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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 80, pp. 1--6.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/80\hfil Some integral inequalities]
{Some nonlinear integral inequalities arising in differential equations}

\author[K. Boukerrioua, A. Guezane-Lakoud\hfil EJDE-2008/80\hfilneg]
{Khaled Boukerrioua, Assia Guezane-Lakoud}  % in alphabetical order

\address{Khaled Boukerrioua \newline
University of Guelma, Guelma, Algeria}
\email{khaledV2004@yahoo.fr}

\address{Assia Guezane-Lakoud \newline
Badji-Mokhtar University, Annaba, Algeria}
\email{a\_guezane@yahoo.fr}

\thanks{Submitted Ocotber 31, 2007. Published May 28, 2008.}
\subjclass[2000]{26D15, 26D20}
\keywords{Integral inequalities; nonlinear function}

\begin{abstract}
 The aim of this paper is to obtain estimates for functions
 satisfying some nonlinear integral inequalities. 
 Using ideas from Pachpatte \cite{p1}, we generalize the
 estimates presented in  \cite{j1,p2}.
\end{abstract}

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

\section{Introduction and main results}

Integral inequalities are a necessary tools in the study of
properties of the solutions of linear and nonlinear differential
equations, such as  boundness, stability, uniqueness, etc.
This justifies the intensive investigation on integral inequalities;
see for example \cite{b1,p3,p4}.
The aim of this paper is to establish some new
generalizations of integral inequalities that have a wide
applications in the study of differential equations. More
precisely, using some ideas from \cite{p1}, we give further
generalizations of the results presented in \cite{j1,p2}.


We begin by giving some material necessary for our study.
We denote by $\mathbb{R}$ the set of real numbers, and by $\mathbb{R}_{+}$ the
nonnegative real numbers

\begin{lemma} \label{lem1}
For $x\in \mathbb{R}_{+}$, $y\in \mathbb{R}_{+}$, $1/p+1/q=1$, we have
$x^{1/p}y^{1/q}\leq x/p+y/q$.
\end{lemma}

Now we state the main results of this work.

\begin{theorem} \label{thm1}
Let $u,a,b,g$ and $h$ \ be real valued nonnegative continuous functions
defined on $\mathbb{R}_{+}$, $p,r,q$  be real  non negative
constants. Assume that the functions
\[
\frac{a(t)+p/r}{b(t)},\quad
\frac{a(t)+r/p}{b(t)},\quad
\frac{a(t)+\min(r/p,q/p)}{b(t)}
\]
 are nondecreasing and that
\begin{equation} \label{e2.1}
u^{p}(t)\leq a(t)+b(t)\int_{0}^{t}[ g(s)u^{q}(s)+h(s)u^{r}(s)] ds.
\end{equation}
(1) If  $0<r<p<q$, then
\begin{equation} \label{e2.2}
u(t) \leq (a(t)+\frac{p}{r})^{1/p}\Big( 1
-(\frac{q}{p}-1)\int_{0}^{t}b(s)(g(s)
+\frac{r}{p}h(s))\big(a(s)+\frac{p}{r}\big)^{\frac{q}{p}-1}ds\Big) ^{\frac{1}{p-q}}
\end{equation}
for  $t\leq \beta _{p,q,r}$, where
\[
\beta _{p,q,r}=\sup \big\{t\in \mathbb{R}_{+}:(\frac{q}{p}
-1)\int_{0}^{t}b(s)(g(s)+\frac{r}{p}h(s))\big(a(s)+\frac{p}{r}\big)^{\frac{q
}{p}-1}ds<1\big\} .
\]
(2) If  $\ 0<p<r<q$, then
\[
u(t) \leq (a(t)+\frac{r}{p})^{1/p}\Big( 1
  -(\frac{q}{p}-1)\int_{0}^{t}b(s)(g(s)+h(s))\big(a(s)+\frac{r}{p}
\big)^{\frac{q}{p}-1}ds\Big) ^{\frac{1}{p-q}}
\]
for $t\leq \beta _{p,q,r}$, where
\[
\beta _{p,q,r}=\sup \big\{ t\in R_{+}:(\frac{q}{p}-1)\int
_{0}^{t}b(s)(g(s)+h(s))\big(a(s)+\frac{r}{p}\big)^{\frac{q}{p}
-1}ds<1\big\} .
\]
(3) If  $0<\ p<q$ and $p<r$, then
\begin{align*}
u(t) &\leq (a(t)+\min (\frac{r}{p},\frac{q}{p}))^{1/p}
\Big(1-(\max (\frac{q}{p},\frac{r}{p})-1)\int
_{0}^{t}b(s)(g(s)+h(s))\big(a(s) \\
&\quad +\min (\frac{r}{p},\frac{q}{p})\big)^{\max (\frac{q}{p},\frac{r}{p})-1}ds
\Big) ^{\frac{1}{p(1-\max \{q/p,r/p\})}}
\end{align*}
for $t\leq \beta _{p,q,r}$, where
\begin{align*}
 \beta _{p,q,r}=\sup \big\{& t\in R_{+}:
 \Big( \max (\frac{q}{p},\frac{r}{p})-1\Big)
 \int_{0}^{t}b(s)(g(s)+h(s))\big(a(s)\\
 &+\min (\frac{r}{p},\frac{q}{p})\big)
 ^{\max (\frac{q}{p},\frac{r}{p})-1}ds<1\big\} .
\end{align*}
\end{theorem}

\begin{theorem} \label{thm2}
Suppose that the hypothesis of Theorem \ref{thm1}  hold and the function
 $b(t)$ is decreasing. Let $c$ be a real valued nonnegative continuous and
nondecreasing function for $t\in \mathbb{R}_{+}$. Also assume that
\begin{equation} \label{e2.5}
u^{p}(t)\leq c^{p}(t)+b(t)\int_{0}^{t}[ g(s)u^{q}(s)+h(s)u^{r}(s)] ds\,.
\end{equation}
(1) If $0<r<p<q$, then
\begin{equation} \label{e2.6}
u(t)\leq c(t)(1+\frac{p}{r})^{1/p}\Big\{ 1-(\frac{q}{p}
-1)\int_{0}^{t}(1+\frac{p}{r})^{\frac{q}{p}-1}b(s)K(s)ds\Big\}
^{\frac{1}{p-q}}
\end{equation}
for  $t\leq \beta _{p,q,r}$, where
\begin{gather*}
\beta _{p,q,r}=\sup \big\{ t\in R_{+}:(\frac{q}{p}-1)\int_{0}^{t}(1+
\frac{p}{r})^{\frac{q}{p}-1}b(s)K(s)ds<1\big\},\\
K(s)=g(s)c(s)^{q-p}+\frac{r}{p}h(s)c(s)^{r-p}.
\end{gather*}
(2) If $t\in \mathbb{R}+$ and $\ 0<p<r<q$, then
\[ %2.7
u(t) \leq c(t)(1+\frac{r}{p})^{1/p}  \\
\Big( 1-(\frac{q}{p}-1)\int_{0}^{t}b(s)K(s)(1+\frac{r}{p})^{\frac{
q}{p}-1}ds\Big)^{\frac{1}{p-q}}
\]
for $t\leq \beta _{p,q,r}$, where
\begin{gather*}
\beta _{p,q,r}=\sup \big\{ t\in R_{+}:(\frac{q}{p}-1)\int
_{0}^{t}b(s)K(s)(1+\frac{r}{p})^{\frac{q}{p}-1}ds<1\big\} ,
\\
K(s)=(g(s)c(s)^{q-p}+h(s)c(s)^{r-p}).
\end{gather*}
(3) If $t\in \mathbb{R}+$ and $0<p<q$, $p<r$, then
\begin{align*} %2.8
u(t) &\leq (1+\min (\frac{r}{p},\frac{q}{p}))^{1/p}
\Big(1-(\max (\frac{q}{p},\frac{r}{p})-1)\int_{0}^{t}b(s)K(s) \\
&\quad\times (1+\min (\frac{r}{p},\frac{q}{p}))
^{\max (\frac{q}{p},\frac{r}{p})-1}ds
\Big) ^{\frac{1}{p(1-\max (\frac{q}{p},\frac{r}{p}))}}
\end{align*}
for  $t\leq \beta _{p,q,r}$, where
\begin{align*}
\beta _{p,q,r}=\sup \Big\{&
t\in R_{+}:\Big( \max (\frac{q}{p},\frac{r}{p})-1)\int_{0}^{t}b(s)K(s)(1\\
&+ \min (\frac{r}{p},\frac{q}{p})\Big)
^{\max (q/p, r/p)-1}ds<1
\end{align*}
and $K(s)=(g(s)c(s)^{q-p}+h(s)c(s)^{r-p})$.
\end{theorem}

Note that in Theorems \ref{thm1} and  \ref{thm2}, we have studied the case $p<q$.
For the case $p>q$, similar results are given in \cite{j1}.


\begin{proof}[Proof of Theorem \ref{thm1}]
(1) Define a function
\[
v(t)=\int_{0}^{t}\left[ g(s)u^{q}(s)+h(s)u^{r}(s)\right] ds\,.
\]
then from inequality \eqref{e2.1} and Lemma \ref{lem1}, we deduce that
\begin{gather}
u^{q}(t) \leq (a(t)+b(t)v(t))^{q/p}, \label{e2.9}\\ 
u^{r}(t) \leq (a(t)+b(t)v(t))^{r/p},  \\ %2.10
u^{r}(t) \leq \frac{r}{p}(a(t)+b(t)v(t))+\frac{p-r}{p},   \\
u^{r}(t) \leq \frac{r}{p}(a(t)+b(t)v(t)+\frac{p-r}{r}),  \\
u^{r}(t) \leq \frac{r}{p}(a(t)+b(t)v(t)+\frac{p}{r}). 
\end{gather}
 Since $\frac{q}{p}>1$,
which implies
\begin{equation} \label{e2.11}
v'(t)\leq \big[ g(t)+\frac{r}{p}h(t)\big] \big[ a(t)+b(t)v(t)+
\frac{p}{r}\big] ^{q/p}.
\end{equation}
Taking into account that the function $\frac{a(t)+\frac{p}{r}}{b(t)}$ is
nondecreasing for $0\leq t\leq \tau $, we have
\[ %2.11
v'(t)\leq M(t)(\frac{a(\tau )+\frac{p}{r}}{b(\tau )}+v(t)),
\]
where
\[ %  \label{e2.12}
M(t)=b(t)(g(t)+\frac{r}{p}h(t))(a(t)+b(t)v(t)+\frac{p}{r})^{\frac{q}{p}-1},
\]
consequently
\[ %2.14
v(t)+\frac{a(\tau )+\frac{p}{r}}{b(\tau )}\leq \frac{a(\tau )+\frac{p}{r}
}{b(\tau )}\exp \int_{0}^{t}M(s)ds.
\]
For $\tau =t$, we can see that
\begin{equation} \label{e2.15}
a(t)+b(t)v(t)+\frac{p}{r}\leq (a(t)+\frac{p}{r})\exp \int_{0}^{t}M(s)ds,
\end{equation}
then the function $M(t)$\ can be estimated as
\begin{equation} \label{e2.16}
M(t)\leq b(t)(g(t)+\frac{r}{p}h(t))(a(t)+\frac{p}{r})^{\frac{q}{p}-1}.\exp
\int_{0}^{t}(\frac{q}{p}-1)M(s)ds.
\end{equation}
Let
\begin{equation} \label{e2.17}
L(t)=(\frac{q}{p}-1)M(t).
\end{equation}
Now we estimate the expression $L(t)\exp (-\int_{0}^{t}L(s)ds)$ by
using \eqref{e2.16} to obtain
\[ %2.18
L(t)\exp (\int_{0}^{t}-L(s)ds)\leq (\frac{q}{p}-1)b(t)(g(t)+\frac{r}{
p}h(t))(a(t)+\frac{p}{r})^{\frac{q}{p}-1}.
\]
Observing that
\begin{align*}
L(t)\exp (\int_{0}^{t}-L(s)ds)
&= \frac{d}{dt}(-\exp(\int_{0}^{t}-L(s)ds)),   \\
&\leq (\frac{q}{p}-1)b(t)(g(t)+\frac{r}{p}h(t))(a(t)+\frac{p}{r})^{\frac{q
}{p}-1}.
\end{align*}
Then integrate  from $0$ to $t$ to obtain
\[ %2.20
(1-\exp \int_{0}^{t}-L(s)ds)\leq \int_{0}^{t}(\frac{q}{p}
-1)b(s)(g(s)+\frac{r}{p}h(s))\big(a(s)+\frac{p}{r}\big)^{\frac{q}{p}-1}ds.
\]
Replacing $L(t)$ by its value in \eqref{e2.17}, we obtain
\[ %2.21
(1-\exp \int_{0}^{t}(1-\frac{q}{p})M(s)ds)\leq \int_{0}^{t}(
\frac{q}{p}-1)b(s)(g(s)+\frac{r}{p}h(s))\big(a(s)+\frac{p}{r}\big)^{\frac{q}{p}
-1}ds,
\]
then
\[ %2.22
\exp \int_{0}^{t}M(s)ds\\
\leq \Big\{ 1-\Big[
\int_{0}^{t}(\frac{q}{p}-1)b(s)(g(s)
+ \frac{r}{p}h(s))\big(a(s)+\frac{p}{r}\big)^{\frac{q}{p}-1}ds
\Big] \Big\} ^{\frac{p}{p-q}}.
\]
Using this inequality, \eqref{e2.15}, and \eqref{e2.1} we obtain \eqref{e2.2}.
This completes  the proof of stament (1).

(2) for $t\in \mathbb{R}+$ and $\ 0<p<r<q$, from \eqref{e2.9} we have
\begin{gather*}
u^{q}(t)\leq \big(a(t)+b(t)v(t)+\frac{r}{p}\big)^{q/p}, \\
v'(t)\leq (g(t)+h(t))\big(a(t)+b(t)v(t)+\frac{r}{p}\big)
^{q/p}.
\end{gather*}
Since $\frac{a(t)+\frac{r}{p}}{b(t)}$ is nondecreasing for $0\leq t\leq
\tau $,
\[
v'(t)\leq M(t)(\frac{a(\tau )+\frac{r}{p}}{b(\tau )}+v(t)),
\]
where
\[
M(t)=b(t)(g(t)+h(t))\big(a(t)+b(t)v(t)+\frac{r}{p}\big)^{\frac{q}{p}-1}.
\]
By the same method as in the proof of the first part, we have
\[
u(t) \leq (a(t)+\frac{r}{p})^{1/p}
\Big( 1-(\frac{q}{p}-1)\int_{0}^{t}b(s)(g(s)+h(s))\big(a(s)+\frac{r}{p}
\big)^{\frac{q}{p}-1}ds\Big) ^{\frac{1}{p-q}},
\]
where
\[
\beta _{p,q,r}=\sup \big\{ t\in R_{+}:(\frac{q}{p}-1)\int
_{0}^{t}b(s)(g(s)+h(s))\big(a(s)+\frac{r}{p}\big)^{\frac{q}{p}
-1}ds<1\big\} .
\]

(3) For $t\in \mathbb{R}+$ and $p<r$, $p<q$, we have
\[
u^{q}(t)\leq (a(t)+b(t)v(t)+\min (\frac{r}{p},\frac{q}{p}))
^{\max (r/p,q/p)},
\]
which gives
\begin{gather*}
v'(t)\leq (g(t)+h(t))(a(t)+b(t)v(t)+\min (\frac{r}{p},\frac{q}{p}
))^{\max (r/p,q/p)}
\\
v'(t)\leq M(t)(\frac{a(\tau )+\min (\frac{r}{p},\frac{q}{p})}{
b(\tau )}+v(t)),
\end{gather*}
where
\[
M(t)=b(t)(g(t)+h(t))(a(t)+b(t)v(t)+\min (\frac{r}{p},\frac{q}{p}))
^{\max (\frac{r}{p},\frac{q}{p})-1}.
\]
Using the proof of the first part of Theorem \ref{thm1}, we get the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
Since $c(t)$ is nonnegative, continuous and nondecreasing, it follows that
\eqref{e2.5} can be written as
\begin{equation}
(\frac{u(t)}{c(t)})^{p}\leq 1+b(t)\int_{0}^{t}\big[ g(s)(\frac{u(s)}{
c(s)})^{q}.c(s)^{q-p}+h(s)(\frac{u(s)}{c(s)})^{r}c(s)^{r-p}\big] ds.
\end{equation}
Then  a direct application of the inequalities established in
Theorem \ref{thm1}
gives the required results.
\end{proof}

\section{Application}

As an application of Theorem \ref{thm1}, consider  the nonlinear differential
equation
\begin{equation} \label{e3.1}
u^{p-1}(t)u'(t)+g(t)u^{q}(t)=l(t,u(t)).
\end{equation}
Assume that $p<q$, $u:\mathbb{R}_{+}\to \mathbb{R}$,
$g:\mathbb{R}_{+}\to \mathbb{R}_{+}$,
$l:\mathbb{R}_{+}\times \mathbb{R} \to \mathbb{R}$,
\begin{equation} \label{e3.2}
| l(t,u(t))| \leq \alpha (t)+h(t)|
u(t)| ^{r},
\end{equation}
$\alpha :\mathbb{R}_{+}\to \mathbb{R}_{+}$,
$h:\mathbb{R}_{+}\longrightarrow \mathbb{R}_{+}$ are continuous functions.

Integrating \eqref{e3.1} from $0$ to $t$, we have
\[
\frac{u^{p}(t)}{p}-\frac{u_{0}^{p}}{p}+\int_{0}^{t}g(s)u^{q}(s)ds=
\int_{0}^{t}l(s)ds.
\]
From this equality and \eqref{e3.2}, we obtain
\[
| u(t)| ^{p}\leq a(t)+p\int_{0}^{t}[
g(s)| u(s)| ^{q}+h(s)| u(s)| ^{r}] ds,
\]
where $a(t)=| u_{0}| ^{p}+p\int_{0}^{t} \alpha (s) ds$.
Applying Theorem \ref{thm1}, we find explicit bounds of the solution $u(t)$ of the
equation \eqref{e3.1} in different cases where $p<r$ and $p>r$.

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\end{thebibliography}



\end{document}