\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 $$\label{e2.1} u^{p}(t)\leq a(t)+b(t)\int_{0}^{t}[ g(s)u^{q}(s)+h(s)u^{r}(s)] ds.$$ (1) If $0q$, 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 $$\label{e2.11} v'(t)\leq \big[ g(t)+\frac{r}{p}h(t)\big] \big[ a(t)+b(t)v(t)+ \frac{p}{r}\big] ^{q/p}.$$ 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 $$\label{e2.15} a(t)+b(t)v(t)+\frac{p}{r}\leq (a(t)+\frac{p}{r})\exp \int_{0}^{t}M(s)ds,$$ then the function $M(t)$\ can be estimated as $$\label{e2.16} M(t)\leq b(t)(g(t)+\frac{r}{p}h(t))(a(t)+\frac{p}{r})^{\frac{q}{p}-1}.\exp \int_{0}^{t}(\frac{q}{p}-1)M(s)ds.$$ Let $$\label{e2.17} L(t)=(\frac{q}{p}-1)M(t).$$ 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 $\ 0r$. \begin{thebibliography}{0} \bibitem{b1} D. Bainov and P. Simeonov; \textit{Integrale Inequalities and application}, Kluwer Academic Publishers, New York, 1992. \bibitem{j1} F. C. Jiang and F. W. Meng; \textit{Explicit bounds on some new nonlinear integral inequalities.} J. Comput. Appl. Math. 205, (2007). 479-486. \bibitem{p1} B. G. Pachpatte; \textit{Inequalities for Differential and integral equation}, Academic Press, New York, 1998. \bibitem{p2} B. G. Pachpatte; \textit{On some new inequalities related to certain inequality arising in the theory of differential equations.} JMAA251, 736-751, 2000. \bibitem{p3} B. G. Pachpatte; \textit{Some new finite difference inequalities}, Comput. Math. Appl. 28, 227-241, 1994. \bibitem{p4} B. G. Pachpatte; \textit{On Some discrete inequalities useful in the theory of certain partial finite difference equations}, Ann. Differential Equations, 12, 1-12, 1996. \end{thebibliography} \end{document}