\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
\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)] $0<t_1<t_{2}\dots <t_{k}\dots $ and
$\lim_{k\to \infty } t_{k}=+\infty$;

\item[(H2)] $x_{0}$ and $x_{0}^{\ast }$ are given
values in a cone $P$ of a Banach space $(X,\| .\|)$ which
defines a partial ordering in $X$  as follows: $x\leq y$
 if and only if $y-x\in P$.
We assume that $x_{0}$ is different from the null vector
$\theta $ of $P$.

\item[(H3)] $F\in C(J\times P\times P\times P\times P,P)$,
$g\in C(J\times J\times P\times P,P)$,
$k\in C(J\times J\times P,P)$,
$h\in C( J\times J\times P,P)$, $I_{k}$,
$\hat{I}_{k}\in C( J\times P\times P,P)$
and
$\delta _{i}\in C(J,J)$ are given functions such that
$\delta_{i}(t)\leq t$, $t\in J$, such that 
$\lim\limits_{t\to\infty}\delta_i(t)=\infty$ for $i=1,\dots 5$.
\end{itemize}

In the sequel we shall use the following spaces:
\\
$\mathcal{PC}(J,X)=\{ x:J\to X:x( t)$
 is continuous at $t\neq t_{k}$, left continuous at
$t=t_{k}$, and $x(t_{k}^{+})$ exists for $k=1,2\dots \}$;
\\
$\mathcal{PC}(J,P)=\{ x\in \mathcal{PC}( J,X):\theta \leq
x(t);\;t\in J\}$
\\
$\mathcal{PC}^{1}(J,X)=\{ x\in \mathcal{PC}( J,X)
:x'(t)$ is continuous at $t\neq t_{k}$, left continuous
at $t=t_{k}$, and $x'(t_{k}^{+})$ exists for
$k=1,2\dots \}$.

Moreover, we introduce the Banach spaces
\[
\mathcal{SPC}(J,X)=\{ x\in \mathcal{PC}(J,X):
\sup_{t\in J}\frac{\| x(t) \|}{t+1}<\infty \} ,
\]
with  the norm  $\| x\| _{S}=\sup\limits_{t\in J}\frac{\| x(t)\| }{t+1}$,
and
\[
\mathcal{SPC}^{1}(J,X)=\{ x\in \mathcal{PC}^{1}( J,X) :
\sup_{t\in J} \frac{\| x(t)\| }{t+1}<\infty  \text{ and }
\sup_{t\in J}\| x'(t)\| <\infty \}
\]
 with the norm $\| x\| _{D}=\max(\| x\| _{S},\| x'\| _{C})$,
where $\| x'\| _{C}=\sup\limits_{t\in J}\| x'(t)\|$. We note that
\[
\mathcal{SPC}(J,P)=\{ x\in \mathcal{SPC}( J,X):\theta \leq
x(t);\text{ }t\in J\}
\]
is a cone in $\mathcal{SPC}(J,X)$, and
\[
\mathcal{SPC}^{1}(J,P)=\{ x\in \mathcal{SPC}^{1}( J,X)
:\theta \leq x(t)\text{ and } \theta\leq x'(t);\;t\in J\}
\]
is a cone in $\mathcal{SPC}^{1}(J,X)$.

We recall that a cone   is said to be \textit{normal}  if
there exists a constant $N>0$ 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<t_{k}<t} \big[ I_{k}(t_{k},x(t_{k})
,x'(t_{k}))+(t-t_{k})\hat{I}
_{k}(t_{k},x(t_{k}),x'(t_{k}))\big] .
\end{aligned}
\end{equation}
\end{lemma}

Now we are in position to state and prove our main results.

\section{Main results}

Besides the mentioned hypotheses (H1) to (H3), we add the
following:
\begin{itemize}
\item[(H4)] $\| g(t,s,x,y)\| \leq m_1(t,s)\| x\| +m_{2}(s)\| y\|$;
 $\| k(t,s,x)\| \leq m_{3}(t,s)\|x\|$
and $\| h(t,s,x)\| \leq m_{4}(t,s)\| x\|$, for every $t,s\in J$  and
$x,y\in P$, where the functions $~m_1$, $m_{3}$, $m_{4}$ and $m_{2}$
satisfy
\begin{gather*}
\sup_{t\in J}\int_{0}^{t}m_1(t,s)ds=m_1^{\ast }<\infty , \quad
\sup_{t\in J}\int_{0}^{+\infty}m_{4}(t,s)ds=m_{4}^{\ast }<\infty,\\
 \sup_{t\in J} \int_{0}^{t}m_{2}(s)\int_{0}^{s}m_{3}(s,\tau )d\tau\,
ds=m_{2}^{\ast }<\infty .
\end{gather*}

\item[(H5)]  $\|F(t,x,y,z,w)\| \leq p_1(t)\| x\|+p_{2}(t)\| y\|
+p_{3}(t)\| z\|+p_{4}(t)\| w\| +q(t)$,
for every $t\in J$  and $x,y,z,w\in P$. The functions
$p_{i}$, $i=1,\dots ,4$, and $q$  satisfy
\[
\int_{0}^{+\infty }(s+1)p(s)ds=p^{\ast }<\infty ,
\int_{0}^{+\infty }q(s)ds=q^{\ast }<\infty \]
with $p(t)=\max\limits_{i=1,2,3,4} p_{i}(t)$.


\item[(H6)] $\| I_{k}(t,x,y)\| \leq a_{k}\| x\| +b_{k}\| y\| +c_{k}$,
 $\| \hat{I}_{k}(t,x,y)\| \leq d_{k}\| x\|+e_{k}\| y\| +f_{k}$,
for every $t\in J$  and $x,y\in P$; $a_{k}$, $b_{k}$, $c_{k}$, $d_{k}$,
$e_{k}$, $f_{k}$; $k=1,2,\dots $, being positive constants such that
\begin{gather*}
\sum_{k=1}^\infty (t_{k}+1)a_{k} =a<\infty;\quad
\sum_{k=1}^\infty b_{k}=b<\infty;\quad
\sum_{k=1}^\infty  c_{k}=c<\infty; \\
\sum_{k=1}^\infty (t_{k}+1)d_{k} =d<\infty;\quad
\sum_{k=1}^\infty e_{k}=e<\infty ;\quad
\sum_{k=1}^\infty f_{k}=f<\infty\,.
\end{gather*}

\item[(H7)] For each $t\in J$ and $x_1$, $x_{2}$,
$y_1$, $y_{2}$, $z_1$, $z_{2}$, $w_1$ and $w_{2}\in P$  such that
\[
x_1\leq x_{2}, \quad y_1\leq y_{2}, \quad z_1\leq z_{2}
\quad\text{and } w_1\leq w_{2}
\]
we have  $I_{k}(t,x_1,y_1)\leq I_{k}(t,x_{2},y_{2})$;
$\hat{I}_{k}(t,x_1,y_1)\leq \hat{I}_{k}(t,x_{2},y_{2})$;
$k=1,2,\dots $
and  $F(t,x_1,y_1,z_1,w_1)\leq F(t,x_{2},y_{2},z_{2},w_{2})$.
\end{itemize}
Now define the operator $B$  as follows
\begin{equation}
\begin{aligned}
Bx(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\\
&\quad +\sum_{0<t_{k}<t} [ I_{k}(t_{k},x(t_{k})
,x'(t_{k}))+(t-t_{k})\hat{I}
_{k}(t_{k},x(t_{k}),x'(t_{k}))] ,
\quad t\in J.
\end{aligned} \label{b}
\end{equation}


\begin{lemma}\label{lem2}
Assume that {\rm (H1)-(H4)} hold. Then $T$
and $S$ are bounded operators from $\mathcal{SPC}(J,X)$ into
$\mathcal{SPC}(J,X)$
and $\ \| T\| \leq T^{\ast };\|S\| \leq S^{\ast }$, where
$T^{\ast }=m_1^{\ast}+m_{2}^{\ast }$ and $S^{\ast }=m_{4}^{\ast }$.
Furthermore, $T$ and $S$ map $\mathcal{SPC}(J,P)$ into itself.
\end{lemma}

\begin{proof}
First, we note that if $x\in \mathcal{SPC}(J,X)$, then
$Tx\in C(J,X)$ and $Sx\in C(J,X)$.
On the other hand,  from (H4), we have the  estimates
\begin{align*}
\frac{\| Tx(t)\| }{1+t}
&\leq \int_{0}^{t}m_1(t,s)\frac{1+\delta _{3}(s)}{1+t}\frac{
\| x(\delta _{3}(s))\| }{1+\delta _{3}(s)}ds \\
&\quad +\int_{0}^{t}m_{2}(s)\int_{0}^{s}m_{3}(s,\tau )\frac{
1+\delta _{4}(\tau )}{1+t}\frac{\| x(\delta _{4}(\tau
))\| }{1+\delta _{4}(\tau
)}d\tau ds \\
&\leq \| x\| _{S}\big[ \int_{0}^{t}m_1(t,s)ds+
\int_{0}^{t}m_{2}(s)\int_{0}^{s}m_{3}(s,\tau )d\tau ds\big] ,
\end{align*}
so that
\[
\| Tx\| _{S}\leq T^{\ast }\| x\| _{S}, \quad  t\in J,
\]
and
\[
\frac{\| Sx(t)\| }{1+t}\leq
\int_{0}^{+\infty }m_{4}(t,s)\frac{1+\delta _{5}(s)}{1+t}
\frac{\| x(\delta _{5}(s))\| }{
1+\delta _{5}(s)}ds\leq \| x\|
_{S}\int_{0}^{+\infty }m_{4}(t,s)ds,
\]
yielding
\[
\| Sx\| _{S}\leq S^{\ast }\| x\| _{S}.
\]
Hence $Tx$ and $Sx\in \mathcal{SPC}(J,X)$ from which we get
$\| T\| \leq T^{\ast }$  and $\| S\|\leq S^{\ast }$.
Finally, it is clear that
$T:\mathcal{SPC}(J,P) \to \mathcal{SPC}(J,P)$ and
$S:\mathcal{SPC}( J,P)\to \mathcal{SPC}(J,P)$, and so, the lemma
is proved.
\end{proof}


\begin{lemma} \label{lem3.2}
If the hypotheses {\rm (H1)-(H6)} are satisfied, then the
operator $B$ maps $\mathcal{SPC}^{1}(J,P)$ into itself and
\begin{equation}
\| Bx\| _{D}\leq \alpha \| x\| _{D}+\beta ,
\quad x\in \mathcal{SPC}^{1}(J,P),  \label{c}
\end{equation}
where
\[
\alpha =(p^{\ast }(2+T^{\ast }+S^{\ast })+a+b+d+e), \\
\beta = 2\max (\| x_{0}\| ,\|x_{0}^{\ast }\| )+q^{\ast }+c+f.
\]
\end{lemma}

\begin{proof}
 From (H5), (H6) we have, for each $x\in \mathcal{SPC}^{1}(J,P)$,
\begin{equation}
\int_{0}^{+\infty }\| F(s,x(\delta _1(s)),x'(\delta
_{2}(s)),Tx(s),Sx( s)) \| ds\leq p^{\ast }(2+T^{\ast
}+S^{\ast })\| x\| _{D}+q^{\ast } \label{7}
\end{equation}
and
\begin{gather}
\sum_{k=1}^\infty \| I_{k}(t_{k},x(t_{k}),x'(t_{k}))\|
\leq ( a+b) \| x\| _{D}+c,  \label{8}
\\
\sum_{k=1}^\infty \|
\hat{I}_{k}(t_{k},x( t_{k}),x'( t_{k}) )\| \leq
(d+e)\| x\| _{D}+f. \label{9}
\end{gather}
Next, for each $x\in \mathcal{SPC}^{1}(J,P)$, we have by virtue of
\eqref{b}, lemma \ref{lem2} and \eqref{7}-\eqref{9},
\begin{align*}
\frac{\| Bx(t)\| }{1+t}
&\leq \frac{1}{1+t}\| x_{0}\| +\frac{t}{1+t}\| x_{0}^{\ast }\|
\\
&\quad +\int_{0}^{t}\frac{t-s}{1+t}\| F(s,x(\delta
_1(s)),x'(\delta _{2}(s)),Tx(s),Sx( s)
)\| ds \\
&\quad +\sum_{0<t_{k}<t}\frac{1}{1+t}\| I_{k}(t_{k},x(
t_{k}),x'(t_{k}))\|\\
&\quad +\sum_{0<t_{k}<t}\frac{t-t_{k}}{1+t}\| \hat{I}
_{k}(t_{k},x(t_{k}),x'(t_{k}))\|
\\
&\leq (p^{\ast }(2+T^{\ast }+S^{\ast })+a+b+d+e)
\| x\| _{D} \\
&\quad +(2\max (\| x_{0}\| ,\| x_{0}^{\ast
}\| )+q^{\ast }+c+f).
\end{align*}
Thus,
\begin{equation}
\| Bx\| _{S}\leq \alpha \| x\| _{D}+\beta ,\quad
 x\in \mathcal{SPC}^{1}(J,P).  \label{10}
\end{equation}
On the other hand, differentiating \eqref{b} we get for each
$t\in J$,
\begin{align*}
(Bx)'(t)&=x_{0}^{^{\ast }}+\int_{0}^{t}F(s,x(\delta
_1(s)),x'(\delta
_{2}(s)),Tx(s),Sx(s))ds \\
&\quad +\underset{0<t_{k}<t}{\sum }\hat{I}_{k}(t_{k},x(t_{k})
,x'(t_{k})).
\end{align*}
Thanks to \eqref{7}-\eqref{9}, we have
\[
\| (Bx)'(t)\| \leq \| x_{0}^{^{\ast
}}\| +p^{\ast }(2+T^{\ast }+S^{\ast })\|
x\| _{D}+q^{\ast }+( d+e)\| x\| _{D}+f.
\]
It follows that
\begin{equation}
\begin{aligned}
\| (Bx)'\| _{C}
&\leq ( p^{\ast }(2+T^{\ast}+S^{\ast })+d+e)\| x\| _{D}+\|
x_{0}^{^{\ast }}\| +q^{\ast }+f \\
&\leq \alpha \| x\| _{D}+\beta ,\quad x\in \mathcal{SPC}
^{1}(J,P),
\end{aligned} \label{11}
\end{equation}
Hence, \eqref{10} and \eqref{11} give \eqref{c}; therefore
$Bx\in \mathcal{SPC}^{1}(J,P)$.
\end{proof}

\begin{theorem}\label{thm}
Assume that $P$ is a fully regular cone and hypotheses
{\rm (H1)-(H7)} are satisfied. If $\alpha <1$, then
problem \eqref{1}-\eqref{3}  has at least one positive solution $x$.
In addition, it satisfies
\begin{equation}
\| x\| _{D}\leq \frac{\beta }{1-\alpha }.  \label{12}
\end{equation}
\end{theorem}

\begin{proof}
First, we need to prove that the impulsive integral equation has
at least one positive solution.
Define the sequence $(x_{n}(t))_{n\geq 1}$ as follows
\begin{equation}
x_{0}(t)=x_{0},\quad x_{n}(t)=Bx_{n-1}( t);\quad n=1,2,\dots  \label{17}
\end{equation}
where $B$  is defined by \eqref{b} and satisfies \eqref{c}. Then
$x_{n}\in \mathcal{SPC}^{1}( J,P)$ and
\[
\| x_{0}\| _{D}\leq \beta ,\quad
\|x_{n}\| _{D}\leq \alpha \| x_{n-1}\| _{D}+\beta;\quad n=1,2,\dots\,.
\]
We obtain by induction
\begin{equation}
\| x_{n}\| _{D}\leq \alpha ^{n}\beta +\alpha
^{n-1}\beta +\dots +\alpha \beta +\beta =\frac{(1-\alpha
^{n+1})}{(1-\alpha )}\beta \leq \frac{\beta }{1-\alpha }.
\label{18}
\end{equation}
On the other hand, the sequence $(x_{n}'( t))_{n\geq 1}$  defined by
\begin{align*}
x_{n}'(t)
&= x_{0}^{^{\ast}}+\int_{0}^{t}F(s,x_{n}(\delta _1(s)),x_{n}'
(\delta_{2}(s)),Tx_{n}(s),Sx_{n}(s))ds \\
&\quad +\sum_{0<t_{k}<t} \hat{I}_{k}(t_{k},x_{n}(t_{k})
,x_{n}'(t_{k})).
\end{align*}
satisfies the  estimate
\begin{equation}
\| x_{n}'\| _{C}\leq \frac{\beta }{1-\alpha }.
\label{19}
\end{equation}
We infer from (H7) that $(x_{n}(t))_{n\geq 1} $ and
$(x_{n}'(t))_{n\geq 1}$, for $t\in J$, satisfy
\begin{gather*}
\theta \leq x_{n}(t)\leq x_{n+1}(t);\quad n=1,2,\dots , \\
\theta \leq x_{n}'(t)\leq x_{n+1}'(t) ;\quad n=1,2,\dots ,
\end{gather*}
 As $P$  is a fully regular cone, it follows from
\eqref{18} and \eqref{19} that
\begin{gather}
\lim_{n\to \infty } x_{n}(t)=x( t);\quad t\in J,
\lim_{n\to \infty } x_{n}'(t) =y(t);\quad t\in J.
\end{gather}

Now, since $x_{n}(t)$ is an equicontinuous function on each
closed interval $J_{k}$; $k=1,2,\dots $ such that
$J_{0}=[ 0,t_1] $, $J_{k}=] t_{k},t_{k+1}] $; $k=1,2,\dots $,  then by
using Ascoli-Arzela theorem we deduce the existence of a
subsequence $( x_{n_{i}}(t))\subset ( x_{n}( t))$ such that
$x_{n_{i}}(t)$ converges uniformly to $x(t)$ on each $J_{k}$;
$k=1,2,\dots $, and so, as the cone is normal,
the sequence $(x_{n}(t))$ converges uniformly to
$x(t)$ on each $J_{k}$; $k=1,2,\dots $.
 Therefore, $x\in \mathcal{PC} (J,P)$, and taking into account
 \eqref{18}, $x\in \mathcal{SPC}(J,P)$ and
$\| x\|_{S}\leq \beta /(1-\alpha )$.

Next, by a double differentiation of $x_{n}(t)$; $n=1,2,\dots $, we
obtain
\begin{equation}
\begin{aligned}
\| x_{n}''(t)\|
&\leq p(t)(1+t)[ 2+T^{\ast}+S^{\ast }] \| x_{n-1}\| _{D}
+q(t); t\neq t_{k},\quad k=1,2,\dots,\\
&\leq p(t)(1+t)[ 2+T^{\ast }+S^{\ast }]
\frac{\beta }{1-\alpha }+q(t)=f(t).
\end{aligned} \label{21}
\end{equation}

We note that the function $f(t)$ is bounded on any finite
interval. We observe from \eqref{21} that $x_{n}'(t)$, $n\geq 1$
are equicontinuous functions on each $J_{k}$; $k=1,2,\dots $.
 We conclude that the sequence $( x_{n}'(t))$ converges uniformly
 on each $J_{k}$; $k=1,2,\dots $, to  $y(t)$.
Hence, $x'(t)$ exists and $x'(t)=$ $y(t)$ on each $J_{k}$;
$k=1,2,\dots $, $x'\in \mathcal{PC}(J,P)$ and
$\|x'\| _{C}\leq \beta /(1-\alpha )$. Consequently,
$x\in \mathcal{SPC}^{1}(J,P)$ and $\| x\|_{D}\leq \beta /(1-\alpha )$.
On the other hand, since
\begin{equation*}
\| k(s,\tau ,x_{n-1}(\delta _{4}(\tau )))-k(s,\tau,x(\delta _{4}(\tau)))\| 
\leq 2m_{3}(t,s)\frac{\beta }{1-\alpha },
\end{equation*}
by applying the dominated convergence theorem we obtain
\[
\lim_{n\to \infty } \int_{0}^{s}k(s,\tau
,x_{n-1}(\delta _{4}(\tau )))d\tau =\int_{0}^{s}k(s,\tau
,x(\delta _{4}(\tau )))d\tau ,\quad s\in J.
\]
By the same reasoning for $g$  and $h$  we obtain
\[
\lim_{n\to\infty} Tx_{n-1}(t)=Tx( t) ,\quad
\lim_{n\to\infty} Sx_{n-1}(t) =Sx(t) ,\quad t\in J
\]
and once again, for $F$,
\begin{equation}
\begin{aligned}
&\lim_{n\to\infty} \int_{0}^{t}F(
s,x_{n-1}(\delta _1(s)),x_{n-1}'(\delta
_{2}(s)),Tx_{n-1}(s),Sx_{n-1}(s))
\\
&=\int_{0}^{t}F(s,x(\delta _1(s)),x'(\delta
_{2}(s)),Tx(s),Sx(s)),\text{ }t\in J.
\end{aligned}\label{22}
\end{equation}
Likewise, $I_{k}$ and $\hat{I}_{k}$ , $k=1,2,\dots $ satisfy
the inequalities
\begin{gather*}
\|
I_{k}(t_{k},x_{n}(t_{k}),x_{n}'(t_{k}))-I_{k}(t_{k},x(t_{k}),x'(
t_{k}))\|  \leq 2(a_{k}+b_{k})\frac{\beta }{1-\alpha }
+2c_{k} \\
\|
\hat{I}_{k}(t_{k},x_{n}(t_{k}),x_{n}'(t_{k}))
-\hat{I}_{k}(t_{k},x(t_{k}),x'(t_{k}))\|
\leq 2(d_{k}+e_{k})\frac{\beta }{1-\alpha }+2f_{k};
\end{gather*}
which give at once the  limits
\begin{gather*}
\lim_{n\to\infty} I_{k}(t_{k},x_{n}(t_{k}),x_{n}'( t_{k}))
=I_{k}(t_{k},x(t_{k}),x'(t_{k})) \\
\lim_{n\to\infty} \hat{I}_{k}(t_{k},x_{n}(
t_{k}),x_{n}'(t_{k})) =\hat{I} _{k}(t_{k},x(t_{k}),x'(t_{k})).
\end{gather*}
Now, since the series
$\sum\limits_{0<t_{k}<t} I_{k}$ and $\sum\limits_{0<t_{k}<t} \hat{I}_{k}$
 converge, we obtain
\begin{equation}
\begin{aligned}
&\lim_{n\to\infty} \sum_{0<t_{k}<t}
[ I_{k}(t_{k},x_{n}(t_{k}),x_{n}'( t_{k})
)+(t-t_{k})\hat{I}_{k}(t_{k},x_{n}( t_{k})
,x_{n}'(t_{k}))]  \\
&=\sum_{0<t_{k}<t} [ I_{k}(t_{k},x(t_{k})
,x'(t_{k}))+(t-t_{k})\hat{I}
_{k}(t_{k},x(t_{k}),x'(t_{k}))] .
\end{aligned}  \label{23}
\end{equation}
Taking the limit of $x_{n}(t)=Bx_{n-1}(t)$, when
$n\to \infty $, we get from \eqref{22} and \eqref{23}, that
$x(t)$ is a solution of \eqref{a}.
Consequently, $x(t)$ is a solution of \eqref{1}-\eqref{3}.
Finally, as $x_{0}\in P\backslash \{ \theta\} $, then $x(t)$
 is positive, which completes the proof.
\end{proof}


\begin{example} \label{exa3.4} \rm
Let us consider the  initial-value problem
\begin{equation}
\begin{gathered}
x''(t)=\frac{1}{(t+1)^{102}}
\big(e^{-t}(t+1)^{102}+x(t)+\frac{x'(t)}{2}
+Tx(t)+Sx(t)\Big),\quad t\neq k;\; k\geq1, \\
\Delta x =I_{k}(k,x,x')=\frac{1}{1000^{k}}\Big( x(k)+\sqrt{x'(k)}
+\cos k\Big),\quad k=1,2\dots , \\
\Delta x'=\hat{I}_{k}(k,x,x')=\frac{1}{1000^{k}}
\Big(\sqrt{x(k)}+x'(k)+\cos k\Big)
,\quad k=1,2\dots , \\
x(0)=\frac{1}{2};\quad x'(0)=\frac{1}{2},
\end{gathered}  \label{x}
\end{equation}
where
\begin{gather*}
Tx(t)=\int_{0}^{t}\Big(\frac{1}{(t+1+s)
^{3/2}}x(s)+\frac{e^{-s}}{\sqrt{s}}\int_{0}^{s}\frac{1}{
(s+\tau )^{1/2}}d\tau \Big)ds, \\
Sx(t)=\int_{0}^{+\infty }\frac{t-s}{t+1}e^{-s}\sqrt{
x(s)}ds.
\end{gather*}
It is easy to check that all the hypotheses cited in the preliminaries
and lemmas  are satisfied.  Hence Theorem \ref{thm}
ensures the existence of at least one positive solution to
\eqref{x} whenever
\[
\alpha =(p^{\ast }(2+T^{\ast }+S^{\ast })+a+b+d+e)<1.
\]
This solution satisfies the impulsive integral equation \eqref{a} with
$\| x\| _{D}\leq \frac{\beta }{1-\alpha }$; where
$\beta =1+q^{\ast }+c+f$. According to the above notations a
straightforward computation gives the following:
\begin{gather*}
p^{\ast }=\int_{0}^{+\infty }\frac{(s+1)}{( s+1)
^{102}}ds=\frac{1}{100};\quad
q^{\ast}=\int_{0}^{+\infty }e^{-s}ds=1
\\
T^{\ast } = m_1^{\ast }+m_{2}^{\ast },\quad
m_1^{\ast } =\sup_{t\in J} \int_{0}^{t}\frac{1}{(t+1+s)^{3/2}}
ds<2, \\
m_{2}^{\ast } =\sup_{t\in J} \int_{0}^{t}\frac{e^{-s}}{
\sqrt{s}}\int_{0}^{s}\frac{1}{(s+\tau )^{1/2}}d\tau
ds=2(\sqrt{2}-1),
\\
 S^{\ast }=m_{4}^{\ast }=\sup_{t\in J} \int_{0}^{+
\infty }\frac{t-s}{t+1}e^{-s}ds=1,
\\
a =d=\sum_{k=1}^\infty (k+1)\frac{1}{
1000^{k}}<\frac{1}{999}+\frac{1}{1000}, \\
b =e=c=f=\sum_{k=1}^\infty \frac{1}{1000^{k}}=\frac{1}{999}.
\end{gather*}
Therefore, $\alpha =\frac{1}{100}(2\sqrt{2}+m_1^{\ast }+2)+
\frac{2}{999}+2a<1$  and $\ \beta =2(1+\frac{1}{999})$.
\end{example}


 \subsection*{Acknowledgments}
The author wishes to thank Professor S. Mazouzi and the anonymous
referee for their valuable suggestions for improving this article.

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\end{document}