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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 169, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/169\hfil
 Nonlinear delay integral inequalities]
{Nonlinear delay integral inequalities for multi-variable
functions}

\author[H. Khellaf, M. Smakdji\hfil EJDE-2011/169\hfilneg]
{Hassane Khellaf, Mohamed el hadi Smakdji}  % in alphabetical order

\address{Hassane Khellaf \newline
Department of Mathematics, Faculty of Exact Sciences,
University of Mentouri, Constantine 25000, Algeria}
\email{khellafhassane@umc.edu.dz, khellaf1973@gmail.com}

\address{Mohamed el hadi Smakdji \newline
Department of Mathematics, Faculty of Exact Sciences,
University of Mentouri, Constantine 25000, Algeria}
\email{smakelhadi71@gmail.com}

\thanks{Submitted August 5, 2011. Published December 18, 2011.}
\subjclass[2000]{26D15, 26D20, 26D10}
\keywords{Delay integral inequality; multi-variable function;
\hfill\break\indent  delay partial differential equation}

\begin{abstract}
 In this article, we establish some nonlinear retarded integral
 inequalities in $n$ independent variables.
 These inequalities represent a generalization of the results
 obtained in \cite{aa,pec,pach1} for function of  one and two
 variables. Our results can be used in the qualitative theory
 of  delay partial differential equations and delay integral
 equations.
\end{abstract}

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

\section{Introduction}

In the study of ordinary differential and integral equations, one
often deals with certain integral inequalities. The Gronwall-Bellman
inequality and its various linear and nonlinear generalizations
are crucial in the discussion of the existence, uniqueness,
continuation, boundedness, oscillation, stability and other
qualitative properties of the solutions of differential and
integral equations. The literature on such inequalities and
their applications is vast; see \cite{9,10,12,13,17} and references
therein.

During the past few years,  investigators have established some useful
and interesting delay integral inequalities in order to achieve various
goals; see \cite{8,11,14,15,16} and the references cited therein.

Let us first list the main results of \cite{aa,pec,pach1}, for
functions with two variables for
$u(x,y)\in (\Delta \in \mathbb{R}_{+}^{2},\mathbb{R}_{+})$:

Inequality by Ma and Pecaric \cite[Theorem 2.1]{pec}:
\begin{equation} \label{1.1}
\begin{split}
u^{p}(x,y) &= k+\sum_{i=1}^{m}\int_{\alpha _{1i}(x_0)}^{\alpha
_{1i}(x)}\int_{\beta _{1i}(y_0)}^{\beta _{1i}(y)}a_i(s,t)
 u^{q}(s,t)\,dt\,ds \\
&\quad  +\sum_{j=1}^n\int_{\alpha _{2j}(x_0)}^{\alpha _{2j}(x)}
\int_{\beta_{2j}(y_0)}^{\beta _{2j}(y)}b_j(s,t)u^{q}(s,t)w(u(s,t))
\,dt\,ds.
\end{split}
\end{equation}

 Pachpatte's inequality \cite[Theorem 4]{pach1};
\begin{equation} \label{1.2}
\begin{split}
u^{p}(x,y) &= k+\int_{x_0}^{x}
\int_{y_0}^{y}a(s,t)g_1(u(s,t))\,dt\,ds  \\
&\quad +\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta (y_0)}^{\beta
(y)}b(s,t)g_2(u(s,t))\,dt\,ds.
\end{split}
\end{equation}

 Cheung's inequality  \cite[Theorem 2.4]{aa}:
\begin{equation} \label{1.3}
\begin{split}
u^{p}(x,y)
&= k+\frac{p}{p-q}\int_{\alpha (x_0)}^{\alpha (x)}\int_{\beta
(y_0)}^{\beta (y)}a(s,t)u^{q}(s,t)\,dt\,ds   \\
&\quad + \int_{\gamma (x_0)}^{\gamma (x)}
 \int_{\gamma (y_0)}^{\delta
(y)}b(s,t)u^{q}(s,t)\varphi (u(s,t))\,dt\,ds.
\end{split}
\end{equation}


However, sometimes we need to study such inequalities with a
function $c(x)$ in place of the constant term $k$.
Our main result, for functions with $n$ independent variables,
is given in the inequality
\begin{equation}
\begin{split}
\varphi (u(x))
&\leq c(x)+\sum_{j=1}^{n_1}d_j(x)\int_{\widetilde{\alpha
}_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)
\Phi (u(t))w_1(u(t))dt  \\
&\quad + \sum_{k=1}^{n_2}l_k(x)
 \int_{\widetilde{\beta }_k(x^0)}
^{\widetilde{\beta }_k(x)}b_k(x,t)\Phi (u(t))w_2(u(t))dt,
\end{split} \label{1.4}
\end{equation}
where $c(x)$ is a function and all the functions which appear in this
inequality are assumed to be real valued of $n$ variables.

It is interesting to note that the results \eqref{1.1}-\eqref{1.3}
can be deduced from our inequality \eqref{1.4} in some special cases.
As applications we give the estimate solution of retarded partial
differential equation.

The main purpose of this article is to establish some nonlinear
retarded integral inequalities for functions of $n$ independent
variables which can  be used as handy tools in the theory of
partial differential and integral equations with time delays.
These new inequalities represent a generalization of the results
obtained by Ma and Pecaric \cite{pec}, Pachpatte \cite{pach1}
and by Cheung \cite{aa} in case of the functions
with one and two variables.
We note that the inequality \eqref{1.4} is also
a generalization of the main results in \cite{lp,sun}.

\section{Main results}

In this article, we denote
 $\mathbb{R}_{+}^n=[ 0,\infty) $ which is a
subset of $\mathbb{R}^n$. All the functions which appear in the
inequalities are assumed to be real valued of $n$-variables which are
nonnegative and continuous. All integrals are assumed to exist on their
domains of definitions.

For $x=(x_1,x_2,\dots, x_n)$,
$t=(t_1,t_2,\dots, t_n)$, $x^0=(x_1^0,x_2^0,\dots ,x_n^0)\in
\mathbb{R}_{+}^n$, we shall denote:
\begin{gather*}
\int_{\widetilde{\alpha }_i(x^0)}^{\widetilde{\alpha }_i(x)}dt
= \int_{\alpha _{j1}(x_1^0)}^{\alpha _{j1}(x_1)}\int_{\alpha
_{j2}(x_2^0)}^{\alpha _{j2}(x_2)}\dots \int_{\alpha
_{jn}(x_n^0)}^{\alpha _{jn}(x_n)}\dots dt_n\dots dt_1,\quad
j=1,2,\dots ,n_1, \\
\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }_k(x)}dt
= \int_{\beta _{k1}(x_1^0)}^{\beta _{k1}(x_1)}\int_{\beta
_{k2}(x_2^0)}^{\beta _{k2}(x_2)}\dots \int_{\beta
_{kn}(x_n^0)}^{\beta _{kn}(x_n)}\dots dt_n\dots dt_1,\quad
 k=1,2,\dots ,n_2,
\end{gather*}
with $n_1,n_2\in \{1,2,\dots ,\}$.
For $x,t\in \mathbb{R}_{+}^n$, we shall write
$t\leq x$ whenever $t_i\leq x_i$, $i=1,2,\dots ,n$ and
$x\geq x_0\geq 0$, for $x,x^0\in \mathbb{R}_{+}^n$.

We denote $D=D_1D_2\dots D_n$, where $D_i=\frac{\partial }{\partial
x_i}$, for $i=1,2,\dots ,n$,
We use the usual convention of writing
$\sum_{s\in\emptyset}u(s)=0$ if $\emptyset$ is the empty set.
\begin{gather*}
\widetilde{\alpha }_j(t)=\big( \alpha _{j1}(t_1),\alpha
_{j2}(t_2),\dots ,\alpha _{jn}(t_n)\big) \in \mathbb{R}_{+}^n\quad
\text{for} j=1,2,\dots ,n_1;\\
\widetilde{\beta }_k(t)=\big( \alpha _{k1}(t_1),\alpha
_{k2}(t_2),\dots ,\alpha _{kn}(t_n)\big) \in \mathbb{R}_{+}^n\quad
\text{for } k=1,2,\dots ,n_1.
\end{gather*}
We denote $\widetilde{\alpha }_j(t)\leq t$ for $j=1,2,\dots ,n_1$
whenever $\alpha _{ji}(t_i)\leq t_i$ for  $i=1,2,\dots ,n$ and
$j=1,2,\dots ,n_1$,
and $\widetilde{\beta }_k(t)\leq t$ for $k=1,2,\dots ,n_2$
whenever $\beta_{ki}(t_i)\leq t_i$ for $i=1,2,\dots ,n$ and
$k=1,2,\dots ,n_2$

Our main results read as the follows.

\begin{theorem}\label{th1}
Let $c\in C(\mathbb{R}_{+}^n,\mathbb{R}_{+})$,
$w_1,w_2\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ be nondecreasing
functions with $w_1(u),w_2(u)>0$ on $(0,\infty )$ and
let $a_j(x,t)$ and
$b_k(x,t)\in C(\mathbb{R}_{+}^n\times \mathbb{R}_{+}^n,
\mathbb{R}_{+})$
be nondecreasing functions in $x$ for every $t$ fixed for any
$j=1,2,\dots ,n_1$, $k=1,2,\dots ,n_2$. Let
$\alpha _{ji},\beta _{ki}\in C^{1}(\mathbb{R}_{+},\mathbb{R}_{+})$
be nondecreasing functions with $\alpha _{ji}(t_i)\leq t_i$ and
$\beta _{ki}(t_i)$ $\leq t_i$ on $\mathbb{ R}_{+}$ for
$i=1,2,\dots ,n$; $j=1,2,\dots ,n_1$, $k=1,2,\dots ,n_2$ and
$p>q\geq 0$.

\textbf{(A1)} If $u\in C(\mathbb{R}_{+}^n,\mathbb{R}_{+})$
and
\begin{equation} \label{2.1}
\begin{split}
u^{p}(x)
&\leq c(x)+\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)u^{q}(t)dt   \\
&\quad + \sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(x,t)u^{q}(t)w_1(u(t))dt,
\end{split}
\end{equation}
for any $x\in \mathbb{R}_{+}^n$ with $x^0\leq t\leq x$,
then there exists $x^{\ast }\in \mathbb{R}_{+}^n$, such as for
all $x^0\leq t\leq x^{\ast }$, we have
\begin{equation}
u(x)\leq \Big( \Psi _1^{-1}\Big[ \Psi _1(p(x))+\frac{p-q}{p}
\sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(x,t)dt\Big] \Big) ^{1/(p-q)}.  \label{2.2}
\end{equation}
Where
\begin{gather}
p(x)=c^{(p-q)/p}(x)+\frac{p-q}{p}\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)\,dt,  \label{2.3}
\\
\Psi _1(\delta )=\int_{\delta _0}^{\delta }
\frac{ds}{w_1(s^{\frac{1}{p-q}})}\,,\quad \delta >\delta _0>0.  \label{2.4}
\end{gather}
Here, $\Psi ^{-1}$ is the inverse function of $\Psi $, and the real
numbers $x^{\ast }$ are chosen so that
$\Psi _1(p(x))+\frac{p-q}{p}\sum_{k=1}^{n_2}
\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(x,t)dt\in\operatorname{dom}(\Psi _1^{-1})$.

\textbf{(A2)} If $u\in C(\mathbb{R}_{+}^n,\mathbb{R}_{+})$
and
\begin{equation} \label{2.5}
\begin{split}
u^{p}(x) &\leq c(x)+\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)u^{q}(t)w_1(u(t))dt
 \\
&\quad + \sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(x,t)u^{q}(t)w_2(u(t))dt.
\end{split}
\end{equation}
(i) In the case $w_2(u)\leq w_1(u)$, for any $x\in \mathbb{R}_{+}^n$
with $x^0\leq t\leq x$, there exists $\xi _1\in \mathbb{R}_{+}^n$,
such as for all $x^0\leq t\leq \xi _1$, we have
\[
u(x)\leq \Big( \Psi _1^{-1}\big( \Psi _1(c^{(p-q)/p}(x))+e(x)\big)
\Big) ^{1/(p-q)}.
\]
(ii) In the case $w_1(u)\leq w_2(u)$, for any $x\in \mathbb{R}_{+}^n$
with $x^0\leq t\leq x$, there exists $\xi _2\in \mathbb{R}_{+}^n$,
such as for all $x^0\leq t\leq \xi _2$, we have
\[
u(x)\leq \Big( \Psi _2^{-1}\big( \Psi _2(c^{(p-q)/p}(x))+e(x)\big)
\Big) ^{1/(p-q)},
\]
where
\begin{gather*}
e(x) = \frac{p-q}{p}\Big[ \sum_{j=1}^{n_1}\int_{\widetilde{\beta }
_k(x^0)}^{\widetilde{\beta }_k(x)}a_j(x,t)dt
+\sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}
^{\widetilde{\beta }_k(x)}b_k(x,t)dt\Big] , \\
\Psi _i(\delta )
= \int_{\delta _0}^{\delta }\frac{ds}{w_i(s^{\frac{1
}{p-q}})},\quad \delta >\delta _0>0,\text{ for } i=1,2.
\end{gather*}
Here, $\Psi _i^{-1}$ is the inverse function of $\Psi _i$  and the
real numbers $\xi _i$ are chosen so that
$\Psi_2(c^{(p-q)/p}(x))+e(x)\in\operatorname{dom}(\Psi _i^{-1})$ for $i=1,2$
 respectively.
\end{theorem}

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

\begin{corollary}\label{c1}
Let the functions $u,c,w_1,a_j,b_k$ $(j=1,2,\dots ,n_1;\,
 k=1,2,\dots ,n_1)$ and the constants $p,q$ be defined as in
Theorem \ref{th1} and
\begin{equation} \label{2.6}
\begin{split}
u^{p}(x,y)
&\leq c(x,y)+\sum_{j=1}^{n_1}\int_{\alpha _j(x_0)}
^{\alpha_j(x)}\int_{\alpha _j(y_0)}^{\alpha_j(y)}
a_j(x,y,s,t)u^{q}(s,t)\,ds\,dt   \\
&\quad + \sum_{k=1}^{n_2}\int_{\beta _k(x_0)}^{\beta _k(x)}
\int_{\beta _k(y_0)}^{\beta _k(y)}b_k(x,y,s,t)u^{q}(t)w_1(u(t))dt,
\end{split}
\end{equation}
for any $(x,y)\in \mathbb{R}_{+}^{2}$ with $x_0\leq s\leq x$ and
$y_0\leq t\leq y$, then there exists
$(x^{\ast },y^{\ast })\in \mathbb{R}_{+}^n$, such as for all
$x_0\leq s\leq x^{\ast }$ and $y_0\leq s\leq y^{\ast }$, then
\begin{equation}
u(x,y)\leq \Big(\Psi ^{-1}\big[ \Psi (p_1(x,y))+\frac{p-q}{p}B_1(x,y)
\big] \Big) ^{1/(p-q)},  \label{2.7}
\end{equation}
where
\begin{gather*}
p_1(x,y) = c^{(p-q)/p}(x,y)+\frac{p-q}{p}A_1(x,y), \\
A_1(x,y) = \sum_{j=1}^{n_1}\int_{\alpha _j(x_0)}^{\alpha
_j(x)}\int_{\alpha _j(y_0)}^{\alpha _j(y)}a_j(x,y,s,t)\,ds\,dt, \\
B_1(x,y) = \sum_{k=1}^{n_2}\int_{\beta _k(x_0)}^{\beta
_k(x)}\int_{\beta _k(y_0)}^{\beta _k(y)}b_k(x,y,s,t)\,ds\,dt,
\end{gather*}
and
\begin{equation}
\Psi (\delta )=\int_{\delta _0}^{\delta }\frac{ds}{w_1(s^{1/(p-q)})
},\quad \delta >\delta _0>0.
\end{equation}
Here, $\Psi ^{-1}$ is the inverse function of $\Psi $, and the real
numbers $(x^{\ast },y^{\ast })$ are chosen so that
$\Psi (p_1(x,y))+\frac{p-q}{p} B_1(x,y)\in\operatorname{dom}(\Psi ^{-1})$.
\end{corollary}

\begin{remark} \label{rmk1} \rm
Setting $a_j(x,y,s,t)=a_j(s,t)$,  $b_k(x,y,s,t)=b_k(s,t)$ and
$c(x,y)=k$ $\geq 0$ in  Corollary \ref{c1}, we obtain
Ma and Pecaric's result \cite[Theorem 2.1]{pec}.
\end{remark}

\begin{remark}
Defining $a_j(x,y,s,t)=\frac{p}{p-q}a_j(s,t),\ b_k(x,y,s,t)=\frac{p}{
p-q}b_k(s,t)\ $\ $c(x,y)=k>0$ (Constant) and $j=k=1$ in
Corollary \ref{c1}, we obtain Cheung's result
\cite[Theorem 2.4]{aa}.

Obviously, \eqref{1.1}--\eqref{1.3} are special cases of
Theorem \ref{th1}.
So our result includes the main results in \cite{pec,pach1,aa}.
\end{remark}

Using Theorem \ref{th1}, we can get some more generalized results
 as follow:

\begin{theorem} \label{th2}
Let the functions $u,c,w_i,a_j,b_k$
($i=1,2$, $j=1,2,\dots ,n_1$, $k=1,2,\dots ,n_1$)
be defined as in Theorem \ref{th1}.
Moreover, let $\varphi \in C(\mathbb{R}_{+},\mathbb{R}_{+})$
be a strictly increasing function so that
$\lim_{x\to \infty }\varphi (x)=\infty $,
and let $\Phi \in C(\mathbb{R}_{+},\mathbb{R}_{+})$ be nondecreasing
function with $\Phi (x)>0$ for all $x\in \mathbb{R}_{+}^n$.

\textbf{(B1)} If $u\in C(\mathbb{R}_{+}^n,\mathbb{R}_{+})$
and
\begin{equation} \label{2.8}
\begin{split}
\varphi (u(x))
&\leq c(x)+\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }_j(x^0)}
 ^{\widetilde{\alpha }_j(x)}a_j(x,t)\Phi (u(t))dt   \\
&\quad + \sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}
 ^{\widetilde{\beta }_k(x)}b_k(x,t)\Phi (u(t))w_1(u(t))dt,
\end{split}
\end{equation}
for any $x\in \mathbb{R}_{+}^n$ with $x^0\leq t\leq x$, then there
exists $x^{\ast }\in \mathbb{R}_{+}^n$, so that for all
$x^0\leq t\leq x^{\ast }$, we have
\begin{equation}
u(x)\leq \varphi ^{-1}\big( G^{-1}[ \Psi _1^{-1}\left( \Psi _1(\pi
(x))+B(x)\right)] \big) ,  \label{2.9}
\end{equation}
where
\begin{gather}
\pi (x) = G(c(x))+A(x),  \label{2.10} \\
A(x) = \sum_{j=1}^{n_1}\int_{\widetilde{\alpha }_j(x^0)}^{\widetilde{
\alpha }_j(x)}a_j(x,t)dt,  \label{2.11} \\
B(x) = \sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{
\beta }_k(x)}b_k(x,t)dt,  \label{2.12}\\
G(x) = \int_{x_0}^{x}\frac{ds}{\Phi (\varphi ^{-1}(s))},\quad
x>x_0>0,  \label{2.13} \\
\Psi _i(\delta ) = \int_{\delta _0}^{\delta }\frac{ds}{w_i(\varphi
^{-1}(G^{-1}(s)))},\quad  \delta >\delta _0>0,\; i=1,2.
\label{2.14}
\end{gather}
The real number $x^{\ast }$ is chosen so that
$\Psi _1(\pi (x))+B(x)\in \operatorname{dom}(\Psi _1^{-1})$.

\textbf{(B2)} If $u\in C(\mathbb{R}_{+}^n,\mathbb{R}_{+})$
and
\begin{align*}
\varphi (u(x))
&\leq  c(x)+\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)\Phi (u(t))w_1(u(t))dt\\
&\quad + \sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}
 ^{\widetilde{\beta }_k(x)}b_k(x,t)\Phi (u(t))w_2(u(t))dt.
\end{align*}
(i) When $w_2(u)\leq w_1(u)$, for any $x\in \mathbb{R}
_{+}^n$ with $x^0\leq t\leq x$,  there exists $\xi _1\in \mathbb{R}
_{+}^n$, so that for all $x^0\leq t\leq \xi _1$, we have
\begin{equation*}
u(x)\leq \varphi ^{-1}\big( G^{-1}[ \Psi _1^{-1}\big( \Psi
_1(G(c(x)))+A(x)+B(x)\big) ] \big) .
\end{equation*}
(ii) When $w_1(u)\leq w_2(u)$, for any $x\in \mathbb{R}_{+}^n$
with $x^0\leq t\leq x$, there exists $\xi _2\in \mathbb{R}_{+}^n$,
so that for all $x^0\leq t\leq \xi _2$, we have
\begin{equation*}
u(x)\leq \varphi ^{-1}\big( G^{-1}[ \Psi _2^{-1}\big( \Psi
_2(G(c(x)))+A(x)+B(x)\big) ] \big) .
\end{equation*}
Where $A,B,G$ and $\Psi _i(i=1,2)$ are defined in
\eqref{2.11}-\eqref{2.14}, $\Psi _i^{-1}$ is the inverse
function of $\Psi _i$  and the real
numbers $\xi _i$ are chosen so that
$\Psi _i(G(c(x)))+A(x)+B(x)\in \operatorname{dom}(\Psi _i^{-1})$
for $i=1,2$ respectively.
\end{theorem}

Many interesting corollaries can also be obtained from the above
theorems (in the case of one or $n$ independent variables).

\begin{corollary}[Inequality in one variable]\label{c2}
Let $p>q\geq 0$, $c>0$ be  constant and $w_1,w_2$ be defined as
in Theorem \ref{th1}. Moreover, let $a_j(x,t)$ and
$b_k(x,t)\in C(\mathbb{R}_{+}\times \mathbb{R}_{+},\mathbb{R}_{+})$
be nondecreasing functions in $x$ for every $t$ fixed and
$\alpha _j,\beta _k\in C^{1}(\mathbb{R}_{+},\mathbb{R}_{+})$
be nondecreasing functions with $\alpha_j(t)\leq t$ and
$\beta _k(t)\leq t_i$ on $\mathbb{R}_{+}$ for
$j=1,2,\dots ,n_1$, $k=1,2,\dots ,n_2$ for any $j=1,2,\dots ,n_1$,
$k=1,2,\dots ,n_2$.

\textbf{(C1)} Let $u\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ and
\begin{align*}
u(x)^{p}
&\leq c^{p/(p-q)}+\frac{p}{p-q}\sum_{j=1}^{n_1}\int_0^{
\alpha _j(x)}a_j(x,t)u(t)^{q}dt \\
&\quad + \frac{p}{p-q}\sum_{k=1}^{n_2}\int_0^{\beta
_k(x)}b_k(x,t)u(t)^{q}w_1(u(t))dt,
\end{align*}
for any $x\in \mathbb{R}_{+}$ with $0\leq t\leq x$. Then there exists
$(x^{\ast })\in \mathbb{R}_{+}$, so that for all
$0\leq t\leq x^{\ast }$, we have
\begin{equation}
u(x)\leq \big( [ \Psi _1^{-1}\big( \Psi _1(\pi (x))+B(x)\big)
] \big) ^{1/(p-q)}.
\end{equation}
Where $\pi (x)=c+A(x)$ and
\begin{gather}
A(x) = \sum_{j=1}^{n_1}\int_0^{\alpha _j(x)}a_j(x,t)dt,
\label{2.15} \\
B(x) = \sum_{k=1}^{n_2}\int_0^{\beta _k(x)}b_k(x,t)dt,  \label{2.16}
\\
\Psi _i(\delta ) = \int_{\delta _0}^{\delta }\frac{ds}{w_i( s^{
\frac{1}{p-q}}) }\quad  \delta >\delta _0>0,\; i=1,2.
\label{2.17}
\end{gather}
Where the real number $x^{\ast }$ is chosen so that
$\Psi _1(\pi (x))+B(x)\in\operatorname{dom}(\Psi _1^{-1})$.

\textbf{(C2)} If $u\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ and
\begin{align*}
u(x)^{p} &\leq c^{p/(p-q)}+\frac{p}{p-q}\sum_{j=1}^{n_1}\int_0^{
\alpha _j(x)}a_j(x,t)u(t)^{q}w_1(u(t))dt   \\
&\quad + \frac{p}{p-q}\sum_{k=1}^{n_2}\int_0^{\beta
_k(x)}b_k(x,t)u(t)^{q}w_2(u(t))dt.
\end{align*}
(i) In the case $w_2(u)\leq w_1(u)$, for any $x,t\in \mathbb{R}_{+}$
with $0\leq t\leq x$, we have
\begin{equation*}
u(x)\leq u(x)\leq \big( [ \Psi _1^{-1}\left( \Psi
_1(c)+A(x)+B(x)\right) ] \big) ^{1/(p-q)}.
\end{equation*}
(ii) In the case $w_1(u)\leq w_2(u)$, for any $x,t\in \mathbb{R}_{+}$
with $0\leq t\leq x$, we have
\begin{equation*}
u(x)\leq u(x)\leq \big( [ \Psi _2^{-1}\left( \Psi
_2(c)+A(x)+B(x)\right) ] \big) ^{1/(p-q)}.
\end{equation*}
Where $\Psi _i, A, B$ ($i=1,2$) are defined in
\eqref{2.15}-\eqref{2.17}.
\end{corollary}

\begin{remark} \rm
(i) Corollary \ref{c2} (C1) reduces to  Sun's inequality
\cite[Theorem 2.1]{sun} in case of one variable ($n=1)$ when
$a_j(x,t)=a_j(t)$, $b_k(x,t)=b_k(t)$, $\beta _k(x)=\alpha _j(x)$
and $j=k=1$.

(ii) Corollary \ref{c2} (C2) reduces to  Sun's inequality
\cite[Theorem 2.2]{sun} in case of one variable ($n=1)$ when
$a_j(x,t)=a_j(t)$,$b_k(x,t)=b_k(t)$, $\beta _k(x)=x$ and
$j=k=1$ and $w_1=w_2$.
\end{remark}

\begin{remark} \rm
Under some suitable conditions in (B1), the inequality \eqref{2.8}
gives a new estimate for the inequality \eqref{2.1} in
(A1).
\end{remark}

\begin{theorem}\label{th3}
Let the functions $u,c,,\varphi ,\Phi ,w_i,a_j, b_k$
($i=1,2$, $j=1,2,\dots ,n_1$, $k=1,2,\dots ,n_1)$ be defined
as in Theorem \ref{th2} and If
\begin{align*}
\varphi (u(x))
&\leq c(x)+\sum_{j=1}^{n_1}d_j(x)\int_{\widetilde{\alpha
}_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)\Phi (u(t))w_1(u(t))dt
\\
&\quad + \sum_{k=1}^{n_2}l_k(x)\int_{\widetilde{\beta }_k(x^0)}^{
\widetilde{\beta }_k(x)}b_k(x,t)\Phi (u(t))w_2(u(t))dt,
\end{align*}
for any $x\in \mathbb{R}_{+}^n$, we have
\begin{equation*}
u(x)\leq \varphi ^{-1}\big( G^{-1}[ \Psi ^{-1}\left( \Psi (G(c(x)))+
\widetilde{A}(x)+\widetilde{B}(x)\right) ] \big) ,
\end{equation*}
where
\begin{gather*}
\widetilde{A}(x) = \sum_{j=1}^{n_1}d_j(x)\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(x,t)dt, \\
\widetilde{B}(x) = \sum_{k=1}^{n_2}l_k(x)\int_{\widetilde{\beta }
_k(x^0)}^{\widetilde{\beta }_k(x)}b_k(x,t)dt.
\end{gather*}
\end{theorem}

\begin{corollary}\label{c3}
If
\begin{equation*}
u^{p}(x)\leq c(x)+\int_0^{\widetilde{\alpha }
(x)}a(t)u^{q}(t)+b(t)u^{p}(t)dt
\end{equation*}
for any $x\in \mathbb{R}_{+}^n$ with $x^0\leq t\leq x$, then there
exists $x^{\ast }\in \mathbb{R}_{+}^n$, so that for all
$x^0\leq t\leq x^{\ast }$, we have
\begin{equation*}
u(x)\leq \frac{p}{p-q}c^{\frac{p-q}{p}}(x)\exp
\Big[ \frac{p}{p-q}\int_0^{\widetilde{\alpha }(x)}a(t)+b(t)dt\Big]
\end{equation*}
\end{corollary}

\begin{remark} \rm
(i)  Theorem \ref{th3} reduced to \cite[Theorem 2.2]{lp} in the
case of one variable, when $\varphi (x)=x$, $b_k(x,t)=0,w_1(t)=1$,
$j=1$ and $n=1$

(ii) Theorem \ref{th3} is also a generalization of the main result
in Lipovan \cite[Theorem 2.1]{lp} in case of one variable,
when $\varphi (x)=x$, $b_k(x,t)=0$, $w_1(t)=1$, $\Phi (t)=1$,
for any $x,t\in \mathbb{R}_{+}(n=1)$ and  for $j=1$.
\end{remark}

\begin{remark} \rm
(i) Under some suitable conditions, Theorem \ref{th3} reduced
to Theorem 2.3 and Theorem 2-4 in case of two variables of the
 main results in Zhang and Meng  \cite{meng}.

(ii) Under some suitable conditions in Theorem \ref{th3}, we can also
obtain other estimations of the Ma and Pecaric's inequality
\eqref{1.1} and the main results in \cite{pec}.
\end{remark}

\begin{remark} \rm
Theorem \ref{th3} further reduces to the main results in
\cite[Theorem 2.1, 2.2, 2.4]{aa} and the results in \cite{pach2}.
\end{remark}

\section{Proof of theorems}

Since the proofs resemble each other, we give the details for
(A1) in Theorem \ref{th3} only; the proofs of the remaining
inequalities can be completed by following the proofs of
the above-mentioned inequalities.

\begin{proof}[Proof of Theorem \ref{th1} (A1)]
Fixing arbitrary numbers $y=(y_1,\dots ,y_n)\in \mathbb{R}_{+}^n$
with $x^0<y\leq x^{\ast }$, we define on $[x^0;y]$ a function $z(x)$
by
\begin{equation}
z(x) = c(y)+\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }_j(x^0)}^{
\widetilde{\alpha }_j(x)}a_j(y,t)u^{q}(t)dt
 + \sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(y,t)u^{q}(t)w_1(u(t))dt.  \label{3.1}
\end{equation}
Then $z(x)$ is a positive and nondecreasing function with
 $z(x^0)=c(y)$, and
\begin{equation}
u(x)\leq z(x)^{1/p},\quad  x\in [x^0;y].   \label{3.2}
\end{equation}
We know that
\begin{equation} \label{3.3}
\begin{split}
D_1D_2\dots D_nz(x)
&= \sum_{j=1}^{n_1}a_j(y,\widetilde{\alpha }
 _j(x))u^{q}(\widetilde{\alpha }_j(x))\alpha _{j1}'\alpha
 _{j2}'\dots \alpha _{jn}'   \\
&\quad + \sum_{k=1}^{n_2}b_j(y,\widetilde{\beta }_j(x))u^{q}(\widetilde{
 \beta }_j(x))w_1(u(\widetilde{\beta }_j(x)))\beta _{k1}'\beta _{k2}'\dots \beta _{kn}'   \\
&\leq z^{q/p}(x)\Big[ \sum_{j=1}^{n_1}a_j(y,\widetilde{\alpha }
_j(x))\alpha _{j1}'(x_1)\alpha _{j2}'(x_2)\dots \alpha _{jn}'(x_n)
   \\
&\quad + \sum_{k=1}^{n_2}b_j(y,\widetilde{\beta }_j(x))w_1(z^{1/p}(
\widetilde{\beta }_j(x)))\beta _{k1}'\beta _{k2}'\dots \beta _{kn}'
\Big] .
\end{split}
\end{equation}
Using the above inequality, we have
\begin{equation}   \label{3.4}
\begin{split}
\frac{D_1D_2\dots D_nz(x)}{z^{q/p}(x)}
&\leq \Big[\sum_{j=1}^{n_1}a_j(y,\widetilde{\alpha }_j(x))
 \alpha _{j1}'(x_1)\alpha _{j2}'(x_2)\dots \alpha _{jn}'(x_n)
 \\
&\quad + \sum_{k=1}^{n_2}b_j(y,\widetilde{\beta }_j(x))w_1(z^{1/p}(
\widetilde{\beta }_j(x)))\beta _{k1}'\beta _{k2}'\dots \beta _{kn}'
\Big] .
\end{split}
\end{equation}
Using $D_1D_2\cdots D_{n-1}z(x)\geq 0$,
$\frac{q}{p}z^{(q-p)/p}(x)\geq 0$,
$D_n(x)\geq 0$ and\eqref{3.4}, we have
\begin{equation} \label{3.5'}
\begin{split}
&D_n\Big( \frac{D_1D_2\dots D_{n-1}z(x)}{z^{q/p}(x)}\Big)\\
&\leq \frac{ D_1D_2\cdots D_nz(x)}{z^{q/p}(x)}   \\
&\leq \sum_{j=1}^{n_1}a_j(y,\widetilde{\alpha }_j(x))\alpha
_{j1}'(x_1)\alpha _{j2}'(x_2)\dots \alpha
_{jn}'(x_n)   \\
&\quad + \sum_{k=1}^{n_2}b_k(y,\widetilde{\beta }_k(x))w_1(z^{1/p}(
\widetilde{\beta }_k(x)))\beta _{k1}'\beta _{k2}'\cdots \beta _{kn}'.
\end{split}
\end{equation}
Fixing $x_1,x_2,\dots ,x_{n-1}$,
setting $x_n=t_n$ and integrating \eqref{3.5'}
from $x_n^0$ to $x_n$, we obtain
\begin{align*}
&\frac{D_1D_2\cdots D_{n-1}z(x)}{z^{q/p}(x)} \\
&\leq \sum_{j=1}^{n_1}\int_{\alpha _{jn}(x_n^0)}^{\alpha
_{jn}(x_n)}a_j\big(y,\alpha _{j1}(x_1),\alpha _{j2}(x_2),\cdots ,\alpha
_{jn-1}(x_{n-1}),\alpha _{jn}(t_n)\big)\\
&\quad\times \alpha _{j1}'\alpha_{j2}'\cdots \alpha _{jn-1}'dt_n \\
&\quad + \sum_{k=1}^{n_2}\int_{\beta _{kn}(x_n^0)}^{\beta
_{kn}(x_n)}b_k(y,\beta _{k1}(x_1),\beta _{k2}(x_2),\cdots ,\beta
_{kn-1}(x_{n-1}),t_n)\\
&\quad\times w_1(z^{1/p}(\beta _{k1},\beta _{k2},\cdots ,\beta
_{kn-1},t_n)) \beta _{k1}'(x_1)\beta _{k2}'(x_2)\cdots \beta
_{kn-1}'(x_{n-1})\,dt_n.
\end{align*}
Using the same method, we obtain
\begin{equation} \label{3.6}
\begin{split}
&\frac{D_1z(x)}{z^{q/p}(x)}   \\
&\leq \sum_{j=1}^{n_1}\Big[ \int_{\alpha _{jn}(x_n^0)}^{\alpha
_{jn}(x_n)}\dots \int_{\alpha _{jn}(x_n^0)}^{\alpha
_{jn}(x_n)}a_j(y,\alpha _{j1}(x_1),t_2,\dots ,t_n)\alpha
_{j1}'(x_1)dt_n\dots dt_2\Big]   \\
&\quad + \sum_{k=1}^{n_2}\quad[ \int_{\beta _{jn}(x_n^0)}^{\beta
_{jn}(x_n)}\dots \int_{\beta _{jn}(x_n^0)}^{\beta
_{jn}(x_n)}b_k(y,\beta _{k1}(x_1),t_2,\dots ,t_n)   \\
&\quad\times  w_1(z^{1/p}(\beta _{k1}(x_1),t_2,\dots ,t_n))\beta
_{k1}'(x_1)dt_n\dots dt_2\Big].
\end{split}
\end{equation}
Integrating \eqref{3.6} form $x_1^0$ to $x_1$, we obtain
\begin{align*}
\frac{p}{p-q}z^{(p-q)/p}(x)
&\leq \frac{p}{p-q}c^{(p-q)/p}(y)+
\sum_{j=1}^{n_1}\int_{\widetilde{\alpha }_j(x^0)}^{\widetilde{\alpha }
_j(y)}a_j(y,t)dt \\
&\quad + \sum_{k=1}^{n_2}
\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(y,t)w_1(z^{1/p}(t))dt,
\end{align*}
for all $x\in [x^0;y]$, which implies that
\begin{equation} \label{3.7}
\begin{split}
z^{(p-q)/p}(x)
&\leq  c^{(p-q)/p}(y)+\frac{p-q}{p}\sum_{j=1}^{n_1}
 \int_{\widetilde{\alpha }_j(x^0)}^{\widetilde{\alpha }_j(y)}
 a_j(y,t)dt \\
&\quad + \frac{p-q}{p}\sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{
\widetilde{\beta }_k(x)}b_k(y,t)w_1(z^{1/p}(t))dt.
\end{split}
\end{equation}
Setting $r_1(x)=z^{(p-q)/p}(x)$, \eqref{3.7} can be rewritten as
\begin{equation*}
r_1(x)\leq p(y)+\frac{p-q}{p}\sum_{k=1}^{n_2}\int_{\widetilde{\beta }
_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(y,t)w_1(r_1^{1/(p-q)}(t))dt.
\end{equation*}
Defining $v(x)$ on $[x^0;y]$, by
\begin{equation}
v(x)=p(y)+\frac{p-q}{p}\sum_{k=1}^{n_2}\int_{\widetilde{\beta }
_k(x^0)}^{\widetilde{\beta }
_k(x)}b_k(y,t)w_1(r_1^{1/(p-q)}(t))dt,  \label{3.8}
\end{equation}
by \eqref{3.8}, we have $v(x^0)=p(y)$ and
\begin{equation}
z^{(p-q)/p}(x)\leq v(x),  \label{3.8'}
\end{equation}
and
\begin{align*}
D_1D_2\cdots D_nv(x)
&= \frac{p-q}{p}\sum_{k=1}^{n_2}b_k(y,\widetilde{
\beta }_k(x))w_1(r_1^{1/(p-q)}(\widetilde{\beta }_k(x)))\beta
_{k1}'\beta _{k2}'\dots \beta _{kn}' \\
&\leq \frac{p-q}{p}\sum_{k=1}^{n_2}b_k(y,\widetilde{\beta }
_k(x))w_1(v^{1/(p-q)}(\widetilde{\beta }_k(x)))\beta _{k1}'\beta _{k2}'\dots \beta _{kn}'.
\end{align*}
Using the same method as above, we obtain
%\label{3.9}
\begin{align*}
&\frac{D_1v(x)}{w_1(v(x)^{1/p-q})}   \\
&\leq \frac{p-q}{p}\sum_{k=1}^{n_2}\Big[ \int_{\beta
_{jn}(x_n^0)}^{\beta _{jn}(x_n)}\dots \int_{\beta
_{jn}(x_n^0)}^{\beta _{jn}(x_n)}b_k(y,\beta
_{k1}(x_1),t_2,\dots ,t_n)\beta _{k1}'(x_1)dt_n\dots dt_2
\Big] .
\end{align*}
Integrating form $x_1^0$ to $x_1$, we obtain
\begin{equation}
\Psi _1(v(x))\leq \Psi _1(p(y))+\frac{p-q}{p}\sum_{k=1}^{n_2}\int_{
\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }_k(x)}b_k(y,t)dt,
\label{3.10}
\end{equation}
from \eqref{3.10}  and for any arbitrary  $y$, we obtain
\begin{equation}
v(y)\leq \Psi _1^{-1}\Big[ \Psi _1(p(y))+\frac{p-q}{p}
\sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(y)}b_k(y,t)dt\Big] .  \label{3.11}
\end{equation}
 From \eqref{3.11} and \eqref{3.8'},
\begin{equation}
z(y)\leq \Big( \Psi _1^{-1}\Big[ \Psi _1(p(y))+\frac{p-q}{p}
\sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(y)}b_k(y,t)dt\Big] \Big) ^{p/(p-q)}.  \label{3.12}
\end{equation}
By \eqref{3.12} and \eqref{3.2},
\begin{equation*}
u(y)\leq \Big( \Psi _1^{-1}\Big[ \Psi _1(p(y))+\frac{p-q}{p}
\sum_{k=1}^{n_2}\int_{\widetilde{\beta }_k(x^0)}^{\widetilde{\beta }
_k(y)}b_k(y,t)dt\Big] \Big) ^{1/(p-q)}.
\end{equation*}
Since $y\leq x^{\ast }$ is arbitrary, the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th3}]
Fixing arbitrary numbers
$\tau =(\tau _1,\dots ,\tau _n)\in \mathbb{R}_{+}^n$ with
$x^0<\tau \leq \xi$, we define on $[x^0;\tau ]$ a function $z(x)$ by
\begin{align*}
z(x) &= c(\tau )+\sum_{j=1}^{n_1}d_j(\tau )\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(\tau ,t)\Phi
(u(t))w_1(u(t))dt \\
&\quad + \sum_{k=1}^{n_2}l_k(\tau )\int_{\widetilde{\beta }_k(x^0)}^{
\widetilde{\beta }_k(x)}b_k(\tau ,t)\Phi (u(t))w_2(u(t))dt.
\end{align*}
Then $z(x)$ is a positive and nondecreasing function with
$z(x^0)=c(\tau )$, and
\begin{equation*}
u(x)\leq \varphi ^{-1}(z(x));\quad x\in [x^0;\tau ].
\end{equation*}
We know that
\begin{align*}
&D_1D_2\cdots D_nz(x)\\
&= \sum_{j=1}^{n_1}d_j(\tau )a_j(\tau ,
\widetilde{\alpha }_j(x))\Phi (u(\widetilde{\alpha }_j(x)))w_1(u(
\widetilde{\alpha }_j(x)))\alpha _{j1}'\alpha _{j2}'\dots \alpha _{jn}' \\
&\quad + \sum_{k=1}^{n_2}l_k(\tau )b_k(\tau ,\widetilde{\beta }_k(x))\Phi
(u(\widetilde{\beta }_k(x)))w_2(u(\widetilde{\beta }_k(x)))\beta
_{k1}'\beta _{k2}'\dots \beta _{kn}', \\
&\leq \Phi (\varphi ^{-1}(z(x))\Big[ \sum_{j=1}^{n_1}d_j(\tau
)a_j(\tau ,\widetilde{\alpha }_j(x))w_1(\varphi ^{-1}(z(\widetilde{
\alpha }_j(x)))\alpha _{j1}'\alpha _{j2}'\dots \alpha
_{jn}' \\
&\quad +  \sum_{k=1}^{n_2}l_k(\tau )b_k(\tau ,\widetilde{\beta }
_k(x))w_2(\varphi ^{-1}(z(\widetilde{\beta }_k(x)))\beta
_{k1}'\beta _{k2}'\dots \beta _{kn}'\Big].
\end{align*}
Using the same method in proof of the Theorem \ref{th1}, and for
all $x\in [x^0;\tau ]$, which implies that
\begin{align*}
z(x) &\leq G^{-1}\Big[ G(c(\tau ))+\sum_{j=1}^{n_1}d_j(\tau )\int_{
\widetilde{\alpha }_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(\tau
,t)w_1(u(t))dt  \\
&\quad +\sum_{k=1}^{n_2}l_k(\tau )\int_{\widetilde{\beta }
_k(x^0)}^{\widetilde{\beta }_k(x)}b_k(\tau ,t)w_2(u(t))dt\Big] .
\end{align*}
Defining $v(x)$ on $[x^0;\tau ]$ by
\begin{align*}
v(x) &= G(c(\tau ))+\sum_{j=1}^{n_1}d_j(\tau )\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(x)}a_j(\tau ,t)w_1(u(t))dt\\
 &\quad + \sum_{k=1}^{n_2}l_k(\tau )\int_{\widetilde{\beta }_k(x^0)}^{
\widetilde{\beta }_k(x)}b_k(\tau ,t)w_2(u(t))dt.
\end{align*}
We have $v(x^0)=G(c(\tau ))$,
$z(x)\leq G^{-1}(v(x))$, and
\begin{equation}
u(x)\leq \varphi ^{-1}(G^{-1}(v(x))).  \label{3.13}
\end{equation}
Then we obtain
\begin{align*}
\frac{D_1D_2\cdots D_nv(x)}{w_1(\varphi ^{-1}(G^{-1}(v(x))))}
&\leq
\Big[ \sum_{j=1}^{n_1}d_j(\tau )a_j(\tau ,\widetilde{\alpha }
_j(x))\alpha _{j1}'(x_1)\alpha _{j2}'(x_2)\dots \alpha _{jn}'(x_n) \\
&\quad +\sum_{k=1}^{n_2}l_k(\tau )b_k(\tau ,\widetilde{\beta }
_k(x))\beta _{k1}'(x_1)\beta _{k2}'(x_2)\dots \beta _{kn}'(x_n)\Big].
\end{align*}
Using the same method as above, we obtain
\[
\Psi _1(v(x))
\leq \Psi _1(G(c(\tau )))+\sum_{j=1}^{n_1}d_j(\tau)
\int_{\widetilde{\alpha }_j(x^0)}^{\widetilde{\alpha }
_j(x)}a_j(\tau ,t)dt \\
 + \sum_{k=1}^{n_2}l_k(\tau )\int_{\widetilde{\beta }_k(x^0)}^{
\widetilde{\beta }_k(x)}b_k(\tau ,t)dt.
\]
From which we have
\begin{equation}
\begin{split}
v(\tau ) &\leq \Psi _1^{-1}\Big[ \Psi _1(G(c(\tau
)))+\sum_{j=1}^{n_1}d_j(\tau )\int_{\widetilde{\alpha }_j(x^0)}^{
\widetilde{\alpha }_j(\tau )}a_j(\tau ,t)dt  \\
&\quad +\sum_{k=1}^{n_2}l_k(\tau )\int_{\widetilde{\beta }
_k(x^0)}^{\widetilde{\beta }_k(\tau )}b_k(\tau ,t)dt\Big] ,
\end{split}\label{3.14}
\end{equation}
for any arbitrary numbers $\tau \in \mathbb{R}_{+}^n$, with
$x^0<\tau \leq \xi $.
From \eqref{3.13} and \eqref{3.14}, we obtain
\begin{align*}
u(\tau )
&\leq \varphi ^{-1}\Big\{ G^{-1}\Big( \Psi _1^{-1}\Big[ \Psi
_1(G(c(\tau )))+\sum_{j=1}^{n_1}d_j(\tau )\int_{\widetilde{\alpha }
_j(x^0)}^{\widetilde{\alpha }_j(\tau )}a_j(\tau ,t)dt \\
&\quad +\sum_{k=1}^{n_2}l_k(\tau )\int_{\widetilde{\beta
}_k(x^0)}^{\widetilde{\beta }_k(\tau )}b_k(\tau ,t)dt\Big] \Big)
\Big\} .
\end{align*}
Since $\tau $ is arbitrary and $\tau \leq \xi $, we obtain
the result in the Theorem \ref{th3}.
\end{proof}

\section{An application}

 In this section we present an immediate application of our
results (Theorem \ref{th1} and Corollary \ref{c3}) to study
boundedness of  solutions of delay partial differential equations.
First we consider the nonlinear partial delay differential
equation in $\mathbb{R}^n$:
\begin{equation}
\begin{gathered}
Du^{p}(x)=h(x,u(x),u(x-\widetilde{\alpha }(x)), \\
u^{p}(0,x_2,x_{3},\dots ,x_n)=c_1(x_1), \\
u^{p}(0,x_2,x_{3},\dots x_{n-1},x_n)=c_n(x_n) \\
u^{p}(\dots ,x_{i-1},0,x_{i+1},\dots )=c_i(x_i)
\text{ for }i=2,3,\dots ,n-1,
\\
c_i(0)=0\quad \text{for }i=1,2,\dots ,n.
\end{gathered}  \label{4.1}
\end{equation}
For $x=(x_1;x_2,\dots ,x_n)\in \mathbb{R}_{+}^n$ and
$\widetilde{\alpha }(x)=\big( \alpha _1(x_1),\alpha _2(x_2),\dots ,
\alpha _n(x_n)\big) \in \mathbb{R}_{+}^n$ for
$\alpha _i,c_i\in C^{1}(\mathbb{R}_{+},\mathbb{R}_{+})$ for
$i=1,2,\dots ,n$. Where
$h:\mathbb{R}_{+}^n\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$,
is a continuous function.
Assume that these functions are defined and continuous on their
respective domains of definition such that
\begin{gather}
\widetilde{\alpha }(x)\leq x,\quad \text{for all }
 x=(x_1;x_2,\dots ,x_n)\in \mathbb{R}_{+}^n,  \label{4.2} \\
| h(x,u,v)| \leq a(x)| v(x)|
^{q}+b(x)| v(x)| ^{p},  \label{4.3}
\end{gather}
for $x\in \mathbb{R}_{+}^n$, where $p>q\geq 0$ is a constants and
$ a(x),b(x)$ are nonnegative, continuous functions defined
for $x\in \mathbb{R} _{+}^n$.
For any solution $u(x)$ of the boundary value problem \eqref{4.1},
\begin{equation}
u^{p}(x)=\sum_{i=1}^nc_i(x_i)+\int_0^{x}h(t,u(t),u(t-\widetilde{
\alpha }(t))dt,  \label{4.4}
\end{equation}
For all $x,t\in \mathbb{R}_{+}^n$ with $0\leq t\leq x$.
Using \eqref{4.1}, \eqref{4.3}  and a suitable change of
variables in \eqref{4.4}, we have
\begin{equation}
\left| u^{p}(x)\right| \leq c(x)+\int_0^{\widetilde{\alpha }(x)}
\widetilde{a}(t)\left| u(t)\right| ^{q}+\widetilde{b}(t)\left|
u(t)\right| ^{p}dt,  \label{4.5}
\end{equation}
with $c(x)=\sum_{i=1}^n| c_i(x_i)|$,
$\widetilde{a},\widetilde{b}\in C^{1}(\mathbb{R}_{+}^n,\mathbb{R}_{+})$.

(E1) Applying (A1) in Theorem \ref{th1} to \eqref{4.5}, when
$\widetilde{\alpha }_j=\widetilde{\beta }_k$,
$a_j(x,t)=\widetilde{a}(t)$, $b_k(x,t)=\widetilde{b}(t)$ with
$j=k=1$ and $w_1(u)=u^{p-q}$, we obtain a bound for the
solution $u(x)$:
\begin{equation}
u(x)\leq \Big( c^{(p-q)/p}(x)+\frac{p-q}{p}
\int_0^{\widetilde{\alpha }(x)}
\widetilde{a}(t)dt\Big) ^{1/(p-q)}
\exp \Big( \frac{1}{p}\int_0^{
\widetilde{\alpha }(x)}\widetilde{b}(t)dt\Big).  \label{4.6}
\end{equation}

(E2) Or by a direct application of Corollary \ref{c3}
to \eqref{4.5},
\begin{equation}
u(x)\leq \frac{p}{p-q}c^{\frac{p-q}{p}}\exp
\Big[ \frac{p}{p-q}\int_0^{
\widetilde{\alpha }(x)}[ \widetilde{a}(t)+\widetilde{b}(t)] dt
\Big] .  \label{4.7}
\end{equation}

\begin{remark} \rm
In the special case ($p=2$ and $q=1$) in the boundary value
problem \eqref{4.1}, we  have

(i) By \eqref{4.6}, we obtain
\begin{equation*}
u(x)\leq \Big( \sqrt{c(x)}+\frac{1}{2}\int_0^{\widetilde{\alpha }(x)}
\widetilde{a}(t)dt\Big) \exp \Big( \frac{1}{2}\int_0^{\widetilde{\alpha
}(x)}\widetilde{b}(t)dt\Big) .
\end{equation*}
(ii) Or by using \eqref{4.7},
\begin{equation*}
u(x)\leq 2\sqrt{c(x)}\exp \Big[ 2\int_0^{\widetilde{\alpha }(x)}[
\widetilde{a}(t)+\widetilde{b}(t)] dt\Big] .
\end{equation*}
\end{remark}

\begin{remark}
Note that the results given here can be very easily generalized
to obtain explicit bounds on integral inequalities involving several
retarded arguments.

Using similar method of those in the proof of the Theorems above,
we can also obtain a new reversed inequalities of our results.
\end{remark}

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\end{document}