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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 98, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/98\hfil Positive solutions and continuous branches]
{Positive solutions and continuous branches for 
boundary-value problems of differential inclusions}

\author[N. T. Hoai, N. V. Loi\hfil EJDE-2007/98\hfilneg]
{Nguyen Thi Hoai, Nguyen Van Loi}  % in alphabetical order

\address{Nguyen Thi Hoai \newline
Faculty of mathematics \\
Voronezh State Pedagogical University, Russia}
\email{nthoai0682@yahoo.com}

\address{Nguyen Van Loi \newline
Faculty of mathematics \\
Voronezh State Pedagogical University, Russia}
\email{loitroc@yahoo.com}

\thanks{Submitted February 16, 2007. Published July 13, 2007.}
\subjclass[2000]{34B16, 34A60, 34B18, 47H04}
\keywords{Boundary value problems; positive solutions;
  multivalued map; \hfill\break\indent differential inclusions}

\begin{abstract}
 In this paper, we consider second order differential
 inclusions with periodic boundary conditions. We obtain the
 existence of positive solutions and of continuous branches
 of positive solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

  Consider the boundary-value problem
\begin{equation} \label{e1.1}
\begin{gathered}
Lu\in\lambda{F(t,u)},\quad 0<t<1\,,\\
\alpha{u(0)}-\beta{u'(0)}=0,\quad \gamma{u(1)}+\delta{u'(1)}=0,
\end{gathered}
\end{equation}
where $Lu=-(ru')'+qu$, $r\in{C^{1}[0,1]}$, $q\in{C[0,1]}$
with $r>0$, $q\geq{0}$ on
$[0,1]$, $\alpha,\beta,\gamma,\delta\quad \geq{0}$ with
$\alpha\delta+\alpha\gamma+\beta\gamma\quad >0$,
$F\colon [0,1]\times{[0,+\infty)\to {P([0,+\infty))}}$, and
$\lambda$ is a positive parameter.

When $F$ is a continuous map, the existence of
positive solutions of  \eqref{e1.1} was studied in \cite{Dang}. In
this paper, the results in \cite{Dang,Loi} will be used to prove the
existence  of positive solutions of \eqref{e1.1}.

First, we recall the following notion (see, e.g. \cite{BGMO,KOZ}).
Let $X,Y$  be two Banach spaces. Let $P(Y)$,
 $K(Y)$, $Kv(Y)$, $C(Y)$, $Cv(Y)$ denote the collections of all nonempty,
nonempty compact, nonempty convex compact, nonempty closed,
nonempty convex closed subsets of $Y$, respectively.

 A multimap $F\colon{X}\to {P(Y)}$ is said to be
upper semicontinuous (u.s.c.) [lower semicontinuous (l.s.c.)] if
the set $F_{+}^{-1}(V)=\{x\in{X}: {F(x)\subset{V}}\}$ is open
[respectively, closed] for every open [respectively, closed]
subset $V\subset{Y}$. $F$ is said to be compact if the set $F(X)$
is relatively compact in $Y$.

Let $A\subset{K(Y)}$ and the
max-normal  and  min-normal be
$$
\|A\|=\max\{\|x\| : x\in{A}\}\quad \text{and}\quad
\|A\|_{0}=\min\{\|z\| : z\in{A}\}.
$$
Let $C_{+}[0,1]\,(L_{+}^{1}[0,1])$ denote the cone of all positive
continuous (respectively, integrable) functions on $[0,1]$.
We will consider the cone $C_{+}[0,1]\quad (L_{+}^{1}[0,1])$ as
subspace of the space $C[0,1]$ (respectively, $L^{1}[0,1]$)
with induced topology.


The nonempty subset $M\subset{L_{+}^{1}[0,1]}$ is said to be decomposable
provided for every $f,g\in{M}$ and each Lebesgue measurable subset
$m\subset{[0,1]}$,
$$
f\chi_{m}+g\chi_{[0,1]\setminus{m}}\in{M},
$$
where $\chi_{m}$ is the characteristic function of the set $m$.

\section{ Existence of positive solutions}

Let $G(t,s)$ be the Green's function for \eqref{e1.1}. Then $u$ is a
solution of \eqref{e1.1} if and only if
$$
u(t)\in\lambda\int_{0}^{1}G(t,s)F(s,u(s))ds.
$$
Recall that
\[
G(t,s)= \begin{cases}
c^{-1}\phi(t)\psi(s) & \text{if $t\leq{s}$}\\
c^{-1}\phi(s)\psi(t) & \text{if $s\leq{t}$},
\end{cases}
\]
where $\phi$ and $\psi$ satisfy
\begin{gather*}
L{\phi}=0,\quad \phi(0)=\beta,\quad \phi'(0)=\alpha, \\
L{\psi}=0,\quad \psi(1)=\delta,\quad \psi'(1)=-\gamma
\end{gather*}
and $c=r(t)(\phi'(t)\psi(t)-\psi'(t)\phi(t))>0$. Note that
$\phi'>0$ on $(0,1]$ and $\psi'<0$ on $[0,1)$. Let
$G=\max\{G(t,s):0\leq{t,s}\leq{1}\}$.
We shall make the following assumptions:
\begin{itemize}
\item[(H1)] For every $x\in{[0,+\infty)}$ the multifunction
$F(\cdot ,x)\colon{[0,1]}\to {Kv([0,+\infty))}$
has a measurable selection,
i.e., there exists a measurable function $f$ such that
$f(t)\in{F(t,x)}$ for a.e. $t\in{[0,1]}$;


\item[(H2)] For a.e. $t\in{[0,1]}$ the multimap
$F(t,\cdot)\colon{[0,+\infty)}\to {Kv([0,+\infty))}$ is
u.s.c.;

\item[(H3)] There exists a positive function $\omega\in{L^{1}[0,1]}$
such that
$$
\|F(t,x)\|\leq\omega(s)(1+x),
$$
for all $x\in{[0,+\infty)}$ and a.e. $t\in{[0,1]}$;

\item[(H4)] The multioperator
$F\colon{[0,1]\times{[0,+\infty)}}\to {K([0,\infty))}$ is
almost lower semicontinuous; i.e., there exists a sequence of
disjoint compact sets $\{I_m\}, {I_m}\subset{{[0,1]}}$ such that:
\begin{itemize}

\item[(i)] $\mathop{\rm meas}([0,1]\setminus\bigcup_{m}I_{m})=0$;

\item[(ii)] the restriction of $F$ on each set ${J_m}={I_m}\times{[0,\infty)}$
is l.s.c.;
\end{itemize}
\end{itemize}
We will use the method in \cite{Loi} to prove the following results.

\begin{theorem} \label{thm1}
 Let (H1)--(H3) hold. If \eqref{e1.1} has no zero solution, then for each
$0<\lambda<\frac{1}{G\int_{0}^{1}\omega(s)ds}$, \eqref{e1.1} has a positive
solution.
\end{theorem}

\begin{theorem} \label{thm2}
Let (H3)-(H4) hold. If \eqref{e1.1} has no zero solution,
then for each $0<\lambda<\frac{1}{G\int_{0}^{1}\omega(s)ds}$,
 \eqref{e1.1} has a positive solution.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thm1}]
From (H1)--(H3) it follows easily that the multioperator superposition
\begin{gather*}
\wp_{F}\colon{C_{+}[0,1]}\to {Cv(L_{+}^{1}[0,1])},\\
\wp_{F}(u)=\{f\in{L_{+}^{1}[0,1]}:f(s)\in{F(s,u(s))}\text{ for a.e. }
 s\in{[0,1]}\}.\notag
\end{gather*}
is defined and closed (see, e.g. \cite{BGMO}). Consider a completely
continuous operator
\[
Q_{\lambda}\colon{L_{+}^{1}[0,1]}\to {C_{+}[0,1]},\quad
Q_{\lambda}(f)(t)=\lambda\int_{0}^{1}G(t,s)f(s)ds,
\]
Let $\Gamma_{\lambda}=Q_{\lambda}\circ\wp_{F}$.
From  \cite[Theorem 1.5.30]{BGMO} it follows that the multioperator
$\Gamma_{\lambda}$ is closed. We can easily prove that for every bounded
subset $U\subset{C_{+}[0,1]}$, the set $\Gamma_{\lambda}(U)$ is relatively
compact in $C_{+}[0,1]$. Hence applying  \cite[Theorem 1.2.48]{BGMO},
we have that the Hammerstein's multioperator
\begin{gather*}
\Gamma_{\lambda}\colon{C_{+}[0,1]}\to {Kv(C_{+}[0,1])},\\
\Gamma_{\lambda}(u)=\lambda\int_{0}^{1}G(t,s)F(s,u(s))ds.
\end{gather*}
is upper semicontinuous.
Let $T_{+}=\{u\in{C_{+}[0,1]}:\|u\|_{C}\leq\rho, \text{ where }\rho>0\}$
For  $u$ in ${T_{+}}$ we have
$$
\big\|\Gamma_{\lambda}(u)\big\|_{C}
=\max \big\{\big\|\lambda\int_{0}^{1}G(t,s)f(s)ds
\big\|_{C}:f\in{\wp_{F}(u)}\big\},
$$
where
$$
\big\|\int_{0}^{1}G(t,s)f(s)ds\big\|_{C}
=\sup_{t\in{[0,1]}} \big\{\int_{0}^{1}G(t,s)f(s)ds\big\}.
$$
Since $f(s)\in{F(s,u(s))}$ for a.e. $s\in{[0,1]}$ and (H3), for a.e.
$s\in{[0,1]}$ we have
$$
f(s)\leq{\|F(s,u(s))\|}\leq\omega(s)(1+u(s))\leq\omega(s)(1+\|u\|_{C})
\leq{\omega(s)(1+\rho)}.
$$
Therefore,
$$
\int_{0}^{1}G(t,s)f(s)ds\leq{G(1+\rho)\int_{0}^{1}\omega(s)ds},
$$
and hence
$$
\big\|\int_{0}^{1}G(t,s)f(s)ds\big\|_{C}\leq
{G(1+\rho)\int_{0}^{1}\omega(s)ds}.
$$
Because the last inequality holds for all $f\in{\wp_{F}(u)}$,
$$
{\|\Gamma_{\lambda}(u)\|}_{C}\leq\lambda{G(1+\rho)\int_{0}^{1}\omega(s)ds}.
$$
Choose
$\rho\geq\frac{\lambda{G\int_{0}^{1}\omega(s)ds}}{1-\lambda{G\int_{0}^{1}
\omega(s)ds}}$ then $\|\Gamma_{\lambda}(u)\|_{C}\leq\rho$, i.e.,
$\Gamma_{\lambda}$ maps the set $T_{+}$ in to itself.
The existence of positive solution of the problem \eqref{e1.1} can be
easily follow from the Bohnenblust-Karlin fixed
point theorem
\end{proof}

For the proof of Theorem \ref{thm2} we need the following result proved in
 \cite{Deim,HuPa}.

\begin{lemma} \label{lem1}
 Let $X$ be a separable metric space; $E$ be a Banach space.
Then every l.s.c. multimap $\tilde{F}\colon{X}\to {P(L^{1}([0,1],E))}$
with closed decomposable values has a continuous selection.
\end{lemma}


\begin{proof}[Proof of theorem \ref{thm2}]
From  conditions (H3)--(H4) it follows that
$$
\wp_{F}\colon{C_{+}[0,1]}\to {C(L_{+}^{1}[0,1])}
$$
is a l.s.c. multioperator with closed decomposable values
(see, e.g. \cite{BGMO,KOZ}).

Consider again the Hammerstein's multioperator
$\Gamma_{\lambda}=Q_{\lambda}\circ\wp_{F}$.  By  Lemma \ref{lem1},
the multioperator superposition $\wp_{F}$ has a continuous selection
\[
\ell\colon{C_{+}[0,1]}\to {L_{+}^{1}[0,1]},\quad
\ell(u)\in\wp_{F}(u).
\]
Hence the operator
\[
\gamma_{\lambda}\colon{C_{+}[0,1]}\to {C_{+}[0,1]},\quad
\gamma_{\lambda}(u)(t)=\lambda\int_{0}^{1}G(t,s)\ell(u)(s)ds,
\]
is a completely continuous selection of the multioperator $\Gamma_{\lambda}$.
As  shown above, for each $0<\lambda<\frac{1}{G\int_{0}^{1}\omega(s)ds}$,
we can choose $\rho>0$ such that the multioperator $\Gamma_{\lambda}$ maps
the set $T_{+}$ in to itself. From the Schauder fixed theorem it follows
that the operator $\gamma_{\lambda}$ has a fixed point in $T_{+}$, i.e.,
 \eqref{e1.1} has a positive solution
\end{proof}

Now we use the result in \cite{Dang} to prove the existence and
multiplicity of positive solutions for \eqref{e1.1}, when $F$ is
lower semicontinuous. Assume that
\begin{itemize}

\item[(F1)] $F\colon{(0,1)\times{[0,+\infty)}}\to {Kv([0,+\infty))}$ is
l.s.c.;


\item[(F2)] For each $M>0$, there exists a continuous function
$g_{M}$ on $(0,1)$ such that $\|F(t,x)\|\leq{g_{M}(t)}$ for
$t\in{(0,1)}\text{,\quad }x\in{[0,M]}$, and
$$
\int_{0}^{1}G(s,s)g_{M}(s)ds<\infty.
$$

\item[(F3)] There exist an interval $I\subset{(0,1)}$ and a
non-zero function $m\in{L^{1}(I)}$ with $m\geq{0}$ such that
for every $b>0$, there exists $r_{b}>0$ such that
$$
\|F(t,x)\|_{0}\geq bm(t)x \quad \text{for } t\in{I}, \;
x\in{(0,r_{b})};
$$

\item[(F4)] There exist an interval $I_{1}\subset{(0,1)}$ and a
non-zero function $m_{1}\in{L^{1}(I_{1})}$ with $m_{1}\geq{0}$
such that for every $c>0$, there exists $R_{c}>0$ such that
$$
\|F(t,x)\|_{0}\geq c\,m_{1}(t)x \quad \text{for }
t\in{I_{1}},\; x\geq{R_{c}};
$$
\end{itemize}

\begin{theorem} \label{thm3}
Let (F1)--(F3) hold.
Then there exists $\lambda_{0}>0$ such that \eqref{e1.1} has a positive
solution for $0<\lambda<\lambda_{0}$. If, in addition, (F4)
holds, then \eqref{e1.1} has at least two positive solutions for
$0<\lambda<\lambda_{0}$
\end{theorem}
For the proof of this we need the following
result (see, e.g. \cite{BGMO,Mic}).

\begin{lemma} \label{lem2}
 Let $X$ be a metric space; $Y$ be a  Banach space.
Then every l.s.c. multi-map $W\colon{X}\to {Cv(Y)}$
has a continuous selection.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm3}]
Let $f\colon{(0,1)\times{[0,+\infty)}}\to {[0,+\infty)}$ be a
continuous selection of $F$, i.e.,
$$
f(t,x)\in{F(t,x)}\quad \text{for all } (t,x)\in{(0,1)\times{[0,+\infty)}}.
$$
It is easy to see that for all $(t,x)\in{(0,1)\times{[0,+\infty)}}$
the following inequality holds
$$
\|F(t,x)\|_{0}\leq{f(t,x)}\leq{\|F(t,x)\|}.
$$
Consider now the problem
\begin{equation} \label{e1.2}
Lu=\lambda{f(t,u)},\quad 0<t<1,
\end{equation}
with the conditions in \eqref{e1.1}.
By (F1)--(F4) we have
\begin{itemize}
\item[(f1)] The map $f\colon{(0,1)\times{[0,+\infty)}}\to {[0,+\infty)}$
is continuous;


\item[(f2)] For each $M>0$, there exists a continuous function
$g_{M}$ on $(0,1)$ such that $f(t,x)\leq{g_{M}(t)}$ for
$t\in{(0,1)}$, $0\leq{x}\leq{M}$ and
\[
\int_{0}^{1}G(s,s)g_{M}(s)ds<\infty.
\]

\item[(f3)] There exist an interval $I\subset{(0,1)}$ and a non-zero function
$m\in{L^{1}(I)}$ with $m\geq{0}$ such that for every $b>0$, there
exists $r_{b}>0$ such that
\[
f(t,x)\geq{bm(t)x},\quad \text{for } t\in{I},\; x\in{(0,r_{b})};
\]
\end{itemize}
If $(F4)$ holds then we have
\begin{itemize}

\item[(f4)] There exist an interval $I_{1}\subset{(0,1)}$ and a
non-zero function $m_{1}\in{L^{1}(I_{1})}$ with $m_{1}\geq{0}$
such that for every $c>0$, there exists $R_{c}>0$ such that
\[
f(t,x)\geq{c\,m_{1}(t)x},\text{\quad for } t\in{I_{1}},\quad
x\geq{R_{c}};
\]
\end{itemize}
 From \cite[Theorem 1.1]{Dang} it follows that if (f1)--(f3) hold then
there exists $\lambda_{0}>0$ such that \eqref{e1.2} has a positive
solution for $0<\lambda<\lambda_{0}$. If, in addition, $(f4)$ holds
then \eqref{e1.2}
has at least two positive solutions for $0<\lambda<\lambda_{0}$.
Hence we obtain our result
\end{proof}

\section{Continuous branch of positive solutions}

A sphere and a ball with center at the point $0$  (the zero function)
and radius $r$ in the cone $C_{+}[0,1]$ will be denoted respectively by
\begin{gather*}
S_{+}(0,r)=\{u\in{C_{+}[0,1]} :\|u\|_{C}=r\},\\
T_{+}(0,r)=\{u\in{C_{+}[0,1]} :\|u\|_{C}\leq{r}\}.
\end{gather*}
Recall the following notion (see, \cite{Bakh1,Bakh2,Kras2}).

\noindent\textbf{Definition} A set $V$ of positive
solutions of \eqref{e1.1} is said to form a continuous branch connecting
the spheres $S_{+}(0,r)$ and $S_{+}(0,R)$, with $0\leq{r}<R\leq\infty$,
if for every nonempty open bounded subset
$$
\Delta\subset{C_{+}[0,1]}: T_{+}(0,r')\subset\Delta
\subset{T_{+}(0,R')},\;r<r'<R'<R
$$
the set $V\cap\partial\Delta$ is nonempty, where
$\partial\Delta$ is a boundary of $\Delta$.
If, in addition, $r=0$ and $R=\infty$ then the set $V$
is said to be a continuous branch with infinite length.


Let $E$ be a Banach space; $\mathbf{K}\subset{E}$ be a cone.

\noindent\textbf{Definition} An operator $A\colon{E}\to {E}$ is
said to be positive, if $A\mathbf{K}\subset\mathbf{K}$.

\begin{lemma}[\cite{Bakh1,Kras1}] \label{lem3}
Let $A$ be a positive completely continuous operator on the cone
$\mathbf{K}$. Assume that on the border $\partial\Xi_{\mathbf{K}}$
of every bounded subset $\Xi_{\mathbf{K}}\ni{0}$ of the cone
$\mathbf{K}$ the following inequality holds
$$
\inf_{x\in{\partial\Xi_{\mathbf{K}}}}\|Ax\|>0.
$$
Then the positive solutions of the equation
$$
Ax=\mu{x},\quad  x\in{\mathbf{K}\setminus\{0\}}
$$
form a continuous branch with infinite length.
\end{lemma}

Let $a$ be a positive constant. Consider now the problem
\eqref{e1.1} with the multimap
\[
F\colon{[0,1]\times{[0,+\infty)}}\to {K([a,+\infty))}
\]
satisfying the following assumptions:
\begin{itemize}

\item[(A1)] $F$ is almost lower semicontinuous;

\item[(A2)] For every nonempty bounded subset
$\Omega\subset{[0,+\infty)}$ there exists a function
$\vartheta_{\Omega}\in{L_{+}^{1}[0,1]}$ such that
$$
\|F(t,x)\|\leq\vartheta_{\Omega}(t),
$$
for all $x\in\Omega$ and a.e. $t\in{[0,1]}$;

\item[(A3)] There exists $q>0$ such that the Green's function
satisfies $G(t,s)\geq{q}$, for all
$0\leq{t,s}\leq{1}$;
\end{itemize}

\begin{theorem} \label{thm4}
Let (A1)--(A3) hold. Then
the positive solutions of \eqref{e1.1} form a continuous branch with
infinite length.
\end{theorem}

\begin{proof}
Note that the condition (H3) is special case of the condition (A2).
As is shown above, from (A1)--(A2) the multioperator $\Gamma_{\lambda}$
has a completely continuous selection $\gamma_{\lambda}$ on the cone
$C_{+}[0,1]$. Let $\Xi\ni{0}$ be an open bounded subset of $C_{+}[0,1]$.
For all $u\in\Xi$, since $\ell(u)(s)\in{F(s,u(s))}$ for a.e.
$s\in{[0,1]}$ we have
$$
\gamma_{\lambda}(u)(t)=\lambda\int_{0}^{1}G(t,s)\ell(u)(s)ds
\geq{\lambda{aq}}>0.
$$
Hence
$$
\inf_{u\in\partial\Xi}\|l(u)\|_{C}\geq{aq}>0,\quad \text{where }
 l=\frac{\gamma_{\lambda}}{\lambda}.
$$
On the cone $C_{+}[0,1]$ consider the equation
\begin{equation} \label{e1.3}
l(u)=\frac{1}{\lambda}{u}
\end{equation}
By  Lemma \ref{lem3}, the positive solutions of \eqref{e1.3} form a continuous
branch with infinite length. And hence we obtain our result
\end{proof}

\section{Examples}

\begin{example} \label{exa1} \rm
 Let $D\subset{[0,1]}$ be a nonmeasurable set;
$$
F\colon{[0,1]\times{[0,+\infty)}}\to {Kv([0,+\infty))}
$$
be the multimap
\[
F(t,x)=\begin{cases}
[0,x+1] & \text{if $x=t$ and $t\in{[0,1]\setminus{D}}$}\\
[0,x+1] & \text{if $x=t+1$ and $t\in{D}$}\\
x+1 & \text{otherwise.}
\end{cases}
\]
Consider the  differential inclusion
\begin{equation} \label{e1.4}
\begin{gathered}
-u''(t)\in{\lambda\,F(t,u(t))},\quad \lambda>0,\quad 0<t<1,\\
u(0)=u(1)=0.
\end{gathered}
\end{equation}
It is easy to see that
\[
G(t,s)=\begin{cases}
t(1-s) & \text{if $0\leq{t}\leq{s}\leq{1}$}\\
s(1-t) & \text{if $0\leq{s}\leq{t}\leq{1}$}
\end{cases}
\]
is a Green's function for the operator $Lu=-u''$.
Note that
$\max\{G(t,s):0\leq{t,s}\leq{1}\}=1$.
Choose a function $\omega\equiv{1}$ then the conditions
(H1)-(H3) hold.
Zero function is not a solution of \eqref{e1.4}.
From Theorem \ref{thm1} it follows that for each $0<\lambda<1$ the
inclusion \eqref{e1.4} has a positive solution
\end{example}

\begin{example} \label{exa2} \rm
Let $\varepsilon\in{(0,1)}$ and
$F\colon{(0,1)\times{[0,+\infty)}}\to {Kv([0,+\infty))}$ be
the  multimap
\[
F(t,x)=\begin{cases}
t(x^{2}+\frac{1}{1+x}) & \text{if $0<t\leq\varepsilon$ and $0\leq{x}\leq{1}$}\\
(t+1)(x^{2}+\frac{1}{x+\varepsilon}) & \text{if
$0<t\leq\varepsilon$ and $2\leq{x}\leq{3}$}\\
[t(x^{2}+\frac{1}{1+x}),\,(t+1)(x^{2}+\frac{1}{x+\varepsilon})] &
\text{otherwise.}
\end{cases}
\]
It is clear that the multimap $F$ is lower semicontinuous.
Consider the  inclusion
\begin{equation} \label{e1.5}
\begin{gathered}
{(-e^{\frac{-{t}^2}{2}}\,u')'+e^{\frac{-{t}^2}{2}}\,u}
\in{\lambda{F(t,u)}},\quad 0<\lambda,0<t<1,\\
u(0)=u(1)=0.
\end{gathered}
\end{equation}
Let
$Lu=(-e^{t^2/2} u')'+e^{-t^2/2}u$.
Then
\[
G(t,s)=\begin{cases}
\frac{e^{t^2/2}}{\int_{0}^{1}e^{-\tau^2/2}d\tau}\,
\int_{s}^{1}e^{-\tau^2/2}d\tau\int_{0}^{t}e^{-\tau^2/2}d\tau,
 & \text{if $0\leq{t}\leq{s}$} \\[4pt]
\frac{e^{t^2/2}}{\int_{0}^{1}e^{-\tau^2/2}d\tau}
\int_{0}^{s}e^{-\tau^2/2}d\tau\int_{t}^{1}
e^{-\tau^2/2}d\tau,
 & \text{if $s\leq{t}\leq 1$}
\end{cases}
\]
is a Green's function for the operator $L$ (see, e.g. \cite{BoHo}).


For each $M>0$, let
$g_{M}(t)=(M^{2}+\frac{1}{\varepsilon})(t+1)$. We have
\[
\|F(t,x)\|\leq(t+1)(x^{2}+\frac{1}{x+\varepsilon})\leq{g_{M}(t)},
\]
for $0<t<1$, $0\leq{x}\leq{M}$ and
\[
\int_{0}^{1}G(s,s)g_{M}(s)ds<+\infty.
\]
Hence the condition (F2) holds.
Let $I=(0,\varepsilon), m(t)=t$. Then for every $b>0$
$$
\|F(t,x)\|_{0}=t(x^{2}+\frac{1}{1+x})\geq{b\,m(t)x}\quad
\text{for } t\in{I},\; x\in{(0,r_{b})},
$$
where
$r_{b}=\min\{\frac{-b+(b^{2}+4b)^{1/2}}{2b},1\}$. The condition
(F3) holds.
For every $c>0$
$$
\|F(t,x)\|_{0}\geq{t(x^{2}+\frac{1}{1+x})}\geq{c\,m(t)x},\quad
\text{for }t\in{I},\;x\geq{c}.
$$
The condition (F4) holds.
By  Theorem \ref{thm3}, there exists $\lambda_{0}>0$ such that
\eqref{e1.5} has at least two positive solutions for $0<\lambda<\lambda_{0}$
\end{example}

\begin{example} \label{exa3} \rm
 Let
$F\colon{[0,1]\times{[0,+\infty)}}\to {K([1,+\infty))}$ be the multimap
\[
F(t,x)= \begin{cases}
(t^{2}+2)(x^{2}+\frac{1}{x+1}) & \text{if $0\leq{t}\leq{1}$,
$0\leq{x}\leq{1}$}\\
(t+2)(x^{2}+\frac{1}{x+1}) & \text{if $0\leq{t}\leq{1}$,
$2\leq{x}\leq{3}$}\\
[(t^{2}+2)(x^{2}+\frac{1}{1+x}),\,(t+2)(x^{2}+\frac{1}{x+1})] &
\text{otherwise.}
\end{cases}
\]
Consider the problem
\begin{equation} \label{e1.6}
\begin{gathered}
-(1+e^{t})u''-e^{t}u'\in{\lambda{F(t,u)}},\quad 0<t<1,\quad 0<\lambda,\\
u(0)-2u'(0)=0,\quad u'(1)=0.
\end{gathered}
\end{equation}
It is clear that
$F$ is lower semicontinuous. Hence the condition (A1) holds.
\[
G(t,s)=
\begin{cases}
x-\ln(1+e^{x})+1+\ln{2} & \text{if $0\leq{t}\leq{s}$}\\
s-\ln(1+e^{s})+1+\ln{2} & \text{if $0\leq{s}\leq{t}$}
\end{cases}
\]
is a Green's function for operator
$Lu=-(1+e^{t})u''-e^{t}u'$ (see, \cite{BoHo}) and
$$
G(t,s)\geq{1},\quad \text{for all }  t,s\in{[0,1]}.
$$
The condition (A3) holds.


For every bounded subset $\Omega\subset{[0,+\infty)}$, let
$\vartheta_{\Omega}(t)=(t+2)(1+\|\Omega\|^{2})$. We have
$$
\|F(t,x)\|\leq{(t+2)(x^{2}+\frac{1}{1+x})}\leq\vartheta_{\Omega},
$$
for all $x\in\Omega$ and all $t\in{[0,1]}$. Therefore the condition
(A2) holds.
From Theorem \ref{thm4} it follows easily that the set of positive
solutions of \eqref{e1.6} forms a continuous branch with infinite
length
\end{example}

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\end{document}