\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 67, pp. 1--11.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/67\hfil Existence results]
{Existence results for nonlocal multivalued boundary-value problems}
\author[P. Candito, G. Molica Bisci\hfil EJDE-2006/67\hfilneg]
{Pasquale Candito, Giovanni Molica Bisci}  % in alphabetical order

\address{Pasquale Candito \newline
Dipartimento di Informatica, Matematica, Elettronica e Trasporti\\
Facolt\`a di Ingegneria, Universit\`a degli
Studi Mediterranea di Reggio Calabria\\
Via Graziella (Feo di Vito), 89100 Reggio Calabria, Italy}
\email{pasquale.candito@unirc.it}

\address{Giovanni Molica Bisci \newline
Dipartimento P.A.U., Universit\`a  degli Studi Mediterranea di
Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio
Calabria, Italy}
 \email{giovanni.molica@ing.unirc.it}

\date{}
\thanks{Submitted January 19, 2006. Published July 6, 2006.}
\subjclass[2000]{34A60, 34B15}
\keywords{Nonlocal boundary-value problems; integral boundary conditions;
\hfill\break\indent  operator inclusions}


\begin{abstract}
 In this paper we establish some existence results for nonlocal
 multivalued boundary-value problems. Our approach is based on
 existence results for operator inclusions involving a suitable
 closed-valued multifunction; see \cite{am1,am}.
 Some applications are given.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

 Let  $(X, \|\cdot\|_X)$ be a separable real Banach space and
let $(\mathbb{R}^n , \|\cdot\|)$ be the
real Euclidean $n$-space with the norm $\|z\|=\max_{1\leq i\leq
n}|z_i|$ and induced metric $d$. Denote by $M([0,1],X)$ the family
of all (equivalence classes of) strongly Lebesgue measurable
functions from $[0,1]$ to $X$. The papers \cite{am1} and \cite{am}
provide some existence results for operator inclusions of the type
\begin{equation} \label{P_F}
\begin{gathered}
    u\in U,  \\
    \Psi(u)(t)\in F(t,\Phi(u)(t)) \quad \text{a.e.  in } [0,1],
  \end{gathered}
\end{equation}
where $U$ is a nonempty set, $F$ is a multifunction from
$[0,1]\times X$ into $\mathbb{R}^n $, and $\Phi:U\to
M([0,1],X)$, $\Psi:U\to L^s([0,1],{\mathbb{R}}^n)$ are two
abstract operators (see Theorems \ref{thm2.1}, \ref{thm2.2},
and \ref{thm2.3} below).
Their approach is chiefly based on the following conditions:
\begin{itemize}
\item[(U1)] $\Psi$ is bijective and for any $v\in L^s([0,1],\mathbb{R}^n )$
and any sequence
$\{v_h\}\subset L^s([0,1], \mathbb{R}^n )$ weakly
converging to $v$ in $L^q([0,1],\mathbb{R}^n )$ there exists a
subsequence of $\{\Phi(\Psi^{-1}(v_h))\}$ which
converges a.e. in $[0,1]$ to $\Phi(\Psi^{-1}(v))$. Furthermore, a
nondecreasing function $\varphi:[0,+\infty[\to
[0,+\infty]$ can be defined in such a way that
\begin{equation}\label{varphi}
\mathop{\rm ess\,sup}_{t\in [0,1]}\left\|\Phi(u)(t)\right\|\leq
\varphi(\left\|\Psi(u)\right\|_{p})\quad  \forall u\in U.
\end{equation}
Here $p, q, s \in [0,+\infty] $ with $ q< +\infty $ and $ q\leq
p\leq s$.

\item[(U2)] To each $\rho\in L^s([0,1],{\mathbb{R}}^+_0)$ there
corresponds a nonnegative measurable function $\rho^*$ so that if
$u\in U$
and $\|\Psi(u)(t)\|\leq \rho(t)$ a.e. in $[0,1]$, then $\Phi(u)$
is Lipschitz continuous with constant $\rho^*(t)$ at almost all
$t\in [0,1]$.

\item[(F1)] There exists $r>0$ such that the function
$$
M(t):=\sup_{\|x\|_X\leq \varphi(r)}d(0,F(t,x)),  \quad  t\in
[0,1],
$$
belongs to $L^s([0,1],{\mathbb{R}}_0^{+})$ and $ \|M\|_{{p}}\leq
r $.
\end{itemize}
Let us denote by $W^{2,s}([0,1],\mathbb{R}^n
)$ the space of all $u\in
C^1([0,1],\mathbb{R}^n )$ such that $u'$ is
absolutely continuous in $[0,1]$ and $u''\in
L^s([0,1],\mathbb{R}^n )$.

The aim of this paper is to establish, under suitable assumptions,
the existence of at least one generalized solution in $W^{2,s}([0,1],
\mathbb{R}^n )$ to the problem
\begin{equation} \label{P^F}
\begin{gathered}
    u''\in F(t,u,u') \quad \text{a.e. } t\in [0,1] \\
    u(0)-k_1u'(0)=H_1(u),  \\
    u(1)+k_2u'(1)=H_2(u),
  \end{gathered}
\end{equation}
 where $F:[0,1]\times \mathbb{R}^n \times
\mathbb{R}^n \to 2^{\mathbb{R}^n }$ is a
multifunction with nonempty closed values, for $i=1,2$, $L_{H_i}$
and $k_i$ are nonnegative constants,
$H_i:W^{2,s}([0,1],\mathbb{R}^n )\to \mathbb{R}^n $ fulfill
\begin{equation}\label{LHi}
 \|H_i(u)-H_i(v)\|\leq L_{H_i}\|u-v\|_{\infty},
  \quad \forall  u, v \in W^{2,s}([0,1], \mathbb{R}^n ).
\end{equation}
 The key to solve problem \eqref{P^F} is to show that the operators $\Phi$,
$\Psi$ and $F$ satisfy the structural hypotheses (U1), (U2)
and (F1). This study is motivated by many nonlocal boundary-value
problems discussed in \cite{ake,bb,bc,k}.
For more details on these topics, see also
\cite{kt1,kt2,kt3}. Recent references are furnished in
\cite{eloe,ruyun,xue}. In our opinion this method
exhibits at least two interesting features: owing to (F1), no
compactness condition on the values of $F$ is required; we can
treat both the case when $F$ takes convex values, $F(\cdot,x,y)$
is measurable, while $F(t,\cdot,\cdot)$ is upper semicontinuous,
and the one where $F$ is measurable and $F(t,\cdot,\cdot)$ is
lower semicontinuos. However, in the latter case, in addition to
(U1), we also need (U2). In all concrete situations, to
verify (U1) and (U2) we exploit only the boundary conditions
of \eqref{P^F} and
we do not use the assumptions on $F$.

 Basic definitions and preliminary results are given in
Section 2. Main results are contained in Section 3, while Section 4
is devoted to some applications.

\section{Basic definitions and preliminary results}

Given a separable real Banach space $(X,\|\cdot\|_X)$, the symbol
$\mathcal{B}(X)$ indicates the Borel $\sigma$-algebra of $X$. If
$W$ is a nonempty subset of $X$, $x_0\in W$ and $\epsilon > 0$, we
write $d(x_0,W):=\inf_{w\in W}\|x_0-w\|_X$ as well as
$$
B(x_0,\epsilon):=\{z\in X:\|x_0-z\|_X\leq \epsilon\},\quad
B^0(x_0,\epsilon):=\{z\in X:\|x_0-z\|_X< \epsilon\}.
$$
A function $\psi$ from $[0,1]$ into $X$ is said to be Lipschitz
continuous at the point $t\in [0,1]$ when there exist a
neighborhood $V_t$ of $t$ and a constant $k_t\geq 0$ such that
$\|\psi(\tau)-\psi(t)\|_X\leq k_t|\tau-t|$ for every $\tau\in
[0,1]\cap V_t$. Given any $p\in [1,+\infty]$, we write $p'$ for
the conjugate exponent of $p$ besides $L^p([0,1],X)$ for the space
of $u\in M([0,1],X)$ satisfying $\|u\|_{p}<+\infty $, where
$$
\|u\|_{p}:=
  \begin{cases}
    \big( \int_{0}^{1}\|u(t)\|^p_Xd\mu\big)^{1/p} & \text{if $p<+\infty$, } \\
    \mathop{\rm ess\,sup}_{t\in [0,1]} \|u(t)\|_X & \text{if $p=+\infty$},
  \end{cases}
$$
 and $\mu$ is the Lebesgue measure on $[0,1]$.
 Let $F$ be a
multifunction from $W$ into $\mathbb{R}^n $ (briefly,
$F:W\to 2^{\mathbb{R}^n }$), namely a function which
assigns to each point $x\in W$ a nonempty subset $F(x)$ of
$\mathbb{R}^n $. If $V\subseteq W$ we write $F(V):=\cup_{x\in
V}F(x)$ and $F|_V$ for the restriction of $F$ to $V$. The graph of
$F$ is the set $\{(x,z)\in W\times \mathbb{R}^n :z\in F(x)\}$.
If $Y\subseteq \mathbb{R}^n $ we define $F^-(Y):=\{x\in
W:F(x)\cap Y\neq \emptyset\}$. If $(W,\mathfrak F)$ is a
measurable space and $F^-(Y)\in \mathfrak F$ for any open subset
$Y$ of $\mathbb{R}^n $, we say that $F$ is $\mathfrak
F$-measurable, or simply measurable as soon as no confusion can
arise. We denote with $\mathcal{L}$ the Lebesgue $\sigma-$algebra
in $\mathbb{R}^n $. We say that $F$ is upper semicontinuous at
the point $x_0\in W$ if to every open set $Y\subseteq
\mathbb{R}^n $ satisfying $F(x_0)\subseteq Y$ there corresponds
a neighborhood $W_0$ of $x_0$ such that $F(W_0)\subseteq Y$. The
multifunction $F$ is called upper semicontinuous when it is upper
semicontinuous at each point of $W$. In such a case its graph is
clearly closed in $W\times \mathbb{R}^n $ provided that $F(x)$
is closed for all $x\in W$. We say that $F$ has a closed graph at
$x_0$ if the condition $\{x_k\}\subseteq W$, $\{z_k\}\subseteq
\mathbb{R}^n $, $lim_{k\to+\infty}x_k=x_0$,
$lim_{k\to+\infty}z_k=z_0$, $z_k\in F(x_k)$, $k\in
{\mathbb{N}}$, imply $z_0\in F(x_0)$. We say that $F$ is lower
semicontinuous at the point $x_0$ if to every open set $Y\subseteq
\mathbb{R}^n $ satisfying $F(x_0)\cap Y\neq \emptyset$ there
corresponds a neighborhood $V_0$ of $x_0$ such that $F(x)\cap
Y\neq \emptyset$, $x\in V_0$. The multifunction $F$ is called
lower semicontinuous when it is lower semicontinuous at each point
of $W$. Finally, for $B\subseteq [0,1]\times X$, $\mathop{\rm proj}_X(B)$
indicates the projection of $B$ onto $X$. We say that a
multifunction $F:B\to 2^{\mathbb{R}^n }$ has the lower
Scorza Dragoni property if to every $\epsilon>0$ there corresponds
a closed subset $I_\epsilon$ of $[0,1]$ such that
$\mu([0,1]\setminus I_\epsilon)<\epsilon$ and
$F|_{(I_\epsilon\times X)\cap B}$ is lower semicontinuous. Let $D$
be a nonempty closed subset of $X$, let $A\subseteq [0,1]\times
D$, and let $C:=([0,1]\times D)\setminus A$. We always suppose
that the set $A$ complies with

\begin{itemize}

\item[(A1)] $A\in \mathcal{L}\otimes \mathcal{B}(X)$
and $A_t=\{x\in D:(t,x)\in A\}$ is open in $D$ for every $t\in [0,1]$.
\end{itemize}
Moreover, let $F$ be a closed-valued multifunction from
$[0,1]\times D$ into $\mathbb{R}^n $, let $m\in
L^s([0,1],{\mathbb{R}}^{+}_0)$, and let $N\in \mathcal{L}$ with
$\mu(N)=0$. The conditions below will be assumed in what follows.

\begin{itemize}
\item[(A2)] $F|_A$ has the lower Scorza Dragoni property.

\item[(A3)] $F(t,x)\cap B^0(0,m(t))\neq \emptyset$ whenever
$(t,x)\in A\cap[([0,1]\setminus N)\times D]$.

\item[(A4)] The set $\{x\in \mathop{\rm proj}_X(C):$ $F|_C(\cdot,x)$ is
measurable$\}$ is dense in $\mathop{\rm proj}_X(C)$.

\item[(A5)] For every $(t,x)\in C\cap [([0,1]\setminus N)\times D]$
the set $F(t,x)$ is convex, $F(t,\cdot)$ has a closed graph at
$x$, and $F(t,x)\cap B(0,m(t))\neq \emptyset$.
\end{itemize}

Combining opportunely the above conditions we point out the
abstract results that we will apply to study problem \eqref{P_F}. More
precisely we have the following results.


\begin{theorem}[{\cite[Theorem 3.1]{am1}}] \label{thm2.1}
Let $\Phi,\Psi,\varphi$ be as in (U1)
and let $F:[0,1]\times X\to 2^{\mathbb{R}^n }$ be a
multifunction with convex closed values satisfying (F1).
Suppose that
\begin{itemize}
\item[(A4')] The set $\{x\in X:$ $F(\cdot,x)$ is
$\mathfrak F$-measurable$\}$ is dense in $X$
\item[(A5')] For almost every $t\in [0,1]$ and every $x\in X$, $F(t,\cdot)$
has closed graph at $x$.
\end{itemize}
Then  \eqref{P_F} has at least one solution $u\in U$ with
$\|\Psi(u)(t)\|\leq M(t)$ a.e. in $[0,1]$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 2.3]{am}}] \label{thm2.2}
Let $r>0$ be such that $\|m\|_p\leq r$, and let
$D=B(0_X,\varphi(r))$. Suppose that $F,\Phi,\Psi$ satisfy
(A2)-(A5), (U1) and (U2). Then problem \eqref{P_F} has at
least one solution $u\in U$ with $\|\Psi(u)(t)\|\leq M(t)$ a.e. in
$[0,1]$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 2.5]{am}}] \label{thm2.3}
Suppose that $H:[0,1]\times \mathbb{R}^n \to 2^{\mathbb{R}^n }$
is a closed-valued multifunction with the
following properties:

\begin{itemize}
\item[(C1)] For almost all $t\in [0,1]$ and for all $x\in
\mathbb{R}^n $, either $H(t,x)$ is convex or $H(t,\cdot)$ is
lower semicontinuous at $x$.

\item[(C2)] The set $\{x\in \mathbb{R}^n :$ $H(\cdot,x)$ is measurable$\}$
is dense in $\mathbb{R}^n $.

\item[(C3)] For almost every $t\in [0,1]$ and every $x\in X$, $H(t,\cdot)$ has
closed graph at $x$.

\item[(C4)] There is $m_1\in L^s([0,1],{\mathbb{R}}^{+}_0)$
such that $H(t,x)\cap B(0,m_1(t))\neq \emptyset$ a.e. in $[0,1]$,
for all $x\in \mathbb{R}^n $.
\end{itemize}

 Then there exists a set $A\subseteq [0,1]\times
\mathbb{R}^n $ and a closed-valued multifunction $F$ from
$[0,1]\times \mathbb{R}^n $ into $\mathbb{R}^n $ satisfying
(A1)-(A5). Moreover, for almost every $t\in [0,1]$ and every
$x\in \mathbb{R}^n $, one has $F(t,x)\subseteq H(t,x)$.
\end{theorem}

Moreover, to problem \eqref{P^F} we associate the following functions:
$G:[0,1]\times [0,1]\to \mathbb{R}$ (the Green function
associated with \eqref{P^F}),
$$
G(t,s):=\frac{g(t,s)}{1+k_1+k_2},\quad\text{with}\quad
g(t,s):=\begin{cases}
(k_1+t)(s-1-k_2), & 0\leq t \leq s \leq 1 \\
(k_1+s)(t-1-k_2), &  0\leq s \leq t \leq 1;
\end{cases}
$$
for every $r\in ]0,+\infty[$, put
$$
\varphi'(r)=m_{\varphi'}r+q_{\varphi'}, \quad
\varphi''(r)=m_{\varphi''}r+q_{\varphi''},\quad
\varphi(r)=\max\{\varphi(r),\varphi'(r)\},
$$
where
\begin{gather*}
m_{\varphi'}:=\frac{\sup_{[0,1]^{2}}|g(t,s)|}{1+k_1(1-L_{H_2})
+k_2(1-L_{H_1})-(L_{H_1}+L_{H_2})},
\\
q_{\varphi'}:=\frac{((1+k_2)\|H_1(0)\|+(1+k_1)\|H_2(0)\|)}{1+k_1(1-L_{H_2})
+k_2(1-L_{H_1})-(L_{H_1}+L_{H_2})},
\\
m_{\varphi''}:=(L_{H_1}+L_{H_2})m_{\varphi'}+\sup_{[0,1]^{2}}|g(t,s)|, \\
q_{\varphi''}:=(L_{H_1}+L_{H_2})q_{\varphi'}+\|H_1(0)\|+\|H_2(0)\|.
\end{gather*}

\section{Main results}

 Let $F$ be a closed-valued multifunction from $[0,1]\times
\mathbb{R}^n \times \mathbb{R}^n \to 2^{\mathbb{R}^n }$ fulfilling (F1).

\begin{theorem}\label{main1}
Suppose that:
\begin{itemize}

\item[(I1)] There exist $c,d\in \mathbb{R}^n $ such that
 $$H_1(ct+d)=d-ck_1,\quad \ H_2(ct+d)=d+c(1+k_2).$$

\item[(I2)]
$$
\frac{(1+k_2)L_{H_1}+(1+k_1)L_{H_2}}{1+k_1+k_2}<1.
$$

\item[(I3)] For almost all $t\in [0,1]$ and all
 $(x,y)\in \mathbb{R}^n \times\mathbb{R}^n $, $F(t,x,y)$ is convex.

\item[(I4)] The set
$\{(x,y)\in \mathbb{R}^n \times\mathbb{R}^n :F(\cdot,x,y)$ is
measurable$\}$ is dense in $\mathbb{R}^n \times\mathbb{R}^n $.

\item[(I5)] For almost every $t\in [0,1]$ the graph of $F(t,\cdot,\cdot)$
 is closed.

\end{itemize}

Then problem \eqref{P^F} admits at least one solution $u\in W^{2,s}([0,1])$
such that
$$
\|u \|_{\infty}\leq \varphi'(r), \quad
\|u' \|_{\infty}\leq \varphi''(r), \quad
\|u''(t)\| \leq M(t) \quad  \text{for a.e. } t\in [0,1].
$$
\end{theorem}

\begin{proof} We apply Theorem 2.1
 by choosing $X=\mathbb{R}^n \times \mathbb{R}^n $, $\Phi(u)(t)=(u(t),u'(t))$
and $\Psi(u)(t)=u''(t)$ for each $u\in U$, where
\begin{align*}
U=&\Big\{ u\in W^{2,s}([0,1],\mathbb{R}^n ):\exists \;
 \sigma\in L^{s}([0,1],\mathbb{R}^n ) \text{ such that } \\
&u(t)= \frac{1}{1+k_1+k_2}\Big[(1+k_2-t)H_1(u)+
(k_1+t)H_2(u)\Big]+\int_{0}^{1}G(t,s) \sigma(s)ds \Big\}.
\end{align*}
 Clearly, since $Y$ has finite dimension and (F1)
holds, we only need to show that $U_1$ is verified. With this aim,
we first observe that $U$ is not empty. Indeed, by (I1), the
function $w(t):=ct+d\in U$. Moreover, for each $u\in U$,
arguing in standard way it results that $u''\equiv \sigma$ and, by
\eqref{LHi}, we get the following inequalities
\begin{gather*}
\|H_i(u)\|\leq \|H_i(0)\|+L_{H_i}\|u\|_{\infty}. \quad \text{for } i=1,2,\\
\begin{aligned}
\|u\|_\infty& \leq \frac{1}{1+k_1+k_2}(1+k_2)[\|H_1(0)\|
 +L_{H_1}\|u\|_{\infty}] \\
&\quad +\frac{1+k_1}{1+k_1+k_2}[L_{H_2}\|u\|_\infty+\|H_2(0)\|]
 +\int_0^1\sup_{[0,1]^2}|G(t,s)|\|u''(s)\|\,ds.
\end{aligned}
\end{gather*}
Hence
\begin{align*}
&\|u\|_\infty\Big(1-\frac{(1+k_2)L_{H_1}+(1+k_1)L_{H_2}}{(1+k_1+k_2)}\Big)\\
&\leq \frac{(1+k_1)\|H_2(0)\|+(1+k_2)\|H_2(0)\|}{1+k_1+k_2}
+\frac{\sup_{[0,1]^2}|g(t,s)|}{1+k_1+k_2}\|u''\|_p,
\end{align*}
which implies
\begin{equation}\label{stima}
\|u\|_{\infty}\leq \varphi'(\|u''\|_{p}).
\end{equation}
Similarly, we have
\begin{align*}
\|u'\|_{\infty}
&\leq \frac{1}{1+k_1+k_2}\Big[(L_{H_1}+L_{H_2})\varphi'(\|u''\|_{p})\\
&\quad +\|H_1(0)\|+\|H_2(0)\|+\sup_{[0,1]^2}\Big|\frac{\partial g }{\partial t}(t,s)
\Big|\|u''\|_p\Big],
\end{align*}
 which clearly ensures,
\begin{equation}\label{stima1}
 \|u'\|_{\infty}\leq \varphi''(\|u''\|_{p}).
\end{equation}
So, \eqref{varphi} holds. Let us next prove that
$\Psi:U\to L^s([0,1],\mathbb{R}^n )$ is injective.
Arguing by contradiction, suppose that there exist two functions
$u,v\in U$ and $B\subseteq [0,1]$ with $\mu(B)>0$, such that
$u(t)\neq v(t)$ for each $t\in B$ and $u''(t)=v''(t)$ a.e. in
$[0,1]$. Pick $t\in [0,1]$. By \eqref{LHi}, one has
$$
\|u(t)-v(t)\|\leq\frac{(1+k_2)L_{H_1}+(1+k_1)L_{H_2}}{1+k_1+k_2}
\|u-v\|_{\infty}.
$$
 From this, taking into account (I2), we have
$\|u-v\|_{\infty}=0$, that is $u(t)=v(t)$ a.e. in $[0,1]$, which
is absurd. Hence we get a contradiction.  Next, fix
$v\in L^s([0,1],\mathbb{R}^n )$ and a sequence $\{v_h\}$ weakly
converging to $v$ in $L^s([0,1],\mathbb{R}^n )$. To simplify
the notation we put $u_h=\Psi^{-1}(v_h)$ and $u=\Psi^{-1}(v)$,
i.e., $u''_h=v_h$, $u''=v$, $u''_h \rightharpoonup u''$ in
$L^s([0,1],\mathbb{R}^n )$ and for
 a.e. $t$ in $[0,1]$ one has
$$
\Phi(\Psi^{-1}(v_h))(t)=(u_h(t),u'_h(t)).
$$
 We claim that
\begin{equation}\label{point}
\Phi(\Psi^{-1}(v_h))(t)\to \Phi(\Psi^{-1}(v))(t),\quad \text{a.e. }
 t\in [0,1].
\end{equation}
 To see this, we first prove that
\begin{equation}
\lim_{h\to +\infty}u_h(t)=u(t) \quad  \text{for all }t\in [0,1].
\end{equation}
 Fix $t\in [0,1]$. By \eqref{LHi} it is easy to show that
\begin{align*}
&\|u_h(t)-u(t)\| \\
&\leq \frac{1}{1+k_1+k_2}\Big((1+k_2)\|H_1(u_h)-H_1(u)\|+
(k_1+1)\|H_2(u_h)-H_2(u)\|\Big)\\
&\quad +\big\|\int_{0}^{1}G(t,s)(u''_h(s)-u''(s)) ds\big\|\\
&\leq \frac{(1+k_2)L_{H_1}+(1+k_1)L_{H_2}}{1+k_1+k_2}
  \|u_h-u\|_{\infty}+\Big\|\int_{0}^{1}G(t,s)(u''_h(s)-u''(s))ds\Big\|.
\end{align*}
 Moreover, since $u''_h$ weakly converges to $ u''$ in
$L^{s}([0,1],\mathbb{R}^n  )$, and $G(t,\cdot)\in L^{s}([0,1])$
if $s\geq1$, it results
$$
\lim_{h\to +\infty}\int_{0}^{1}G(t,s)(u''_h(s)-u''(s))ds=0.
$$
 Therefore, we have
$$
\limsup_{h\to\infty}\|u_h-u\|_{\infty}\leq
\frac{(1+k_2)L_{H_1}+(1+k_1)L_{H_2}}{1+k_1+k_2}
\limsup_{h\to\infty}\|u_h-u\|_{\infty}.
$$
 Furthermore, since the sequence
$\{\|u_h\|_{p}\}$ is bounded, from \eqref{stima}, it
is easy to show that
$\limsup_{h\to\infty}\|u_h-u\|_{\infty}<+\infty$. Hence,
on account of (I2), the preceding inequality provides
\begin{equation}\label{uniforme}
\lim_{h\to +\infty}\|u_h-u\|_\infty =0.
\end{equation}
 Now we prove that
\[
\lim_{h\to +\infty}u_h'(t)=u'(t) \quad \text{a.e. in } [0,1].
\]
 To do this, we observe that if $u\in U$, then an easy computation
 ensures that
$$
u'(t)=\frac{1}{1+k_1+k_2}(H_2(u)-H_1(u))+\int_0^1\frac{\partial
G(t,s)}{\partial t}u''(s)ds.
$$
 Hence, for every $t\in [0,1]$,  one has
\begin{align*}
\|u'_h(t)-u'(t)\|
&\leq \frac{1}{1+k_1+k_2}\Big(\|H_2(u_h)-H_2(u)\|\\
&\quad +\|H_1(u_h)-H_1(u)\|\Big)+\Big\|\int_0^1\frac{\partial
G(t,s)}{\partial t}(u''_h(s)-u''(s))ds\Big\|\\
&\leq \frac{L_{H_1}+L_{H_2}}{1+k_1+k_2}\|u_h-u\|_{\infty}
+\Big\|\int_0^1\frac{\partial G(t,s)}{\partial t}(u''_h(s)-u''(s))ds\Big\|.
\end{align*}
 Thus by \eqref{uniforme} and taking into account that
$\frac{\partial G(t,\cdot)}{\partial t}\in L^s([0,1])$, exploiting
again that $u''_h\rightharpoonup u''$ we also get
\begin{equation}\label{uniformeprime}
\lim_{h\to +\infty}\|u'_h(t)-u'(t)\|= 0.
\end{equation}
From \eqref{uniforme} and \eqref{uniformeprime},
 we conclude that $(\ref{point})$ holds, and
the proof is complete.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
If we take $k_1=k_2=0$ and
$H_1(\cdot)=H_2(\cdot)=0$, problem \eqref{P^F} reduces to homogeneous
Dirichlet problem and Theorem \ref{main1} takes the form of
\cite[Theorem 2.1]{ma} provided
$\mathbb{R}^n $ is equipped with the norm used here.
\end{remark}

\begin{theorem}\label{lower}
Suppose that (I1) and (I2) hold. In addition, assume that:
\begin{itemize}
    \item[(A1')] $F$ has the lower Scorza Dragoni property.
    \item[(A2')] There exist two positive constants $r$ and
    $\delta$ with $\delta<r$ such that $\|M\|_{p}\leq r-\delta$.
\end{itemize}
Then the conclusion of Theorem \ref{main1} holds.
\end{theorem}

 \begin{proof} We apply Theorem \ref{thm2.2} by putting
$A=D=[0,1]\times B(0, \varphi(r))$ and $m=M+\delta$. Clearly,
(A2') yields $\|m\|_{p}\leq r$ and since $C=\emptyset$,
(A2)-(A5) hold. Moreover the same arguments used in the
previous proof ensure (U1). Hence, to achieve the conclusion we
only need to verify (U2). To do this, let $\rho\in
L^{s}([0,1],{\mathbb{R}}_0^{+})$ be such that $\|u''(t)\|\leq
\rho(t)$ a.e. in $[0,1]$ and let $u\in U$, $t_1,t_2\in [0,1]$ with
$t_1<t_2, (t_2<t_1)$. We observe that
$$
\|u_j'(t_1)-u_j'(t_2)\|=\|u''(\xi)\||t_1-t_2|\leq
\rho(\xi)|t_1-t_2|\quad \, \xi\in(t_1,t_2), ((t_2,t_1)).
$$
 Further, by \eqref{stima1}, for every $j=1,\dots ,n$, one
has
$$
|u_j(t_1)-u_j(t_2)|\leq \varphi''(r)|t_1-t_2|,\quad \forall
t_1,t_2\in [0,1].
$$
 Hence, putting
$\rho^{*}(t)=\max\{\rho(t),\varphi''(r)\}$, for all $t\in [0,1]$,
one has
$$
\|\Phi(u)(t_1)-\Phi(u)(t_2)\|=\max_{j=1,\dots ,n}\{|u_j(t_1)-u_j(t_2)|,|u_j'(t_1)-u_j'(t_2)|\}\leq \rho^{*}(t)|t_1-t_2|.
$$
 So the proof is complete.
 \end{proof}

 Finally, we have the result.

\begin{theorem}\label{LU}
Assume that $H_1$ and $H_2$ are two bounded functions satisfying
\eqref{LHi} and replace (I3) in Theorem \ref{main1} and (F1),
respectively, with
\begin{itemize}
\item [(Q1)] For almost every $t\in [0,1]$ and for all
$(x,y)\in \mathbb{R}^n \times \mathbb{R}^n $, $F(t,x,y)$
is convex or $F(t,\cdot,\cdot)$ is lower semicontinuous at $(x,y)$.
\item[(Q2)] The function
$$ m'(t):=\sup\Big\{d(0,F(t,x,y)), (x,y) \in \mathbb{R}^n
\times \mathbb{R}^n  \Big\} \quad  \text{ for a.e. } t\in [0,1],
$$
lies in $ L^s([0,1],{\mathbb{R}}^{+}_{0})$.
\end{itemize}
 Then problem \eqref{P^F} admits at least
one generalized solution in $W^{2,s}([0,1],\mathbb{R}^n )$.
\end{theorem}

 \begin{proof} Arguing as in the proof of Theorem \ref{thm2.3}
we obtain a closed-valued multifunction $F':[0,1]\times
\mathbb{R}^n \times \mathbb{R}^n \to 2^{\mathbb{R}^n }$ satisfying
properties (A1)-(A5). Moreover, in this framework, by
\cite[Theorem 2.1]{am}, we can associate to $F'$ a convex
closed-valued multifunction $G:[0,1]\times \mathbb{R}^n \times
\mathbb{R}^n \to 2^{\mathbb{R}^n }$ fulfilling the assumptions of
Theorem \ref{thm2.2} and such that any solution of problem
\eqref{P_F}, with $F=G$, is also solution of \eqref{P_F} with $F'$
instead of $F$. Bearing in mind that $H_1$ and $H_2$ are bounded
and that a weakly convergent sequence in $L^p$ is bounded, arguing
as above it is easy to show that (U1) and (U2) hold. Then due to
Theorem \ref{main1}, problem \eqref{P^F} with $G$ instead of $F$
 has a solution, thereby implying that problem \eqref{P^F} with
 $F'$ instead of $F$ has a solution.
Therefore the conclusion follows taking into account
that $F'(t,x,y)\subseteq H(t,x,y)$.
\end{proof}

\section{Applications}

 This section is devoted to study some boundary-value problems by means
of the results obtained before. In this order of
ideas, let $T:[0,1]\times {\mathbb{R}}\to 2^{\mathbb{R}}$
be a closed-valued multifunction such that $T(\cdot,x)$ is
measurable for every $x\in {\mathbb{R}}$ and let
$h_1,h_2:\mathbb{R}\to \mathbb{R}$ be two  Lipschitz
continuous functions with constants $L_1$ and $L_2$ respectively.
For every $u\in W^{2,s}([0,1]$, put
$$
H_i(u)=\int_0^1h_i(u(s))ds, \quad  \ i=1,2.
$$
We first study a second order differential
inclusion with boundary integral conditions.

\begin{theorem}\label{integral}
Assume that:
\begin{itemize}
\item[(C1)] There exist two constants $c,d\in \mathbb{R}$ such
that
$$
\int_d^{d+c}h_1(s)ds=cd-c^2k_1,\quad \int_{d}^{d+c}h_2(s)ds=c^2(1+k_2)^2+cd
$$
or
$h_1(d)=h_2(d)=d$ according to whether $c\neq 0$ or $c=0$.

\item[(C2)] $$\frac{(1+k_2)L_1+(1+k_1)L_2}{1+k_1+k_2}<1.$$

\item[(C3)] For almost every $t\in [0,1]$, $T(t,\cdot)$ is
 upper semicontinuous and takes convex values.

\item[(C4)] There exist $\alpha$, $\beta\in L^{1}([0,1])$ with
$\|\alpha\|_{1}m_{\varphi'}<1$ such that for a.e. $t\in [0,1]$ one has
$$
m(t):=\sup\Big\{d(0,T(t,x)): |x|\leq
\frac{\varphi'(\|\beta\|_{1})}{1
-\|\alpha\|_{1}m_{\varphi'}}\Big\}\leq \alpha(t)|x|+\beta(t).
$$
\end{itemize}
Then problem
\begin{equation} \label{T}
\begin{gathered}
    u''\in T(t,u) \quad \text{a.e. }t\in [0,1] \\
    u(0)-k_1u'(0)=\int_0^1h_1(u(s))ds,  \\
    u(1)+k_2u'(1)=\int_0^1h_2(u(s))ds,
  \end{gathered}
\end{equation}
admits at least one generalized solution
$u\in W^{2,1}([0,1],{\mathbb{R}}^{2})$ such that
$$
\|u\|_\infty\leq \frac{\varphi'(\|\beta\|_{1})}{1-\|\alpha\|_{1}m_{\varphi'}}
\quad \text{and} \quad
 \|u''(t)\|\leq m(t) \quad\text{a.e in } [0,1].
$$
\end{theorem}

 \begin{proof} Taking $r=\frac{\|\alpha\|_{1}q_{\varphi'}
 + \|\beta\|_{1}}{1-\|\alpha\|_{1} m_{\varphi'}}$,
an easy computation shows that
$$
\varphi'(r)=\frac{\varphi'(\|\beta\|_{1})}{1-\|\alpha\|_{1}}.
$$
Moreover, by (C4), one has $\|M \|_{1}\leq r$.
Then the conclusion follows at once from Theorem \ref{main1}.
\end{proof}


\begin{remark} \label{rmk4.1} \rm
 We point out that Theorem \ref{integral} and
 \cite[Theorem 3.3]{bb} are mutually independent.
 Indeed, here, $T$ does not take compact values, whereas there
$h_1$ and $h_2$ are two continuous and bounded functions.
\end{remark}

 Arguing as above and using Theorem \ref{LU} it is easy to
verify the following result.

\begin{theorem}\label{integral2}
Assume that $h_1$ and $h_2$ are bounded. Let
$T:[0,1]\times {\mathbb{R}}\to 2^{\mathbb{R}}$ be a closed-valued
multifunction satisfying (C1), (C2) such that $T(\cdot,x)$ is measurable for
every $x\in {\mathbb{R}}$. Further, we require
\begin{itemize}
\item[(C5)] For almost every $t\in [0,1]$, and for all
$u\in \mathbb{R}$, $T(t,x)$ is convex or $T(t,\cdot)$ is lower semicontinuous;

\item[(C6)] There exist $\alpha$, $\beta\in L^{1}([0,1])$ with
$\|\alpha\|_{1}m_{\varphi'}<1$, such that for a.e. $t\in [0,1]$
and for every $x\in {\mathbb{R}}$ one has
$$
d(0,T(t,x))\leq \alpha(t)|x|+\beta(t).
$$
\end{itemize}
Then problem $(T)$ admits at least one solution $u\in W^{2,1}([0,1])$.
\end{theorem}

\begin{remark} \label{rmk4.2} \rm
The following is a sufficient condition for
(C5) and (C6) hold true:
\begin{itemize}
    \item [(C6')]
    There exist $\alpha$ and $\beta\in L^{1}([0,1])$ such that for almost
    every $t\in [0,1]$ one has
    $$
    d_{H}(T(t,x), T(t,y))\leq \alpha(t)|x-y| \quad \text{and} \quad
    d_{H}(0, T(t,0))\leq \beta(t),$$
\end{itemize}
where $d_{H}(T(t,x), T(t,y))=\max \{\sup_{z\in T(t,x)} d(z,
T(t,y)), \sup_{w\in T(t,y)} d(w, T(t,x))\}$ indicates the
Hausdorff distance in $\mathbb{R}^n .$ Here, with respect to
\cite[Theorem 3.5]{bb}, $h_1$ and $h_2$ are bounded. However, in
this framework on the data we only require that (C2) is satisfied.
To be precise, there they need the following condition
$$
\frac{(1+k_2)L_{H_1}+(1+k_1)L_{H_2}}{1+k_1+k_2}+\sup_{[0,1]^2}|G(t,s)|
\|\beta\|_1<1.
$$
\end{remark}

 Now as consequence of Theorem \ref{lower} we have the following theorem.

\begin{theorem}\label{mixed}
Let $Q: [0,1] \times {\mathbb{R}}^{2}\to 2^{{\mathbb{R}}}$ be a
closed-valued multifunction fulfilling the Scorza Dragoni property.
Assume that
\begin{itemize}
    \item [(C7)] There exist $\alpha$, $\beta$, $\gamma \in L^{1}([0,1])$
with $ \|\alpha\|_{1}+\|\beta\|_{1} <1/4 $ end a positive constant $\rho$,
such that for almost every $t\in [0,1]$ and for all
$(x,y) \in {\mathbb{R}}^{2}$ with
$$
\max\{|x|,|y|\} \leq 4 \Big(\frac{\rho+
\|\gamma\|_{1}}{1-4(\|\alpha\|_{1} + \|\beta \|_{1})}\Big),
$$
    one has
$ M(t)\leq \alpha(t)|x|+\beta(t)|y|+\gamma(t)$.
\end{itemize}
Then the Nicoletti problem
\begin{equation} \label{Q}
 \begin{gathered}
    u''\in Q(t,u,u') \quad \text{a.e. } t\in [0,1] \\
    u(0)=0, \quad   u'(1)=0,
  \end{gathered}
\end{equation}
admits at least one generalized solution
$u\in W^{2,1}([0,1],{\mathbb{R}}^{2})$ such that $\|u''(t)\|\leq M(t)$
a.e in $[0,1]$,
$$
\|u\|_\infty\leq 2\Big(\frac{\rho+ \|\gamma \|_1}{1-4(\|\alpha\|_1
+ \|\beta \|_1)}\Big)\quad  \text{and} \quad \|u'\|_{\infty}\leq
4\Big(\frac{\rho+ \|\gamma\|_1}{1-4(\|\alpha\|_1 + \|\beta
\|_1)}\Big) .
$$
\end{theorem}

 \begin{proof} Choose $H_1=0$, $H_2(u)=u(1)$, $k_1=0$, $k_2=1$ and
 $$
r=\frac{\rho+\| \gamma \|_1}{1-(\| \alpha \|_1+\| \beta \|_1)}.
$$
Taking into account that $\varphi(r)\leq 4r$,
the conclusion follows immediately from Theorem \ref{lower}.
\end{proof}


 The last application is devoted to study the following three-point
boundary-value problem with nonlinear boundary conditions
\begin{equation} \label{Pfg}
\begin{gathered}
    u''=f(t,u) \quad \text{a.e. in }[0,1]  \\
    u(0)=a, \quad    u(1)=g(u(\eta)), \quad  \eta\in ]0,1[,
  \end{gathered}
\end{equation}
where $f:[0,1]\times {\mathbb{R}}\to \mathbb{R}$ is a
Carath\'{e}odory function, $a\in {\mathbb{R}}$, and
$g:\mathbb{R}\to \mathbb{R}$ is a Lipschitz continuous
function with Lipschitz constant $L_g$. Obviously, problem
\eqref{Pfg} can be rewritten in the framework of problem \eqref{P_F}.
Due to Theorem \ref{main1} and by means of arguments similar to
those used above we obtain the following result.

\begin{theorem}\label{three}
Assume that $L_g <1$ and
\begin{itemize}
\item [(C8)] There exists $\xi\in \mathbb{R}$ such that
$g(\xi)=a+\frac{\xi-a}{\eta}$.
\item [(C9)] There is $r\in ]0,+\infty[$ such that for a.e.
$t\in [0,1]$ and all $x\in {\mathbb{R}}$ with
$|x|\leq \frac{1}{1-L_g}(|a|+|g(0)|+\frac{r}{4})$ one has
$|f(t,x)|\leq r$.
\end{itemize}
Then problem \eqref{Pfg} admits at least one generalized solution
$u\in W^{2,1}([0,1])$ such that
$$
\|u\|_\infty\leq \frac{1}{1-L_g}\Big(|a|+|g(0)|+\frac{r}{4}\Big)
\quad \text{and} \quad  \|u''\|_{\infty}\leq r.
$$
\end{theorem}

\subsection*{Acknowledgements}
The authors are gratefully to the anonymous referee for her/his careful
reading of the manuscript and suggestions to improve this paper.

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