\documentclass[reqno]{amsart}
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\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
\begin{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}
\end{equation}
subject to the impulsive conditions
\begin{equation}
\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}
\end{equation}
and the initial conditions
\begin{equation}
x(0)=x_{0},\quad x'(0)=x_{0}^{\ast }, \label{3}
\end{equation}
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{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