\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 39, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/39\hfil Existence of positive solutions] {Existence of positive solutions to some impulsive second-order integrodifferential equations} \author[R. Atmania\hfil EJDE-2010/39\hfilneg] {Rahima Atmania} \address{Rahima Atmania \newline Laboratory of Applied Mathematics (LMA)\\ Department of Mathematics, University of Badji Mokhtar Annaba \\ P.O. Box 12, Annaba 23000, Algeria} \email{atmanira@yahoo.fr} \thanks{Submitted January 27, 2010. Published March 16, 2010.} \thanks{Supported by the LMA lab, University of Badji Mokhtar Annaba, Algeria} \subjclass[2000]{34A37, 34G20} \keywords{Integrodifferential equation; impulses; positive solution; cone theory; \hfill\break\indent Arzela-Ascoli theorem} \begin{abstract} In this article, we consider an initial-value problem for second-order nonlinear integrodifferential equations with impulses in a Banach space. By using the monotone iterative technique in a cone together with Arzela-Ascoli theorem and the dominated convergence theorem, we establish the existence of positive solutions of such a problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} It is well known that the theory of impulsive differential equations is an important area of research which has been investigated in the last few years by many authors in several directions. So, a great deal of techniques and methods have been used in the study of the second order impulsive differential equations to obtain some quantitative or qualitative results regarding the solutions of such new problems, see for instance \cite{guo,nie,zha} . We recall that the impulsive differential equations can model natural phenomena and evolving processes which are subject to abrupt changes such as shocks, food shortenings, natural disasters and so on. Thus, we may treat these short-term perturbations as impulses that affect later on the behavior of the solutions. To learn more about the recent developments of the theory of impulsive equations we refer the reader to the works of Benchohra et al \cite{ben} without forgetting to quote the book by Lakshmikantam et al. \cite{lak}, we recall that the latter is considered as one of the basic references in this domain. Our contribution in this paper is the investigation of positive solutions to the following second order nonlinear integrodifferential equation $$x''(t)=F(t,x(\delta _1(t)),x'(\delta _{2}(t)),Tx(t),Sx(t)), \quad t\in J\backslash \{ t_{k};\text{ }k=1,2\dots \} \label{1}$$ subject to the impulsive conditions $$\begin{gathered} \Delta x=x(t^{+})-x(t^{-})=I_{k}(t,x,x'),\quad t=t_{k};\; k=1,2\dots , \\ \Delta x'=x'(t^{+})-x'(t^{-})=\hat{I}_{k}(t,x,x'),\quad t=t_{k};\; k=1,2\dots , \end{gathered} \label{2}$$ and the initial conditions $$x(0)=x_{0},\quad x'(0)=x_{0}^{\ast }, \label{3}$$ where for $x\in X$, a given Banach space, and $t\in J=[0,+\infty )$, the functionals $T$ and $S$ are defined as follows: \begin{gather*} Tx(t)= \int_{0}^{t}g(t,s,x(\delta _{3}( s)),\int_{0}^{s}k(s,\tau ,x( \delta _{4}(\tau )))d\tau )ds, \\ Sx(t)=\int_{0}^{+\infty }h(t,s,x(\delta _{5}( s) ))ds. \end{gather*} So, inspired by the results in \cite{guo} devoted to the existence of positive solutions to the corresponding problem for $Sx( t)=0$, $Tx(t)=\int_{0}^{t}K( t,s)x(s)ds$ and $\delta _{i}(t)=t$, $i=1,\dots 4$, we have established the existence of positive solutions for problem \eqref{1}-\eqref{3} by using the monotone iterative technique in a cone of a Banach space $X$ together with Ascoli-Arzela theorem and the dominated convergence theorem on an infinite time interval with the presence of an infinite number of impulses. \section{Preliminaries} We first set the following assumptions: \begin{itemize} \item[(H1)] $00$ such that $\theta \leq x\leq y$ implies $\| x\| \leq N\| y\|$, and a cone is said to be \textit{regular} (resp. \textit{fully regular}) if $x_1 \leq \dots \leq x_n \leq \dots \leq y$ for some $y\in X$ (resp. $\| x_1\| \leq \dots \leq \| x_{n}\| \leq \dots \leq \sup_{n}\| x_{n}\| <\infty$) implies that there is $x_{n}\in X$, such that $\|x_n-x\|\to 0$ as $n\to \infty$. Of course, the full regularity of a cone implies its regularity which in turn implies its normality. By a \textit{positive solution} to the problem \eqref{1}-\eqref{3}, we mean a function $x\in C^{2}(J\setminus \{t_{k}\} _{k\geq 1},X) \cap \mathcal{SPC}^{1}(J,P)$ satisfying \eqref{1}-\eqref{3} and $x(t)\in P\backslash \{ \theta \}$, for every $t\in J$. We need the following lemma whose proof can be handled without any difficulty. \begin{lemma} \label{lem2.1} A function $x\in C^{2}(J\setminus \{ t_{k}\} _{k\geq 1},X)\cap \mathcal{SPC}^{1}(J,P)$ is a solution to the problem \eqref{1}-\eqref{3} if and only if $x\in \mathcal{PC}(J,P)$ satisfies the impulsive integral equation \begin{aligned} x(t)&= x_{0}+tx_{0}^{^{\ast }}+\int_{0}^{t}(t-s)F(s,x(\delta _1(s)),x'(\delta _{2}(s)),Tx(s) ,Sx(s))ds \label{a} \\ &\quad +\sum_{0