\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 123, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2003 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE--2003/123\hfil Integral inequalities in several variables]
{On integral inequalities for functions of several independent variables} 

\author[Hassane Khellaf\hfil EJDE--2003/123\hfilneg]
{Hassane Khellaf}  

\address{University of Badji Mokhtar, 
Faculty of Science,
Department of Mathematics, 
B. P. 12, Annaba 23000, Algeria}
\email{khellafhassane@yahoo.fr} 

\date{}
\thanks{Submitted September 15, 2003. Published December 16, 2003.}
\subjclass[2000]{45H40, 45K05}
\keywords{Integral inequality, subadditive and submultiplicative function}

\begin{abstract}
  This paper presents some non-linear integral inequalities for 
  functions of $n$ independent variables. These results extend 
  the Gronwall type inequalities obtained for two variables by 
  Dragomir and Kim \cite{2}
\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}

Integral inequalities play a significant role in the study of differential
and integral equations. One of the most useful inequalities of Gronwall type
is given in the following lemma (see \cite{1,2}).

\begin{lemma}
\label{l1} Let $u(t)$ and $k(t)$ be continuous, $a(t)$ and $b(t)$ Riemann
integrable function on $J=[\alpha ,\beta ]\subset \mathbb{R}$ and 
$t\in \mathbb{R}$ with $b(t)$ and $k(t)$ nonnegative on $J$. 
If $u(t)\leq a(t)+b(t)\int_{\alpha }^{t}k(s)u(s)ds$ for $t\in J$, then 
\begin{equation}
u(t)\leq a(t)+b(t)\int_{\alpha }^{t}a(s)k(s)\exp \Big( \int_{s}^{t}b(\tau
)k(\tau )d\tau \Big) ds,\quad t\in J,  \label{1.1}
\end{equation}
If $u(t)\leq a(t)+b(t)\int_{t}^{\beta }k(s)u(s)ds$ for $t\in J$, then 
\begin{equation}
u(t)\leq a(t)+b(t)\int_{t}^{\beta }a(s)k(s)\exp \Big( \int_{t}^{s}b(\tau
)k(\tau )d\tau \Big) ds,\quad t\in J.  \label{1.2}
\end{equation}
\end{lemma}

In the past few years, these inequalities have been generalized to more than
one variable. Many authors have established Gronwall type integral
inequalities in two or more independent variables; see for example \cite
{3,4,5,6,7}. The results obtained have generated a lot of research interests
due to its usefulness in the theory of differential and integral equations.
Dragomir and Kim \cite{2} considered integral inequalities for functions
with two independent variables. The purpose of this paper is to generalize
their results by obtaining new integral inequalities in $n$ independent
variables.

In what follows we denote by $\mathbb{R}$ the set of real numbers and $%
\mathbb{R}_{+}=[ 0,\infty)$. All the functions appearing in the inequalities
are assumed to be real valued of $n$-variables which are nonnegative and
continuous. All integrals exist on their domains of definitions.

Throughout this paper, we shall assume that $x=(x_{1},x_{2},\dots x_{n})$
and $x^{0}=(x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ are in 
$\mathbb{R}_{+}^{n} $. We shall denote 
\begin{equation*}
\int_{x^{0}}^{x}dt=\int_{x_{1}^{0}}^{x_{1}}\int_{x_{2}^{0}}^{x_{2}}\dots
\int_{x_{n}^{0}}^{x_{n}}\dots dt_{n}\dots dt_{1}
\end{equation*}
and $D_{i}=\frac{\partial}{\partial x_{i}}$ for $i=1,2,\dots ,n$. For 
$x,t\in \mathbb{R}_{+}^{n}$, we shall write $t\leq x$ whenever 
$t_{i}\leq x_{i}$, $i=1,2,\dots ,n$.

\section{Results}

\begin{lemma}
\label{l2} Let $u(x),a(x)$ and $b(x)$ be nonnegative continuous functions,
defined for $x\in \mathbb{R}_{+}^{n}$.

\noindent (1) Assume that $a(x)$ is positive, continuous function,
nondecreasing in each of the variables $x\in \mathbb{R}_{+}^{n}$. Suppose
that 
\begin{equation}
u(x)\leq a(x)+\int_{x^{0}}^{x}b(t)u(t)dt  \label{2.1}
\end{equation}
holds for all $x\in \mathbb{R}_{+}^{n}$ with $x\geq x^{0}$, then 
\begin{equation}
u(x)\leq a(x)\exp \Big( \int_{x^{0}}^{x}b(t)dt\Big) ,  \label{2.2}
\end{equation}
(2) Assume that $a(x)$ is positive, continuous function, non-increasing in
each of the variables $x\in \mathbb{R}_{+}^{n}$. Suppose that 
\begin{equation}
u(x)\leq a(x)+\int_{x}^{x^{0}}b(t)u(t)dt  \label{2.3}
\end{equation}
holds for all $x\in \mathbb{R}_{+}^{n}$ with $x\leq x^{0}$, then 
\begin{equation}
u(x)\leq a(x)\exp \Big( \int_{x}^{x^{0}}b(t)dt\Big) .  \label{2.4}
\end{equation}
\end{lemma}

\begin{proof}
The proof of (1) is similar to the proof of (2), so we present the proof of
(2) and refer the reader to {\cite[p. 112]{1}} for more details.

\noindent (2) Since $a(x)$ is positive, non-increasing in each of the
variables $x\in \mathbb{R}_{+}^{n}$, with $x\leq x^{0}$, then 
\begin{equation}
\frac{u(x)}{a(x)}\leq 1+\int_{x}^{x^{0}}b(t)\frac{u(t)}{a(t)}dt,  \label{2.5}
\end{equation}
Setting 
\begin{equation}
v(x)=\frac{u(x)}{a(x)},  \label{2.6}
\end{equation}
we have 
\begin{equation}
v(x)\leq 1+\int_{x}^{x^{0}}b(t)v(t)dt,  \label{2.7}
\end{equation}
Let 
\begin{equation}
r(x)=1+\int_{x}^{x^{0}}b(t)v(t)dt,  \label{2.8}
\end{equation}
Then $r(x_{1}^{0},x_{2},\dots ,x_{n})=1$, and $v(x)\leq r(x)$, $r(x)$ is
positive and nonincreasing in each of the variables 
$x_{2},\dots ,x_{n}\in \mathbb{R}_{+}$. Hence 
\begin{equation}
\begin{aligned}
D_{1}r(x)
&=\int_{x_{2}}^{x_{2}^{0}}\int_{x_{3}}^{x_{3}^{0}}\dots 
\int_{x_{n}}^{x_{n}^{0}}b(x_{1,}t_{2},\dots ,t_{n})v(x_{1,}t_{2},\dots ,
t_{n})dt_{n}\dots dt_{2} \\
&\leq \int_{x_{2}}^{x_{2}^{0}}\int_{x_{3}}^{x_{3}^{0}}\dots 
\int_{x_{n}}^{x_{n}^{0}}b(x_{1,}t_{2},\dots ,t_{n})r(x_{1,}t_{2},\dots ,
t_{n})dt_{n}\dots dt_{2} \\
&\leq r(x)\int_{x_{2}}^{x_{2}^{0}}\int_{x_{3}}^{x_{3}^{0}}\dots 
\int_{x_{n}}^{x_{n}^{0}}b(x_{1,}t_{2},\dots ,t_{n})dt_{n}\dots dt_{2},  
\end{aligned}\label{2.9}
\end{equation}
Dividing both sides of (\ref{2.9}) by $r(x)$ we get 
\begin{equation}
\frac{D_{1}r(x)}{r(x)}\leq
\int_{x_{2}}^{x_{2}^{0}}\int_{x_{3}}^{x_{3}^{0}}\dots
\int_{x_{n}}^{x_{n}^{0}}b(x_{1,}t_{2},\dots ,t_{n})dt_{n}\dots dt_{2}.
\label{2.10}
\end{equation}
Integrating with respect to $t_{1}$ from $x_{1}$ to $x_{1}^{0}$, we have 
\begin{equation}
r(x)\leq \exp \Big( \int_{x^{0}}^{x^{0}}b(t)dt\Big) ,  \label{2.11}
\end{equation}
Hence 
\begin{equation}
v(x)\leq \exp \Big( \int_{x}^{x_{0}}b(t)dt\Big) .  \label{2.12}
\end{equation}
Substituting (\ref{2.12}) into (\ref{2.6}), we have the result (\ref{2.4}).
\end{proof}

\begin{theorem}
\label{t1} Let $u(x)$, $a(x)$, $b(x)$, $c(x)$, $d(x)$, $f(x)$ be real-valued
non-negative continuous functions defined for $x\in \mathbb{R}_{+}^{n}$. Let 
$W(u(x))$ be real-valued, positive, continuous, strictly non-decreasing,
subadditive, and submultiplicative function for $u(x)\geq 0$, and let 
$H(u(x))$ be real-valued, positive, continuous, and non-decreasing function
defined for $x\in \mathbb{R}_{+}^{n}$. Assume that $a(x)$, $f(x)$ are
nondecreasing in the first variable $x_{1}$ for $x_{1}\in \mathbb{R}_{+}$.
If 
\begin{equation}
\begin{aligned}
u(x) &\leq a(x)+b(x)\int_{\alpha}^{x_{1}}c(s,x_{2},\dots ,x_{n})u(s,x_{2},\dots ,
x_{n})ds \\ 
&\quad +f(x)H\Big( \int_{x^{_{0}}}^{x}d(t)W(u(t))dt\Big) ,  
\end{aligned}\label{2.13}
\end{equation}
for $\alpha \geq 0$, $x,t\in \mathbb{R}_{+}^{n}$ with $\alpha \leq x_{1}$
and $x^{_{0}}\leq t\leq x$, then 
\begin{equation}
u(x)\leq p(x)\Big\{ a(x)+f(x)H\Big[ G^{-1}\Big( G(A(t))+%
\int_{x^{_{0}}}^{x}d(t)W(p(t)f(t))dt\Big) \Big]\Big\},  \label{2.14}
\end{equation}
for $\alpha \geq 0$, $x\in \mathbb{R}_{+}^{n}$ with $\alpha \leq x_{1}$,
where 
\begin{gather}
p(x)=1+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots ,x_{n})\exp \Big( %
\int_{\alpha }^{x_{1}}b(\tau ,x_{2},\dots ,x_{n})c(\tau ,x_{2},\dots
,x_{n})d\tau \Big) ds,  \label{2.15} \\
A(t)=\int_{x^{_{0}}}^{\infty }d(t)W(a(t)p(t))dt,  \label{2.16} \\
G(z)=\int_{z^{0}}^{z}\frac{ds}{W(H(s))},\quad z\geq z^{0}>0\,.  \label{2.17}
\end{gather}
Here $G^{-1}$ is the inverse function of $G$ and 
\begin{equation*}
G\Big( \int_{x^{_{0}}}^{\infty }d(t)W(a(t)p(t))dt\Big) %
+\int_{x^{_{0}}}^{x}d(t)W(p(t)f(t))dt,
\end{equation*}
is in the domain of $G^{-1}$ for $x\in \mathbb{R}_{+}^{n}$.
\end{theorem}

\begin{proof}
Define a function 
\begin{equation}
z(x)=a(x)+f(x)H\Big( \int_{x^{_{0}}}^{x}d(t)W(u(t))dt\Big) ,  \label{2.18}
\end{equation}
Then (\ref{2.13}) can be restated as 
\begin{equation}
u(x)\leq z(x)+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots
,x_{n})u(s,x_{2},\dots ,x_{n})ds.  \label{2.19}
\end{equation}
Clearly $z(x)$ is a nonnegative and continuous in $x_{1}\in \mathbb{R}%
_{+}.\;x_{2},x_{3,}\dots x_{n}\in \mathbb{R}_{+}$fixed in (\ref{2.19}) and
using (1) of lemma \ref{l1} to (\ref{2.19}), we get 
\begin{align*}
u(x)& \leq z(x)+b(x)\int_{\alpha }^{x_{1}}z(s,x_{2},\dots
,x_{n})c(s,x_{2},\dots ,x_{n}) \\
& \quad \times \exp \Big( \int_{\alpha }^{x_{1}}b(\tau ,x_{2},\dots
,x_{n})c(\tau ,x_{2},\dots ,x_{n})d\tau \Big) ds,
\end{align*}
Moreover, $z(x)$ is nondecreasing in $x_{1},x_{1}\in R_{+}$, we obtain 
\begin{equation}
u(x)\leq z(x)p(x),  \label{2.20}
\end{equation}
where $p(x)$ is defined by (\ref{2.15}). From (\ref{2.18}) we have 
\begin{equation}
u(x)\leq \left( a(x)+f(x)H(v(x))\right) p(x),  \label{2.21}
\end{equation}
where $v(x)=\int_{x^{_{0}}}^{x}d(t)W(u(t))dt$. From (\ref{2.21}), we observe
that 
\begin{equation} \label{2.22}
\begin{aligned}
v(x) &\leq \int_{x^{_{0}}}^{x}d(t)W\left( \left( a(t)+f(t)H(v(t))\right)
p(t)\right) dt   \\
&\leq \int_{x^{_{0}}}^{x}d(t)W(a(t)p(t))dt  
 +\int_{x^{_{0}}}^{x}d(t)W\left( p(t)f(t)\right) W\left( H(v(t))\right) dt, \\
&\leq \int_{x^{_{0}}}^{\infty }d(t)W(a(t)p(t))dt  
+\int_{x^{_{0}}}^{x}d(t)W\left( p(t)f(t)\right) W\left( H(v(t))\right) dt,
\end{aligned}
\end{equation}
Since $W$ is subadditive and submultiplicative function. Define $r(x)$ as
the right side of (\ref{2.22}), then $r(x_{0}^{1},x_{2},\dots
,x_{n})=\int_{x^{_{0}}}^{\infty }d(t)W(a(t)p(t))dt$, $v(x)\leq r(x)$, $r(x)$
is positive nondecreasing in each of the variables 
$x_{2},\dots ,x_{n}\in \mathbb{R}_{+}$ and 
\begin{equation} \label{2.23}
\begin{aligned}
D_{1}r(x)
&=\int_{x_{2}^{0}}^{x_{2}}\int_{x_{3}^{0}}^{x_{3}}\dots 
\int_{x_{n}^{0}}^{x_{n}}d(x_{1,}t_{2},\dots ,t_{n})  \ \\
&\quad\times W\left( p(x_{1,}t_{2},\dots ,t_{n})f(x_{1,}t_{2},\dots ,t_{n})\right)
W\left( H(v(x_{1,}t_{2},\dots ,t_{n}))\right) dt_{n}\dots dt_{2}  \\
&\leq \int_{x_{2}^{0}}^{x_{2}}\int_{x_{3}^{0}}^{x_{3}}\dots 
\int_{x_{n}^{0}}^{x_{n}}d(x_{1,}t_{2},\dots ,t_{n})  \\
&\quad\times W\left( p(x_{1,}t_{2},\dots ,t_{n})f(x_{1,}t_{2},\dots ,t_{n})\right)
W\left( H(r(x_{1,}t_{2},\dots ,t_{n}))\right) dt_{n}\dots dt_{2}   \\
&\leq W\left( H(r(x))\right)
\int_{x_{2}^{0}}^{x_{2}}\int_{x_{3}^{0}}^{x_{3}}\dots 
\int_{x_{n}^{0}}^{x_{n}}d(x_{1,}t_{2},\dots ,t_{n})   \\
&\quad \times W\left( p(x_{1,}t_{2},\dots ,t_{n})f(x_{1,}t_{2},\dots ,t_{n})\right)
dt_{n}\dots dt_{2}. 
\end{aligned} 
\end{equation}
Dividing both sides of (\ref{2.23}) by $W(H(r(x)))$ we get 
\begin{equation}
\begin{aligned}
\frac{D_{1}r(x)}{W(H(r(x)))} 
&\leq \int_{x_{2}^{0}}^{x_{2}}\int_{x_{3}^{0}}^{x_{3}}\dots 
\int_{x_{n}^{0}}^{x_{n}}d(x_{1,}t_{2},\dots ,t_{n})  \\
&\quad \times W\left( p(x_{1,}t_{2},\dots ,t_{n})f(x_{1,}t_{2},\dots ,t_{n})\right)
dt_{n}\dots dt_{2},  
\end{aligned}\label{2.24}
\end{equation}
Note that for 
\begin{equation}
G(z)=\int_{z^{0}}^{z}\frac{ds}{W(H(s))},\quad z\geq z^{0}>0  \label{2.25}
\end{equation}
it follows that 
\begin{equation}
D_{1}G(r(x))=\frac{D_{1}r(x)}{W(H(r(x)))},  \label{2.26}
\end{equation}
 From (\ref{2.25}) , (\ref{2.26}) and (\ref{2.24}), we have 
\begin{equation}
\begin{aligned}
D_{1}G(r(x)) &\leq
\int_{x_{2}^{0}}^{x_{2}}\int_{x_{3}^{0}}^{x_{3}}\dots 
\int_{x_{n}^{0}}^{x_{n}}d(x_{1,}t_{2},\dots ,t_{n})  \\\
&\quad \times W\left( p(x_{1,}t_{2},\dots ,t_{n})f(x_{1,}t_{2},\dots ,t_{n})\right)
dt_{n}\dots dt_{2}, 
\end{aligned} \label{2.27}
\end{equation}
Now setting $x_{1}=s$ in (\ref{2.27}) and then integrating with respect to 
$x_{1}^{0}$ to $x_{1}$, we obtain 
\begin{equation}
G(r(x))\leq G(r(x_{1}^{0},x_{2},\dots
,x_{n}))+\int_{x_{0}}^{x}d(t)W(p(t)f(t))dt  \label{2.28}
\end{equation}
Noting that $r(x_{1}^{0},x_{2},\dots ,x_{n})=\int_{x_{0}}^{\infty
}d(t)W(a(t)p(t))dt$, we have 
\begin{equation}
r(x)\leq G^{-1}\Big[ G\Big( \int_{x^{_{0}}}^{\infty }d(t)W(a(t)p(t))dt\Big) %
+\int_{x^{_{0}}}^{x}d(t)W(p(t)f(t))dt\Big] .  \label{2.29}
\end{equation}
The required inequality in (\ref{2.14}) follows from the fact 
$v(x)\leq r(x)$, (\ref{2.19}) and (\ref{2.29})
\end{proof}

\begin{theorem}
\label{t2} Let $u(x)$, $a(x)$, $b(x)$, $c(x)$, $d(x)$, $f(x)$, $W(u(x))$,
and $H(u(x))$ be as defined in theorem \ref{t1}. Assume that $a(x),f(x)$ are
non-increasing in the first variable $x_{1}$, for $x_{1}\in \mathbb{R}_{+}$.
If 
\begin{equation}
\begin{aligned}
u(x) &\leq a(x)+b(x)\int_{x_{1}}^{\beta}c(s,x_{2},\dots ,x_{n})u(s,x_{2},\dots ,
x_{n})ds  \\
&\quad +f(x)H\left( \int_{x}^{x_{0}}d(t)W(u(t))dt\right) ,
\end{aligned}
\end{equation}
for $\beta \geq 0,\;x\in \mathbb{R}_{+}^{n}$ with $\beta \geq x_{1}$ and 
$x\leq x^{_{0}}$. Then 
\begin{equation*}
u(x)\leq \overline{p}(x)\Big\{ a(x)+f(x)H\Big( G^{-1}\Big[ G(\overline{A}%
(t))+\int_{x}^{x_{0}}d(t)W(p(t)f(t))dt\Big] \Big) \Big\} ,
\end{equation*}
for $\beta \geq 0,\;x\in \mathbb{R}_{+}^{n}$ with $\beta \geq x_{1}$, where 
\begin{gather*}
\overline{p}(x)=1+b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots ,x_{n})\exp 
\Big( \int_{x_{1}}^{s}b(\tau ,x_{2},\dots ,x_{n})c(\tau ,x_{2},\dots ,x_{n})
d\tau \Big) ds, \\
\overline{A}(t)=\int_{0}^{x^{_{0}}}d(t)W(a(t)\overline{p}(t))dt, \\
G(z)=\int_{z^{0}}^{z}\frac{ds}{W(H(s))},\quad z\geq z^{0}>0\,.
\end{gather*}
Here $G^{-1}$ is the inverse function of $G$ and 
\begin{equation*}
G\Big( \int_{0}^{x^{_{0}}}d(t)W(a(t)p(t))dt\Big) 
+\int_{x}^{x^{_{0}}}d(t)W(p(t)f(t))dt,
\end{equation*}
is in the domain of $G^{-1}$ for $x\in \mathbb{R}_{+}^{n}$.
\end{theorem}

The proof is similar to the proof of Theorem \ref{t1} and so it is omitted.

\begin{remark}
\textrm{We note that in the special case $n=2$ (integral inequalities in two
independent variables) $x\in \mathbb{R}_{+}^{2}$ and 
$x_{0}=(x_{1}^{0},x_{2}^{0})=(\infty ,\infty )$ in theorem \ref{t2}. our
estimate reduces to Theorem 2.4 obtained by S. S. Dragomir and Y. H. Kim 
\cite{2}. }
\end{remark}

\begin{theorem}
\label{t3} Let $u(x),a(x),b(x),c(x)$ and $f(x)$ be real-valued nonnegative
continuous functions defined for $x\in \mathbb{R}_{+}^{n}$ and $L:\mathbb{R}%
_{+}^{n+1}\to \mathbb{R}_{+}^{*}$ be a continuous functions which satisfies
the condition 
\begin{equation}
0\leq L(x,u)-L(x,v)\leq M(x,v)\Phi ^{-1}(u-v),  \label{2.30}
\end{equation}
for $u\geq v\geq 0$, where $M(x,v)$ is a real-valued nonnegative continuous
function defined for $x\in \mathbb{R}_{+}^{n},v\in \mathbb{R}_{+}\mathbb{.}$
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 
\begin{equation}
\Phi ^{-1}(uv)\leq \Phi ^{-1}(u)\Phi ^{-1}(v),  \label{2.31}
\end{equation}
for $u,v\in \mathbb{R}_{+}$, Assume that $a(x),f(x)$ are nondecreasing in
the first variable $x_{1}$ for $x_{1}\in \mathbb{R}_{+}$. If 
\begin{equation}
u(x)\leq a(x)+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots
,x_{n})u(s,x_{2},\dots ,x_{n})ds+f(x)\Phi \Big( \int_{x_{0}}^{x}L(t,u(t))dt%
\Big) ,  \label{2.32}
\end{equation}
for $\alpha \geq 0,\;x\in \mathbb{R}_{+}^{n}$ with $\alpha \leq x_{1}$ and $%
x^{_{0}}<x$. Then 
\begin{equation}
u(x)\leq p(x)\Big\{ a(x)+f(x)\Phi \Big[ e(x)\exp \Big(%
\int_{x^{_{0}}}^{x}M(t,p(t)a(t))\Phi ^{-1}\left( p(t)f(t)\right) dt\Big) %
\Big] \Big\}  \label{2.33}
\end{equation}
for $\alpha \geq 0,\;x\in \mathbb{R}_{+}^{n}$ with $\alpha \leq x_{1}\;$and $%
x^{_{0}}<x$, where 
\begin{gather}
p(x)=1+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots ,x_{n})\exp \Big( %
\int_{s}^{x_{1}}b(\tau ,x_{2},\dots ,x_{n})c(\tau ,x_{2},\dots ,x_{n})d\tau %
\Big) ds,  \label{2.34} \\
e(x)=\int_{x^{_{0}}}^{x}L(t,p(t)a(t))dt.  \label{2.35}
\end{gather}
\end{theorem}

\begin{proof}
Define the function 
\begin{equation}
z(x)=a(x)+f(x)\Phi \Big( \int_{x_{0}}^{x}L(t,u(t))dt\Big) ,  \label{2.36}
\end{equation}
Then (\ref{2.32}) can be restated as 
\begin{equation}
u(x)\leq z(x)+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},x_{3},\dots
,x_{n})u(s,x_{2},x_{3},\dots ,x_{n})ds.  \label{2.37}
\end{equation}
Clearly $z(x)$ is nonnegative and continuous in $x_{1}\in \mathbb{R}_{+}$,
where $x_{2},x_{3,}\dots x_{n}\in \mathbb{R}_{+}$fixed in (\ref{2.37}) and
using 1 of lemma \ref{l1} to (\ref{2.37}), we get 
\begin{align*}
u(x)& \leq z(x)+b(x)\int_{\alpha }^{x_{1}}z(s,x_{2},\dots
,x_{n})c(s,x_{2},\dots ,x_{n}) \\
& \times \exp \Big( \int_{s}^{x_{1}}b(\tau ,x_{2},\dots ,x_{n})c(\tau
,x_{2},\dots ,x_{n})d\tau \Big) ds
\end{align*}
Moreover, $z(x)$ is nondecreasing in $x_{1},x_{1}\in \mathbb{R}_{+}$, we
obtain 
\begin{equation}
u(x)\leq z(x)p(x),  \label{2.38}
\end{equation}
Where $p(x)$ is defined by (\ref{2.34}). From (\ref{2.36}) and (\ref{2.38})
we have 
\begin{equation}
u(x)\leq p(x)\left[ a(x)+f(x)\Phi (v(x))\right] ,  \label{2.39}
\end{equation}
where 
\begin{equation*}
v(x)=\int_{x^{_{0}}}^{x}L(t,u(t))dt,
\end{equation*}
 From (\ref{2.39}), and the hypotheses on $L$ and $\Phi $, we observe that 
\begin{equation} \label{2.40}
\begin{aligned}
v(x) &\leq \int_{x^{_{0}}}^{x}\left( L\left( t,p(t)\left[ a(t)+f(t)\Phi
(v(t))\right] \right)  -L\left( t,p(t)a(t)\right) +L\left( t,p(t)a(t)\right) \right) dt, 
\\
&\leq \int_{x^{_{0}}}^{x}L\left( t,p(t)a(t)\right) dt  
+\int_{x^{_{0}}}^{x}M(t,p(t)a(t))\Phi ^{-1}\left( p(t)f(t)\Phi
(v(t))\right) dt,   \\
&\leq e(x)+\int_{x^{_{0}}}^{x}M(t,p(t)a(t))\Phi ^{-1}\left( p(t)f(t)\right)
v(t)dt,  
\end{aligned}
\end{equation}
where $e(x)$ is defined by (\ref{2.35}). Clearly, $e(x)$ is positive,
continuous, nondecreasing in each of the variables $x,x\in \mathbb{R}%
_{+}^{n} $. Now, by part (1) of lemma \ref{l2}, 
\begin{equation}
v(x)\leq e(x)\exp \Big( \int_{x_{0}}^{x}M(t,p(t)a(t))\Phi ^{-1}\left(
p(t)f(t)\right) dt\Big) .  \label{2.41}
\end{equation}
Using (\ref{2.39}) in (\ref{2.41}), we get the required inequality in 
(\ref{2.33}).
\end{proof}

\begin{theorem}
\label{t4} Let $u(x)$, $a(x)$, $b(x)$, $c(x)$, $f(x)$, $L$, $M$, $\Phi $,
and $\Phi ^{-1}$ be as defined in theorem \ref{t3}. Assume that $a(x),f(x)$
are non-increasing in the first variable $x_{1}$ for $x_{1}\in \mathbb{R}%
_{+} $. If 
\begin{equation}
u(x)\leq a(x)+b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots
,x_{n})u(s,x_{2},\dots ,x_{n})ds+f(x)\Phi \Big( %
\int_{x}^{x^{_{0}}}L(t,u(t))dt\Big) ,  \label{2.42}
\end{equation}
for $\beta \geq 0$, $x\in \mathbb{R}_{+}^{n}$ with $\beta \geq
x_{1},\;x<x^{_{0}}$. Then 
\begin{equation*}
u(x)\leq \overline{p}(x)\Big\{ a(x)+f(x)\Phi \Big[ \overline{e}(x)\exp \Big(%
\int_{x}^{x_{0}}M(t,p(t)a(t))\Phi ^{-1}\big( p(t)f(t)\big) dt\Big) \Big] %
\Big\} ,
\end{equation*}
for $\beta \geq 0$, $x\in \mathbb{R}_{+}^{n}$ with $\beta \geq x_{1}$, $%
x<x^{_{0}}$, where 
\begin{gather}
\overline{p}(x)=1+b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots ,x_{n})
\exp \Big(\int_{x_{1}}^{s}b(\tau ,x_{2},\dots ,x_{n})c(\tau ,x_{2},\dots ,
x_{n})d\tau \Big) ds  \notag \\
\overline{e}(x)=\int_{x}^{x^{_{0}}}L(t,\overline{p}(t)a(t))dt.  \label{2.43}
\end{gather}
\end{theorem}

The proof is similar to the proof of Theorem \ref{t3} and so it is omitted.

\begin{remark}
\textrm{We note that in the special case $n=2$ , $x\in \mathbb{R}_{+}^{2}$
and $x^{0}=(x_{1}^{0},x_{2}^{0})=(\infty ,\infty )$ in theorem \ref{t4}. Our
estimate reduces to Theorem 2.6 obtained by Dragomir and Kim \cite{2}. }
\end{remark}

\begin{remark}
\textrm{(1) The preceding results remaining valid if we replace\newline
$b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots ,x_{n})u(s,x_{2},\dots ,x_{n})ds$
by the general case\newline
$b_{i}(x)\int_{\alpha _{i}}^{x_{i}}c_{i}(x_{1,.}\dots
,x_{i-1},s_{i},x_{i+1},\dots ,x_{n})u(x_{1,.}\dots
,x_{i-1},s_{i},x_{i+1},\dots ,x)ds_{i}$, for any $i=2,\dots ,n$ fixed , and $%
\alpha _{i}\geq 0$, $x=(x_{1},\dots x_{n})\in \mathbb{R}_{+}^{n}$ with $%
\alpha _{i}\leq s_{i}\leq x_{i}$, $x_{i},s_{i}\in \mathbb{R}_{+}$. }

\textrm{\noindent (2) The preceding results are also valid if $%
b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots ,x_{n})u(s,x_{2},\dots ,x_{n})ds$
is replaced by the general case \newline
$b_{i}(x)\int_{x_{i}}^{\beta _{i}}c_{i}(x_{1,.}\dots
,x_{i-1},s_{i},x_{i+1},\dots ,x_{n})g(u(x_{1,.}\dots
,x_{i-1},s_{i},x_{i+1},\dots ,x_{n}))ds_{i}$, for any $i=2,\dots ,n$ fixed ,
and $\alpha _{i}\geq 0$, $x=(x_{1},\dots x_{n})\in \mathbb{R}_{+}^{n}$ with $%
\alpha _{i}\leq s_{i}\leq x_{i}$, $x_{i},s_{i}\in \mathbb{R}_{+}$. where $%
b_{i}(x)$ and $c_{i}(x)$ be real-valued nonnegative continuous function
defined for $x\in \mathbb{R}_{+}^{n}$, For any $i=2,\dots ,n$. }
\end{remark}

\section{Further Inequalities}

In this section we require the class of function $S$ as defined in \cite{2}.
A function $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$ is said to belong to the
class $S$ if it satisfies the following conditions:

\begin{enumerate}
\item  $g(u)$ is positive, nondecreasing and continuous for $u\geq 0$

\item  $(1/v)g(u)\leq g(u/v)$, $u>0$, $v\geq 1$.
\end{enumerate}

\begin{theorem}
\label{t5}Let $u(x)$, $a(x)$, $b(x)$, $c(x)$, $d(x)$, $f(x)$ be real-valued
nonnegative continuous function defined for $x\in \mathbb{R}_{+}^{n}$ and
let $g\in S$. Also let $W(u(x))$ be real-valued, positive, continuous,
strictly nondecreasing, subadditive, and submultiplicative function for $%
u(x)\geq 0$ and let $H(u(x))$ be a real-valued, continuous, positive, and
nondecreasing function defined for $x\in \mathbb{R}_{+}^{n},$and $b(x)$
nonincreasing in the first variable $x_{1}$. Assume that a function $m(x)$
is nondecreasing in the first variable $x_{1}$ and $m(x)\geq 1$, which is
defined by 
\begin{equation}
m(x)=a(x)+f(x)H\Big( \int_{x^{_{0}}}^{x}d(t)W(u(t))dt\Big) ,  \label{3.1}
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$, $x>x^{0}\geq 0$. If 
\begin{equation}
u(x)\leq m(x)+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots
,x_{n})g(u(s,x_{2},\dots ,x_{n}))ds,  \label{3.2}
\end{equation}
for $\alpha \geq 0$, $x\in \mathbb{R}_{+}^{n}$ with $\alpha \leq x_{1}$,
then 
\begin{equation}
u(x)\leq F(x)\Big\{ a(x)+f(x)H\Big[ G^{-1}\Big( G(B(t))+%
\int_{x^{_{0}}}^{x}d(t)W(F(t)f(t))dt\Big) \Big] \Big\},  \label{3.3}
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$ , where 
\begin{gather}
F(x)=\Omega ^{-1}\Big( \Omega (1)+\int_{\alpha }^{x_{1}}b(s,x_{2},\dots
,x_{n})c(s,x_{2},\dots ,x_{n})ds\Big) ,  \label{3.4} \\
B(t)=\int_{x^{_{0}}}^{\infty }d(t)W(a(t)F(t))dt,  \label{3.5} \\
\Omega (\delta )=\int_{\varepsilon }^{\delta }\frac{ds}{g(s)},\quad \delta
\geq \varepsilon >0.  \label{3.6}
\end{gather}
Here $\Omega ^{-1}$ is the inverse function of $\Omega $, and $G,G^{-1}$ are
defined in Theorem \ref{t1}, and $\Omega (1)+\int_{\alpha
}^{x_{1}}b(s,x_{2},\dots ,x_{n})c(s,x_{2},\dots ,x_{n})ds$ is in the domain
of $\Omega ^{-1}$, and 
\begin{equation*}
G\Big( \int_{x^{_{0}}}^{\infty
}d(t)W(a(t)F(t))dt)+\int_{x^{_{0}}}^{x}d(t)W(F(t)f(t)dt\Big),
\end{equation*}
is in the domain of $G^{-1}$ for $x\in \mathbb{R}_{+}^{n}$.
\end{theorem}

\begin{proof}
We have $m(x)$ be a positive, continuous, nondecreasing in $x_{1}$ and $g\in
S$, and $b(x)$ non-increasing in the first variable $x_{1}$. Then can be
restated as 
\begin{equation}
\frac{u(x)}{m(x)}\leq 1+\int_{\alpha }^{x_{1}}b(s,x_{2},x_{3},\dots
,x_{n})c(s,x_{2},x_{3},\dots ,x_{n})g(\frac{u(s,x_{2},x_{3},\dots ,x_{n})}{%
m(s,x_{2},x_{3},\dots ,x_{n})})ds  \label{3.7}
\end{equation}
The inequality (\ref{3.7}) may be treated as one-dimensional Bihari-Lasalle
inequality the inequality type was given by Gyori \cite{3} (see \cite{1}),
for any fixed $x_{2},x_{3},\dots ,x_{n}$, which implies 
\begin{equation}
u(x)\leq F(x)m(x).  \label{3.8}
\end{equation}
Here $F(x)$ is defined by (\ref{3.4}), by (\ref{3.1}) and (\ref{3.8}) we get 
\begin{equation}
u(x)\leq F(x)\left\{ a(x)+f(x)H(v(x))\right\} ,  \label{3.9}
\end{equation}
where $v(x)$ is defined by 
\begin{equation*}
v(x)=\int_{x^{_{0}}}^{x}d(t)W(u(t))dt.
\end{equation*}
Using the last argument in the proof of Theorem \ref{t1}, we obtain desired
inequality in (\ref{3.3}).
\end{proof}

\begin{theorem}
\label{t6} Let $u(x)$, $a(x)$, $c(x)$, $d(x)$, $f(x)$, $W(u(x)$, and $%
H(u(x)) $ be as defined in the theorem \ref{t5} and let $g\in S$ and $b(x)$
be nonnegative continuous functions, nondecreasing in the first variable $%
x_{1}$. Assume that a function $\overline{m}(x)$ is non-increasing in the
first variable $x_{1}$ and $\overline{m}(x)\geq 1$, which is defined by 
\begin{equation}
\overline{m}(x)=a(x)+f(x)H\Big( \int_{x}^{x^{0}}d(t)W(u(t))dt\Big)
\label{3.10}
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$, $x^{0}\geq x$. If 
\begin{equation}
u(x)\leq \overline{m}(x)+b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots
,x_{n})g(u(s,x_{2},\dots ,x_{n}))ds,  \label{3.11}
\end{equation}
for $\beta \geq 0$, $x\in \mathbb{R}_{+}^{n}$ with $\beta \geq x_{1}$, then 
\begin{equation}
u(x)\leq \overline{F}(x)\Big\{ a(x)+f(x)H\Big[ G^{-1}\Big( G(\overline{B}%
(t))+\int_{x}^{x^{0}}d(t)W(\overline{F}(t)f(t))dt\Big) \Big]\Big\} ,
\label{3.12}
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$. Here 
\begin{gather}
\overline{F}(x)=\Omega ^{-1}\Big( \Omega (1)+\int_{x_{1}}^{\beta
}b(s,x_{2},\dots ,x_{n})c(s,x_{2},\dots ,x_{n})ds\Big),  \label{3.13} \\
\overline{B}(t)=\int_{0}^{x^{0}}d(t)W(a(t)\overline{F}(t))dt,  \label{3.14}
\end{gather}
and $\Omega $ is defined in (\ref{3.6}). Here $\Omega ^{-1}$ is the inverse
function of $\Omega $, and $G,G^{-1}$ are defined in theorem \ref{t1}, and $%
\Omega (1)+\int_{x_{1}}^{\beta }b(s,x_{2},\dots ,x_{n})c(s,x_{2},\dots
,x_{n})ds$ is in the domain of $\Omega ^{-1}$, and 
\begin{equation*}
G(\int_{0}^{x^{0}}d(t)W(a(t)\overline{F}(t))dt)+\int_{x}^{x^{0}}d(t)W(%
\overline{F}(t)f(t))dt
\end{equation*}
is in the domain of $G^{-1}$ for $x\in \mathbb{R}_{+}^{n}$.
\end{theorem}

\begin{proof}
We have $\overline{m}(x)$ positive, continuous, nonincreasing in $x_{1}$.
Also $g\in S$ and $b(x)$ nondecreasing in the first variable $x_{1}$. Then (%
\ref{3.11}) can be restated as 
\begin{equation}
\frac{u(x)}{\overline{m}(x)}\leq 1+\int_{x_{1}}^{\beta
}b(s,x_{2},x_{3},\dots ,x_{n})c(s,x_{2},x_{3},\dots ,x_{n})g\big(\frac{%
u(s,x_{2},\dots ,x_{n})}{\overline{m}(s,x_{2},\dots ,x_{n})}\big)ds
\label{3.15}
\end{equation}
This inequality can be treated as one-dimensional Bihari-Lasalle inequality 
\cite{3} for a fixed $x_{2},x_{3},\dots ,x_{n}$, which implies 
\begin{equation}
u(x)\leq \overline{F}(x)\overline{m}(x)  \label{3.16}
\end{equation}
where $\overline{F}(x)$ is defined by (\ref{3.13}). Now , by following last
argument as in the proof of Theorem \ref{t2} , we obtain desired inequality
in (\ref{3.12})
\end{proof}

\begin{corollary}
\label{c1} If $b(x)=1$ for $x\in R_{+}^{n}$, then from 
\begin{equation*}
u(x)\leq \overline{m}(x)+\int_{x_{1}}^{\beta }c(s,x_{2},\dots
,x_{n})g(u(s,x_{2},\dots ,x_{n}))ds
\end{equation*}
with $\beta \geq x_{1}$, it follows that 
\begin{equation*}
u(x)\leq \overline{F}(x)\Big\{ a(x)+f(x)H\Big[ G^{-1}\Big( G(\overline{B}%
(t))+\int_{x}^{x^{0}}d(t)W(\overline{F}(t)f(t))dt\Big) \Big]\Big\}
\end{equation*}
for $x\in \mathbb{R}_{+}^{n}$ , where 
\begin{gather*}
\overline{F}(x)=\Omega ^{-1}\Big( \Omega (1)+\int_{x_{1}}^{\beta
}c(s,x_{2},\dots ,x_{n})ds\Big) \\
\overline{B}(t)=\int_{0}^{x^{0}}d(t)W(a(t)\overline{F}(t))dt
\end{gather*}
\end{corollary}

\begin{remark}
\textrm{We note that in the special case $n=2$ ,$x=(x_{1},x_{2})\in %
\mathbb{R}_{+}^{2}$, and $x^{_{0}}=(\infty ,\infty )$ in corollary \ref{c1}.
Our estimate reduces to Theorem 3.2 obtained by Dragomir and Kim \cite{2}. }
\end{remark}

\begin{theorem}
\label{t7} Let $u(x)$, $a(x)$, $b(x)$, $c(x)$, $f(x)$, $L$, $M$, $\Phi $,
and $\Phi ^{-1}$ be as defined in theorem \ref{t3}. Let $g\in S$ and $b(x)$
nonincreasing in the first variable $x_{1}$. Assume that a function $n(x)$
is nondecreasing in the first variable $x_{1}$ and $n(x)\geq 1$ which is
defined by 
\begin{equation}
n(x)=a(x)+f(x)\Phi \Big( \int_{x_{0}}^{x}L(t,u(t))dt\Big)  \label{3.17}
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$, $x\geq x_{0}\geq 0$. If 
\begin{equation}
u(x)\leq n(x)+b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},x_{3},\dots
,x_{n})g(u(s,x_{2},x_{3},\dots ,x_{n}))ds  \label{3.18}
\end{equation}
for $\alpha \geq 0,\;x\in \mathbb{R}_{+}^{n}$ with $\alpha \leq x_{1}$, then 
\begin{equation}
u(x)\leq F(x)\Big\{ a(x)+f(x)\Phi \Big[ e(x)\exp \Big( %
\int_{x^{0}}^{x}M(t,a(t)F(t))\Phi ^{-1}\big( f(t)F(t)\big) dt\Big) \Big] %
\Big\}  \label{3.19}
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$ , where $F(x)$ is defined in (\ref{3.4}), $%
e(x) $ is defined in (\ref{2.35}), $\Omega $ is defined in (\ref{3.6}), Here 
$\Omega ^{-1}$ is the inverse function of $\Omega $, and \newline
$\Omega (1)+\int_{\alpha }^{x_{1}}b(s,x_{2},\dots ,x_{n})c(s,x_{2},\dots
,x_{n})ds$ is in the domain of $\Omega $ for $x\in \mathbb{R}_{+}^{n}$.
\end{theorem}

\begin{proof}
We follow an argument similar to that of Theorem \ref{t5}. We have $n(x)$ be
a positive, continuous, nondecreasing in $x_{1}$ and $g\in S$, and $b(x)$
nonincreasing in the first variable $x_{1}$. Then can (\ref{3.18}) be
restated as 
\begin{equation}
\frac{u(x)}{n(x)}\leq 1+\int_{\alpha }^{x_{1}}b(s,x_{2},x_{3},\dots
,x_{n})c(s,x_{2},x_{3},\dots ,x_{n})g\big(\frac{u(s,x_{2},\dots ,x_{n})}{%
n(s,x_{2},\dots ,x_{n})}\big)ds.  \label{3.20}
\end{equation}
The inequality (\ref{3.20}) may be treated as one-dimensional Bihari-Lasalle
inequality, for any fixed $x_{2},x_{3},\dots ,x_{n}$, which implies 
\begin{equation}
u(x)\leq F(x)n(x)  \label{3.21}
\end{equation}
where $F(x)$ is defined by (\ref{3.4}). From (\ref{3.17}) and (\ref{3.21})
we get 
\begin{equation}
u(x)\leq F(x)\left[ a(x)+f(x)H\Big(\int_{x^{0}}^{x}L(t,u(t))dt\Big)\right]
\label{3.22}
\end{equation}
Following the last argument in the proof of Theorem \ref{t3}, we obtain the
desired inequality in (\ref{3.19}).
\end{proof}

\begin{theorem}
Let $u(x)$, $a(x)$, $b(x)$, $c(x)$, $f(x)$, $L$, $M$, $\Phi $, and $\Phi
^{-1}$ be as defined in theorem \ref{t3}. Let $g\in S$ and $b(x)$ be
nondecreasing in the first variable $x_{1}$. Assume that a function $%
\overline{n}(x)$ is nonincreasing in the first variable $x_{1}$ and $%
\overline{n}(x)\geq 1$, which is defined by 
\begin{equation}
\overline{n}(x)=a(x)+f(x)\Phi \Big(\int_{x}^{x^{0}}L(t,u(t))dt\Big)
\end{equation}
for $x\in \mathbb{R}_{+}^{n}$, $x^{0}\geq x\geq 0$. If 
\begin{equation}
u(x)\leq \overline{n}(x)+b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots
,x_{n})g(u(s,x_{2},\dots ,x_{n}))ds
\end{equation}
for $\beta \geq 0,\;x\in \mathbb{R}_{+}^{n}$ with $\beta \geq x_{1}$, then 
\begin{equation*}
u(x)\leq \overline{F}(x)\Big\{ a(x)+f(x)\Phi \Big[ \overline{e}(x)
\exp \Big(\int_{x}^{x^{0}}M(t,a(t)\overline{F}(t))\Phi ^{-1}\big( f(t)
\overline{F}(t)\big) dt\Big) \Big]\Big\}
\end{equation*}
for $x\in \mathbb{R}_{+}^{n}$, where $\overline{F}(x)$ is defined in (\ref
{3.13}), $\overline{e}(x)$ is defined in (\ref{2.43}), $\Omega $ is defined
in (\ref{3.6}). Here $\Omega ^{-1}$ is the inverse function of $\Omega $,
and \newline
$\Omega (1)+\int_{x_{1}}^{\beta }b(s,x_{2},\dots ,x_{n})c(s,x_{2},\dots
,x_{n})ds$ is in the domain of $\Omega $ for $x\in \mathbb{R}_{+}^{n}$.
\end{theorem}

The proof of this theorem follows by an argument similar to that of Theorem 
\ref{t7}; therefore, we omit it.

\begin{corollary}
\label{c2} if $b(x)=1$ for $x\in R_{+}^{n}$, then from 
\begin{equation*}
u(x)\leq \overline{n}(x)+\int_{x_{1}}^{\beta }c(s,x_{2},\dots
,x_{n})g(u(s,x_{2},\dots ,x_{n}))ds,
\end{equation*}
for $\beta \geq 0$ with $\beta \geq x_{1}$, then it follows that 
\begin{equation*}
u(x)\leq \overline{F}(x)\Big\{ a(x)+f(x)\Phi \Big[ \overline{e}(x)\exp 
\Big(\int_{x}^{x^{0}}M(t,a(t)\overline{F}(t))\Phi ^{-1}\big( f(t)
\overline{F}(t)\big) dt\Big) \Big]\Big\}
\end{equation*}
for $x\in \mathbb{R}_{+}^{n}$, where 
\begin{gather*}
\overline{F}(x)=\Omega ^{-1}\Big( \Omega (1)+\int_{x_{1}}^{\beta
}c(s,x_{2},\dots ,x_{n})ds\Big),\\
\overline{e}(x)=\int_{x}^{x^{0}}L(t,\overline{p}(t)a(t))dt, \\
\overline{p}(x)=1+\int_{x_{1}}^{\beta }c(s,x_{2},\dots ,x_{n})
\exp \Big(\int_{x_{1}}^{s}c(\tau ,x_{2},\dots ,x_{n})d\tau \Big) ds,
\end{gather*}
for $x\in \mathbb{R}_{+}^{n}.\Omega $ is defined in (\ref{3.6}) , where 
$\Omega ^{-1}$ is the inverse function of $\Omega $, and $\Omega
(1)+\int_{x_{1}}^{\beta }c(s,x_{2},\dots ,x_{n})ds$ is in the domain of 
$\Omega $ for $x\in \mathbb{R}_{+}^{n}$.
\end{corollary}

\begin{remark}
\textrm{We note that in the special case $n=2$, $x=(x_{1},x_{2})\in %
\mathbb{R}_{+}^{2}$, and $x^{0}=(\infty ,\infty )$ in corollary \ref{c2}.
our estimate reduces to Theorem 3.4 obtained by Dragomir and Kim \cite{2}.}
\end{remark}

\begin{remark} \rm
(1) All the preceding results remain valid when\newline
$b(x)\int_{\alpha }^{x_{1}}c(s,x_{2},\dots ,x_{n})g(u(s,x_{2},\dots
,x_{n}))ds$ is replaced by the general function \newline
$b_{i}(x)\int_{\alpha
_{i}}^{x_{i}}c_{i}(x_{1,.}dots,x_{i-1},s_{i},x_{i+1},\dots
,x_{n})g(u(x_{1,.}\dots ,x_{i-1},s_{i},x_{i+1},\dots ,x_{n}))ds_{i}$,\newline
with $i=2,\dots ,n$ fixed, and $\alpha _{i}\geq 0$, $x=(x_{1},\dots
x_{n})\in \mathbb{R}_{+}^{n}$ and with $\alpha _{i}\leq s_{i}\leq x_{i}$, 
$x_{i},s_{i}\in \mathbb{R}_{+}$,

\noindent (2) The above results remain valid when\newline
$b(x)\int_{x_{1}}^{\beta }c(s,x_{2},\dots ,x_{n})g(u(s,x_{2},\dots
,x_{n}))ds $ is replaced by the general function\newline
$b_{i}(x)\int_{x_{i}}^{\beta _{i}}c_{i}(x_{1,.}\dots
,x_{i-1},s_{i},x_{i+1},\dots ,x_{n})g(u(x_{1,.}\dots
,x_{i-1},s_{i},x_{i+1},\dots ,x_{n}))ds_{i}$,\newline
with $i=2,\dots ,n$ fixed, and $\alpha _{i}\geq 0$, $x=(x_{1},\dots
x_{n})\in \mathbb{R}_{+}^{n}$ and with $\alpha _{i}\leq s_{i}\leq x_{i}$, $%
x_{i},s_{i}\in \mathbb{R}_{+}$, where $b_{i}(x)$ and $c_{i}(x)$ be
real-valued nonnegative continuous function defined for 
$x\in \mathbb{R}_{+}^{n}$, for all $i=2,\dots ,n$.
\end{remark}

In a future work, we will present some applications for the results obtained
in this work.

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