\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 06, pp. 1--10.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/06\hfil BVP with integral boundary conditions]
{Existence results for nonlinear boundary-value problems \\
with integral boundary conditions}

\author[A. Belarbi,  M. Benchohra\hfil EJDE-2005/06\hfilneg]
{Abdelkader Belarbi, Mouffak Benchohra} % in alphabetical order

\address{Laboratoire de Math\'ematiques,
Universit\'e de Sidi Bel Abb\`es,\\
BP 89, 22000, Sidi Bel Abb\`es, Alg\'erie}
\email[A. Belarbi]{aek\_belarbi@yahoo.fr}
\email[M. Benchohra]{benchohra@univ-sba.dz}

\date{}
\thanks{Submitted July 19, 2004. Published January 4, 2005.}
\subjclass[2000]{34A60, 34B15}
\keywords{Nonlinear boundary-value problem; integral boundary
conditions;
\hfill\break\indent contraction; fixed point}

\begin{abstract}
 In this paper, we investigate the existence of solutions
 for a second order nonlinear boundary-value problem with
 integral boundary conditions. By using suitable fixed point
 theorems, we study the cases when the right hand side has
 convex and nonconvex values.
\end{abstract}

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

\section{Introduction}

 This paper  concerns the existence of solutions of  a nonlinear
boundary-value
problem with integral boundary conditions. More precisely, in Section
3,
we consider the  nonlinear boundary-value problem
\begin{gather}\label{e1}
x''(t)\in F(t,x(t)), \quad \hbox{a.e. }  t\in [0,1], \\
\label{e2}
x(0)-k_1x'(0)=\int_0^1 h_1(x(s)) ds, \\
\label{e3}
x(1)+k_2x'(1)=\int_0^1 h_2(x(s))ds,
\end{gather}
where $F: [0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a
compact valued
multivalued map, $\mathcal{P}(\mathbb{R})$ is the family of all subsets
of
$\mathbb{R}$, $h_i:\mathbb{R}\to \mathbb{R}$  are continuous functions
and
$k_i$  are nonnegative constants ($i=1,2$). Boundary-value
problems with integral boundary conditions constitute a very
interesting and important class of problems. They include two,
three, multipoint and nonlocal boundary-value problems as special
cases. For boundary-value problems with integral boundary
conditions and comments on their importance, we refer the reader
to the papers by Gallardo \cite{ Gall}, Karakostas and Tsamatos
\cite{KaTs}, Lomtatidze and Malaguti \cite{LoMa} and the
references therein. Moreover, boundary-value problems with
integral boundary conditions have been studied by a number of
authors, for instance, Brykalov \cite{Bry}, Denche and Marhoune
\cite{DenM}, Jankowskii \cite{Jan} and Krall \cite{Kra} and Rahmat
and Bashir \cite{RaBa}. The present paper is motivated by a
recent one due to  Rahman \cite{Rah} in which the generalized
method of quasilinearization was applied to a class of
 second order boundary-value problem with integral  boundary conditions
of the form (\ref{e2})
 and (\ref{e3}).
 In this paper, we shall present three existence results for the problem
(\ref{e1})-(\ref{e3}) when the right hand side is convex as well as
nonconvex valued. The first
one relies on the nonlinear alternative of Leray-Schauder type. In the
second one, we shall use
the fixed point theorem for contraction multivalued maps due to Covitz
and Nadler, while in the
third one, we shall combine the nonlinear alternative of Leray-Schauder
type for single-valued
maps with a selection theorem due to Bressan and Colombo for lower
semicontinuous multivalued
maps with nonempty closed and decomposables values. These results
extend to the multivalued case
some ones considered in the literature.

\section{Preliminaries}

 In this section, we introduce notations, definitions,
and preliminary facts from multivalued analysis which are used
throughout this paper. $C([0,1],\mathbb{R})$ is the Banach space  of
all
continuous functions from $[0,1]$ into $\mathbb{R}$ with the norm
$$
\|x\|_{\infty}=\sup\{|x(t)|: 0\leq t\leq 1\}.
$$
$L^1 ([0,1],\mathbb{R})$ denotes the Banach space of measurable
functions
$x:[0,1]\to \mathbb{R}$ which are Lebesgue integrable normed
by
$$
\|x\|_{L^1 }=\int_0^1 |x(t)|dt \quad  \hbox{for all }
x\in L^1 ([0,1],\mathbb{R}).
$$
$AC^1((0,1),\mathbb{R})$ is the space of differentiable functions
$x:(0,1) \to \mathbb{R}$,
whose first derivative, $x'$, is absolutely continuous.
Let $(X,|\cdot|)$ be a normed space, $P_{cl}(X)=\{Y\in \mathcal{P}(X):
Y$
closed$\}$,  $P_{b}(X)=\{Y\in \mathcal{P}(X): Y$ bounded$\}$,
$P_{cp}(X)=\{Y\in \mathcal{P}(X): Y$ compact$\}$ and
$P_{cp,c}(X)=\{Y\in
\mathcal{P}(X): Y$ compact and convex$\}$. A multivalued map $G:X\to
P(X)$ is convex (closed) valued if $G(x)$ is convex (closed) for
all $x\in X$. $G$ is bounded on bounded sets if
$G(B)=\cup_{x\in B}G(x)$ is bounded in $X$ for all $B\in P_{b}(X)$
(i.e. $\sup_{x\in B}\{\sup\{|y|: y\in G(x) \}\}<\infty). \, G$ is
called upper semi-continuous (u.s.c.) on $X$ if for each $x_0\in
X$ the set $G(x_0)$ is a nonempty closed subset of $X$ and if
for each open set $N$ of $X$ containing $G(x_0)$, there exists
an open neighbourhood $N_0$ of $x_0$ such that $G(N_0)\subseteq
N. \, G$ is said to be completely continuous if $G(\mathcal{B})$ is
relatively compact for every $\mathcal{B}\in P_{b}(X)$. If the
multivalued map $G$ is completely continuous with nonempty compact
values, then $G$ is u.s.c. if and only if $G$ has a closed graph
(i.e. $x_{n}\to x_{*}, \ y_{n}\to y_{*}, \
y_{n}\in G(x_{n})$ imply $y_{*}\in G(x_{*})$). $G$ has a fixed
point if there is $x\in X$ such that $x\in G(x)$. The fixed
point set of the multivalued operator $G$ will be denoted by $Fix
G$. A multivalued map $G:[0,1]\to P_{cl}(\mathbb{R})$ is said to be
measurable if for every $y\in \mathbb{R}$, the function $t\mapsto
d(y,G(t))=\inf\{|y-z|: z\in G(t) \}$ is measurable. For more
details on multivalued maps see the books of Aubin and Cellina
\cite{AuCe}, Aubin and Frankowska \cite{AuFr}, Deimling \cite{Dei}
and Hu and Papageorgiou \cite{HuPa} .

\begin{definition} \rm
A multivalued map $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$
is said to be
$L^1 $-Carath\'eodory if
\begin{itemize}
\item[(i)] $t\mapsto F(t,x)$ is  measurable for each $x\in \mathbb{R}$;
\item[(ii)] $x\mapsto F(t,x)$ is upper semicontinuous for almost
all $t\in [0,1]$;
\item[(iii)] for each $q>0$, there exists $\varphi_{q}
\in L^1 ([0,1],\mathbb{R}_{+})$ such that
$$
\|F(t,x)\|=\sup\{|v|: v\in F(t,x)\}\leq \varphi_{q}(t) \quad \hbox{for
all }
|x|\leq q \mbox{ and for }  a.e. \; t\in [0,1].
$$
\end{itemize}
For each $x\in C([0,1],\mathbb{R})$, define the set of selections of
$F$ by
$$
S_{F,x}=\{v\in L^1([0,1],\mathbb{R}):
v(t)\in F(t,x(t))\, \, a.e. \, \,  t\in [0,1]\}.
$$
\end{definition}

Let $E$ be a Banach space,  $X$  a nonempty closed
subset of $E$ and $G:X\to \mathcal{P}(E)$  a multivalued operator
with  nonempty closed values.
$G$ is lower semi-continuous (l.s.c.) if
the set $ \{x\in X: G(x)\cap B\not=\emptyset\}$ is
open for any open set  $ B$ in  $E$.
Let $A$ be a subset of $[0,1]\times \mathbb{R}$.
$A$ is $\mathcal{L}\otimes\mathcal{B}$ measurable if $A$ belongs
to the $\sigma$-algebra generated by all sets of the
form $\mathcal{J}\times  D$, where $\mathcal{J}$ is Lebesgue
measurable in $[0,1]$ and $ D$ is Borel measurable in $\mathbb{R}$.
A subset $A$ of $L^1([0,1],\mathbb{R})$\ is decomposable if
for all $u,v\in A$ and $\mathcal{J}\subset [0,1]$ measurable,
the function $u\chi_\mathcal{J}+v\chi_{J-\mathcal{J}}\in A$, where
$\chi_{\mathcal{J}}$ stands for the characteristic function of
$\mathcal{J}$.

\begin{definition} \rm
Let $Y$ be a separable metric space and let
$N: Y\to \mathcal{P}(L^1([0,1],\mathbb{R}))$ be a multivalued
operator. We say  $N$ has property (BC) if
\begin{itemize}
\item[1)] $N$ is lower semi-continuous (l.s.c.);
\item[2)] $N$ has nonempty closed and decomposable values.
\end{itemize}
\end{definition}

Let $F: [0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ be a
multivalued map with
nonempty compact values. Assign to  $F$  the multivalued operator
$$
\mathcal{F}: C([0,1],\mathbb{R})\to \mathcal{P}(L^1([0,1],\mathbb{R}))
 $$
by letting
$$
 \mathcal{F}(x)=\{w\in L^1([0,1],\mathbb{R}): w(t)\in F(t, x(t)) \quad
\hbox{for  a.e. } t\in[0,1]\}.
$$
The operator $\mathcal{F}$
is called the Nymetzki operator associated with $F$.

\begin{definition} \rm
Let $F: [0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ be a
multivalued function
with nonempty compact values. We say $F$ is of lower
semi-continuous type (l.s.c. type) if its associated Nymetzki
operator $\mathcal{F}$ is lower semi-continuous and has nonempty
closed and decomposable values.
\end{definition}

Let $(X,d)$ be a metric space induced from the normed space
$(X,|\cdot |)$. Consider
$H_{d}:\mathcal{P}(X)\times
\mathcal{P}(X)\to\mathbb{R}_{+}\cup\{\infty\}$
given by
$$
H_{d}(A,B)=\max\{\sup_{a\in A}d(a,B),\sup_{b\in B}d(A,b)\},
$$
where $d(A,b)=\inf_{a\in A}d(a,b), \
d(a,B)=\inf_{b\in B}d(a,b)$.
Then $( P_{b,cl}(X),H_{d})$ is a metric space and
$(P_{cl}(X),H_{d})$ is a
generalized metric space (see \cite{Kis}).

\begin{definition}\rm
A multivalued operator $N:X\to  P_{cl}(X)$ is called
\begin{itemize}
\item[a)] $\gamma$-Lipschitz if and only if there
exists $\gamma>0$ such that
$$
H_d(N(x),N(y))\leq \gamma d(x,y), \quad \hbox{for each}
\ x,\ y\in X,$$
\item[b)] a contraction if and only if it is
$\gamma$-Lipschitz with
$\gamma<1$.
\end{itemize}
\end{definition}

\begin{lemma}[\cite{LaOp}] \label{l1}
Let $X$ be a Banach space. Let $F:[0,1]\times X\to
P_{cp,c}(X)$ be an $L^1 $-Carath\'eodory multivalued map and let
$\Gamma$ be a linear continuous
mapping from $L^1 ([0,1],X)$ to $C([0,1],X)$, then the operator
\begin{gather*}
\Gamma \circ S_{F}:C([0,1],X) \to  P_{cp,c}(C([0,1],X)),\\
 x  \mapsto (\Gamma \circ S_{F})(x):=\Gamma(S_{F,x})
\end{gather*}
is a closed graph operator in $C([0,1],X)\times C([0,1],X)$.
\end{lemma}

\begin{lemma}[\cite{BrCo}]\label{l2}
Let $Y$ be a separable metric space and
$N: Y\to \mathcal{P}(L^1([0,1],\mathbb{R}))$ be a multivalued operator
which has property (BC). Then $N$ has a continuous selection; i.e.,
there exists a
continuous function (single-valued) $g:Y\to L^1([0,1],\mathbb{R})$ such
that $g(x)\in N(x)$\ for every $x\in Y$.
\end{lemma}

\begin{lemma}[\cite{CoNa}]\label{l4}
Let $(X,d)$ be a complete metric space. If $N: X\to P_{cl}(X)$ is
a contraction, then $Fix N \not= \emptyset$.
\end{lemma}

\section{Main Results}

In this section, we are concerned with the existence of
solutions for the problem
(\ref{e1})-(\ref{e3}) when the right hand side has convex as well as
nonconvex values.
Initially, we assume that $F$ is a compact and convex
valued  multivalued map.

\begin{definition} \rm
A function $x\in AC^1((0,1),\mathbb{R})$ is  said to be a
solution of  (\ref{e1})-(\ref{e3}) if there exists a function
$v\in L^1([0,1],\mathbb{R})$ with
$v(t)\in F(t,x(t))$ for a.e. $t\in [0,1]$ such that
$x''(t)= v(t)$ a.e. on $[0,1]$
and the function $x$
satisfies the conditions (\ref{e2}) and (\ref{e3}).
\end{definition}

We need the following auxiliary result. Its proof uses a standard
argument.

\begin{lemma}\label{l5}
For any $\sigma (t)$, $\rho_1(t)$, $\rho_2(t)\in C([0,1],\mathbb{R})$,
the nonhomogeneous linear problem
\begin{gather*}
x''(t)=\sigma(t), \quad \hbox{a.e. }  t\in [0,1],\\
x(0)-k_1x'(0)= \int_0^1 \rho_1(s)ds, \\
x(1)+k_2x'(1)=\int_0^1 \rho_2(s)ds,
\end{gather*}
has a unique solution $x\in AC^1((0,1),\mathbb{R})$ given by
$$
x(t)=P(t)+\int_0^1 G(t,s)\sigma(s)ds,
$$
where
$$
P(t)=\frac{1}{1+k_1+k_2}\{(1-t+k_2)\int_0^1 \rho_1(s)ds
+(k_1+t)\int_0^1 \rho_2(s)ds\}
$$
is the unique solution of the problem
\begin{gather*}
x''(t)=0,\quad \hbox{a.e. } t\in [0,1],\\
x(0)-k_1x'(0)=\int_0^1 \rho_1(s)ds,  \\
x(1)+k_2x'(1)=\int_0^1 \rho_2(s)ds,
\end{gather*}
and
$$
G(t,s)=\frac{-1}{k_1+k_2+1}\begin{cases}
(k_1+t)(1-s+k_2),& 0\leq t<s\leq 1,\\
 (k_1+s)(1-t+k_2),&  0\leq s<t\leq 1
\end{cases}
$$
is the Green's function of the problem. We note that $G(t,s)<0$ on
$(0,1)\times (0,1)$.
\end{lemma}

Let us introduce the following hypotheses which are assumed hereafter:
\begin{itemize}
\item[(H1)] The function $F: [0,1]\times\mathbb{R}\to
P_{cp,c}(\mathbb{R})$
is $L^1$-Carath\'eodory;
\item[(H2)] there exist two nonnegative
constants $c_{1}$ and $c_2$  such that
$$
|h_{1}(x)|\leq c_{1},\quad \mbox{and}\quad   |{h}_{2}(x)|\leq
\bar c_{2} \quad  \mbox{for all }  x\in\mathbb{R};
$$
\item[(H3)] there exist a continuous non-decreasing
function $\psi:[0,\infty)\to (0,\infty)$, a function
$p\in L^1([0,1],\mathbb{R}_{+})$ such that
$$
\|F(t,x)\|\leq p(t)\psi(|x|)\quad \hbox{for each } (t,x)\in
[0,1]\times\mathbb{R},
$$
\item[(H4)] there exists a number $M>0$ such that
$$
\frac{M}{\frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}+\sup_{(t,s)\in
[0,1]\times [0,1]}|G(t,s)|\psi(M)\int_0^1 p(s)ds }>1.
$$
\end{itemize}

\begin{theorem} \label{t1}
Suppose hypotheses (H1)--(H4) are satisfied. Then the
boundary-value problem (\ref{e1})-(\ref{e3}) has at least one solution.
\end{theorem}

\begin{proof} We transform  (\ref{e1})-(\ref{e3}) into a
fixed point problem. Consider the operator
$N:C([0,1],\mathbb{R})\to \mathcal{P}(C([0,1],\mathbb{R}))$ defined by
$$
N(x)=\{h\in C([0,1],\mathbb{R}):  h(t)=P(t)+\int_0^1G(t,s)v(s)ds, \;
v\in S_{F,x}\},
$$
where
$$
P(t)=\frac{1}{1+k_1+k_2}\{(1-t+k_2)\int_0^1 h_1(x(s))ds+(k_1+t)
\int_0^1 h_2(x(s))ds\}.
$$
\begin{remark} \rm
 Clearly, from Lemma \ref{l5}, the fixed
points of $N$ are solutions to (\ref{e1})-(\ref{e3}).
\end{remark}
We shall show that $N$ satisfies the assumptions of the nonlinear
alternative of Leray-Schauder type. The proof will be given in
several steps. \smallskip

\noindent {\bf Step 1:} $N(x)$ is convex for each $x\in
C([0,1],\mathbb{R})$.
 Indeed, if $h_{1},\ h_{2}$
belong to $N(x)$, then there exist $v_{1}, v_{2}\in S_{F,x}$ such
that  for each $t\in [0,1]$ we have
$$
h_{i}(t)=P(t)+\int_0^1 G(t,s)v_i(s)ds, \, (i=1,2). $$ Let
$0\leq d\leq 1$. Then, for each $t\in [0,1]$ we have $$
(dh_{1}+(1-d)h_{2})(t)=P(t)+\int_0^1G(t,s)[dv_{1}(s)+(1-d)v_{2}(s)]ds.
$$
Since $S_{F,x}$ is convex (because $F$ has convex values),
then
$$
dh_{1}+(1-d)h_{2}\in N(x).
$$

\noindent {\bf Step 2}: {\em $N$ maps bounded sets into bounded sets
in $C([0,1],\mathbb{R})$.}
  Let $B_{q}=\{x\in C([0,1],\mathbb{R}): \|x\|_{\infty}\leq
q \}$ be a bounded set in $C([0,1],\mathbb{R})$ and $x\in B_{q}$, then
for each $h\in N(x)$, there exists $v\in S_{F,x}$ such that
$$
h(t)=P(t)+\int_0^1G(t,s)v(s)ds. $$ From (H2) and (H3) we have
\begin{align*}
& |h(t)|\\
&\leq |P(t)|+\int_0^1|G(t,s)||v(s)|ds \\
&\leq |P(t)|+\sup_{(t,s)\in [0,1]\times
 [0,1]}|G(t,s)|\int_0^1|v(s)|ds \\
&\leq \frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}
+\sup_{(t,s)\in
[0,1]\times [0,1]}|G(t,s)|\psi(q)\int_0^1 p(s)ds \\
&\leq \frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}+\sup_{(t,s)\in
[0,1]\times [0,1]}|G(t,s)|\psi(q)\|p\|_{L^1}.
\end{align*}
{\bf Step 3}: {\em $N$ maps bounded sets into equicontinuous
sets of $C([0,1],\mathbb{R})$.}
Let $r_{1}, r_{2}\in [0,1]$, $r_{1}<r_{2}$ and $B_{q}$ be a bounded set
of $C([0,1],\mathbb{R})$ as
in Step 2 and $x\in B_{q}$. For each $h\in N(x)$
\begin{align*}
|h(r_{2})-h(r_{1})|
&\leq |P(r_2)-P(r_1)|+\int_0^1 |G(r_2,s)-G(r_1,s)||v(s)|ds\\
&\leq |P(r_2)-P(r_1)|+\psi(q)\int_0^1|G(r_2,s)-G(r_1,s)| p(s)ds.
\end{align*}
 The right hand side tends to zero as $r_2-r_1\to 0$.
As a consequence of Steps 1 to 3 together with the Arzel\'a-Ascoli
Theorem, we can conclude that \ $N:C([0,1],\mathbb{R})\to
\mathcal{P}(C([0,1],\mathbb{R}))$ is completely continuous. \smallskip

\noindent{\bf Step 4}: {\em $N$ has a closed graph.}
 Let
$x_{n}\to x_{*}$,  $h_{n}\in N(x_{n})$ and  $h_{n} \to h_{*}$.
We need to show that $h_{*}\in N(x_{*})$. \\ $h_{n}\in N(x_{n})$
means that there exists $v_{n}\in S_{F, x_{n}}$ such that for each
$t\in[0,1]$
$$
h_{n}(t)=P_n(t)+\int_0^1 G(t,s)v_n(s)ds,
$$
where
$$
P_n(t)=\frac{1}{1+k_1+k_2}[(1-t+k_2)\int_0^1h_1(x_n(s))ds
+(k_1+t)\int_0^1h_2(x_n(s))ds].
$$
 We must show that there exists $h_{*}\in S_{F, x_{*}}$ such
that for each $t\in [0,1]$
$$
h_{*}(t)=P_*(t)+\int_0^1 G(t,s)v_*(s) ds,
$$
where
$$
P_*(t)=\frac{1}{1+k_1+k_2}[(1-t+k_2)\int_0^1h_1(x_*(s))ds+(k_1+t)
\int_0^1h_2(x_*(s))ds].
$$
Clearly  we have
$\|(h_{n}-P_n)-(h_{*}-P_*)\|_{\infty}\to 0$ as $n\to\infty$.
Consider the continuous linear operator
$\Gamma:L^1 ([0,1],\mathbb{R})\to C([0,1],\mathbb{R})$
defined by
$$
v\mapsto (\Gamma v)(t)= \int_0^1 G(t,s)v(s)ds.
$$
 From Lemma \ref{l1}, it
follows that $\Gamma\circ S_{F}$ is a closed graph operator.
Moreover, we have
$$
\big(h_{n}(t)-P_n(t) \big) \in
\Gamma(S_{F,x_{n}}).
$$
Since $x_{n}\to x_{*}$, it follows from
Lemma \ref{l1} that
$$
h_{*}(t)=P_*(t)+\int_0^1 G(t,s)v_*(s)ds
$$
for some $v_{*}\in S_{F,x_{*}}$. \smallskip

\noindent{\bf Step 5:} {\em A priori bounds on solutions.}
Let $x$ be a possible
solution of the problem (\ref{e1})-(\ref{e3}). Then,
 there exists $v\in L^1([0,1],\mathbb{R})$ with  $v\in S_{F,x}$ such
that for
 each  $t\in [0,1]$
$$
x(t)=P(t)+\int_0^1 G(t,s)v(s)ds.
$$
 This implies by (H2) and (H3) that for each $t\in[0,1]$ we
have
\begin{align*}
&|x(t)|\\
&\leq \frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}
 +\!\sup_{(t,s)\in
[0,1]\times [0,1]}|G(t,s)|\int_0^1 p(s)\psi(|x(s)|)ds \\
&\leq \frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}
+\!\!\!\sup_{(t,s)\in
[0,1]\times [0,1]}|G(t,s)|\psi(\|x\|_{\infty})\int_0^1p(s)ds.
\end{align*}
Consequently
\begin{align*}
&\|x\|_{\infty}\Big(
\frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}\\
&\quad+\sup_{(t,s)\in [0,1]\times
[0,1]}|G(t,s)|\psi(\|x\|_{\infty})\int_0^1p(s)ds \Big)^{-1}
\leq 1.
\end{align*}
Then
by (H4), there exists  $M$ such that $\|x\|_{\infty} \not=M$.
Let
$$
U=\{x\in C([0,1],\mathbb{R}): \|x\|_{\infty}<M\}.
$$
The operator $N:
\overline U\to \mathcal{P}(C([0,1],\mathbb{R}))$ is upper
semicontinuous and
completely continuous. From the choice of $U$, there is no $x\in
\partial U$ such that $x\in \lambda N(x)$ for some
$\lambda\in(0,1)$. As a consequence of the nonlinear alternative
of Leray-Schauder type \cite{DuGr}, we deduce that $N$
 has a fixed point $x$ in $\overline U$ which is a solution
of the problem (\ref{e1})-(\ref{e3}).
\end{proof}

We present now a result for the problem (\ref{e1})-(\ref{e3}) with
a nonconvex valued right hand side.
Our considerations are based on the fixed point theorem for
multivalued map given by Govitz and Nadler \cite{CoNa}. We need the
following  hypotheses:
\begin{itemize}
\item[(H5)] $F:[0,1]\times \mathbb{R} \to  P_{cp}(\mathbb{R})$
has the property that $F(\cdot,x): [0,1]\to P_{cp}(\mathbb{R})$ is
measurable for each $x\in \mathbb{R}$;
\item[(H6)]
$H_{d}(F(t,x),F(t,\overline x))\leq l(t)|x-\overline x|$ for almost all
$t\in [0,1]$ and $x,\overline x\in \mathbb{R}$ where $l\in L^1
([0,1],\mathbb{R})$ and
$d(0,F(t,0))\leq l(t)$ for almost each $t\in[0,1]$;
\item[(H7)] there exist two  nonnegative constants $d_1$ and $d_2$ such
that
$|h_1(x)-h_1(\overline x)|\leq d_1|x-\overline x|$ and
$|h_2(x)-h_2(\overline x)|\leq d_2|x-\overline x|$
for all $x, \,\overline x$ in $\mathbb{R}$.
\end{itemize}

\begin{theorem} \label{t2} Assume that (H5)-(H7)
are satisfied. If $$
\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2]+\sup_{(t,s)\in
[0,1]\times [0,1] }|G(t,s)|\|l\|_{L^1}<1,
$$
then (\ref{e1})-(\ref{e3}) has at least one solution.
\end{theorem}

\begin{remark}  \rm
 For each $x\in C([0,1],\mathbb{R})$, the
set $S_{F,x}$ is nonempty since by (H5), $F$ has  a measurable
selection (see \cite{CaVa}, Theorem III.6).
\end{remark}

\begin{proof}
We shall show that $N$ satisfies the assumptions of Lemma
\ref{l4}. The proof will be given in two steps. \smallskip

\noindent{\bf Step 1}: {\em $N(x)\in  P_{cl}(C([0,1],\mathbb{R}))$ for
each
$x\in C([0,1],\mathbb{R})$.}
 Indeed, let $(x_{n})_{n\geq 0}\in N(x)$ such that
$x_{n}\to \tilde x$ in $ C([0,1],\mathbb{R})$. Then, $\tilde x\in
C([0,1],\mathbb{R})$
and  there exists $v_n\in S_{F,x}$ such that for each $t\in
[0,1]$
$$
x_{n}(t)=P(t)+ \int_0^1G(t,s)v_n(s)ds .
$$
Using the fact that $F$ has compact values and from (H6), we may
pass to a subsequence if necessary to get that $v_n$ converge to
$v$ in $L^1([0,1],\mathbb{R})$ and hence $v\in S_{F,x}$. Then, for each
$t\in [0,1]$
$$
x_{n}(t)\to  \tilde x(t)= P(t)+\int_0^1G(t,s)v(s)ds .
$$
 So, $\tilde x\in N(x)$. \smallskip

\noindent {\bf Step 2}:  {\em There exists $\gamma < 1$ such
that $H_d(N(x),N(\overline x))\leq \gamma\|x-\overline x\|_{\infty}$
for
each $x, \overline x\in C([0,1],\mathbb{R})$.}
Let $x,\overline x \in C([0,1],\mathbb{R})$ and $h_{1}\in N(x)$. Then,
there
exists  $v_{1}(t)\in F(t,x(t))$ such that for each $t\in [0,1]$
$$
h_{1}(t)=P(t)+\int_0^1G(t,s)v_1(s)ds .
$$
>From (H6) it follows that
$$
H_d(F(t,x(t)), F(t,\overline x(t)))\leq l(t)|x(t)-\overline x(t)|.
$$
Hence, there exists $w\in F(t,\overline x(t))$ such that
$$
|v_{1}(t)-w|\leq l(t)|x(t)-\overline x(t)|, \ t\in [0,1].
$$
Consider $U:[0,1]\to \mathcal{P}(\mathbb{R})$ given by
$$
U(t)=\{w\in \mathbb{R}: |v_{1}(t)-w|\leq l(t)|x(t)-\overline x(t)|\}.
$$
Since the multivalued operator $V(t)=U(t)\cap F(t,\overline x(t))$
is measurable \cite[Proposition III.4]{CaVa}, there exists
a function $v_{2}(t)$ which is a measurable selection for $V$.
So, $v_{2}(t)\in F(t,\overline x(t))$ and for
each $t\in [0,1]$
$$
|v_{1}(t)-v_{2}(t)|\leq l(t)|x(t)-\overline x(t)|.
$$
Let us define  for each $t\in [0,1]$
$$
h_{2}(t)=\bar P(t)+\int_0^1G(t,s)v_2(s)ds,
$$
where
$$
\bar P(t)=\frac{1}{1+k_1+k_2}[(1-t+k_2)\int_0^1h_1(\overline x(s))ds
+(1+k_1)\int_0^1h_2(\overline x(s))ds].
$$
We have
\begin{align*}
&|h_{1}(t)-h_{2}(t)|\\
&\leq |P(t)-\bar P(t)|+\int_0^1 |G(t,s)||v_1(s)-v_2(s)|ds \\
 &\leq\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2]\|x-\bar x\|_{\infty}
  +\int_0^1 |G(t,s)|l(s)|x(s)-\overline x(s)|ds.
\end{align*}
Thus,
\begin{align*}
&\|h_{1}-h_{2}\|_{\infty}\\
&\leq \Big(\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2]
+\sup_{(t,s)\in [0,1]\times [0,1]
}|G(t,s)|\|l\|_{L^1}\Big)\|x-\overline x\|_{\infty}.
\end{align*}
By an
analogous relation, obtained by interchanging the roles of $x$ and
$\overline x$, it follows that
\begin{align*}
&H_d(N(x),N(\overline x))\\
&\leq \Big(\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2]
+\sup_{(t,s)\in [0,1]\times [0,1]}|G(t,s)|\|l\|_{L^1}\Big)
\|x-\overline x\|_{\infty}.
\end{align*}
So, $N$ is a contraction and thus, by Lemma \ref{l4}, $N$ has a fixed
point $x$
which is solution to (\ref{e1})-(\ref{e3}).
\end{proof}

In this part, by using the nonlinear alternative of  Leray Schauder
type
 combined with the selection theorem
of Bresssan and Colombo for semi-continuous maps with decomposable
values, we shall establish an existence result for the
problem (\ref{e1})-(\ref{e3}). We need the following hypothesis:
\begin{itemize}
\item[(H8)]
$F:[0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ is a nonempty
compact-valued multivalued map such that: \\
a) $(t,x)\mapsto F(t,x)$ is
$\mathcal{L}\otimes\mathcal{
B}$ measurable; \\
b)$x\mapsto F(t,x)$ is lower semi-continuous for
each $t\in [0,1]$;
\end{itemize}
The following lemma is of great importance in  the proof of our
next result.

\begin{lemma}[\cite{FrGr1}] \label{l6}
Let $F: [0,1]\times \mathbb{R}\to\mathcal{P}(\mathbb{R})$
be a multivalued map with nonempty compact values.
Assume (H3) and (H8) hold. Then $F$ is of l.s.c. type.
\end{lemma}

\begin{theorem}\label{t3}
Assume that (H2), (H3), (H4) and (H8) hold. Then the BVP
(\ref{e1})-(\ref{e3}) has at least one solution.
\end{theorem}

\begin{proof}
Note that (H3), (H8) and Lemma \ref{l6} imply that $F$ is  of l.s.c.
type. Then from Lemma \ref{l2}, there exists a continuous function
$f: C([0,1],\mathbb{R})\to L^1([0,1],\mathbb{R})$ such that $f(x)\in
\mathcal{F}(x)$
for all $x\in  C([0,1],\mathbb{R})$.
Consider the problem
\begin{gather}\label{eq1}
x''(t)= f(x(t)), \quad \hbox{a.e. }  t\in [0,1], \\
\label{eq2}
x(0)-k_1x'(0)=\int_0^1 h_1(x(s)) ds, \\
\label{eq3}
x(1)+k_2x'(1)=\int_0^1 h_2(x(s))ds.
\end{gather}
It is clear that if $x\in  AC^1 ((0,1),\mathbb{R})$ is a solution of
(\ref{eq1})-(\ref{eq3}), then $x$ is a solution to the problem
(\ref{e1})-(\ref{e3}). Transform the problem
(\ref{eq1})-(\ref{eq3}) into a fixed point theorem. Consider the
operator $\bar N$ defined by
$$
\bar N(x)(t)=P(t)+\int_0^1G(t,s)f(x(s))ds.
$$
We can easily show
that $\bar N$ is continuous and completely continuous. The remaning of
the proof is similar to that of Theorem \ref{t1}.
\end{proof}

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