\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2003(2003), No. 32, pp.1--10.
\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/32\hfil Boundary-value problems ]
{Boundary-value problems for first and second order functional
differential inclusions}
\author[Shihuang Hong \hfil EJDE--2003/32\hfilneg]
{Shihuang Hong}
\address{Shihuang Hong\hfill\break
College of science and engineering, Hainan university,
Haikou, 570228, China}
\email{hongshh@hotmail.com}
\date{}
\thanks{Submitted January 10, 2003. Published March 28, 2003.}
\subjclass[2000]{34A60, 34K10}
\keywords{Functional differential inclusions, boundary value problems,
\hfill\break\indent
completely continuous multivalued map, cone, Banach space.}
\begin{abstract}
This paper presents sufficient conditions for the existence of
solutions to boundary-value problems of first and second order
multi-valued differential equations in Banach spaces.
Our results obtained using fixed point theorems, and
lead to new existence principles.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
This paper is concerned with the existence of
solutions for the multi-valued functional differential systems
\begin{equation} \label{e1}
\begin{gathered}
x'\in F(t,x_t),\quad \mbox{a.e. }t\in [0,T]\\
x_0=x_T
\end{gathered}
\end{equation}
and
\begin{equation} \label{e3}
\begin{gathered}
x''\in F(t,x_t,x'(t)),\quad \mbox{a.e. } t\in [0,T]\\
x(t)=\varphi (t),\quad t\in [-r,0],\quad x(T)=\eta,
\end{gathered}
\end{equation}
where $F:J\times C([-r,0],E)\to \mathcal{P}$$(E)$ is a
multi-valued map, $J=[0,T]$ is a compact real interval, $E$ is a
Banach space with norm $|\cdot|$, $\varphi \in C([-r,0],E)$,
$\eta\in E$ and $\mathcal{P}(E)$ is the family of all subsets of
$E$.
For a continuous function $x$ defined on the interval $[-r,T]$
and any $t\in J$, we denote by $x_t$ the element of $C([-r,0],E)$
defined by
\[
x_t(s)=x(t+s),\quad s\in [-r,0].
\]
Here $x_t(\cdot)$ represents the history of the state from time
$t-r$ to the time $t$.
The existence of solutions for functional differential equations in
Banach space has been widely investigated. We refer for instance to
[4-6, 9, 10]. Existence results for functional differential
inclusions
received much attention in the recent years. We refer to [1-3].
For instance, Benchohra and Ntouyas have studied initial
and boundary problems for functional differential inclusions in
[1] on a compact interval with the
map $F$ satisfying Lipschitz's contractive conditions of multivalued
map and
for Neutral functional differential and integrodifferential
inclusions in [2].
This paper is organized as follows. In section 2, we introduce
some definitions and preliminary facts from multivalued analysis
which are used later. In section 3, we give existence results
of positive and negative solutions on compact intervals for the first order boundary value problem
\eqref{e1}. In section 4, some existence theorems are given for the
second order boundary value problem \eqref{e3}.
The fundamental tools used in the existence proofs of all the
above mentioned works are essentially fixed point theorems: Covitz
and Nadler's in [1], Martelli's in [2]. Here we use a fixed point
theorem (Lemma \ref{lm1})
in ordered Banach space. However, the
hypotheses imposed on the multivalued map $F$ and methods of the proof in this paper are different from all the above
cited works.
\section{Preliminaries}
In this section, we introduce notations, definitions and
preliminaries facts from multi-valued analysis which are used
throughout this paper.
Let $(E,|\cdot|)$ be a Banach space with a partial order
introduced by a cone $P$ of $E$, that is, $x\le y$ if and only if $y-x\in P$,
$x0$ such that $|x|\le N|y|$ for any
$x,y\in P$ with $x\le y$.
The set $C([-r,0],E)$ is a Banach space consisting of all continuous
functions from $[-r,0]$ to $E$ with the norm
\[
\|x\|=\sup\{|x(t)|: -r\le t\le 0\}.
\]
For any $x,y\in C([-r,b],E)$ for $b\ge 0$, define $x\le y$ if and
only if $x(t)\le y(t)$ for each $t\in [-r,b]$, $x1\}
\]
is bounded, then $G$ has a fixed point.
\end{theorem}
\begin{lemma} \label{lm1}
Let $P$ be a closed and convex cone of $E$ and
$G: P\to bcf(P)$ a u.s.c. and completely continuous multi-valued map.
If
\begin {equation}
\label{5}
\alpha:=\sup\{|x|:x\in P\;{\rm and\;there\; exists}\;\lambda\in
(0,1)\;{\rm such \;that}\;x\in \lambda G(x)\}<\infty,
\end{equation}
where $\lambda G(x)=\{\lambda g:g\in G(x)\}$, then $G$ has a fixed
point $x\in P$.
\end{lemma}
\begin{proof}
Define the map $\tilde{G}:E\to bcf(P)$ by
$$\tilde{G}(x)=\begin{cases}
G(x)& \hbox{if }x\in P,\\
G(\theta)&\mbox{if }x\notin P\,.
\end{cases}$$
Evidently, $\tilde{G}$ is u.s.c. and completely continuous on $E$,
therefore, $\tilde{G}$ is condensing. Let $\beta=\sup\{|y|:y\in
G(\theta)\}$, then for any $y$ belongs to $M$ given in the above
theorem, we have $y\in \lambda \tilde{G}(y)$ for some $\lambda\in
(0,1)$. If $y\in P$, then $|y|\le \alpha$. Otherwise, $y\in
\lambda G(\theta)$, which yields that $|y|\le \beta$. Hence, $M$
is bounded. By the theorem $\tilde{G}$ has a fixed point $x$. From
$\tilde{G}(x)\in bcf(P)$ it follows that $x\in P$.
\end{proof}
\section{First Order Boundary Value Problems}
In this section we consider the existence of
positive and negative solutions for first order boundary value
problems of the functional
differential inclusion \eqref{e1}.
\noindent{\bf Definition} A function $x:[-r,0]\to E$ is
a solution of \eqref{e1} if $x\in C([-r,T],E)\cap AC([0,T],E)$ and
satisfies the differential inclusion \eqref{e1} a.e. on $[0,T]$.
Let us impose the following hypotheses on the multi-valued map
$F: J\times E\to bcf(E) $.
\begin{itemize}
\item[(H1)] $(t,u)\to F(t,u)$ is measurable with respect to
$t$ for each $u\in C([-r,0],E)$, u.s.c. with respect to $u$ for
each $t\in J$ and for each fixed $u\in C([-r,0],E)$ the set
\[
S_{F(u)}=\{g\in L^1(J,E):g(t)\in F(t,u)\quad\mbox{a.e. } t\in J\}
\]
is nonempty.
\item[(H2)] There exist functions $\alpha\in
L^1(J,\mathbb{R}_+)$,$\quad \beta \in L^1(J,E)$ and $\delta\in
[0,1]$ such that $|\alpha(t)|>0$ for all $t\in J$ and
\[
\beta(t)[|\psi(0)|^{\delta} +1]\le f\le \alpha (t)\psi (0)
\]
for all $t\in J$, $\psi \in C([-r,0], P)$ and $f\in S_{F(\psi)}$.
\item[(H3)] There exists a real number $k>0$ such that
$\int_{0}^{T}f(t)dt>\theta$ for any $f\in S_{F(x)}$ with
\[
S_{F(x)}=\{g\in L^1(J,E):g(t)\in F(t,x_t)\quad\mbox{a.e. }t\in J\}
\]
for all $x\in C([-r,T],P)$ with $\sup_{t\in J}|x(t)|>k$.
\item[(H4)] For each bounded $B\subset C([-r,T],E),\;u\in B$ and
$t\in J$ the set
\[
\Big\{\int_{0}^{T}[f(s)-\alpha(s)u(s)]ds:f\in S_{F(u)}\Big\}
\]
is relatively compact.
\end{itemize}
\begin{remark} \label{rmk1} \rm
If $F$ is measurable, then the function
$Y:J\to \mathbb{R}$, defined by
\[
Y(t)=\inf\{|v|:v\in F(t,u)\},
\]
belongs to $L^1(J,\mathsf(R))$. So $S_{F(u)}$ is nonempty [12].
\end{remark}
\begin{remark} \label{rmk2} \rm
Let $E=\mathbb{R}$, $h(t)\in L^1(J,\mathbb{R}_+)$,
$G:\mathbb{R}\to$$ bcf([0,\;\rho])$,
$F(t,\psi)=h(t)\psi(0)G(\psi(-r))$ for $t\in J, \; \psi\in
C([-r,0],\mathbb{R})$ and $G$ continuous, then $F$ satisfies the
condition (H1). Take $\alpha(t)=\rho
h(t),\;\beta(t)=0,\;\delta=0$, then $F$ satisfies the condition
(H2). For any $f\in S_{F(x)}$, by Fatou's theorem we obtain that
condition (H3) holds.
\end{remark}
\begin{theorem} \label{thm1}
Assume that closed convex cone $P$ is normal. If the conditions
(H1)--(H4) hold, then \eqref{e1} has at least one (positive) solution
$x$ on $[-r,T]$ with $x(t)\in P$.
\end{theorem}
\begin{proof} \textit{Step 1.} Let $X=\{x\in C(J,E):x(0)=x(T)\}$
with the norm
\[
\|x\|_J=\sup\{|x(t)|: 0\le t\le T\}
\]
and $X_+=\{x\in X:x(t)\in P\;{\rm for}\;t\in J\}$.
It is obvious that $X$ is a Banach space and $X_+$ is a closed convex cone of
$X$, moreover, $x\in X_+$ if $x(t)\ge \theta$ for every $t\in J$.
Let us introduce the differential operator $L: AC(J,X)\to
L^1(J,E)$ by
\[
Lx=x'-\alpha(t)x
\]
with $\alpha$ given in (H2). From the well known results of
ordinary differential equations it follows that for any $y\in
L^1(J,E)$ the boundary value problem
\[
Lx(t)=y(t),\quad x(0)=x(T)
\]
has an unique solution $x:=Ky\in AC(J,X)$ with the operator $K$
defined by
\begin{equation}
\label{6}
(Ky)(t)=\int_{0}^{T}G(t,s)y(s)ds\quad {\rm for}\;t\in J,
\end{equation}
where the Green function $G(t,s)$ satisfies
\begin{equation}
\label{7}
(\tilde{\alpha}(T)-1)\tilde{\alpha}(t)G(t,s)
=\begin{cases}\tilde{\alpha}(T)\tilde{\alpha}(s), & s\le t\\
\tilde{\alpha}(s),
& s>t\end{cases}
\end{equation}
with $\tilde{\alpha}(t)=\exp (-\int_{0}^{t}\alpha(s)ds)$. Thus we
have that $K=L^{-1}$ and \eqref{6} guarantees that $K$ is a bounded
linear operator from $L^1(J,E)$ to $X$.
\noindent\textit{Step 2}. For any $x\in X_+$, from $x(0)=x(T)$ it follows
that $x$ can uniquely be extended to a $T-$periodic function on
$\mathbb{R}$, written as $x^*$. Let $\tilde{x}=x^*|_{[-r,T]}$ and
$x_t=\tilde{x}_t$ for each $t\in J$. It immediately follows that
$x_t\in C([-r,0],P),\;x_0=x_T,\;x_t(0)=x(t)$, $\|x_t\|\le \|x\|_J$
and $t\to x_t$ is continuous for $t\in J$.
For any $x\in X_+$, define the multi-valued map as follows:
\[
H(t,x)=\{f(t)-\alpha(t)x(t): f\in S_{F(x)},\;t\in J\}
\]
with $S_{F(x)}$ given in (H3). By (H1) we have that $H(t,x)$ is
measurable with respect to $t$. For each $g\in H(t,x)$, by (H2) we
have that
\begin{equation}
\label{8} \beta(t)[|x(t)|^{\delta}+1]-\alpha(t)x(t)\le g(t)\le
\theta,
\end{equation}
This inequality and the normality of $P$ imply that
\begin{equation}
\label{9}
|g(t)|\le
N|\beta(t)|[|x(t)|^{\delta}+1]+N\alpha(t)|x(t)|\le
N|\beta(t)|(\|x\|_J^{\delta}+1)+N\alpha(t)\|x\|_J,
\end{equation}
here $N$ is the normal constant of $P$. This implies that
$H:X_+\to \mathcal{P}$$(L^1(J,E))$ is bounded.
\noindent\textit{Step 3}. Let $A=KH$ be a multivalued map from $X_+$ to $X$
defined by
\[
A(t,x)=KH(t,x)=\big\{\int_{0}^{T}G(t,s)g(s)ds:g\in H(t,x)\big\}
\]
for $x\in X_+$ and $t\in J$. It is clear that $A$ is bounded.
Moreover, for any $x\in X_+$ and $h(t)\in A(t,x)$, by \eqref{6}, \eqref{8},
and $G(t,s)\le 0$, there exists $g(t)\in H(t,x)$ for $t\in J$ such
that
\[
h(t)=\int_{0}^{T}G(t,s)g(s)ds\ge \theta.
\]
This implies that $h\in X_+$, i.e., $A(t,x)\subset X_+$. It is
easy to see that $A(t,x)\subset AC(J,X)$. Thus, $AX_+\subset
AC(J,X)\cap X_+$.
Now, we are in a position to prove that $A$ is a u.s.c. and
completely continuous multi-valued map with convex closed values.
$A(t,x)$ is convex for each $x\in X_+$. In fact, if $h_1,h_2\in
A(t,x)$, then there exist $f_1,f_2\in S_{F(x)}$ such that for each
$t\in J$ we have
\begin{gather*}
h_1(t)=\int_{0}^{T}G(t,s)[f_1(s)-\alpha(s)x(s)]ds\,, \\
h_2(t)=\int_{0}^{T}G(t,s)[f_2(s)-\alpha(s)x(s)]ds\,.
\end{gather*}
Let $0\le k\le 1$. Then for each $t\in J$ we have
\[
(kh_1+(1-k)h_2)(t)=\int_{0}^{T}G(t,s)[kf_1(s)+(1-k)f_2(s)-\alpha(s)x(s)]ds.
\]
Since $S_{F(x)}$ is convex (because $F$ has convex values), so
$kh_1+(1-k)h_2\in A(t,x)$.
We next shall prove that $A$ is a completely continuous operator.
For any bounded set $M\subset X_+$, let
$Q=AM$, $m=\sup_{x\in M}\|x\|_J$, $q=\sup_{z\in Q}\|z\|_J$. For any
$t,\tau \in [0,T]$ with $t<\tau$ and $x\in M$,
if $z\in A(t,x)$, then there exists $g\in H(t,x)$ such that
$z=\int_{0}^{T}G(t,s)g(s)ds$, hence $z'=\alpha(t)z+g$. By means of
\eqref{9}, we have
\begin{align*}
|z(\tau)-z(t)|&\le \int_{t}^{\tau}|z'(s)|ds \\
&\le \int_{t}^{\tau}[\alpha(s)\|z\|_J+|g(s)|\;]ds \\
&\le
\int_{t}^{\tau}[(m+Nq)\alpha(s)+N|\beta(s)|(M^{\delta}+1)]ds,
\end{align*}
which shows that $Q$ is equi-continuous on $J$. In virtue of (H4)
together with the Ascoli-Arzela theorem we can conclude that $Q$
is a relatively compact subset in $X$, therefore, $A$ is
completely continuous.
Finally, similar to [8] we can prove that $A$ has closed graph.
Hence, $A$ is u.s.c. (see [8]).
\noindent\textit{Step 4}. To prove that the equations \eqref{e1} has
solutions, we show that $A$ satisfies \eqref{5}. Suppose that this is
not the case, then there exist $(\lambda_n,x_n)\in (0,1)\times
X_+$ such that $x_n\in \lambda_nA(t,x_n),\; \mu_n=\|x_n\|_J\ge n$
for $n=1,2,\dots$. In Step 3 we proved that $x_n\in AC(J,X)\cap
X_+$. There exist $f_n\in S_{F(x_n)}$ such that
$x_n=\lambda_nK(f_n-\alpha x_n)$, i.e. $Lx_n=\lambda_n(f_n-\alpha
x_n)$ for $n=1,2,\dots$, that is,
\begin{equation}
\label{10}
x'_n(t)=(1-\lambda_n)\alpha(t)x_n(t)+\lambda_nf_n(t,(x_n)_t).
\end{equation}
By integrating \eqref{10} with respect to $t$ we obtain
\[
\theta
=(1-\lambda_n)\int_0^T\alpha(t)x_n(t)dt+\lambda_n\int_0^Tf_n(t,(x_n)_t)dt,
\]
so
\begin{equation}
\label{11}
\int_0^Tf_n(t,(x_n)_t)dt=\frac{\lambda_n-1}{\lambda_n}\int_0^T\alpha(t)x_n(t)dt\le
\theta\quad\mbox{for } n=1,2,\dots.
\end{equation}
On the other hand, from the condition (H3), it follows that
\[
\int_0^Tf_n(t,(x_n)_t)dt>\theta
\]
for large enough $n$. This contradicts \eqref{11}, which completes the
proof of \eqref{5}.
By Lemma \ref{lm1}, $A$ has a fixed point $x\in X_+$, which is a solution
to \eqref{e1}. The proof is complete.
\end{proof}
Similarly we can prove the next theorem under the following assumptions:
\begin{itemize}
\item[(H'2)] There exist functions $\alpha\in
L^1(J,\mathbb{R}_+)$, $\beta \in L^1(J,E)$ and
$\delta\in [0,1]$ satisfy that $|\alpha(t)|>0$ for all $t\in J$ and
\[
\beta(t)[|\psi(0)|^{\delta} +1]\le f\le -\alpha (t)\psi (0)
\]
for all $t\in J$, $-\psi \in C([-r,0], P)$ and $\in S_F(\psi)$
\item[(H'3)] There exists a real number $k>0$ such that
$\int_{0}^{T}f(t)dt>\theta$ for any $f\in S_{F(x)}$ (see (H3)) if
$x=-y$ with $\sup_{t\in J}|x(t)|>k$, where $y \in
C([-r,T],P)$ .
\end{itemize}
\begin{theorem} \label{thm2}
Let the closed convex cone $P$ be normal. Assume
conditions (H1)--(H4) and (H'2), (H'3) hold.
Then \eqref{e1} has at least one (negative) solution $x$ on $[-r,T]$ with
$-x(t)\in P$.
\end{theorem}
\begin{remark} \label{rmk3}\rm
Let $r=0$, $E=\mathbb{R}$, and $F$ be a single
valued function, then results for ordinary differential equations
in [11] can be deduced from Theorems \ref{thm1} and \ref{thm2}. Therefore, the
results presented in this section are the generalization and
improvement of the corresponding results in [11].
\end{remark}
\section{Second order Boundary Value Problems}
In this section, we consider existence of solutions
for \eqref{e3}. A function $x\in C([-r,T],E)$ is called the solution
if $x_0=\varphi$ $x(T)=\eta$, for any $t\in J$, $x'(t)$
exists and is absolutely continuous and \eqref{e3} is satisfied. The
following hypotheses will be used.
\begin{itemize}
\item[(H'1)] The mapping $(t,\psi,u)\to F(t,\psi,u)$ is measurable
with respect to $t$ for each $(\psi,u)\in C([-r,0],E)\times E$,
u.s.c. with respect to $(\psi,u)$ for each $t\in J$ and for each
fixed $(\psi,u)\in C([-r,0],E)\times E$ the set
\[
S_{F(\psi,u)}=\{g\in L^1(J,E):g(t)\in F(t,\psi,u)\;{\rm
for\;a.e.}\;t\in J\}
\]
is nonempty.
\item[(H'4)] For each bounded set $B\subset C([-r,T],E)\times E$,
$x\in C([-r,T],E)$ with $(x_t,x'(t))\in B$ for $t\in J$, then the
set
\[
\Big\{\int_{0}^{T}[f(s)-\alpha(s)x(s)]ds:f\in S_{F(x)}\Big\}
\]
is relatively compact, where
\[
S_{F(x)}=\{g\in L^1(J,E):g(t)\in F(t,x_t,x'(t))\quad{for\; a.e.
}\;t\in J\}.
\]
\item[(H5)] There exist $\alpha >0$,
$\beta\in L^1(J, E),\;\delta\in[0,1)$ such that
\[
\beta(t)[|\psi(0)-\xi (t)|^{\delta}+|y-\mu|^{\delta}+1]\le f\le
\alpha [\psi(0)-\xi (t)],
\]
where $f\in S_{F(\psi,y)}$,
$\xi
(t)=\varphi(0)+\frac{t}{T}[\eta-\varphi(0)],\;\mu =\frac{\eta
-\varphi(0)}{T}$, $t\in J,\;\psi \in C([-r,0],E),\;y\in E, \;
\psi(0)\ge \xi(t)$.
\end{itemize}
\begin{theorem} \label{thm3}
Assume that the closed convex cone $P$ is normal. If
the conditions (H'1), (H3),(H'4) and (H5) hold, then \eqref{e3}
has at least one solution $x$ on $[-r,T]$ with $x(t)\ge \xi(t)$
($t\in J$).
\end{theorem}
\begin{proof} \textit{Step 1.}
Let $z=x-\xi$, then \eqref{e3} is
transformed into
\begin{gather*}
z''(t)\in F(t,z_t+\xi_t,z'(t)+\mu)
:=\tilde{F}(t,z_t,z'(t))\quad t\in J,\\
z_0=\varphi -\xi_0 :=\hat{\varphi},\quad z(T)=\theta.
\end{gather*}
Here $\hat{\varphi}\in C([-r,0],E)$ and
$\hat{\varphi}(0)=\theta$. The condition (H5) implies that
$\tilde{F}(t,\psi,y)=F(t,\psi+\xi_t,y+\mu)$ satisfies
\[
\beta(t)[|\psi(0)|^{\delta}+|y|^{\delta}+1]\le
\tilde{F}(t,\psi,y)\le \alpha \psi(0)\quad \psi\in
C([-r,0],E),\;\psi (0)\in P.
\]
Since $x(t)\ge \xi(t)$ is equivalent to $z(t)\ge \theta$, for the
sake of convenience, we assume that $\varphi(0)=\eta=\theta$,
which shows that $\xi(t)\equiv \theta,\;\mu=\theta$.
Let $X=\{x\in C^1(J,E):x(0)=x(T)=\theta,\;x'(0)=x'(T)\}$ with the
norm $\|x\|_X=\|x\|_J+\|x'\|_J$, $X_+=\{x\in X:x\in P\}$,
$Y=L^1(J,E)$, $Z=\{x\in X:x'\mbox{ is absolutely continuous}\}$.
Defining
\[
L:Z\to Y,\quad x\to x''-\alpha x,
\]
where $\alpha$ is given in (H5). Similar to the proof of Theorem
\ref{thm1}, there exists the operator $K=L^{-1}$ defined by
\[
(Ky)(t)=\int_{0}^{T}G(t,s)y(s)ds\quad {\rm for}\;t\in J,\;y\in Y,
\]
where Green's function $G(t,s)$ satisfies
\[
\sqrt{\alpha}\mathop{\rm sh} \sqrt{\alpha}G(t,s)
=\begin{cases}\mathop{\rm sh}\sqrt{\alpha}(t-T)
\mathop{\rm sh}\sqrt{\alpha}\,s, & s\le t,\\
\mathop{\rm sh}\sqrt{\alpha}(s-T)\mathop{\rm sh}\sqrt{\alpha}\,t,
& s>t.\end{cases}
\]
\textit {Step 2.} For $x\in X_+$, $t\in J$, let
\[
x_t(s)=\begin{cases}x(t+s),&\max\{-r,-t\}\le s\le 0,\\
\varphi (t+s), &-r\le s\le -t.\end{cases}
\]
Since $x(0)=\varphi(0)=\theta$, we have that $x_t\in C([-r,0],E)$
and $\|x_t\|\le \|x\|_J+\|\varphi\|$, $t\to x_t$ is
continuous for $(t\in J)$.
For $x\in X_+$, define the multi-valued map
\[
H(t,x)=\{f(t)-\alpha x(t):f\in S_{F(x)},\;t\in J\}
\]
with $S_{F(x)}$ given in (H'4). (H'1) guarantees that
$H(t,x)$ is measurable with respect to $t\in J$. For each $g\in
H(t,x)$, from the condition (H5) it follows that
\[
\beta(t)[|x(t)|^{\delta}+|x'(t)|^{\delta}+1]-\alpha x(t)\le
g(t)\le \theta.
\]
This, and the normality of $P$, implies that $|g(t)|\le
N|\beta(t)|(2\|x\|_X^{\delta}+1)+N\alpha \|x\|_X.$ This shows that
$H:X_+\to 2^Y$ is bounded. Let $A=KH$ be a multi-valued map
from $X_+$ to $X$ defined by
\[
A(t,x)=KH(t,x)=\big\{\int_{0}^{T}G(t,s)g(s)ds:g\in H(t,x)\big\}
\]
for $x\in X_+$ and $t\in J$. It is clear that $A$ is bounded.
Similar to Theorem \ref{thm1} we can prove that $AX_+\subset AC(J,X)\cap
X_+$ and $A$ is u.s.c., completely continuous and has convex
closed values.
\noindent\textit{Step 3}. We will now show that $A$ satisfies \eqref{5}.
Suppose that this is not the case, then there exist $\lambda_n\in (0,1),\;
x_n\in \lambda_nA(t,x_n)\in Z\cap X_+$ such that
$\mu_n=\|x_n\|_X\ge n$ for $n=1,2,\dots$. In Step 2 we proved
that $x_n\in AC(J,X)\cap X_+$. There exists $f_n\in S_{F(x_n)}$
such that $x_n=\lambda_nK(f_n-\alpha x_n)$, i.e.
$Lx_n=\lambda_n(f_n-\alpha x_n)$ for $n=1,2,\dots$, that is,
\begin{equation}
\label{12} x''_n(t)=(1-\lambda_n)\alpha
x_n(t)+\lambda_nf_n(t,(x_n)_t,x'_n(t)).
\end{equation}
By integrating this expression with respect to $t$ we obtain
\[
\theta =(1-\lambda_n)\int_0^T\alpha
x_n(t)dt+\lambda_n\int_0^Tf_n(t,(x_n)_t,x'_n(t))dt,
\]
so
\begin{equation}
\label{13}
\int_0^Tf_n(t,(x_n)_t,x'(t))dt=\frac{\lambda_n-1}{\lambda_n}\int_0^T\alpha
x_n(t)dt\le \theta\quad{\rm for}\;n=1,2,\dots.
\end{equation}
On the other hand, the condition (H3) guarantees that
\[
\int_0^Tf_n(t,(x_n)_t,x'(t))dt>\theta
\]
for large enough $n$. This contradicts \eqref{13},
which completes the proof of \eqref{5}. By Lemma \ref{lm1}, $A$
has a fixed point $x\in X_+$, which is a solution to \eqref{e3}.
The proof is completed.
\end{proof}
\begin{remark} \label{rmk4}
In fact, we can allow that $0\le \delta \le 1$ in
(H5).
\end{remark}
\begin{itemize}
\item[(H6)] If $B\subset W(t):=\{x(t): x\in C^1(J,E),
x(0)=x(T)=\theta\}$ is bounded, then $B$ is relatively compact.
\end{itemize}
\begin{theorem} \label{thm4}
Assume that the closed convex cone $P$ is normal. If
the conditions (H'1), (H'4), (H5) and (H6) hold, then \eqref{e3}
has at least one solution $x$ on $[-r,T]$ with $x(t)\ge \xi(t)$
($t\in J$).
\end{theorem}
\begin{proof} According to Theorem \ref{thm3} it suffices
to prove that \eqref{5} is true. Suppose that this is not the case, then
there exist $\lambda_n\in (0,1),\; x_n\in \lambda_nA(t,x_n)\in
Z\cap X_+$ such that $\mu_n=\|x_n\|_X\ge n$ for $n=1,2,\dots$.
Let $y_n=\frac{1}{\mu_n}x_n,\;\rho_n=\frac{\lambda_n}{\mu_n},$
then $\|y_n\|_X=1$. Similar to (12) we obtain
\begin{equation}
\label{14} y''_n(t)=(1-\lambda_n)\alpha
y_n(t)+\rho_nf_n(t,(x_n)_t,x'_n(t)).
\end{equation}
This and (H5) guarantee
\[
\rho_n\beta
(t)[|x_n(t)|^{\delta}+|x'_n(t)|^{\delta}+1]\le
y''_n(t)-(1-\lambda_n)\alpha y_n(t)\le \lambda_n\alpha y_n(t),
\]
that is
\begin{equation} \label{15}
\begin{aligned}
&|y''_n(t)|\\
&\le N(1-\lambda_n)\alpha
|y_n(t)|+N\lambda_n\alpha |y_n(t)|+2N\rho_n|\beta
(t)|[|x_n(t)|^{\delta}+|x'_n(t)|^{\delta}+1] \\
&\le N\alpha +6N|\beta(t)|.
\end{aligned}
\end{equation}
This inequality implies that $\{y'_n\}_{n=1}^{\infty}$ is equicontinuous on
$J$. Note that $|y_n(t)-y_n(s)|\le |t-s|\;\|y'_n\|_J\quad (t,s\in
J)$, we have that $\{y_n\}_{n=1}^{\infty}$ is also equicontinuous
on $J$. For each $t\in J$, since $\{y_n(t)\}_{n=1}^{\infty}\subset
W(t)$ is bounded, by (H6) $\{y_n(t)\}_{n=1}^{\infty}$ is
relatively compact. By Arzel\'a -Ascoli's theorem, one has
that $\{y_n\}_{n=1}^{\infty}$ is relatively compact in $X$.
Without loss of generality, let $y_n\to y $ with some
$y\in X$ and $\lambda_n \to \lambda \in J$ for
$n\to \infty$.
Integrating \eqref{14} with respect to $t$, we obtain
\[
y'_n(t) =y_n(0)+(1-\lambda_n)\alpha \int_0^t
y_n(s)ds+\rho_n\int_0^tf_n(s,(x_n)_s,x'_n(s))ds.
\]
Letting $n$ approach $\infty$, we obtain
\begin{equation}
\label{16} y'(t) =y(0)+(1-\lambda)\alpha \int_0^t y(s)ds+g(t)
\end{equation}
with $g(t)=\lim_{n\to \infty}
\rho_n\int_0^tf_n(s,(x_n)_s,x'_n(s))ds$, which exists for $0\le t<\tau \le T$.
By (H5) we have that
\[
\rho_n\int_t^{\tau}f_n(s,(x_n)_s,x'_n(s))ds\ge
\rho_n\int_t^{\tau}\beta(s)[|(x_n)(s)|^{\delta}+|x'_n(s)|^{\delta}+1)]ds
\]
and
\begin{align*}
&|\rho_n\int_t^{\tau}\beta(s)[|(x_n)(s)|^{\delta}+|x'_n(s)|^{\delta}+1)]ds|\\
&\le \rho_n\int_t^{\tau}|\beta(s)|(2\mu_n^{\delta}+1)ds \\
&\le (2\mu_n^{\delta-1}+\rho_n)\int_t^\tau |\beta(s)|\;ds\to 0\quad
(n\to \infty),\nonumber
\end{align*}
which yields $g(\tau)-g(t)\ge \theta,$ that is, $g(t)$ is monotone
increasing on $J$. Especially, $g(t)\ge g(0)= \theta$ for each
$t\in J$. For any $t\in J$, $y(t)\in X_+$ deduces that $y(t)\ge
\theta =y(0)$, which implies that $y'(0)\ge \theta$. Summing up,
from \eqref{16}, it follows that $y'(t)\ge \theta\;(t\in J)$. This
shows that $y$ is a increasing function on $J$. Note that
$y(0)=y(T)=\theta$, we have $y(t)\equiv \theta$ on $J$, which
contradicts $\|y\|_X=1$. The proof is completed.
\end{proof}
\begin{corollary} \label{coro1}
Let $E=\mathbb{R}^n$. If the conditions (H'1)
and (H5) hold, then \eqref{e3} has at least one solution $x$ on
$[-r,T]$ with $x(t)\ge \xi (t)$ $(t\in J)$.
\end{corollary}
\begin{corollary} \label{coro2}. Let $r=0$ and
$F:J\times \mathbb{R}^{n}\times \mathbb{R}^n \to
2^{\mathbb{R}^n}$. If $F$ satisfies the condition (H'1), and there
exist $\alpha>0$, $\beta\in L^1(J,\mathbb{R}^n)$,
$\delta\in [0,1)$ such that
\[
\beta(t)\Big[|x-\xi(t)|^{\delta}+\big|y-\frac{B-A}{T}\big|^{\delta}+1\Big]
\le f(t,x,y)\le \alpha[x-\xi(t)]
\]
for $t\in J$, $x,y\in \mathbb{R}^{n}$, $ x\ge \xi(t):=A+(t/T)(B-A)$ and
$f\in \{g\in L^1(J,E):g(t)\in F(t,x,y)\;{\rm for\;a.e.}\;t\in J\}$,
then second order ordinary differential inclusion
\begin{gather*}
x''(t)\in F(t,x(t),x'(t))\;(t\in J),\\
x(0)=A,\quad x(T)=B,
\end{gather*}
has at least a solution $x\in C^1(J,\mathbb{R}^n),$ with $x(t)\ge
\xi(t)$ on $J$.
\end{corollary}
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\end{document}