\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2004(2004), No. 18, pp. 1--12.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/18\hfil Existence of solutions]
{Existence of solutions to second order ordinary differential
equations having finite limits at $\pm \infty$}

\author[Cezar Avramescu \& Cristian Vladimirescu \hfil EJDE-2004/18\hfilneg]
{Cezar Avramescu \& Cristian Vladimirescu} % in alphabetical order

\address{Cezar Avramescu \hfill\break
Centre for Nonlinear Analysis and its Applications, University of
Craiova, Al.I. Cuza Street, No. 13, Craiova RO-200585, Romania}
\email{zarce@central.ucv.ro, cezaravramescu@hotmail.com}

\address{Cristian Vladimirescu \hfill\break
Department of Mathematics, University of Craiova, Al.I. Cuza
Street, No. 13, Craiova RO-200585, Romania}
\email{cvladi@central.ucv.ro,  vladimirescucris@hotmail.com}

\date{}
\thanks{Submitted February 14,  2003. Published February 9, 2004.}
\subjclass[2000]{34B15, 34B40, 34C37, 54C60} 
\keywords{Nonlinear boundary-value problem, set-valued mappings, 
\hfill\break\indent
 boundary-value problems on infinite intervals}

\begin{abstract}
 In this article, we study the boundary-value problem
 $$
 \ddot{x}=f(t,x,\dot{x}), \quad x(-\infty )=x(+\infty ), \quad
 \dot{x}(-\infty) = \dot{x}(+\infty ) \,.
 $$
 Under adequate hypotheses and using the Bohnenblust-Karlin 
 fixed point theorem for multivalued mappings, we establish 
 the existence of solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

Let $f:\mathbb{R}^{3}\to \mathbb{R}$ be a continuous mapping. Consider the
infinite boundary-value problem
\begin{gather}\label{ec1.1}
\ddot{x}=f(t,x,\dot{x}) ,\\
\label{ec1.2}
x(-\infty )=x(+\infty ) ,\quad \dot{x}(-\infty ) =\dot{x}(+\infty ) ,
\end{gather}
where $x(\pm \infty ) $ and $\dot{x}(\pm \infty ) $
denote the limits
\begin{equation}\label{ec1.3}
x(\pm \infty ) =\lim_{t\to \pm \infty }x(t)\quad\mbox{and}\quad
\dot{x}(\pm \infty ) =\lim_{t\to \pm \infty } \dot{x}(t),
\end{equation}
which are assumed to be finite.
Problem \eqref{ec1.1}-\eqref{ec1.2} may be considered as a
gene\-ralization of problem \eqref{ec1.1} with boundary condtions
\begin{equation}\label{ec1.4}
x(a) =x(b) ,\quad\dot{x}(a) =\dot{x}(b) ,
\end{equation}
as $a\to -\infty $ and $b\to +\infty $. The bilocal boundary-value problem
\eqref{ec1.1}-\eqref{ec1.4} is closely related to
the problem of finding periodic solutions to \eqref{ec1.1}.
The reader is referred to \cite{17,19,20} where extensive use of topological degree theory is made to study this problem.

Problem \eqref{ec1.1}-\eqref{ec1.2} is
related to the so-called {\it convergent solutions}, i.e. the solutions
defined on $\mathbb{R}_{+}=[0,+\infty )$ (or $\mathbb{R}$) and having finite limit to $
+\infty $ (respectively $-\infty$), see
\cite{4,5,14,15,16}.
For studies on \eqref{ec1.1}-\eqref{ec1.2}
using variational methods, we refer the reader to \cite{1,2,3,13,20,21}.
In \cite{12} the
existence of the solutions to the equation \eqref{ec1.1} with the
boundary conditions $x(\infty)=\dot x(\infty)=0$
is studied for $f(t,u,v)=g(t)v-u+h(t,u)$.
Through the Schauder-Tychonoff and Banach fixed point Theorems
estimates for the solutions are found.

When $f$ is a differentiable function, \eqref{ec1.1} can be
written as
\begin{equation}\label{ec1.5}
\ddot{x}=a(t,x,\dot{x}) \dot{x}+b(t,x,\dot{x})
x+c(t) ,
\end{equation}
where $a$, $b:\mathbb{R}^{3}\to \mathbb{R}$, $c:\mathbb{R}\to \mathbb{R}$,
$a(t,u,v) :=\int_{0}^{1}\frac{\partial f}{\partial u}(t,su,sv)\,ds$,
$b(t,u,v) :=$ $\int_{0}^{1}\frac{\partial f}{\partial v}(t,su,sv)\,ds$ and
$c(t) :=f(t,0,0) $, for all $t$, $u$, $v\in \mathbb{R}$.

Sufficient conditions for the existence of solutions to the linear
problem
\begin{equation}\label{ec1.6}
\ddot{x}=a(t) \dot{x}+b(t) x+c(t) ,
\end{equation}
with boundary condition \eqref{ec1.2}, were given in \cite{11}.
By using this result, in the real Banach space
\[
X:=\big\{ x\in C^{2}(\mathbb{R}) : (\exists )\,x(\pm \infty ),
\;(\exists ) \,\dot{x}(\pm \infty ) \big\}
\]
endowed with the uniform convergence topology on $\mathbb{R}$ one defines an
operator $T:X\to 2^{X}$ which maps $u\in X$ into the set of
the solutions to the problem \eqref{ec1.7}-\eqref{ec1.2}, where
\begin{equation}\label{ec1.7}
\ddot{x}=a(t,u(t) ,\dot{u}(t) ) \dot{x}+b(t,u(t) ,\dot{u}(t) ) x+c(t) .
\end{equation}

Next one considers the restriction of $T$ to a bounded, convex and
closed set $M$, conveniently chosen so that the
Bohnenblust-Karlin Theorem can be applied. The compactness of $T(M)$ is
established by using a characterization developed by the
the first author in \cite{4,6}.

The use of a multivalued operator $T$ is motivated by the
fact that one cannot determine a solution to the problem \eqref{ec1.7}-\eqref{ec1.2}
through an ``initial'' condition independent of $u$.

\section{Main result}

Let $a$, $b:\mathbb{R}^{3}\to \mathbb{R}$, $c:\mathbb{R}\to \mathbb{R}$ be
continuous functions, and let
\begin{gather*}
\alpha _{1}(t) :=\inf_{u,v\in \mathbb{R}}\big\{ a(t,u,v) \big\} ,\quad
\alpha _{2}(t) :=\sup_{u,v\in \mathbb{R}}\big\{ a(t,u,v) \big\} , \\
\beta (t) :=\sup_{u,v\in \mathbb{R}}\big\{ b(t,u,v)\big\} ,\quad
A_{i}(t) :=\exp \Big(\int_{0}^{t}\alpha_{i}(s)\, ds\Big) ,
\end{gather*}
for $i\in \{ 1,2\}$ and $t\in \mathbb{R}$. We shall assume that
$\alpha _{1}$, $\alpha _{2}$, $\beta $ are defined on $\mathbb{R}$.

Consider the following hypotheses, where  the integrals are considered in the
Riemann sense:
\begin{itemize}
\item[(A1)] The mappings $\alpha _{1}$ and $\alpha _{2}$ are bounded on $\mathbb{R}$,
and $\lim_{t\to \pm \infty }\alpha _{i}(t) =0$,
for $i\in \{ 1,2\}$

\item[(A2)] $\lim_{t\to \pm \infty }A_{i}(t) =0$
for $i\in \{ 1,2\}$

\item[(B1)] $0\leq b(t,u,v)$ for every $t$, $u$, $v\in \mathbb{R}$ and
$\lim_{t\to \pm \infty }\beta (t) =0$

\item[(B2)]
$ \int_{-\infty }^{+\infty }\big(A_{i}(t) \cdot \int_{0}^{t}
\frac{\beta (s) }{A_{i}(s) }ds\big) dt\in \mathbb{R}$
for  $i\in \{ 1,2\}$

\item[(B3)]
$\int_{-\infty }^{+\infty }\frac{\beta (t) }{A_{i}(t)}dt<+\infty$,
for $i\in \{ 1,2\}$

\item[(C1)] $\int_{-\infty }^{+\infty }|c(t)| dt<+\infty$

\item[(C2)] $\int_{-\infty }^{+\infty}\Big(\int_{-t}^{t}\frac{|c(s)|}{A_{i}(s)}
ds\Big) dt\in \mathbb{R}$ for $i\in \{ 1,2\} $.
\end{itemize}
Our main result is as follows:

\begin{theorem}\label{main}
If the hypotheses (A1)--(A2), (B1)--(B3), (C1)--(C2) are
satisfied, then \eqref{ec1.5}-\eqref{ec1.2} has a solution.
\end{theorem}

Since
\[
\lim_{t\to \pm \infty }\frac{A_{i}(t) }{A_{i}(
t) \cdot \int_{0}^{t}\frac{\beta (s) }{A_{i}(
s) }ds}=\lim_{t\to \pm \infty }\frac{1}{\int_{0}^{t}\frac{
\beta (s) }{A_{i}(s) }ds}
\]
is a real number by hypothesis (B3), it follows by hypothesis (B2), via a well known convergence criterion for Riemann integrals, that for each $i\in \{ 1,2\} $,
\begin{equation}\label{ec2.1}
\int_{-\infty }^{+\infty }A_{i}(t) dt<+\infty \,.
\end{equation}
Similarly, by hypothesis (A2),
\[
\lim_{t\to \pm \infty }\frac{\beta (t) }{\frac{\beta
(t) }{A_{i}(t) }}=0,
\quad
\lim_{t\to \pm \infty }\frac{|c(t)| }{\frac{|c(t)|}{A_{i}(t) }}=0,
\]
it follows, by hypothesis (B3), that
\begin{equation}\label{ec2.2}
\int_{-\infty }^{+\infty }\beta (t) dt<+\infty ,
\end{equation}
and, by hypothesis (C1),
\begin{equation}\label{ec2.3}
\int_{-\infty }^{+\infty }\frac{|c(t)|}{A_{i}(t) } dt<+\infty ,
\end{equation}
for each $i\in \{ 1,2\} $.

\begin{remark}\label{Remark2.1} \rm
(i) One can replace the hypothesis (B2) by
\begin{itemize}
\item[(B2')] $\int_{-\infty}^{+\infty}A_{i}(t)dt< +\infty$\,.
\end{itemize}
(ii) Assumption (B2') does not imply (C2).
\end{remark}

\noindent(i) Indeed, since (B3) implies the boundedness of the mapping
$\int_{0}^{(\cdot)} \frac{\beta(s)}{A_{i}(s)}ds$ and therefore, (B2')
implies (B2).

\noindent (ii) It is sufficient to choose $c(t)=A_{i}(t) $, for all $t\in
\mathbb{R}$, where $i=1$ or $i=2$.
\smallskip

For proving our main result we  use the following theorem.

\begin{theorem}[{Bohnenblust-Karlin \cite[p. 452]{22}}] \label{Theorem2.1}
Let $X$ be a
Banach space and $M\subset X$ be a convex closed subset of it.
Suppose that $T:X\to 2^{X}$ is a multivalued
operator on $X$ satisfying the following hypotheses:
\begin{itemize}
\item[(a)] $T(M) \subset 2^{M}$ and $T$ is upper semicontinuous
\item[(b)] the set $T(M) $ is relatively compact
\item[(c)] for every $x\in M$, $T(x) $ is a non-empty convex closed set.
\end{itemize}
Then $T$ admits fixed points.
\end{theorem}

Recall that $T:M\to 2^{M}$ is \emph{upper semicontinuous} if for
every closed subset $A$ of $M$, the set
\[
T^{-1}(A) :=\big\{ x\in M : T(x) \cap A\neq \emptyset \big\}
\]
is also a closed subset of $M$.
Another useful result is the following Lemma.

\begin{lemma}[Barb\u{a}lat] \label{Lemma2.1}
If  $f:[0,+\infty )\to \mathbb{R}$ satisfies:
(a) $f$ is uniformly continuous and
(b) the integral $\int_{0}^{+\infty }f(t)\,dt$
exists and is finite,
then $\lim_{t\to +\infty }f(t) =0$.
\end{lemma}

The main idea of this paper is to build a multivalued operator $T$
defined on an adequate space which satisfies the hypotheses of the
Bohnenblust-Karlin Theorem.
We define
\[
X:=\big\{ x\in C^{2}(\mathbb{R}) : (\exists )
\;x(\pm \infty ) \mbox{ and }\dot{x}(\pm \infty) \big\} ,
\]
which, endowed with the usual norm,
\[
\|x\| :=\sup_{t\in \mathbb{R}}\max \big\{ | x(t) | ,| \dot{x}(t) | \big\},
\]
becomes a real Banach space. The relative compactness of the set
$T(M) $ be will be proved by using  the following Proposition.

\begin{proposition}[Avramescu \cite{4,6}] \label{Proposition2.1}
 A set $\mathcal{A}\subset X$ is relatively compact if and only if
the following conditions are fulfilled:
\begin{itemize}
\item[(a)] There exist $h_{1}$, $h_{2}\geq 0$ such that for every
$x\in \mathcal{A}$ and $t\in \mathbb{R}$, we have $| x(t) | \leq
h_{1}$ and $| \dot{x}(t) | \leq h_{2}$

\item[(b)] For every $K=[ a,b] \subset \mathbb{R}$ and $\varepsilon>0 $ there
exists $\delta =\delta (K,\varepsilon ) >0$ such that
for every $x\in \mathcal{A}$ and $t_{1}$, $t_{2}\in K$ with
$| t_{1}-t_{2}| <\delta $, we have
$| x(t_{1}) -x(t_{2}) | <\varepsilon$ and $| \dot{x}(t_{1}) -\dot{x}(t_{2})| <\varepsilon$

\item[(c)] For every $\varepsilon >0$ there exists $T=T(\varepsilon) >0$ such
that for every $t_{1}$, $t_{2}$ with $|t_{1}| $, $| t_{2}| >T$ and
$t_{1}\cdot t_{2}>0$, and for every $x\in \mathcal{A}$, we have
$| x(t_{1}) -x(t_{2}) | <\varepsilon$ and
$| \dot{x}(t_{1}) -\dot{x}(t_{2})| <\varepsilon $.
\end{itemize}
\end{proposition}

\section{Construction of the multivalued operator $T$}

Let $u\in C^{2}(\mathbb{R}) $ be arbitrary. Consider the problem
\begin{equation}\label{ec3.1}
\begin{gathered}
\ddot{x}=a_{u}(t) \dot{x}+b_{u}(t) x+c(t) \\
x(+\infty ) =x(-\infty ) ,\quad \dot{x}(+\infty ) =\dot{x}(-\infty ),
\end{gathered}
\end{equation}
where
$a_{u}(t) :=a(t,u(t) ,\dot{u}(t))$ and $b_{u}(t) =b(t,u(t) ,\dot{u}(t) )$.
Consider the homogeneous problem
\begin{equation}\label{ec3.2}
\begin{gathered}
\ddot{x}=a_{u}(t) \dot{x}+b_{u}(t) x \\
x(+\infty ) =x(-\infty ) ,\quad \dot{x}(+\infty ) =\dot{x}(-\infty ).
\end{gathered}
\end{equation}
Since
\[
x(t) =\exp \Big(\int_{0}^{t}y(s) ds\Big) ,
\quad t\in \mathbb{R}
\]
is a solution to $\ddot{x}=a_{u}(t) \dot{x}
+b_{u}(t) x$ if and only if $y$ is a solution to
\begin{equation}\label{ec3.3}
\dot{y}=a_{u}y+b_{u}-y^{2},
\end{equation}
we have
$a_{u}(t) y-y^{2}\leq \dot{y}\leq a_{u}(t)
y+b_{u}(t) ,\mbox{ for every }t\in \mathbb{R}$.

Let $v$, $w$ satisfy
\begin{equation}\label{ec3.4}
\begin{gathered}
\dot{v}=a_{u}(t) v-v^{2} \\
v(0) =\xi
\end{gathered}
\end{equation}
and
\begin{equation}\label{ec3.5}
\begin{gathered}
\dot{w}=a_{u}(t) w+b_{u}(t) \\
w(0) =\xi\,.
\end{gathered}
\end{equation}
Hence
\begin{gather*}
\dot{y}=a_{u}y+b_{u}-y^{2} \\
y(0) =\xi,
\end{gather*}
which implies
\begin{gather*}
v(t) \leq y(t) \leq w(t) ,\quad \mbox{if } t\geq 0, \\
w(t) \leq y(t) \leq v(t) ,\quad \mbox{if } t\leq 0.
\end{gather*}
Let
$\alpha _{u}(t) :=\exp \big(\int_{0}^{t}a_{u}(s) ds\big)$,
for every $t\in \mathbb{R}$. Thus
\begin{equation}\label{ec3.6}
\begin{gathered}
v(t) =\frac{\xi \alpha _{u}(t) }{1+\xi
\int_{0}^{t}\alpha _{u}(s) ds} \\
w(t) =\alpha _{u}(t) \big[ \xi +\int_{0}^{t}\frac{b_{u}(s) }{\alpha _{u}(s) }ds\big].
\end{gathered}
\end{equation}
Therefore,
\begin{gather*}
\frac{\xi \alpha _{u}(t) }{1+\xi \int_{0}^{t}\alpha _{u}(s) ds}
 \leq y(t) \leq \alpha _{u}(t)
\Big[\xi +\int_{0}^{t}\frac{b_{u}(s) }{\alpha _{u}(s) }ds
\Big] ,\quad \mbox{if }t\geq 0, \\
\alpha _{u}(t) \Big[ \xi +\int_{0}^{t}\frac{b_{u}(s) }{\alpha _{u}(s) }ds\Big]
\leq y(t) \leq \frac{\xi \alpha _{u}(t) }{1+\xi \int_{0}^{t}\alpha
_{u}(s) ds},\quad \mbox{if }t\leq 0.
\end{gather*}
We write
\begin{equation}\label{ec3.7}
g_{u}(t) \leq y(t) \leq G_{u}(t) ,\quad \mbox{for }t\in \mathbb{R},
\end{equation}
where
\begin{equation}\label{ec3.8}
g_{u}(t) :=\begin{cases}
\frac{\xi \alpha _{u}(t) }{1+\xi \int_{0}^{t}\alpha _{u}(
s) ds},&\mbox{if }t\geq 0 \\
\alpha _{u}(t) \Big[ \xi +\int_{0}^{t}\frac{b_{u}(s) }{\alpha _{u}(s) }ds\Big] ,
&\mbox{if }t\leq 0
\end{cases}
\end{equation}
and
\begin{equation}\label{ec3.9}
G_{u}(t) :=\begin{cases}
\alpha _{u}(t) \Big[ \xi +\int_{0}^{t}\frac{b_{u}(s) }{\alpha _{u}(s) }ds\Big] ,
&\mbox{if }t\geq 0 \\
\frac{\xi \alpha _{u}(t) }{1+\xi \int_{0}^{t}\alpha _{u}(s) ds},&\mbox{if }t\leq 0\,.
\end{cases}
\end{equation}
Let $y_{u}$ denote the solution to the equation \eqref{ec3.3} with the
initial condition
\[y_{u}(0) =\xi\,.
\]
Hence, $g_{u}(t) \leq y_{u}(t) \leq G_{u}(t)$, for every $t\in \mathbb{R}$.
 From \eqref{ec3.6} we see that
$y_{u}$ is defined for all $t \in \mathbb{R}$ if and only if
\[
\xi \in \Big(-\frac{1}{\int_{0}^{+\infty }\alpha _{u}(s) ds},
\frac{1}{\int_{-\infty }^{0}\alpha _{u}(s) ds}\Big)
:=(\lambda _{u},\mu _{u}) .
\]
We let $\lambda :=\sup_{u\in C^{2}(\mathbb{R}) }\left\{
\lambda _{u}\right\} $ and $\mu :=\inf_{u\in C^{2}(\mathbb{R})
}\left\{ \mu _{u}\right\} $.
Since
\begin{equation}\label{ec3.10}
\begin{gathered}
A_{1}(t) \leq \alpha _{u}(t) \leq A_{2}(
t) ,\quad \mbox{for every }t\geq 0\mbox{ and }u\in C^{2}(\mathbb{R})
\\
A_{2}(t) \leq \alpha _{u}(t) \leq A_{1}(
t) ,\quad \mbox{for every }t\leq 0\mbox{ and }u\in C^{2}(\mathbb{R})
\end{gathered}
\end{equation}
it follows that
\[
-\frac{1}{\int_{0}^{+\infty }A_{1}(t) dt}\leq -\frac{1}{
\int_{0}^{+\infty }\alpha _{u}(s) ds}\leq -\frac{1}{
\int_{0}^{+\infty }A_{2}(t) dt}:=\lambda
\]
and
\[
\mu :=\frac{1}{\int_{-\infty }^{0}A_{1}(t) dt}\leq \frac{1}{
\int_{-\infty }^{0}\alpha _{u}(s) ds}\leq \frac{1}{\int_{-\infty
}^{0}A_{2}(t) dt}.
\]
Therefore,
\begin{equation}\label{ec3.11}
-\frac{1}{\int_{0}^{+\infty }A_{2}(t) dt}:=\lambda <0<\mu :=
\frac{1}{\int_{-\infty }^{0}A_{1}(t) dt}.
\end{equation}
Let
\[
g(t) :=\inf_{u\in C^{2}(\mathbb{R}) }g_{u}(t)\quad
\mbox{and}\quad G(t) :=\sup_{u\in C^{2}(\mathbb{R})
}G_{u}(t) ,\mbox{ for }t\in \mathbb{R}.
\]
For $t\leq 0$, we have
\[
g_{u}(t) \geq \alpha _{u}(t) \Big[ \lambda
+\int_{0}^{t}\frac{b_{u}(s) }{\alpha _{u}(s) }ds\Big] \geq A_{1}(t)
\Big[ \lambda +\int_{0}^{t}\frac{\beta (s) }{A_{2}(s) }ds\Big]
\]
and for $t\geq 0$,
\[
g_{u}(t) \geq \frac{\lambda \alpha _{u}(t) }{
1+\lambda \int_{0}^{t}\alpha _{u}(s) ds}\geq \frac{\lambda
A_{2}(t) }{1+\lambda \int_{0}^{t}A_{2}(s) ds}.
\]
Thus
\begin{equation}\label{ec3.12}
g(t) :=\begin{cases}
\frac{\lambda A_{2}(t) }{1+\lambda \int_{0}^{t}A_{2}(s) ds},&\mbox{if }t\geq 0 \\
A_{1}(t) \Big[ \lambda +\int_{0}^{t}\frac{\beta (s)
}{A_{2}(s) }ds\Big] ,&\mbox{ if }t\leq 0\,.
\end{cases}
\end{equation}
Similarly
\begin{equation}\label{ec3.13}
G(t) :=\begin{cases}
A_{2}(t) \Big[ \mu +\int_{0}^{t}\frac{\beta (s) }{
A_{1}(s) }ds\Big] ,&\mbox{ if }t\geq 0 \\
\frac{\mu A_{1}(t) }{1+\mu \int_{0}^{t}A_{1}(s) ds},
&\mbox{ if }t\leq 0\,.
\end{cases}
\end{equation}
By hypothesis (A2), one has $g(\pm \infty )=G(\pm \infty ) =0$.
Thus for every $\xi \in (\lambda ,\mu ) $ and for
every $y$ solution to the equation \eqref{ec3.3} with the initial
condition $y(0) =\xi $, we have
\begin{equation}\label{ec3.14}
g(t) \leq y(t) \leq G(t) ,
\quad \mbox{for every }t\in \mathbb{R}.
\end{equation}
Let $\xi _{1}, \mbox{ }\xi _{2}\in (\lambda ,\mu ) $, $\xi _{1}\neq \xi
_{2}$ be arbitrary, and $y_{i}^{u}$ be the solution to the problem
\begin{gather*}
\dot{y}=a_{u}(t) y+b_{u}(t) -y^{2} \\
y(0) =\xi _{i}
\end{gather*}
where $i\in \{ 1,2\}$ and $u\in C^{2}(\mathbb{R})$.
Let $x_{i}^{u}(t) :=\exp (\int_{0}^{t}y_{i}^{u}(s) ds) $, for  $t\in \mathbb{R}$,
$i\in \{ 1,2\} $ and $u\in C^{2}(\mathbb{R})$. Then $x_{i}^{u}(0) =1$,
$\dot{x}_{i}^{u}(0) =\xi _{i}$,
$\dot{x}_{i}^{u}(t)=y_{i}^{u}(t) \cdot x_{i}^{u}(t) $,
for $t\in \mathbb{R}$, $i\in \{ 1,2\} $ and $u\in C^{2}(\mathbb{R})$.

Let us prove that, for every $i\in \{ 1,2\} $
and $u\in C^{2}(\mathbb{R}) $,  $x_{i}^{u}(\pm \infty ) $,
$\dot{x}_{i}^{u}(\pm \infty )$, exist and are finite.
Indeed, by relation \eqref{ec2.1},
\begin{align*}
x_{i}^{u}(+\infty ) &=\exp \Big(\int_{0}^{+\infty
}y_{i}^{u}(t) dt\Big) \\
&\leq \exp \Big(\int_{0}^{+\infty }A_{2}(t) \big[ \mu
+\int_{0}^{t}\frac{\beta (s) }{A_{1}(s) }ds\big] dt\Big) \\
&\leq \exp \Big\{ (\int_{0}^{+\infty }A_{2}(t) dt\Big)
\cdot \Big[ \mu +\int_{0}^{+\infty }\frac{\beta (s) }{A_{1}(s) }ds\Big] \Big\}
<+\infty ,
\end{align*}
and
\begin{align*}
x_{i}^{u}(-\infty ) &=\exp \Big(\int_{0}^{-\infty }y_{i}^{u}(t) dt\Big) \\
&\leq \exp \Big\{ \Big(\int_{0}^{-\infty }A_{1}(t) dt\Big)
\cdot \Big[ \lambda +\int_{0}^{-\infty }\frac{\beta (s) }{
A_{1}(s) }ds\Big] \Big\} <+\infty ,
\end{align*}
for every $i\in \{ 1,2\} $ and $u\in C^{2}(\mathbb{R}) $.
For $i\in \{ 1,2\} $ and $u\in C^{2}(\mathbb{R}) $,
\[
| x_{i}^{u}(t) | =\exp \Big(\int_{0}^{t}y_{i}^{u}(s) ds\Big) .
\]
Hence, for $t\geq 0$,
\[
\exp \Big(\int_{0}^{t}y_{i}^{u}(s) ds\Big) \leq \exp
\Big\{(\int_{0}^{+\infty }A_{2}(t) dt) \cdot \Big[ \mu
+\int_{0}^{+\infty }\frac{\beta (s) }{A_{1}(s) }ds \Big] \Big\} =:\delta _{1}
\]
and for $t\leq 0$,
\[
\exp \Big(\int_{0}^{t}y_{i}^{u}(s) ds\Big) \leq \exp \Big\{
(\int_{0}^{-\infty }A_{1}(t) dt) \cdot \Big[
\lambda +\int_{0}^{-\infty }\frac{\beta (s) }{A_{1}(
s) }ds\Big] \Big\} =:\delta _{2}.
\]
Therefore, taking $M_{1}:=\max \left\{ \delta _{1},\delta
_{2}\right\} >0$, we have
$| x_{i}^{u}(t) | \leq M_{1}$, for every $t\in \mathbb{R}$, $i\in \{ 1,2\} $,
and $u\in C^{2}(\mathbb{R})$.

Since $g$ and $G$ are continuous with $g(\pm \infty ) =G(
\pm \infty ) =0$ it follows that they are bounded on $\mathbb{R}$. But
\[
g(t) \leq y_{i}^{u}(t) \leq G(t) ,\quad
\mbox{for every }t\in \mathbb{R},\;i\in \{ 1,2\} \mbox{ and }u\in
C^{2}(\mathbb{R}) .
\]
Hence, there exists a constant $\delta _{3}>0$ such that
\[
| y_{i}^{u}(t) | \leq \delta _{3},\quad \mbox{for }
t\in \mathbb{R},\;i\in \{ 1,2\} \mbox{ and }u\in C^{2}(\mathbb{R})
\]
and so
\[
| \dot{x}_{i}^{u}(t) | \leq M_{1}\cdot \delta
_{3}=: M_{2},\quad \mbox{for }t\in \mathbb{R},\; i\in \{ 1,2\}
\mbox{ and }u\in C^{2}(\mathbb{R}) .
\]
For $u\in C^{2}(\mathbb{R}) $ the general solution to the
nonhomogeneous equation
\begin{equation}\label{ec3.15}
\ddot{x}=a_{u}(t) \dot{x}+b_{u}(t) x+c(t)
\end{equation}
is
\begin{equation}\label{ec3.16}
\begin{aligned}
x(t) &=\gamma _{1}^{u}x_{1}^{u}(t) +\gamma_{2}^{u}x_{2}^{u}(t)
+x_{2}^{u}(t) \cdot \int_{0}^{t}x_{1}^{u}(s) \cdot
\frac{c(s) }{(\xi _{2}-\xi _{1}) \alpha _{u}(s) }ds \\
&\quad -x_{1}^{u}(t) \cdot \int_{0}^{t}x_{2}^{u}(s) \cdot
\frac{c(s) }{(\xi _{2}-\xi _{1}) \alpha _{u}(s) }ds,
\end{aligned}
\end{equation}
with $\gamma _{1}^{u}$, $\gamma _{2}^{u}\in \mathbb{R}$.
 From the condition $x(+\infty ) =x(-\infty )$,
we have
\begin{equation}\label{ec3.17}
\begin{aligned}
&\gamma _{1}^{u}\cdot \left[ x_{1}^{u}(+\infty )
-x_{1}^{u}(-\infty ) \right] +\gamma _{2}^{u}\cdot \left[
x_{2}^{u}(+\infty ) -x_{2}^{u}(-\infty ) \right] \\
&=x_{1}^{u}(+\infty ) \cdot \int_{0}^{+\infty }x_{2}^{u}(
 s) \cdot \frac{c(s) }{(\xi _{2}-\xi _{1})\alpha _{u}(s) }\,ds\\
&\quad -x_{1}^{u}(-\infty ) \cdot \int_{0}^{-\infty }x_{2}^{u}(s) \cdot
\frac{c(s) }{(\xi _{2}-\xi _{1}) \alpha _{u}(s) }\,ds \\
&\quad +x_{2}^{u}(-\infty ) \cdot \int_{0}^{-\infty }x_{1}^{u}(s) \cdot
\frac{c(s) }{(\xi _{2}-\xi _{1})\alpha _{u}(s) }\,ds   \\
&\quad -x_{2}^{u}(+\infty ) \cdot \int_{0}^{+\infty }x_{1}^{u}(s) \cdot
\frac{c(s) }{(\xi _{2}-\xi _{1})\alpha _{u}(s) }\,ds.
\end{aligned}
\end{equation}
Now we prove that the relation \eqref{ec3.17} is satisfied
by infinitely many pairs $(\gamma _{1}^{u},\gamma _{2}^{u}) $, $
u\in C^{2}(\mathbb{R}) $.
Indeed, if we denote by
\[
d_{1}:=x_{1}^{u}(+\infty ) -x_{1}^{u}(-\infty ) ,
\quad d_{2}:=x_{2}^{u}(+\infty ) -x_{2}^{u}(-\infty ),
\]
and $d_{3}$ the right hand side of \eqref{ec3.17}, then we have to
consider only three cases.

\noindent {\it Case 1}. If $d_{1}\neq 0$ and $d_{2}=0$, it follows that
$\gamma _{1}^{u}=\frac{d_{3}}{d_{1}}$  and $\gamma _{2}^{u}\in \mathbb{R}$;
similarly, if $d_{1}=0$ and $d_{2}\neq 0$, it follows that
$\gamma _{1}^{u}\in \mathbb{R}\mbox{ and }\gamma _{2}^{u}=\frac{d_{3}}{d_{2}}$.

\noindent {\it Case 2}. If $d_{1}\neq 0$ and $d_{2}\neq 0$, it follows that
\[
\gamma _{1}^{u}=\frac{d_{3}-d_{2}\gamma _{2}^{u}}{d_{1}}\quad\mbox{and}\quad
\gamma_{2}^{u}\in \mathbb{R}.
\]
{\it Case 3}. If $d_{1}=0$ and $d_{2}=0$, we show that $d_{3}=0$ (and so the
solutions are $\gamma _{1}^{u}$, $\gamma _{2}^{u}\in \mathbb{R}$).

Indeed, in this case, $x_{1}^{u}(+\infty ) =x_{1}^{u}(
-\infty ) $ and $x_{2}^{u}(+\infty ) =x_{2}^{u}(
-\infty ) $, and we have to prove that
\begin{equation}\label{ec3.18}
x_{1}^{u}(+\infty ) \cdot \int_{-\infty }^{+\infty
}x_{2}^{u}(s) \cdot \frac{c(s) }{\alpha _{u}(
s) }ds=x_{2}^{u}(+\infty ) \cdot \int_{-\infty }^{+\infty
}x_{1}^{u}(s) \cdot \frac{c(s) }{\alpha _{u}(s) }ds.
\end{equation}
To prove \eqref{ec3.18} we shall apply Lemma \ref{Lemma2.1} to the
mapping $f:[0,+\infty )\to \mathbb{R}$, defined by
\[
f(t) :=x_{1}^{u}(t) \cdot
\int_{-t}^{+t}x_{2}^{u}(s) \cdot \frac{c(s) }{\alpha
_{u}(s) }ds-x_{2}^{u}(t) \cdot
\int_{-t}^{+t}x_{1}^{u}(s) \cdot \frac{c(s) }{\alpha
_{u}(s) }ds.
\]
Thus
\begin{align*}
\frac{df}{dt}(t)
&=\dot{x}_{1}^{u}(t) \cdot \int_{-t}^{+t}x_{2}^{u}(s) \cdot
\frac{c(s) }{\alpha_{u}(s) }\,ds-\dot{x}_{2}^{u}(t) \cdot
\int_{-t}^{+t}x_{1}^{u}(s) \cdot \frac{c(s) }{\alpha_{u}(s) }\,ds \\
&\quad +\frac{c(-t) }{\alpha _{u}(-t) }\left[
x_{1}^{u}(t) \cdot x_{2}^{u}(-t) -x_{2}^{u}(t) \cdot x_{1}^{u}(-t) \right] .
\end{align*}
Since $\dot{x}_{i}^{u}(\pm \infty ) =x_{i}^{u}(\pm \infty
) \cdot y_{i}^{u}(\pm \infty ) =0$, $i\in \{ 1,2\}$, the mapping
$\frac{c}{\alpha _{u}}$ is bounded on $\mathbb{R}$ (see hypothesis (C2)),
and
\[
\lim_{t\to +\infty }\left[ x_{1}^{u}(t) \cdot x_{2}^{u}(-t) -x_{2}^{u}(t)
\cdot x_{1}^{u}(-t) \right] =0,
\]
it follows that $\lim_{t\to +\infty }\frac{df}{dt}(t) =0$.
Therefore $f$ is uniformly continuous on $[0,+\infty )$, being Lipschitz
on $[0,+\infty )$.
Since $x_{i}^{u}$, $i\in \{ 1,2\} $ are bounded, from (C2) it follows that
$\int_{0}^{+\infty }f(t)\,dt$ exists and is finite.
Hence, by Lemma \ref{Lemma2.1} we obtain
\[
\lim_{t\to +\infty }f(t) =0.
\]
Now we define the multivalued operator $T:X\to 2^{X}$, by
\begin{align*}
Tu &:=\Big\{ \gamma _{1}^{u}x_{1}^{u}(\cdot ) +\gamma_{2}^{u}x_{2}^{u}(\cdot )
+x_{2}^{u}(\cdot ) \cdot \int_{0}^{(\cdot )
}x_{1}^{u}(s) \cdot \frac{c(s) }{(\xi _{2}-\xi
_{1}) \alpha _{u}(s) }\,ds \\
&\quad -x_{1}^{u}(\cdot ) \cdot \int_{0}^{(\cdot )
}x_{2}^{u}(s) \cdot \frac{c(s) }{(\xi _{2}-\xi
_{1}) \alpha _{u}(s) }\,ds, \\
&\quad \mbox{with }| \gamma _{1}^{u}| +| \gamma_{2}^{u}| \leq 1,\;
\gamma _{1}^{u},\;\gamma _{2}^{u}\mbox{ satisfying }\eqref{ec3.17} \Big\} ,
\end{align*}
for every $u\in X$.
By \eqref{ec3.15}-\eqref{ec3.16} we have
\[
| x(t) | \leq 2M_{1}+\frac{M_{1}}{| \xi_{2}-\xi _{1}| }
\Big(\big | \int_{0}^{t}x_{1}^{u}(s)\frac{c(s) }{\alpha _{u}(s) }ds\big|
+\big| \int_{0}^{t}x_{2}^{u}(s) \frac{c(s)}{\alpha _{u}(s) }ds\big| \Big).
\]
Hence
$| x(t) | \leq k_{1}$, for every $t\in \mathbb{R}$,
where
\[
k_{1} :=\max \Big\{ 2M_{1}+\frac{2M_{1}^{2}}{| \xi _{2}-\xi
_{1}| }\int_{0}^{+\infty } \frac{|c(s)| }{A_{1}(s) } ds,
 2M_{1}+\frac{2M_{1}^{2}}{| \xi _{2}-\xi _{1}| }
\int_{-\infty }^{0} \frac{|c(s)| }{A_{2}(s) } ds\Big\}.
\]
Similarly
\[
| \dot{x}(t) | =\Big| \gamma _{1}^{u}\dot{x}
_{1}^{u}(t) +\gamma _{2}^{u}\dot{x}_{2}^{u}(t) +\dot{
x}_{2}^{u}(t) \int_{0}^{t}x_{1}^{u}(s) \frac{c(s) }{\alpha _{u}(s) }ds
-\dot{x}_{1}^{u}(t) \int_{0}^{t}x_{2}^{u}(s) \frac{c(s) }{\alpha _{u}(s) }ds\Big| ,
\]
and there exists another constant $k_{2}\geq 0$,
\[
k_{2} :=\max \Big\{ 2M_{2}+\frac{2M_{1}M_{2}}{| \xi _{2}-\xi
_{1}| }\int_{0}^{+\infty } \frac{|c(s)| }{A_{1}(s) } ds,
2M_{2}+\frac{2M_{1}M_{2}}{| \xi _{2}-\xi _{1}| }
\int_{-\infty }^{0} \frac{|c(s)| }{A_{2}(s) } ds\Big\} ,
\]
such that $|\dot  x(t) | \leq k_{2}$, for every $t\in \mathbb{R}$. Remark that, by relation \eqref{ec2.3}, $k_{1}$, $k_{2}$ are finite.
We let $k:=\max \{ k_{1},\;k_{2}\}$, and
\[
M:=\left\{ x\in C^{2}(\mathbb{R}) ,\;| x(t)
| \leq k,\;| \dot{x}(t) | \leq k,
\mbox{ for every }t\in \mathbb{R}\right\} .
\]

\section{Proof of main result}

To prove Theorem \ref{main} it is sufficient to prove that
the operator $T$ has a fixed point. We do this
in three steps.

\noindent{\it Step 1:  For every $u\in M$,  $T(u) $ is a non-empty convex closed set}. Let $u\in M$ be arbitrary.

From the definition of $T$ we see that $T(u)$ is non-empty
and convex.

Let  $(x^{n}) _{n\in
\mathbb{N}}\subset T(u) $ be such that $x^{n}\to x$ and
$\dot{x}^{n}\to \dot{x}$ uniformly on $\mathbb{R}$  as
$n\to\infty$. We have
\[
x^{n}(t) :=\gamma _{1,n}^{u}x_{1}^{u}(t) +\gamma
_{2,n}^{u}x_{2}^{u}(t) +H^{u}(t) ,
\]
 for every $n\in \mathbb{N}$, with
$| \gamma _{1,n}^{u}| +| \gamma _{2,n}^{u}| \leq 1$,
$\gamma _{1,n}^{u}, \gamma _{2,n}^{u}$ satisfying \eqref{ec3.17},
and
\[
H^{u}(t) :=x_{2}^{u}(t) \cdot
\int_{0}^{t}x_{1}^{u}(s) \cdot \frac{c(s) }{(
\xi _{2}-\xi _{1}) \alpha _{u}(s) }ds
-x_{1}^{u}(t) \cdot \int_{0}^{t}x_{2}^{u}(s) \cdot
\frac{c(s) }{(\xi _{2}-\xi _{1}) \alpha _{u}(s)}\,ds.
\]
Then there exist subsequences such that $\gamma _{1,k_{n}}^{u}\to
\gamma _{1}^{u}$ and $\gamma _{2,k_{n}}^{u}\to \gamma _{2}^{u}$, as
$n\to \infty$.

Since  $(x^{k_{n}}) _{n\in \mathbb{N}}$ converges uniformly to $y:=\gamma
_{1}^{u}x_{1}^{u}+\gamma _{2}^{u}x_{2}^{u}+H^{u}$, it follows that
$x = y$.
Also
\[
\dot{x}^{k_{n}}\to \dot{y}=\dot{x},\quad \mbox{ as }n\to \infty.
\]
So $x\in T(u) $, that is $T(u) $ is a closed set.
\smallskip

\noindent{\it Step 2: $T(M)$ is relatively compact.}
 The relative compactness of $T(M) $ will be proved by
using Proposition \ref{Proposition2.1}.

 From the definitions of
$T$ and $M$ we see that  $| x(t)| \leq k$, $| \dot{x}(t) | \leq
k$, for all $t\in \mathbb{R}$. Thus the first condition of
Proposition \ref{Proposition2.1} is fulfilled with
$h_{1}=h_{2}=k$.

Conditions (b) and (c) of Proposition \ref{Proposition2.1} are
implied by the following assumption:

(d) {\it There exist }$f_{1}$, $f_{2}:\mathbb{R}\to \mathbb{R}_{+}$ {\it
integrable on }$\mathbb{R}$ {\it such that for every }$x\in \mathcal{A}$
\[
| \dot{x}(t) | \leq f_{1}(t) \quad \mbox{{\it and}}\quad
| \ddot{x}(t) | \leq f_{2}(t) ,\quad
\mbox{{\it for} }t\in \mathbb{R}.
\]
This last assertion follows from the fact that,
for every $t_{1},t_{2}\in \mathbb{R}$,
\[
x(t_{1}) -x(t_{2}) =\int_{t_{1}}^{t_{2}}\dot{x}
(t) dt\quad \mbox{and}\quad
\dot{x}(t_{1}) -\dot{x}(t_{2}) =\int_{t_{1}}^{t_{2}}\ddot{x}(t)\, dt\,.
\]
For $i\in \{ 1,2\} $ let
\[
g_{1i}(t) :=\begin{cases}
\max \Big\{ A_{2}(t) \big[ \mu +\int_{0}^{t}\frac{\beta (
s) }{A_{1}(s) }ds\big] ,\frac{| \xi _{i}|
A_{2}(t) }{| 1+\xi _{i}\int_{0}^{t}A_{2}(s)
ds| }\Big\} , & t\geq 0 \\
\max \Big\{ \frac{| \xi _{i}| A_{1}(t) }{|
1+\xi _{i}\int_{0}^{t}A_{1}(s) ds| },A_{1}(t)
\big[ -\lambda +\int_{t}^{0}\frac{\beta (s) }{A_{2}(
s) }ds\big] \Big\} ,& t\leq 0\,.
\end{cases}
\]
Hence $| \dot{x}_{i}^{u}| $ is bounded by the integrable
function $M_{1}\cdot g_{1i}$, $i\in \{ 1,2\} $.
Furthermore, since
\[
\big| \int_{0}^{t}x_{2}^{u}(s) \cdot \frac{c(s) }{(
\xi _{2}-\xi _{1}) \cdot \alpha _{u}(s) }ds\big|
\]
is bounded (on the positive semiaxis by
$\frac{M_{1}}{| \xi _{2}-\xi_{1}| }\cdot \int_{0}^{+\infty }\frac{| c(s) |
}{A_{1}(s) }\,ds$ and on the negative semiaxis by
$\frac{M_{1}}{| \xi _{2}-\xi _{1}| }\cdot \int_{-\infty }^{0}\frac{|
c(s) | }{A_{1}(s) }\,ds$), and $|\dot{x}_{1}^{u}| $ is bounded by an integrable
function, we see that
\[
\big| \dot{x}_{1}^{u}\cdot \int_{0}^{(\cdot ) }x_{2}^{u}(
s) \cdot \frac{c(s) }{(\xi _{2}-\xi _{1}) \cdot
\alpha _{u}(s) }ds\big|
\]
is bounded by an integrable function. Similarly,
\[
\big| \dot{x}_{2}^{u}\cdot \int_{0}^{(\cdot ) }x_{1}^{u}(
s) \cdot \frac{c(s) }{(\xi _{2}-\xi _{1}) \cdot
\alpha _{u}(s) }ds\big|
\]
is bounded by an integrable function.
Therefore, the existence of $f_{1}$ in assertion (d) follows.
Now, since
\[
\ddot{x}(t) =a_{u}(t) \dot{x}(t)
+b_{u}(t) x+c(t) ,
\]
$a_{u}$ is bounded (hypothesis (A1)), $| \dot{x}| $ is
bounded by an integrable function, $| x| $ is bounded
(by $k$), $b_{u}$ is integrable on $\mathbb{R}$ (by
relation \eqref{ec2.2}, hypothesis (B1), and $| c| $ is
integrable on $\mathbb{R}$ (by hypothesis (C1)), we
see that $| \ddot{x}| $ is bounded by an integrable function.
This proves the existence of $f_{2}$, and hence assertion (d) is verified.
\smallskip

\noindent {\it Step 3: $T$ is upper semicontinuous.}
Let $A$ be a closed subset of $M$. Hence if $(u_{n}) _{n}\subset A$ such that
$u_{n}\to u$ and $\dot{u}_{n}\to \dot{u}$ uniformly on $\mathbb{R}$, as
$n\to \infty $, it follows that $u\in A$.

Let $z_{n}\in T^{-1}(A) $ be such that $z_{n}\to z$ and $
\dot{z}_{n}\to \dot{z}$ uniformly on $\mathbb{R}$, as $n\to \infty$.
We have to prove that $z\in T^{-1}(A) $.
Since $z_{n}\in T^{-1}(A) $ there exists
$x_{n}\in A$, $x_{n}\in Tz_{n}$. Thus
\begin{equation}\label{ec4.1}
\ddot{x}_{n}=a(t,z_{n},\dot{z}_{n}) \dot{x}_{n}+x(t,z_{n},
\dot{z}_{n}) x_{n}+c(t) ,\quad n\in \mathbb{N}
\end{equation}
and
\begin{equation}\label{ec4.2}
x_{n}(+\infty ) =x_{n}(-\infty ) ,\quad
\dot{x}_{n}(+\infty ) =\dot{x}_{n}(-\infty ) ,\quad n\in \mathbb{N}.
\end{equation}
Since $x_{n}\in T(M) $ and  $ T(M) $ is relatively compact,
the sequence
$x_{n}$ contains subsequence converging in $C^{2}$ to some $x$. One can
assume that $x_{n}\to x$, $\dot{x}_{n}\to \dot{x}$
uniformly on $\mathbb{R}$, as $n\to \infty $.

Since
$a(t,z_{n}(t) ,\dot{z}_{n}(t) )\to a(t,z(t) ,\dot{z}(t) )$
and
$b(t,z_{n}(t) ,\dot{z}_{n}(t) )\to b(t,z(t) ,\dot{z}(t) )$,
uniformly  on compact subsets of $\mathbb{R}$, it follows that $
x$ is solution to the equation
\[
\ddot{x}=a(t,z(t) ,\dot{z}(t) ) \dot{z}+b(t,z(t) ,\dot{z}(t) ) z+c(t) ,
\]
with
\[
x(0) =\lim_{n\to \infty }x_{n}(0)\quad \mbox{and}\quad
\dot{x}(0) =\lim_{n\to \infty }\dot{x}_{n}(0) .
\]
Furthermore, by \eqref{ec4.2} we find, by passing to the limit as $n\to \infty $,
\[
x(+\infty ) =x(-\infty ) \quad\mbox{and}\quad
\dot{x}(+\infty ) =\dot{x}(-\infty ) .
\]
Since the set $A$ is closed, $x\in A$. Therefore,
$z\in T^{-1}(A)$, which completes the proof of Theorem \ref{main}.


\begin{thebibliography}{00}
\bibitem[1]{1}  {\sc S. Adly, D. Goeleven and D. Motreanu, }{\it Periodic and
homoclinic solutions for a class of unilateral problems}, Discrete Cont.
Dyn. Systems, {\bf 3}(1997), pp. 579-590.

\bibitem[2]{2}  {\sc S. Adly and D. Goeleven, }{\it Homoclinic orbits for a
class of hemivariational inequalities}, Applicable Analysis, {\bf 58}(1995),
pp. 229-240.

\bibitem[3]{3} {\sc A. Ambrosetti and M. L. Bertotti, }{\it Homoclinics for
second order conservative systems}, in: Partial differential
equations and related subjects (Proc. conf. in honor of L.
Nirenberg), Longman Scientific Technical, Harbour, England, 1992,
pp. 21-37.

\bibitem[4]{4}  {\sc C. Avramescu, }{\it Sur l'existence des solutions
convergentes des syst\`{e}mes d'\'{e}quations diff\'{e}rentielles ordinaires}, Ann. Mat. Purra ed Appl., {\bf LXXXI}(IV)(1969), pp. 147-168.

\bibitem[5]{5}  {\sc C. Avramescu, }{\it Sur l'existence des solutions des
\'{e}quations int\'{e}grales dans certains espaces fonctionnels}, Ann. Univ.
Sc. Budapest, {\bf XIII}(1970), pp. 19-34.

\bibitem[6]{6}  {\sc C. Avramescu, }{\it Existence problems for homoclinic
solutions}, Abstract and Applied Analysis, {\bf 7}(1)(2002), pp. 1-27.

\bibitem[7]{7}  {\sc C. Avramescu, }{\it Continuation theorems and }$\omega $
{\it -closed orbits for ordinary differential equations}, University of the
West Timi\c{s}oara, Preprint Series in Math., Seminar of Math. Analysis, {\bf 131}(2001).

\bibitem[8]{8}  {\sc C. Avramescu, }{\it Evanescent solutions for linear
ordinary differential equations}, Electronic Journal of Qualitative Theory
of Differential Equations, {\bf 9}(2002), pp.1-12.

\bibitem[9]{9}  {\sc C. Avramescu and C. Vladimirescu, }{\it Homoclinic
solutions for li\-ne\-ar and linearizable ordinary differential equations},
Abstract and A\-pplied Analysis, {\bf 5}(2)(2000), pp. 65-85.

\bibitem[10]{10}  {\sc C. Avramescu and C. Vladimirescu, }{\it Existence results
for ge\-ne\-ra\-lized bilocal boundary-value problems}, Ann. Univ. Craiova, {\bf
XXVI}(1999), pp. 5-14.

\bibitem[11]{11}  {\sc C. Avramescu and C. Vladimirescu, }{\it Homoclinic
solutions for second order linear di\-ffe\-ren\-ti\-al equations}, Ann. Univ.
Craiova, {\bf XXVII}(2000), pp. 14-23.

\bibitem[12]{12}  {\sc C. Avramescu and C. Vladimirescu, }{\it Limits of solutions
of a perturbed linear differential equation},
Electronic Journal of Qualitative Theory of Differential Equations, {\bf 3}(2002), pp. 1-11.

\bibitem[13]{13}  {\sc V. Coti Zelati and P. H. Rabinovici}, {\it Homoclinic orbits for
second order Hamiltonian systems possessing superquadratic potentials}, J.
Amer. Math. Soc., {\bf 4}(1992), pp. 693-727.

\bibitem[14]{14}  {\sc G. Th. Hallam, }{\it Convergence of solutions of perturbed
nonlinear di\-ffe\-ren\-tial equations}, J. Diff. Eqs., {\bf 12}(1972), pp. 63-80.

\bibitem[15]{15}  {\sc A.G. Kartsatos, }{\it Convergence in perturbed nonlinear
systems}, Tohoku Math. J., {\bf 4}(1972), pp. 1-7.

\bibitem[16]{16}  {\sc A. G. Kartsatos, }{\it Advanced Ordinary Differential
Equations}, Mariner Publishing Co., Inc., 1980.

\bibitem[17]{17}  {\sc J. Mawhin, }{\it Topological degree methods in nonlinear
boundary-value problems}, CBMS Regional Conf. Soc. Math., {\bf
40}, Amer. Math. Soc., Providence, R. I., 1979.

\bibitem[18]{18}  {\sc J. Mawhin, }{\it Continuation theorems and periodic
solutions of ordinary differential equations}, in: Topological
Methods in Differential Equations and Inclusion (Montreal 1996),
Kluwer, 1997.

\bibitem[19]{19}  {\sc J. Mawhin, }{\it Topological degree and boundary-value problems for non\-li\-ne\-ar differential equations}, Lectures Notes
in Math., {\bf 1537}, Springer-Verlag, Berlin, 1993.

\bibitem[20]{20}  {\sc P. H. Rabinowitz, }{\it Homoclinic orbits for a class of
Hamiltonian systems}, Proc. Royal E\-din\-burg, {\bf 114 A}(1990), pp. 33-38.

\bibitem[21]{21}  {\sc D. Yanheng and M. Girardi, }{\it Periodic and homoclinic
solutions to a class of Hamiltonian systems with the potentials changing sign}, Dyn. Syst. Appl., {\bf 2}(1993), pp. 131-145.

\bibitem[22]{22}  {\sc E. Zeidler, }{\it Nonlinear Functional Analysis and its
Applications, I, Fixed - Point Theorems}, Springer-Verlag,
Berlin, 1993.
\end{thebibliography}

\end{document}