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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 13, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/13\hfil Multiplicity of solutions]
{Multiplicity of solutions for some fourth-order m-point
 boundary-value problems}

\author[H. Li, Y. Liu\hfil EJDE-2010/13\hfilneg]
{Haitao Li, Yansheng Liu}  % in alphabetical order

\address{Haitao Li \newline
Department of Mathematics, Shandong Normal University, Jinan,
250014,  China}
\email{haitaoli09@gmail.com}

\address{Yansheng Liu \newline
 Department of Mathematics, Shandong
Normal University, Jinan, 250014,  China}
\email{yanshliu@gmail.com}


\thanks{Submitted December 17, 2009. Published January 21, 2010.}
\thanks{Supported by grants 209072 from Key Project of Chinese
Ministry of Education, \hfill\break\indent
and ZR2009AM006 from the Natural Science
Foundation of Shandong Province.}
\subjclass[2000]{34B16}
\keywords{Fixed point index; Leray-Schauder degree;  fourth order;
 \hfill\break\indent $m$-point boundary-value problems;
           sign-changing solution}

\begin{abstract}
 Using the theory of the fixed point index in a cone and
 the Leray-Schauder degree, this paper investigates the
 existence and multiplicity of nontrivial solutions for a 
 class of fourth order $m$-point boundary-value problems.
\end{abstract}

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

\section{Introduction}

Consider the following fourth order $m$-point boundary-value
problem
\begin{equation}
\begin{gathered}
u^{(4)}(t) = f(u(t),-u''(t)), \quad t\in (0, 1);\\
u'(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_{i}u(\eta_{i});\\
u'''(0)=0,\quad u''(1)=\sum_{i=1}^{m-2}\alpha_{i}u''(\eta_i),
\end{gathered} \label{e1.1}
\end{equation}
where $f: \mathbb{R}\times\mathbb{R}\to \mathbb{R}$ is a given
sign-changing continuous
function, $m \geq 3$, $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$
and $\alpha_i > 0$ for $i = 1, \dots, m - 2$, with
\begin{equation}
\sum_{i=1}^{m-2}\alpha_{i}<1.\label{e1.2}
\end{equation}

The multi-point boundary-value problems for ordinary differential
equations arise in many areas of applied mathematics and physics.
The existence of solutions of the fourth order two-point
boundary-value problems and the second order $m$-point
boundary-value problems have been studied intensively because of
their interest to physics(see [1,2,6,7,9-11] and [5,8,13,14],
resp.). However, to our best knowledge, the multiplicity of
nontrivial solutions of the nonlinear multi-point boundary-value
problems for fourth order differential equations has not been
studied intensively.

Recently in [12], Wei and Pang investigated the existence and
multiplicity of nontrivial solutions for the following fourth order
$m$-point boundary-value problems
\begin{equation}
\begin{gathered}
x^{(4)}(t) = f(x(t),-x''(t)), \quad t\in (0, 1);\\
x(0)=0,\quad x(1)=\sum_{i=1}^{m-2}\alpha_{i}x(\eta_{i});\\
x''(0)=0,\quad x''(1)=\sum_{i=1}^{m-2}\alpha_{i}x''(\eta_i),
\end{gathered}\label{e1.3}
\end{equation}
where $m \geq 3$, $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$ are constants
and $\alpha_i \in (0, 1)$ for $i = 1, \dots, m - 2$ satisfies \eqref{e1.2}.
$f: \mathbb{R}\times\mathbb{R}\to \mathbb{R}$ satisfies the
following conditions:
\begin{itemize}
\item[(S0)]  The sequence of positive solutions of
$$
\sin(\sqrt{s})=\sum_{i=1}^{m-2}\alpha_{i}\sin(\eta_{i}\sqrt{s})
$$
is $0<\lambda_1<\lambda_{2}<\dots<\lambda_n<\lambda_{n+1}<\dots$.

\item[(S1)] $f(0,0)=0$; and for $u>0$, $v>0$, $f(u,v)\geq0$; for $u<0$,
$v<0$, $f(u,v)\leq0$; for $uv>0$, $f(u,v)$ does not vanished.

\item[(S2)] $f(u,v)$ has a continuous partial derivative at the point
$(0,0)$, and there exists a positive integer $n_{0}$ such that
$\mu_{2n_{0}}<1<\mu_{2n_{0}+1}$, where
$\mu_n=\frac{\lambda_n^{2}}{a_{0}+b_{0}\lambda_n}$,
$a_{0}=f_{u}'(0,0)>0$, $b_{0}=f_{v}'(0,0)>0$.

\item[(S3)] There exist $a_1>0$, $b_1>0$ such that
$$
\lim_{|u|+|v|\to +\infty}\frac{|f(u,v)-a_1u-b_1v|}{|u|+|v|}=0,
$$
and there exists a positive integer $n_1$ such that
$\gamma_{2n_1}<1<\gamma_{2n_1+1}$, where
$\gamma_n=\frac{\lambda_n^{2}}{a_1+b_1\lambda_n}$.

\item[(S4)] There exists a constant $T>0$ such that
$|f(u,v)|<W^{-1}T$, for all $0<|u|\leq T$, $0<|v|\leq T$, where
$$
W=\frac{1}{2}+\frac{\sum\limits_{i=1}^{m-2}\alpha_{i}}{6(1-\sum\limits_{i=1}^{m-2}
\alpha_{i}\eta_{i})}.
$$

\end{itemize}
Using the theory of the fixed point index in a cone and the
Leray-Schauder degree, we obtain the following results.

\begin{theorem} \label{thmA1}
 Suppose {\rm (S0)--(S4)} hold. Then
\eqref{e1.3} has at least six  nontrivial solutions: Two
positive solutions, two sign-changing solutions, and two negative
solutions.
\end{theorem}

\begin{theorem} \label{thmA2}
Suppose {\rm (S0)--(S4)} hold, and
$f$ is odd. Then \eqref{e1.3} has at least eight  nontrivial
solutions.
\end{theorem}

Motivated by [12], we investigate the existence and multiplicity of
nontrivial solutions for \eqref{e1.1}. Let $X=C[0,1]$ with the norm
$\|u\|_{0}=\max\limits_{t\in [0,\ 1]}|u(t)|$,
\[
Y=\{u\in C^{2}[0,1]: u'(0)=0,u(1)=\sum_{i=1}^{m-2}\alpha_{i}u(\eta_{i})\}
\]
with the norm $\|u\|=\max\{ \|u\|_{0},\|u'\|_{0},
\|u''\|_{0}\}$,
\[
E=\{u\in C^{3}[0,1]\cap Y: u'''(0)=0,
u''(1)=\sum_{i=1}^{m-2}\alpha_{i}u''(\eta_{i})\}
\]
with the norm $ \|u\|=\max\{\|u\|_{0},\| u'\|_{0},
\|u''\|_{0},\|u'''\|_{0}\}$. Then
$X, Y, E$ are Banach spaces. We define a cone in $E$ as
\[
P=\{x\in E: x(t)\geq 0, -x''(t)\geq 0, \forall  t\in
[0, 1]\}.
\]
Let
$$
\Gamma(s)=\cos(\sqrt{s})-\sum_{i=1}^{m-2}\alpha_{i}
\cos(\eta_i\sqrt{s}),\quad s\in\mathbb{R}.
$$
Then we can list the sequence of positive solutions of the equation
$\Gamma(s)=0$ as follows:
$$
0<s_1<s_{2}<\dots<s_n<s_{n+1}<\dots.
$$
Regarding the nonlinearity $f(u,v)$, we assume that it satisfies
the following conditions:
\begin{itemize}
\item[(H1)]
 $f(0,0)=0$; and for $u>0$, $v>0$, $f(u,v)\geq0$; for $u<0$,
$v<0$, $f(u,v)\leq0$; for $uv>0$, $f(u,v)$ does not vanish.

\item[(H2)] There exist  $a_0>0$, $b_0>0$,  such that
$$
f(u, v)= a_0u+b_0v +o(|(u, v)|),\quad \text{as } |(u, v)|\to 0,
$$
where  $(u, v)\in\mathbb{R}\times\mathbb{R}$, and
$|(u, v)|:=\max\{|x|,\ |y|\}$. And there exists a positive
integer $n_{0}$
such that $\mu_{n_{0}}<1<\mu_{n_{0}+1}$, where
$\mu_n=\frac{s_n^{2}}{a_{0}+b_{0}s_n}$.

\item[(H3)] There exist  $a_1>0$, $b_1>0$,  such that
$$
f(u, v)= a_1u+b_1v +o(|(u, v)|),\quad \text{as }
 |(u, v)|\to +\infty,
$$
where  $(u, v)\in\mathbb{R}\times\mathbb{R}$, and
$|(u, v)|:=\max\{|x|,\ |y|\}$. And there exists a positive integer
$n_1$ such that $\gamma_{n_1}<1<\gamma_{n_1+1}$, where
$\gamma_n=\frac{s_n^{2}}{a_1+b_1s_n}$.

\item[(H4)] There exists a constant $T>0$ such that
$|f(u,v)|<M^{-1}T$, for all $(u,v)$ satisfying $0<|u|\leq T$,
$0<|v|\leq T$, where $M=\max\{1,N,N^{2}\}$ and
$N=\frac{1}{2}(1+\frac{\sum\limits_{i=1}^{m-2}\alpha_{i}}{1-\sum\limits_{i=1}
^{m-2}\alpha_{i}\eta_{i}})$.

\end{itemize}

This paper is organized as follows. In Section 2, we present some
basic properties of the fixed point index, and make use of these
properties to obtain some important lemmas. In Section 3, we shall
give our main results and their proofs.

\section{Preliminaries}

 Let us list some properties of the fixed point index
in a cone (for details, \cite{g1,g2}).

\begin{lemma}[\cite{g2}] \label{lem2.1}
Let $P$ be a cone of the Banach
space $E$, and $A: P\to P$ be completely continuous, suppose
that $A$ is differential at $\theta$ and $\infty$ along $P$ and $1$
is not an eigenvalue of $A'_{+}(\theta)$ and
$A'_{+}(\infty)$ corresponding to a positive eigenfunction.

(i) If $A'_{+}(\theta)$ has a positive eigenfunction
corresponding to an eigenvalue greater than 1, and $A\theta=\theta$,
then there exists $\tau>0$ such that $i(A,P\cap B(\theta,r),P)=0$
for any $0<r<\tau$.

(ii) If $A'_{+}(\infty)$ has a positive eigenfunction which
corresponds to an eigenvalue greater than 1, then there exists
$\zeta>0$ such that $i(A,P\cap B(\theta,R),P)=0$ for any $R>\zeta$.
\end{lemma}

\begin{lemma}[\cite{g2}] \label{lem2.2}
Let $\theta\in\Omega$ and $A:P\cap\overline{\Omega}\to P$ be condensing.
Suppose that
$Ax\neq\mu x$,  for all $x\in P\cap\partial\Omega$ and  $\mu\geq 1$.
Then $i(A,P\cap\Omega,P)=1$.
\end{lemma}

We first transform \eqref{e1.1} into another form.
Suppose $u(t)$ is a
solution of \eqref{e1.1}. Let $v(t)=-u''(t)$. Note that
\begin{equation}
\begin{gathered}
u''(t)+v(t)=0, \quad t\in (0, 1);\\
u'(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_{i}u^(\eta_i),
\end{gathered} \label{e2.1}
\end{equation}
thus $u(t)$ can be written as
\begin{equation}
u(t)=Lv(t),  \label{e2.2}
\end{equation}
where the operator $L$ is defined by
$Lv(t)=\int_{0}^{1}H(t,s)v(s)ds$, for all $v\in Y$, and
\begin{gather*}
H(t, s)=G(t,s)+\frac{\sum\limits_{i=1}^{m-2}\alpha_{i}G(\eta_{i},
s)}{1-\sum\limits_{i=1}^{m-2}\alpha_{i}\eta_{i}}t, \\
 G(t, s)= \begin{cases}
1-t,& 0\leq s\leq t\leq 1; \\
1-s, & 0\leq t\leq s\leq 1.
\end{cases}
\end{gather*}
Therefore, we obtain the following equivalent form of \eqref{e1.1}:
\begin{equation}
\begin{gathered}
v''(t)+f((Lv)(t),v(t))=0, \quad t\in (0, 1);\\
v'(0)=0,\quad v(1)=\sum_{i=1}^{m-2}\alpha_{i}v^(\eta_i).
\end{gathered} \label{e2.3}
\end{equation}
Similar to \eqref{e2.1} and \eqref{e2.2}, $v(t)$ can be written as
\begin{equation}
 v(t)=(LF)u(t),  \label{e2.4}
\end{equation}
where $(Fu)(t)=f(u(t)$, $-u''(t))$, $t\in (0, 1)$,
for all $u\in E$. From \eqref{e2.2} and \eqref{e2.4} we obtain
$u(t)=(L^{2}F)u(t)$. Define $A=L^{2}F$, then it is easy to get the
following lemma.

\begin{lemma} \label{lem2.3}
$u(t)$ is a solution of  \eqref{e1.1}  if and
only if $u(t)$ is a solution of the operator equation
\begin{equation}
u(t)=Au(t). \label{e2.5}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.4}
Suppose (H1) holds. Then $A:P\to P $ is completely continuous.
\end{lemma}

\begin{proof}
 By the continuity of $f$, it is easy to see that
$A: E\to E $ is completely continuous.
Suppose $x(t)\in P$, condition (H1) implies
$$
Ax(t)=(L^{2}F)x(t)\geq 0,\quad
-(Ax)''(t)=(LF)x(t)\geq 0,\quad \forall t\in [0, 1].
$$
Therefore ,$Ax(t)\in P$.
\end{proof}

\begin{remark} \label{rmk2.1}\rm
Similarly to the above, if $f$ satisfies
(H1), then $A: -P\to -P $ is completely continuous.
\end{remark}

Set
\begin{gather}
Kx(t)=L^{2}x(t), \label{e2.6} \\
Qx(t)=L^{2}(-x'')(t). \label{e2.7}
\end{gather}

\begin{lemma} \label{lem2.5}
\begin{itemize}
\item[(i)] $K: C[0,1]\to E $ is a
completely continuous linear operator;

\item[(ii)] $F: E\to  C[0,1]$ is a continuous bounded operator,
and $A=KF$;

\item[(iii)] $Q: E\to E $ is a completely continuous linear
operator;

\item[(iv)] the sequences of all eigenvalues of the operators
$a_{0}K+b_{0}Q$ and $a_1K+b_1Q$ are $\{\frac{1}{\mu_n}\}$,
 and $\{\frac{1}{\gamma{n}}\}$,
respectively, where $\mu_n$ and $\gamma_n$ are respectively
defined by {\rm (H2)} and {\rm (H3)}.
\end{itemize}
\end{lemma}

\begin{proof}
Items (i)-(iii) have obvious proofs.
To prove (iv), let $\mu$ be a positive eigenvalue of the
linear operator $a_{0}K+b_{0}Q$, and $y\in E\setminus \{\theta\}$ be an
eigenfunction corresponding to the eigenvalue $\mu$.
By \eqref{e2.6} and \eqref{e2.7}, we have
\begin{equation}
\begin{gathered}
\mu y^{(4)} = a_{0}y+b_{0}(-y'');\\
y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_{i}y(\eta_i);\\
y'''(0)=0,\quad
y''(1)=\sum_{i=1}^{m-2}\alpha_{i}y''(\eta_i).
\end{gathered} \label{e2.8}
\end{equation}
Define $D=\frac{d}{dt}$, $G=\mu D^{4}-a_{0}+b_{0}D^{2}$, then there
exist complex constants $r_1, r_{2}$ such that
$$
Gu=\mu (D^{2}+r_1)(D^{2}+r_{2})u.
$$
By the properties of differential operators,
if \eqref{e2.8} has a nonzero
solution, then there exists $r_{s}, s\in \{1, 2\}$ such that
$r_{s}=s_k, k\in N_{+}$. In this case, $\cos t\sqrt{s_k}$ is a
nonzero solution of \eqref{e2.8}.
On substituting this solution into \eqref{e2.8},
we have
$$
\mu s_k^{2}-(a_{0}+b_{0}s_k)=0.
$$
Hence, $\{\frac{a_{0}+b_{0}s_k}{s_k^{2}}=\frac{1}{\mu_k}\}$,
$k=1, 2, \dots$ is the sequence of  eigenvalues of the operator
$a_{0}K+b_{0}Q$. Then $\mu$ is one of the values
$$
\frac{1}{\mu_1}>\frac{1}{\mu_{2}}>\dots
>\frac{1}{\mu_n}>\dots
$$
and the eigenfunction
corresponding to the eigenvalue $1/\mu_n$ is
$$
y_n(t)=C\cos (t\sqrt{s_n}),\quad t\in [0,1],
$$
where $C$ is a nonzero constant.

Similarly, we can show that the sequence of eigenvalues of the
operator $a_1K+b_1Q$ is $\{1/\mu_n\}$, $n=1, 2,\dots$.
\end{proof}

\begin{lemma} \label{lem2.6}
 Suppose (H2) and (H3) hold. Then
the operator $A$ is Frechet differentiable at $\theta$ and $\infty$.
Moreover, $A'(\theta)=a_{0}K+b_{0}Q$ and
$A'(\infty)=a_1K+b_1Q$.
\end{lemma}

\begin{proof}
For any $x\in E$, we have
\begin{gather}
\begin{aligned}{}
[Ax-A\theta-(a_{0}Kx+b_{0}Qx)](t)
&=L^{2}[f(x(t),-x''(t))-(\alpha_{0}x(t)-\beta_{0}x''(t))]\\
&=L^{2}Bx(t), \quad  \forall  t\in [0, 1]
\end{aligned} \label{e2.9}\\
[Ax-A\theta-(a_{0}Kx+b_{0}Qx)]'(t)
 =-\int^{t}_{0}LBx(s)ds, \label{e2.10}\\
[Ax-A\theta-(a_{0}Kx+b_{0}Qx)]''(t)=-LBx(t), \label{e2.11}\\
[Ax-A\theta-(a_{0}Kx+b_{0}Qx)]'''(t)=\int^{t}_{0}Bx(s)ds,
 \label{e2.12}
\end{gather}
where $Bx(t)=f(x(t),-x''(t))-(a_{0}x(t)-b_{0}x''(t))$.

For each $\varepsilon > 0$, by (H2), there exists a $\delta > 0$
such that for any $0<|u|,\ |v|<\delta$,
$$
|\frac{f(u,v)-a_{0}u-b_{0}v}{\sqrt{u^{2}+v^{2}}}|<\varepsilon.
$$
This means
\begin{equation}
|f(u,v)-(a_{0}u+b_{0}v)|<\varepsilon  \sqrt{u^{2}+v^{2}},\quad
 \forall 0<|u|,\; |v|<\delta. \label{e2.13}
\end{equation}
Then, for any $x\in E$ with $\| x\|<\delta$, by
\eqref{e2.9}-\eqref{e2.13}, we get
\begin{equation}
\|Ax-A\theta-(a_{0}Kx+b_{0}Qx)\|
\leq \sqrt{2}M\varepsilon\|x\|. \label{e2.14}
\end{equation}
Consequently,
\[
\lim_{\|x\|\to 0}\frac{\|Ax-A\theta-(a_{0}Kx+b_{0}Qx)\|}{\|x\|}=0.
\]
Therefore, $A$ is Frechet differentiable at $\theta$, and
$A'(\theta)=a_{0}K+b_{0}Q$.

 For each $\varepsilon > 0$, by (H3), there exists a constant
$R_1> 0$ such that
$|f(u,v)-a_1u-b_1v|<\varepsilon(|u|+|v|)$, for $|u|+|v|>R_1$.
Let
\[
b=\max_{|u|+|v|\leq R_1}|f(u,v)-a_1u-b_1v|,
\]
then we have
$$
|f(u,v)-a_1u-b_1v|\leq \varepsilon(|u|+|v|)+b,\quad
\forall u,\ v\in\mathbb{R}.
$$
By a consideration similar to \eqref{e2.14}, we get
$$
\|Ax-(a_1Kx+b_1Qx)\|\leq(2\varepsilon\|x\|+b)M, \quad
 \forall x\in E.
$$
Consequently, $\lim\limits_{\|x\|\to
\infty}\frac{\|Ax-(a_1Kx+b_1Qx)\|}{\|x\|}=0$. This implies that $A$
is Frechet differentiable at $\infty$, and $A'(\infty)=a_1K+b_1Q$.
\end{proof}

\begin{lemma} \label{lem2.7}
 Suppose that {\rm (H1)--(H3)} hold. Then
\begin{itemize}
\item[(i)] there exists a constant $r_{0}$ such that $0<r_{0}<T$,
and for any $0< r\leq r_{0}$, $i(A,P\cap B(\theta,r),P)=0$,
$i(A,-P\cap B(\theta,r),-P)=0$;

\item[(ii)] there exists a constant $R_{0}>T$ such that for any
$R\geq R_{0}$, $i(A,P\cap B(\theta,R),P)=0$,
$i(A,-P\cap B(\theta,R),-P)=0$.
\end{itemize}
\end{lemma}

\begin{proof}
We prove conclusion (i) only; conclusion
(ii) can be proved in the same way.
By Lemma \ref{lem2.6}, $A: P\to P$ is completely continuous and
Frechet differentiable along $P$ at $\theta$, and
$A'_{+}(\theta)=a_{0}K+b_{0}Q, A\theta=\theta$. By Lemma \ref{lem2.5}
and (H2), $A'_{+}(\theta)$ has an eigenvalue
$\frac{1}{\mu_1}=\frac{a_{0}+b_{0}s_1}{s_1^{2}}>1$, and
$\frac{1}{\mu_1}>\frac{1}{\mu_{2}}>\dots
>\frac{1}{\mu_{n_{0}}}>1>\frac{1}{\mu_{n_{0}+1}}>\dots>0$, so $1$
is not an eigenvalue of $A'_{+}(\theta)$ corresponding to a
positive eigenfunction.

The eigenfunction corresponding to $\frac{1}{\mu_1}$ is
$y(t)=\cos t\sqrt{s_1}, t\in[0,1]$, where $s_1$ is the smallest
positive solution of the equation
$$
\cos (\sqrt{s})=\sum_{i=1}^{m-2}\alpha_{i}\cos
(\eta_{i}\sqrt{s}).
$$
Since
\begin{gather*}
\cos (\sqrt{0})-\sum_{i=1}^{m-2}\alpha_{i}\cos
0=1-\sum_{i=1}^{m-2}\alpha_{i}>0,\\
\cos (\sqrt{(\pi/2)^{2}})-\sum_{i=1}^{m-2}\alpha_{i}\cos
(\eta_{i}\sqrt{(\pi/2)^{2}})
=-\sum_{i=1}^{m-2}\alpha_{i}\cos
(\frac{\pi}{2}\eta_{i})<0.
\end{gather*}
Then by the mean-value theorem,
$s_1\in(0,(\frac{\pi}{2})^{2})$. Consequently
$$
y(t)=\cos (t\sqrt{s_1})\geq 0, \quad t\in[0,1].
$$
And then it follows from Lemma \ref{lem2.1} that there exists an $\tau_{0}>0$
such that $i(A,P\cap B(\theta,r),P)=0$ for any $0<r\leq\tau_{0}$.

Similarly, we can show that there exists an $\tau_1>0$ such that
$i(A,-P\cap B(\theta,r),-P)=0$ for any $0<r\leq\tau_1$.
Let $r_{0}<\min\{T,\tau_{0},\tau_1\}$, then the conclusion (i)
holds and the proof is complete.
\end{proof}

\section{Main Results}

 Now we are ready to give our main results.

\begin{theorem} \label{thm3.1}
 Suppose {\rm (H1)--(H4)} hold. Then
\eqref{e1.1} has at least four  nontrivial solutions: Two
positive solutions, and two negative solutions.
\end{theorem}

\begin{proof} For  $x\in E$, we have
\begin{gather*}
Ax(t)=L^{2}Fx(t), \quad
(Ax)'(t)=-\int^{t}_{0}LFx(s)ds;\\
(Ax)''(t)=-LFx(t), \quad
(Ax)''(t)=\int^{t}_{0}Fx(s)ds.
\end{gather*}
As for \eqref{e2.14} we have
\begin{equation}
\|Ax\|\leq M\|x\|. \label{e3.1}
\end{equation}
Therefore, for any $x\in E$, $\|x\|=T$, by (H4) and \eqref{e3.1},
$\|Ax\|<T=\|x\|$.
Then by Lemma \ref{lem2.2}, we have
\begin{gather}
i(A,P\cap B(\theta,T),P)=1, \label{e3.2}\\
i(A,-P\cap B(\theta,T),-P)=1. \label{e3.3}
\end{gather}
By Lemma \ref{lem2.7}, there exists two constants $r_{0}$, $R_{0}$,
$0<r_{0}<T<R_{0}$, such that
\begin{gather}
i(A,P\cap B(\theta,r_{0}),P)=0, \label{e3.4}\\
i(A,-P\cap B(\theta,r_{0}),-P)=0, \label{e3.5}\\
i(A,P\cap B(\theta,R_{0}),P)=0, \label{e3.6}\\
i(A,-P\cap B(\theta,R_{0}),-P)=0. \label{e3.7}
\end{gather}
Thus by \eqref{e3.2}, \eqref{e3.4} and \eqref{e3.6} we have
\begin{gather}
i(A,P\cap (B(\theta,T)\setminus\overline{B(\theta,r_{0})}),P)=1,
  \label{e3.8}\\
i(A,P\cap (B(\theta,R_{0})\setminus\overline{B(\theta,T)}),P)=-1.
\label{e3.9}
\end{gather}
Therefore, the operator $A$ has at least two fixed points
 $x_1\in P\cap (B(\theta,R_{0})\setminus\overline{B(\theta,T)})$ and
$x_{2}\in P\cap (B(\theta,T)\setminus\overline{B(\theta,r_{0})})$,
respectively. By Lemma \ref{lem2.3}, $x_1$ and $x_{2}$ are positive
solutions of  \eqref{e1.1}.

Similarly, by \eqref{e3.3}, \eqref{e3.5} and \eqref{e3.7} we have
\begin{gather}
i(A,-P\cap (B(\theta,R_{0})\setminus\overline{B(\theta,T)}),-P)=-1,
 \label{e3.10}\\
i(A,-P\cap (B(\theta,T)\setminus\overline{B(\theta,r_{0})}),-P)=1.
 \label{e3.11}
\end{gather}
Thus, the operator $A$ has at least two fixed points
$x_{3}\in (-P)\cap (B(\theta,T)\setminus\overline{B(\theta,r_{0})})$ and
$x_{4}\in (-P)\cap (B(\theta,R_{0})\setminus\overline{B(\theta,T)})$,
respectively.
Obviously by Lemma \ref{lem2.3}, $x_{3}$ and $x_{4}$ are negative solutions
of  \eqref{e1.1}.
\end{proof}

By the method used in the proof of Theorem \ref{thm3.1}, it is easy to show
the following corollaries.

\begin{corollary} \label{coro3.1}
Equation \eqref{e1.1} has at least two different
nontrivial solutions: One positive  and one negative,
 provided that {\rm (H1), (H2), (H4)} hold.
\end{corollary}

\begin{corollary} \label{coro3.2}
 Suppose that {\rm (H1), (H3), (H4)}  hold.
Then \eqref{e1.1} has at least two
different nontrivial solutions: One positive  and one
negative.
\end{corollary}

If the nonlinearity  $f$ does not depend on the second order
derivative., then \eqref{e1.1} becomes the following fourth-order $m$-point
boundary-value problem
\begin{equation}
\begin{gathered}
u^{(4)}(t) = f(u(t)), \quad t\in (0, 1);\\
u'(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_{i}u(\eta_{i});\\
u'''(0)=0,\quad
u''(1)=\sum_{i=1}^{m-2}\alpha_{i}u''(\eta_i).
\end{gathered} \label{e3.12}
\end{equation}

We have the following corollary.

\begin{corollary} \label{coro3.3}
 If $f$ satisfies
\begin{itemize}
\item[(H1')] $f\in C(\mathbb{R},\mathbb{R})$, $f(0)=0$; and
$xf(x)\geq 0$, for $x\in\mathbb{R}$;

\item[(H2')]  there exists a positive integer $n_{0}$ such
that $s_{n_{0}}^{2}<a_{0}<s_{n_{0}+1}^{2}$, where
$a_{0}=\lim\limits_{x\to 0}\frac{f(x)}{x}$;

\item[(H3')]  there exists a positive integer $n_1$ such
that $s_{n_1}^{2}<a_1<s_{n_1+1}^{2}$, where $a_1=\lim\limits_{x\to
\infty}\frac{f(x)}{x}$;

\item[(H4')] There exists a constant $T>0$ such that
$|f(x)|<M^{-1}T$, for all $0<|x|\leq T$, where $M$ is defined as in
(H4).
\end{itemize}
Then  \eqref{e3.12} has at least four  nontrivial
solutions.
\end{corollary}

\subsection*{Acknowledgements}
The authors want to thank the anonymous referee for the suggestions
on this paper.

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