_4(x,y),  \label{3.12}
\end{equation}
for $(x,y)\in \Delta$, where $Q_4(x,y)$ is defined as in \eqref{2.16}.

By \eqref{3.12} and \eqref{3.10}, we obtain
\begin{equation}
z(x,y)\leq c(x,y)+\sum_{i=1}^{n_1}\int_{\alpha (x_0)}^{\alpha
(x)}a_{i}(s,y)Q_4^p(s,y)z^{p/q}(s,y)ds.  \label{3.13}
\end{equation}
Keeping $y$ fixed in \eqref{3.13}, an estimation of $z(x,y)$ can be obtained
by a suitable application of Lemma \ref{lm1}  to \eqref{3.13}, after that,
we obtain
\begin{equation*}
z(x,y)\leq c(x,y)E_4^{q}(x,y),
\end{equation*}
for $(x,y)\in \Delta ,$where $E_4(x,y)$ is defined as in \eqref{2.17}.

Finally, substituting the last inequality into \eqref{3.12}, the desired
inequality \eqref{2.15} follows immediately.\newline
\end{proof}

\begin{proof} [Proof of Theorem \ref{th3}]
If $c(x,y)>0$ . Setting
\begin{equation}
M_1(x,y)=c(x,y)+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha
(x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(\phi
(s,t))\,ds\,dt,  \label{3.14}
\end{equation}
inequality \eqref{2.18} can be restated as
\begin{equation}
\phi (x,y)\leq M_1(x,y)+\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)\phi
(s,y)ds+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)\phi ^p(x,t)dt,  \label{3.15}
\end{equation}
Clearly, $M_1(x,y)$ is nonnegative and nondecreasing function in each in 
$x$ and $y$ . Now a suitable application of the inequality \eqref{2.4} in
Theorem \ref{th1} to \eqref{3.15}, yields
\begin{equation}
\phi (x,y)\leq M_1(x,y)E_1(x,y)\widetilde{Q}_1(x,y),  \label{3.16}
\end{equation}
where $E_1(x,y)$, $\widetilde{Q}_1(x,y)$ are defined in \eqref{2.6} and 
\eqref{2.22}.
From \eqref{3.14} and \eqref{3.16} and by using the fact that $w$ is
submultiplicative, we have
\begin{equation}
\begin{aligned}
M_1(x,y)
&\leq c(x,y)+\sum_{k=1}^{n_3}v_{k}(x,y)
 \int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)\\
&\quad \times g\big( E_1(s,t)\widetilde{Q}_1(s,t)\big)
g(M_1(s,t))\,ds\,dt,  \label{3.17}
\end{aligned}
\end{equation}
for $(x,y)\in \Delta $.
Fixing any numbers $\widetilde{x}_1$ and $\widetilde{y}_1$ with
$0< \widetilde{x}_1\leq x_1$ and $0<\widetilde{y}_1\leq y_1$, from
\eqref{3.17} we have
\begin{align*}
M_1(x,y)
&\leq c(\widetilde{x}_1,\widetilde{y}_1)
 +\sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,\widetilde{y}_1) \\
&\quad\times \int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(\widetilde{x}_1,\widetilde{y}_1,s,t)
 g\big( E_1(s,t) \widetilde{Q}_1(s,t)\big) g(M_1(s,t))\,ds\,dt
\end{align*}
for $x_0\leq x\leq \widetilde{x}_1,y_0\leq y\leq \widetilde{y}_1$.

Defining $r_1(x,y)$ as the right-hand side of the last inequality, then 
$r_1(x_0,y)=r_1(x,y_0)=c(\widetilde{x}_1,\widetilde{y}_1)$, and
\begin{equation}
M_1(x,y)\leq r_1(x,y),  \label{3.18}
\end{equation}
with $r_1(x,y)$ is positive and nondecreasing in 
$y\in [y_0,\widetilde{y}_0]$, and
\begin{equation}
\begin{aligned}
D_1r_1(x,y) &= \sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,
\widetilde{y}_1)\alpha '(x)
 \int_{\beta (y_0)}^{\beta (y)}d_{k}(\widetilde{x}_1,\widetilde{y
}_1,\alpha (x),t)\\
&\quad\times g\big( E_1(\alpha (x),t)\widetilde{Q}_1(\alpha
(x),t)\big) g(M_1(\alpha (x),t))dt,
  \\
&\leq \sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,\widetilde{y}_1)\alpha '(x)
\int_{\beta (y_0)}^{\beta (y)}d_{k}(\widetilde{x}_1,\widetilde{y
}_1,\alpha (x),t)\\
&\quad\times g\big( E_1(\alpha (x),t)\widetilde{Q}_1(\alpha
(x),t)\big) g(r_1(\alpha (x),t))dt.   \\
&\leq w(r_1(x,y))\sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,
\widetilde{y}_1)\alpha '(x)   \\
&\quad \times \int_{\beta (y_0)}^{\beta (y)}d_{k}(\widetilde{x}_1,\widetilde{y
}_1,\alpha (x),t)g\big( E_1(\alpha (x),t)\widetilde{Q}_1(\alpha
(x),t)\big) dt.
\end{aligned}\label{3.19}
\end{equation}
Dividing both sides of \eqref{3.19} by $w(r_1(x,y))$, we obtain
\begin{equation}
\begin{aligned}
\frac{D_1r_1(x,y)}{g(r_1(x,y))}
 &\leq sum_{k=1}^{n_3}v_{k}(
\widetilde{x}_1,\widetilde{y}_1)\alpha '(x)
 \int_{\beta (y_0)}^{\beta (y)}d_{k}(\widetilde{x}_1,\widetilde{y
}_1,\alpha (x),t)\\
&\quad\times g\big( E_1(\alpha (x),t)\widetilde{Q}_1(\alpha
(x),t)\big) dt,
\end{aligned} \label{3.20}
\end{equation}
from \eqref{2.21} and \eqref{3.20}, we have
\begin{equation}
\begin{aligned}
D_1G(r_1(x,y))
&\leq \sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,
\widetilde{y}_1)\alpha '(x)   \\
&\quad\times \int_{\beta (y_0)}^{\beta (y)}d_{k}(\widetilde{x}_1,\widetilde{y
}_1,\alpha (x),t)g\left( E_1(\alpha (x),t)\widetilde{Q}_1(\alpha
(x),t)\right) dt,.
\end{aligned}  \label{3.21}
\end{equation}
Now setting $x=s$ in \eqref{3.21} and then integrating with respect to $s$
from $x_0$ to $x$, we obtain
\begin{align*}
G(r_1(x,y))
&\leq  G(r_1(x_0,y))+\sum_{k=1}^{n_3}v_{k}(
\widetilde{x}_1,\widetilde{y}_1) \\
&\quad \times \int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(\widetilde{x}_1,\widetilde{y}_1,s,t)g\big( E_1(s,t)
\widetilde{Q}_1(s,t)\big) \,ds\,dt.
\end{align*}
Noting $G(r_1(x_0,y))=G(c(\widetilde{x}_1,\widetilde{y}_1))$, we
have
\begin{align*}
G(r_1(x,y))
&\leq G(c(\widetilde{x}_1,\widetilde{y}_1))
 +\sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,\widetilde{y}_1) \\
&\quad \times \int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(\widetilde{x}_1,\widetilde{y}_1,s,t)g\big( E_1(s,t)
\widetilde{Q}_1(s,t)\big) \,ds\,dt.
\end{align*}
Taking $x=\widetilde{x}_1$, $y=\widetilde{y}_1$ in \eqref{3.18} and the
last inequality, we obtain
\begin{equation}
M_1(\widetilde{x}_1,\widetilde{y}_1)\leq r_1(\widetilde{x}_1,
\widetilde{y}_1),  \label{3.22}
\end{equation}
and
\begin{equation}
\begin{aligned}
G(r_1(\widetilde{x}_1,\widetilde{y}_1))
&\leq G(c(\widetilde{x}_1,
\widetilde{y}_1))+\sum_{k=1}^{n_3}v_{k}(\widetilde{x}_1,
\widetilde{y}_1)   \\
&\quad \times \int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(\widetilde{x}_1,\widetilde{y}_1,s,t)w\left( E_1(s,t)
\widetilde{Q}_1(s,t)\right) \,ds\,dt.  \label{3.23}
\end{aligned}
\end{equation}
Since $0<\widetilde{x}_1\leq x_1$ and $0<\widetilde{y}_1\leq y_1$
are arbitrary, from \eqref{3.22} and \eqref{3.23}, we have
\begin{equation}
M_1(x,y)\leq r_1(x,y),  \label{3.24}
\end{equation}
and
\begin{equation}
\begin{aligned}
r_1(x,y)&\leq G^{-1}\Big[ G(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)
\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)\\
&\quad g\big( E_1(s,t)\widetilde{Q}_1(s,t)\big) \,ds\,dt\Big].
\end{aligned} \label{3.25}
\end{equation}
for all $x_0<x\leq x_1,y_0<y\leq y_1$.  Hence by \eqref{3.24} and
\eqref{3.25}, we obtain
\begin{equation}
\begin{aligned}
M_1(x,y)&\leq G^{-1}\Big[ G(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)
\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)\\
&\quad\times g\big( E_1(s,t)\widetilde{Q}_1(s,t)\big) \,ds\,dt\Big],
\end{aligned} \label{3.26}
\end{equation}
for all $x_0<x\leq x_1$, $y_0<y\leq y_1$. By \eqref{2.18}, \eqref{3.26}
holds also when $x=x_{0,}y=y_0$.

Finally, substituting the last inequality into \eqref{3.16}, the desired
inequality \eqref{2.19} follows immediately.

If $c(x,y)\geq 0$ is nonnegative, we carry out the above procedure in
the proof of Theorem \ref{th3}(1) with 
$c(x,y)+\varepsilon $ instead of $c(x,y)$, where $\varepsilon >0$ is an
arbitrary small constant, and subsequently pass to the limit as 
$\varepsilon \to 0$ to obtain \eqref{2.19} This completes the proof.

The proof of Theorem \ref{th3}(2) is similar
to the argument in the proof of  Theorem \ref{th3}(1)
 with suitable modifications. We omit the details here.
\end{proof}


\begin{proof}[Proof of Theorem \ref{th4}]
Setting
\begin{equation}
N_2(x,y)=c(x,y)+\sum_{k=1}^{n_3}v_{k}(x,y)\int_{\alpha
(x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta (y)}d_{k}(x,y,s,t)g(\phi
(s,t))\,ds\,dt,  \label{3.27}
\end{equation}
 inequality \eqref{2.32'} can be restated as
\begin{equation}
\phi ^{q}(x,y)\leq N_2(x,y)+\sum_{i=1}^{n_1}\int_{\alpha
(x_0)}^{\alpha (x)}a_{i}(s,y)\phi
^p(s,y)ds+\sum_{j=1}^{n_2}\int_{\beta (y_0)}^{\beta
(y)}b_{j}(x,t)\phi ^p(x,t)dt,  \label{3.28}
\end{equation}
Clearly, $N_2(x,y)$ is nonnegative and nondecreasing function in each in 
$x$ and $y$. Now a suitable application of  inequality \eqref{2.15} in
Theorem \ref{th2}, to \eqref{3.28}, yields
\begin{equation}
\phi (x,y)\leq N_2^{1/q}(x,y)\widetilde{E}_4(x,y)\widetilde{Q}_4(x,y),
\label{3.29}
\end{equation}
where $\widetilde{E}_4(x,y)$, $\widetilde{Q}_4(x,y)$ are defined in 
\eqref{2.38} and \eqref{2.37}.
From \eqref{3.27} and \eqref{3.29} and by using the fact that $w$ is
submultiplicative, we have
\begin{equation}
\begin{aligned}
N_2(x,y) &\leq c(x,y)+\sum_{k=1}^{n_3}v_{k}(x,y) 
 \int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}d_{k}(x,y,s,t)\\
&\quad\times g\big( \widetilde{E}_4(s,t)\widetilde{Q}
_4(s,t)\big) g(N_1^{1/q}(s,t))\,ds\,dt,
\end{aligned} \label{3.30}
\end{equation}
for $(x,y)\in \Delta $. 

By following the same steps from \eqref{3.17}-\eqref{3.26} in \eqref{3.30},
we obtain
\begin{align*}
N_2(x,y)&\leq H^{-1}\Big[ H(c(x,y))+\sum_{k=1}^{n_3}v_{k}(x,y)
\int_{x_0}^{x}\int_{y_0}^{y}d_{k}(x,y,s,t)
 w\big( \widetilde{E}_4(s,t)\big) \\
&\quad\times w(\widetilde{Q}_4(s,t))\,ds\,dt\Big] ,
\end{align*}
for all $x_0\leq x\leq x_{41},y_0\leq y\leq y_4$.
Finally, substituting the last inequality in \eqref{3.29}, the desired
inequality \eqref{2.35} follows immediately.
\end{proof}

\section{An application}

 In this section we present an application of the inequality 
\eqref{2.32'} given in Theorem \ref{th4} to study the boundedness of the
solutions of the initial boundary value problem for hyperbolic partial delay
differential equations of the form
\begin{equation}
\begin{gathered}
\begin{aligned}
D_1D_2\phi ^p(x,y)
&=h\big(x,y,\phi (x-\alpha (x),y-\beta (y))\big)
+D_1g_1\big(x,y,\phi (x-\alpha (x),y)\big) \\
&\quad +D_2g_2\big(x,y,\phi (x,y-\beta (y))\big),
\end{aligned} \\
\phi (x,y_0)=\sigma _1(x),\quad \phi (x_0,y)=\sigma _2(y),\quad
\phi (x_0,y_0)=k,
\end{gathered}  \label{5.1}
\end{equation}
for all $(x,y)\in \Delta $.

 Where $h,g_1,g_2\in C(\Delta \times \mathbb{R},\mathbb{R})$ and
$\sigma _1,\sigma _2\in C(\mathbb{R}_{+},\mathbb{R}_{+}),k,p>0$ are 
constants, $\alpha \in C^1(I,\mathbb{R)}, \beta \in C^1(J,\mathbb{R})$
nondecreasing functions such that $\alpha (x)\leq x$ on 
$I$, $\beta (y)\leq y $ on $J$,  and $\alpha '(x)<1$, $\beta '(y)<1$ with 
$\alpha (x_0)=0$ and $\beta (y_0)=0$.

\begin{theorem} \label{th5} Suppose that
\begin{gather}
| h(x,y,\phi )| \leq d_1(x,y)\phi ^{r},  \label{5.2}\\
| g_1(x,y,\phi )| \leq b_1(x,y)\phi ^p,\label{5.3} \\
| g_2(x,y,\phi )| \leq a_1(x,y)\phi ^p, \label{5.4} \\
| c_1(x)+c_2(y)-k|\leq c,\quad  c\geq 0\text{ (constant).}  \label{5.5}
\end{gather}
Where $1>p>r>0$, $b_1(x,y),a_1(x,y)$ are as in Theorem \ref{th4} and 
$d_1(x,y)\in C(\Delta ,\mathbb{R}_{+})$ be nondecreasing  function, with
\begin{gather*}
c_1(x) = \sigma _1(x)-\int_{x_0}^{x}g_2(s,x_0,\sigma _1(s))ds,\\
c_2(y) = \sigma _2(y)-\int_{y_0}^{y}g_1(y_0,t,\sigma _2(t))dt.
\end{gather*}
If $\phi (x,y)$ is any solution of \eqref{5.1}, then
\begin{equation}
\begin{aligned}
\phi (x,y) &\leq \big( \tilde{e}(x,y)\tilde{q}(x,y)\big) ^{1/p}
\Big[ c^{(p-r)/p}\\
&\quad +\frac{p-r}{p}\int_{\psi (x_0)}^{\psi
(x)}\int_{\Omega (y_0)}^{\Omega (y)}\bar{d}_1(s,t)\big( \tilde{e}(x,y)
\tilde{q}(x,y)\big) ^{r/p}\,ds\,dt\Big] ^{\frac{1}{p-r}},
\end{aligned} \label{6.1}
\end{equation}
for $(x,y)\in \Delta$, in which $\psi (x)=x-\alpha (x)$ on $I$ and
$\Omega (y)=y-\beta (y)$ on $J$, and
\begin{gather}
\tilde{e}(x,y)=\exp \Big( \int_{\psi (x_0)}^{\psi (x)}\bar{a}
_1(s,y)ds\Big) ,  \label{5.5'} \\
\tilde{q}(x,y)=\exp \Big( \int_{\Omega (y_0)}^{\Omega (y)}\bar{b}_1(x,t)
\tilde{e}(x,t)dt\Big) .  \label{5.5"}
\end{gather}
For all $(x,y)\in \Delta $, where
\begin{gather}
\bar{a}_1(\delta ,t) = \xi _1\cdot a_1(\delta +\alpha (s),t), \label{6.2} \\
\bar{b}_1(s,\tau ) = \xi _2\cdot b_1(s,\tau +\beta (t)),  \label{6.3}\\
\bar{d}_1(\delta ,\tau ) = \xi _1\xi _2\cdot d_1(\delta +\alpha
(s),\tau +\beta (t)),  \label{6.4}
\end{gather}
for all $s,\delta \in I$ and $t,\tau \in J$; and
\[
\xi _1=\underset{x\in I}{Max}\frac{1}{1-\alpha '(x)},\quad
\xi _2=\underset{y\in I}{Max}\frac{1}{1-\beta '(y)}
\]
\end{theorem}

\begin{proof}
It is easy to see that, the solution $\phi (x,y)$ of problem \eqref{5.1}
satisfies the equivalent integral equation
\begin{equation}
\begin{aligned}
\phi ^p(x,y)
&= \sigma _1(x)+\sigma
_2(y)-k+\int_0^{x}\int_0^{y}h(s,t,\phi
(s,t))\,ds\,dt+\int_0^{y}g_1(x,t,\phi (x,t))dt   \\
&\quad +\int_0^{x}g_2(s,y,\phi (s,y)ds-\int_0^{x}g_2(s,0,\sigma
_1(s))ds
-\int_0^{y}g_1(0,t,\sigma _2(t))dt.
\end{aligned}  \label{6.5}
\end{equation}
From \eqref{6.5} and with \eqref{5.2}-\eqref{6.5}, we have
\begin{equation}
\begin{aligned}
\phi ^p(x,y)
&\leq c+\int_0^{x}a_1(s,y)\phi ^p(s-\alpha (s),y)ds
 +\int_0^{y}b_1(x,t)\phi ^p(x,t-\beta (t))dt   \\
&\quad +\int_0^{x}\int_0^{y}d_1(s,t)\phi ^{r}(s-\alpha (s),t-\beta (t))\,ds\,dt,
\end{aligned}\label{6.6}
\end{equation}
Using \eqref{6.6} and making a change of variables, we have
\begin{equation}
\begin{aligned}
\phi ^p(x,y)
&\leq  c+\int_{\psi (x_0)}^{\psi (x)}\bar{a}_1(s,y)\phi
^p(s,y)ds+\int_{\Omega (y_0)}^{\Omega (y)}\bar{b}_1(x,t)\phi
^p(x,t)dt   \\
&\quad +\int_{\psi (x_0)}^{\psi (x)}\int_{\Omega (y_0)}^{\Omega (y)}\bar{d}
_1(s,t)\phi ^{r}(s,t)\,ds\,dt,
\end{aligned} \label{5.8}
\end{equation}
for $(x,y)\in \Delta$, with $p>r>0$.  Where $\psi (x)=x-\alpha (x)$
on $I $ and $\Omega (y)=y-\beta (y)$ on $J$, and
$\bar{a}_1,\bar{b}_1$ and $\bar{d}_1$ are defined in \eqref{6.2}, \eqref{6.2}
and \eqref{6.3}

Now, a suitable application of Theorem \ref{th4}(1)
to \eqref{5.8}, with $g(s)=s^{r}$, $c(x,y)=c$, $v_1(x,y)=1$,
$d_1(x,y,s,t)=\bar{d}_1(s,t)$, $n_1=n_2=n_3=1$, yields
\begin{align*}
\phi (x,y)
&\leq \big( \tilde{e}(x,y)\tilde{q}(x,y)\big) ^{1/p}
\Big[ c^{(p-r)/p}\\
&\quad +\frac{p-r}{p}\int_{\psi (x_0)}^{\psi (x)}\int_{\Omega
(y_0)}^{\Omega (y)}\bar{d}_1(s,t)\big( \tilde{e}(x,y)\tilde{q}
(x,y)\big) ^{r/p}\,ds\,dt\Big] ^{\frac{1}{p-r}}
\end{align*}
for all $(x,y)\in \Delta$, where $\tilde{e}(x,y)$ and $\tilde{q}(x,y)$ are
defined in \eqref{5.5'} and \eqref{5.5"}.
\end{proof}

We also note that the inequalities established in Theorems \ref{th4} and 
\ref{th4} and the applications given in Theorems \ref{th4} can be extended very
easily to functions involving many independent variables.

Finally, we note that under some suitable conditions, the uniqueness and
continuous dependence of the solutions of \eqref{5.1}, can also be discussed
using our results.

\subsection*{Acknowledgment}
This research is supported by ANDRU, Algeria (Project
PNR nos. U250/R25).


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\end{document}