\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 217, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/217\hfil Sixth-order m-point boundary-value
problems]
{Nodal solutions for sixth-order m-point boundary-value
problems using \\ bifurcation methods}

\author[Y. Ji, Y. Guo, Y. Yao, Y. Feng \hfil EJDE-2012/217\hfilneg]
{Yude Ji, Yanping Guo, Yukun Yao, Yingjie Feng } 

\address{Yude Ji \newline
Hebei University of Science and Technology,
Shijiazhuang, 050018, Hebei, China}
\email{jiyude-1980@163.com}

\address{Yanping Guo \newline
Hebei University of Science and Technology,
Shijiazhuang, 050018, Hebei, China}
\email{guoyanping65@sohu.com}

\address{Yukun Yao \newline
Hebei Medical University, Shijiazhuang, 050200,
Hebei, China}
\email{yaoyukun126@126.com}

\address{Yingjie Feng \newline
 Hebei Vocational Technical College of Chemical and
Medicine, Shijiazhuang, 050026, Hebei, China}
\email{fengyingjie126@126.com}

\thanks{Submitted June 20, 2012. Published November 29, 2012.}
\subjclass[2000]{34B15}
\keywords{Nonlinear boundary value problems; nodal
solution; eigenvalues; \hfill\break\indent  bifurcation methods}

\begin{abstract}
 We consider the sixth-order $m$-point boundary-value problem
 \begin{gather*}
 u^{(6)}(t)=f\big(u(t), u''(t), u^{(4)}(t)\big),\quad t\in(0,1),\\
 u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i),\\
 u''(0)=0, \quad u''(1)=\sum_{i=1}^{m-2}a_iu''(\eta_i),\\
 u^{(4)}(0)=0, \quad u^{(4)}(1)=\sum_{i=1}^{m-2}a_iu^{(4)}(\eta_i),
 \end{gather*}
 where $f: \mathbb{R}\times \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ is
 a sign-changing continuous function, $m\geq3$,
 $\eta_i\in(0,1)$, and $a_i>0$ for $i=1,2,\dots,m-2$ with
 $\sum_{i=1}^{m-2}a_i<1$. We first show that the spectral
 properties of the linearisation of this problem are similar to the
 well-known properties of the standard Sturm-Liouville problem with
 separated boundary conditions. These spectral properties are then
 used to prove a Rabinowitz-type global bifurcation theorem for a
 bifurcation problem related to the above problem. Finally, we obtain
 the existence of nodal solutions for the problem, under various
 conditions on the asymptotic behaviour of nonlinearity $f$ by using
 the global bifurcation theorem.
\end{abstract}

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

\section{Introduction}

 Multi-point boundary value problems for ordinary differential
equations arise in different areas of applied mathematics and
physics. The existence of solutions for second order and high order
multi-point boundary value problems has been studied by many authors
and the methods used are the nonlinear alternative of
Leray-Schauder, coincidence degree theory, fixed point theorems in
cones and global bifurcation techniques; see
\cite{Feng1997,Guo1,Gupta1998,JI1,Liu,Liu2009,Ma2008,Ma2001,Sun2008,Webb2001,
Xu2004}
and the references therein.

 We consider the  sixth order $m$-point boundary value problem
(BVP, for short)
\begin{equation} \label{eq11}
 \begin{gathered}
    u^{(6)}(t)=f\big(u(t), u''(t), u^{(4)}(t)\big),\quad t\in(0,1),\\
    u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i),\\
    u''(0)=0, \quad u''(1)=\sum_{i=1}^{m-2}a_iu''(\eta_i),\\
    u^{(4)}(0)=0, \quad u^{(4)}(1)=\sum_{i=1}^{m-2}a_iu^{(4)}(\eta_i),
     \end{gathered}
\end{equation}
where $f: \mathbb{R}\times \mathbb{R}\times \mathbb{R} \to
\mathbb{R}$ is a  sign-changing continuous function, $m\geq3$,
$\eta_i\in(0,1)$, and $a_i>0$ for $i=1,2,\dots,m-2$ with
\begin{equation} \label{eq12}
\sum_{i=1}^{m-2}a_i<1.
\end{equation}

Ma and O'Regan \cite{Ma20061} investigated the existence of
nodal solutions of $m$-point boundary value problem
\begin{equation} \label{eq13}
    \begin{gathered}
      u''(t)+f(u)=0,\quad t\in(0,1),\\
      u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i).
     \end{gathered}
\end{equation}
where $\eta_i\in \mathbb{Q}(i=1,2,\dots,m-2)$ with
$0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$,
and $\alpha_i\in \mathbb{R}$ $(i=1,2,\dots,m-2)$ with $a_i>0$ and
$\sum_{i=1}^{m-2}a_i\leq1$. They obtained some results on
the spectrum of the linear operator corresponding to \eqref{eq13}
and gave conditions on the ratio $f(s)/s$ at infinity and
zero that guarantee the existence of nodal solutions. The proofs of
main results are based on bifurcation techniques.

Recently, An and Ma \cite{An2008} extended this result, they
considered the nonlinear eigenvalue problems
\begin{equation}\label{eq14}
    \begin{gathered}
      u''(t)+rf(u)=0,\quad 0<t<1,\\
      u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i).
     \end{gathered}
\end{equation}
under the following conditions:
\begin{itemize}
\item[(A0)] $a_i>0$ for $i=1,2,\dots,m-2$ with
$0<\sum_{i=1}^{m-2}a_i<1$, $r\in \mathbb{R}$;

\item[(A1)] $f\in C^{1}(\mathbb{R}, \mathbb{R})$ and there exist two
constants $s_2<0<s_1$ such that $f(s_1)=f(s_2)=f(0)=0$;

\item[(A2)] There exist $f_0, f_{\infty}\in(0,\infty)$ such that
\[
f_0:=\lim_{|u|\to0}\frac{f(u)}{u},\quad
f_{\infty}:=\lim_{|u|\to\infty}\frac{f(u)}{u}.
\]
\end{itemize}
Using Rabinowitz global bifurcation theorem, An and Ma established
the following theorem.

\begin{theorem}\label{the11} 
Let {\rm (A0), (A1), (A2)} hold. Assume that for
some $k\in \mathbb{N}$,
\[
\frac{\lambda_k}{f_{\infty}}<\frac{\lambda_k}{f_0}\quad
\big(\text{resp., }
\frac{\lambda_k}{f_0}<\frac{\lambda_k}{f_{\infty}}\big)
\]
Then
\begin{itemize}
\item[(i)]  if $r\in(\frac{\lambda_k}{f_{\infty}},
\frac{\lambda_k}{f_0}]$ (resp., $r\in(\frac{\lambda_k}{f_0},
\frac{\lambda_k}{f_{\infty}}]$)
 then \eqref{eq14} has at
least two solutions $u_{k,\infty}^{\pm}$ (resp.,$ u_{k,0}^{\pm}$), such
that $u_{k,\infty}^{+}\in T_k^{+}$ and $u_{k,\infty}^{-}\in
T_k^{-}$ (resp., $u_{k,0}^{+}\in T_k^{+}$ and $u_{k,0}^{-}\in
T_k^{-}$),

\item[(ii)]  if $r\in(\frac{\lambda_k}{f_0}, \infty)$
(resp., $r\in(\frac{\lambda_k}{f_{\infty}},\infty))$ then
\eqref{eq14} has at least four solutions $u_{k,\infty}^{\pm}$ and
 $ u_{k,0}^{\pm}$, such that $u_{k,\infty}^{+}, u_{k,0}^{+}\in
T_k^{+}$, and $u_{k,\infty}^{-}, u_{k,0}^{-}\in T_k^{-}$.
\end{itemize}
Where $\lambda_k$ is the $k$th eigenvalue of
\[
u''(t)+\lambda u(t)=0,\quad 0<t<1, \quad u(0)=0,\quad
u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i).
\]
\end{theorem}

\begin{remark}\label{rem12}\rm
Comparing results in
\cite{Ma20061} and the above theorem, we see that as $f$ has two
zeros $s_1, s_2: s_2<0<s_1$, the bifurcation structure of
the nodal solutions of \eqref{eq14} becomes more complicated: the
component of the solutions of \eqref{eq14} from the trivial solution
at $(\frac{\lambda_k}{f_0}, 0)$ and the component of
the solutions of \eqref{eq14} from infinity at
$(\frac{\lambda_k}{f_{\infty}}, \infty)$ are disjoint;
two new nodal solutions are born when
$r>\max\{\frac{\lambda_k}{f_0},\frac{\lambda_k}{f_{\infty}}\}$.
\end{remark}

Liu and O'Regan \cite{Liu2009} studied the existence and
multiplicity of nodal solutions for fourth order $m$-point BVPs:
    \begin{gather*}
    u^{(4)}(t)=f(u(t), u''(t)),\quad t\in(0,1),\\
    u'(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i),\\
    u'''(0)=0,\quad u''(1)=\sum_{i=1}^{m-2}a_iu''(\eta_i),
     \end{gather*}
where $f: \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ is a
given sign-changing continuous function, $m\geq3$, $\eta_i\in(0,1)$,
and $a_i>0$ for $i=1,2,\dots,m-2$ satisfies \eqref{eq12}. The
main tool be used is global results on bifurcation from infinity,
while in \cite{Ma20062} is results on bifurcation coming from the
trivial solutions.

Motivated by \cite{An2008, Liu2009, Ma20062}, in this paper we
consider the existence of nodal solutions (that is, sign-changing
solutions with a specified number of zeros) of BVP \eqref{eq11}. To
the best of our best knowledge, only \cite{ Liu2009, Wei2007} seems
to have considered the existence of nontrivial or positive solutions
of the nonlinear multi-point BVPs for fourth order differential
equations. Of course an interesting question is, as for sixth order
$m$-point BVPs, whether we can obtain some new results which are
similar to \cite{An2008, Liu2009}. Using the global bifurcation
techniques, we study the global behavior of the components of nodal
solutions of BVP \eqref{eq11} and give a positive answer to the
above question. However, when $m$-point boundary value condition of
\eqref{eq11} is concerned, the discussion is more difficult since
the problem is nonsymmetric and the corresponding operator is
disconjugate. Although the paper \cite{Guo1} has also obtained
sign-changing solutions of \eqref{eq11}, but no information is
obtained regarding the number of zeros of the solution, and the
method of proof is entirely different (relying on degree theory in
cones).

The article is organized as follows. Section 2 gives some
preliminaries. The results we obtain are similar to the standard
spectral theory of the linear, separated Sturm-Liouville problem,
with a slight difference in the nodal counting method used, to deal
with the multi-point boundary conditions. We also show that the
standard counting method is inadequate in this situation, and that
the condition \eqref{eq12} is optimal for the results in Section 2
to hold. This section deals entirely with the linear eigenvalue
problem. In Section 3 we first consider a bifurcation problem
related to \eqref{eq11}, and prove a Rabinowitz-type global
bifurcation theorem for this problem. The proof uses the spectral
properties of the linearisation obtained in Section 2. Finally, we
use the global bifurcation theorem to obtain nodal solutions of
\eqref{eq11}, under various hypotheses on the asymptotic behaviour
of $f$. Specifically, we consider the cases where $f$ is
asymptotically linear.


To conclude this section we give some notation and state four
lemmas, which will be used in Section 3. Following the notation of
Rabinowitz, let $F: \mathbb{R}\times E\to E$ where $E$ is real
Banach spaces and $F$ is continuous. Suppose the equation $F(U)=0$
possesses a simple curve of solutions $\mathscr{C}$ given by
$\{U(t)|t\in[a, b]\}$. If for some $\tau\in(a, b)$, $F$ possesses
zeroes not lying on $\mathscr{C}$ in every neighborhood of
$U(\tau)$, then $U(\tau)$ is said to be a bifurcation point for $F$
with respect to the curve $\mathscr{C}$.

A special family of such equations has the form
\begin{equation} \label{eq01}
u=G(\lambda, u).
\end{equation}
where $\lambda\in\mathbb{R},u\in E$, a real Banach space with norm
$\|\cdot\|$ and $G: \mathbb{R}\times E\to E$ is compact and
continuous. Equations of the form \eqref{eq01} are usually called
nonlinear eigenvalue problems and arise in many contexts in
mathematical physics. It is therefore of interest to investigate the
structure of the set of their solutions.

 Bifurcation phenomena occur
in many parts of physics and have been intensively studied. It is
often the case in applications that $F(\lambda,u)=u-(\lambda
Lu+H(\lambda, u))$ where $L: E\to E$ is a compact linear
operator and $H: \mathbb{R}\times E\to E$ is compact(i.e.,
continuous and maps bounded sets into relatively compact sets) with
$H(\lambda,u)=o(\|u\|)$ at $u=0$ uniformly on bounded
$\lambda$ intervals. The zeros
$\mathscr{R}=\{(\lambda,0):\lambda\in \mathbb{R}\}$ of
$F$ are then called the line of trivial solutions of
\begin{equation} \label{eq17}
u=\lambda Lu+H(\lambda, u).
\end{equation}

If there exists $\mu\in\mathbb{R}$ and $0\neq v\in E$ such that
$v=\mu Lv$, $\mu$ is said to be a real characteristic value of $L$.
The set of real characteristic values of $L$ will be denoted by
$r(L)$. The multiplicity of $\mu\in r(L)$ is
\[
\dim\cup_{j=1}^{\infty}N((I-\mu L)^{j}),
\]
where $I$ is the identity map on $E$ and $N(A)$ denotes the null
space of $A$. It is well known that if $\mu\in\mathbb{R}$, a
necessary condition for $(\mu,0)$ to be a bifurcation point of
\eqref{eq17} with respect to $\mathscr{R}$ is that $\mu\in r(L)$. If
$\mu$ is a simple characteristic value of $L$, let $v$ denote the
eigenvector of $L$ corresponding to $\mu$ normalized so $\|v\|=1$.
By $\mathscr{S}$ we denote the closure of the set of nontrivial
solutions of \eqref{eq17}. A component of $\mathscr{S}$ is a maximal
closed connected subset. The following are global results for
\eqref{eq17} on bifurcation from the trivial solutions (see,
Rabinowitz \cite[Theorems 1.3, 1.25, 1.27]{Rabinowitz1971}).


\begin{lemma}\label{lem13} 
If $\mu\in r(L)$ is of odd multiplicity, then $\mathscr{S}$
contains a component $\mathscr{C}_{\mu}$ which can be decomposed
into two subcontinua $\mathscr{C}_{\mu}^{+}, \mathscr{C}_{\mu}^{-}$
such that each of $\mathscr{C}_{\mu}^{+}, \mathscr{C}_{\mu}^{-}$
either
\begin{itemize}
\item[(i)] meets infinity in $\mathscr{S}$, or

\item[(ii)] meets $(\widehat{\mu}, 0)$ where
$\mu\neq\widehat{\mu}\in\sigma(L)$, or

\item[(iii)] contains a pair of points $(\lambda, u), (\lambda,  -u),
u\neq0$.
\end{itemize}
\end{lemma}

\begin{lemma}\label{lem14} 
If $\mu\in r(L)$ is simple, then $\mathscr{S}$
contains a component $\mathscr{C}_{\mu}$ which can be decomposed
into two subcontinua $\mathscr{C}_{\mu}^{+}, \mathscr{C}_{\mu}^{-}$
such that for some neighborhood $\mathscr{N}$ of $(\mu, 0)$,
\[
(\lambda,u)\in \mathscr{C}_{\mu}^{+}(\mathscr{C}_{\mu}^{-})\cap
\mathscr{N}, \quad(\lambda, u)\neq (\mu, 0)
\]
implies $(\lambda,u)=(\lambda, \alpha v+w)$ where $\alpha>0(\alpha<
0)$ and $|\lambda-\mu|=o(1), \|w\|=o(|\alpha|)$ at $\alpha=0$.
\end{lemma}

\begin{remark}\label{rem13} \rm
We say a continuum $\mathscr{C}$ of $\mathscr{S}$ meets infinity if
$\mathscr{C}$ is not bounded.
\end{remark}

\begin{remark}\label{rem14}\rm
 Lemmas \ref{lem13} is the first important result on the existence of
a subcontinuum of solutions for nonlinear equations by degree
theoretic method. When using Lemmas \ref{lem13} to study
multiplicity of nodal solutions one needs to first study the
spectrum structure of the linear operator $L$ corresponding to the
nonlinear eigenvalue problem. Fortunately, the spectrum structure of
linear operators corresponding to most known nonlinear boundary
value problems have been studied systematically. However, for the
case of multi-point boundary value problems, a complete study of the
spectrum structure is not available yet.
\end{remark}


Rabinowitz  showed in \cite[Theorems 1.6, Corollary 1.8]{Rabinowitz1973} 
that analogues of Lemmas \ref{lem13} and
\ref{lem14} when one is dealing with $\infty$ rather than 0. Let $L$
be as above and $K: \mathbb{R}\times E\to E$ be continuous with
$K(\lambda, u)=o(\|u\|)$ at $u=\infty$ uniformly on bounded
$\lambda$ intervals. Consider the equation
\begin{equation} \label{eq110}
u=\lambda L u+K(\lambda,u)
\end{equation}
Let $\mathscr{T}$ denote the closure of the set of nontrivial
solutions of \eqref{eq110}. The following are global results for
\eqref{eq110} on bifurcation from infinity.

\begin{lemma}\label{lem15}  
Suppose $L$ is compact and linear,
$K(\lambda,u)$ is continuous on $\mathbb{R}\times E$,
$K(\lambda,u)=o(\|u\|)$ at $u=\infty$ uniformly on bounded $\lambda$
intervals, and $\|u\|^2K(\lambda,\frac{u}{\|u\|^2})$ is compact.
If $\mu\in r(L)$ is of odd multiplicity, then $\mathscr{T}$
possesses an unbounded component $\mathscr{D}_{\mu}$ which meets
$(\mu, \infty)$. Moreover if $\Lambda\subset\mathbb{R}$ is an
interval such that $\Lambda\cap r(L)=\{\mu\}$ and $\mathscr{M}$
is a neighborhood of $(\mu, \infty)$ whose projection on
$\mathbb{R}$ lies in $\Lambda$ and whose projection on $E$ is
bounded away from 0, then either
\begin{itemize}
\item[(i)] $\mathscr{D}_{\mu}\backslash\mathscr{M}$ is bounded in
$\mathbb{R}\times E$ in which case
$\mathscr{D}_{\mu}\backslash\mathscr{M}$ meets
$\mathscr{R}=\{(\lambda,0)|\lambda\in \mathbb{R}\}$ or

\item[(ii)] $\mathscr{D}_{\mu}\backslash\mathscr{M}$ is unbounded.
\end{itemize}
If (ii) occurs and  $\mathscr{D}_{\mu}\backslash\mathscr{M}$ has a
bounded projection on $\mathbb{R}$, then
$\mathscr{D}_{\mu}\backslash\mathscr{M}$ meets $(\widehat{\mu},
\infty)$ where $\mu\neq\widehat{\mu}\in\sigma(L)$.
\end{lemma}

\begin{remark}\label{rem13b} \rm
A continuum $\mathscr{D}_{\mu}\subset\mathscr{T}$ of solution of
\eqref{eq110} meets $(\lambda_k, \infty)$ which means that there
exists a sequence
 $\{(\lambda_n, u_n)\}\subset\mathscr{D}_{\mu}$ such that
$\|u_n\|\to\infty$ and  $\lambda_n\to\lambda_k$.
\end{remark}

\begin{lemma}\label{lem16} 
 Suppose the assumptions of Lemma \ref{lem15} hold. If
$\mu\in r(L)$ is simple, then $\mathscr{D}_{\mu}$ can be decomposed
into two subcontinua $\mathscr{D}_{\mu}^{+}, \mathscr{D}_{\mu}^{-}$
and there exists a neighborhood $\mathscr{O}\subset\mathscr{M}$ of
$(\mu, \infty)$ such that $(\lambda,u)\in
\mathscr{D}_{\mu}^{+}(\mathscr{D}_{\mu}^{-})\cap\mathscr{O}$ and
$(\lambda,u)\neq(\mu, \infty)$ implies $(\lambda,u)=(\lambda, \alpha
v+w)$ where $\alpha>0(\alpha< 0)$ and $|\lambda-\mu|=o(1),
\|w\|=o(|\alpha|)$ at $\alpha=\infty$.
\end{lemma}

\begin{remark}\label{rem17} \rm
We say $(\mu, \infty)$ is a bifurcation point for \eqref{eq110} if
every neighborhood of $(\mu, \infty)$  contains solutions of
\eqref{eq110}; i.e., there exists a sequence $(\lambda_n, u_n)$
of solutions of \eqref{eq110} such that $\lambda_n\to\mu$ and
$\|u_n\|\to\infty$.
\end{remark}


\section{Preliminaries}\label{s2}

Let $X=C[0,1]$ with the norm $\|u\|=\max_{t\in[0,1]}|u(t)|$.
Let
\begin{gather*}
Y=\{u\in C^{1}[0,1]|u(0)=0,u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i)\},\\
Z=\{u\in C^2[0,1]|u(0)=0,u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i)\}
\end{gather*}
with the norm
\[
\|u\|_1=\max\{\|u\|,\|u'\|\},\quad 
\|u\|_2=\max\{\|u\|,\|u'\|,\|u''\|\},
\]
respectively. Then $X, Y$ and $Z$ are Banach spaces.

For any $C^{1}$ function $u$, if $u(t_0)=0$, then $t_0$ is a
simple zero of $u$ if $u'(t_0)\neq0$. For any integer $k\in
\mathbb{N}$ and any $\nu\in\{\pm\}$, as in \cite{Rynne}, define sets
$S_k^{\nu}$, $T_k^{\nu}$ subsets of  $C^2[0,1]$ consisting of 
of functions $u\in C^2[0,1]$ satisfying the following conditions:

$S_k^{\nu}$: 
(i) $u(0)=0, \nu u'(0)>0$;
(ii) $u$ has only simple zeros in $[0,1]$ and has exactly $k-1$
zeros in $(0,1)$;

$T_k^{\nu}$:
(i) $u(0)=0, \nu u'(0)>0$, and $u'(1)\neq0$;
(ii) $u'$ has only simple zeros in $(0,1)$ and has exactly $k$ zeros
in $(0,1)$;
(iii) $u$ has a zero strictly between each two consecutive zeros of
$u'$.

\begin{remark}\label{rem21} \rm
(i) If $u\in T_k^{\nu}$, then $u$
 has exactly one zero between each two consecutive zeros of $u'$, 
and all zeros of  $u$ are simple. Thus, $u$ has at least $k-1$ zeros
 in $(0,1)$, and at most $k$ zeros
in $(0,1]$; i.e., $u\in S_k^{\nu}$ or $u\in S_{k+1}^{\nu}$.

(ii) The sets $T_k^{\nu}$ are open in $Z$ and disjoint.

(iii) Note $T_k^{-}=-T_k^{+}$ and let $T_k=T_k^{-}\cup
T_k^{+}$. It is easy to see that the sets $T_k^{+}$ and
$T_k^{-}$ are disjoint and open in $Z$.
\end{remark}

\begin{remark}\label{rem22}\rm
One could regard the sets $S_k^{\nu}$ as counting zeros of $u$,
while the sets $T_k^{\nu}$ count 'bumps'. The nodal properties of
solutions of nonlinear Sturm-Liouville problems with separated
boundary conditions are usually described in terms of sets similar
to $S_k^{\nu}$ (with an additional condition at $x=1$ to
incorporate the boundary condition there); see 
\cite{An2008,Guo1,JI1,Ma2008,Webb2001}.
However, Rynne \cite{Rynne} stated that $T_k^{\nu}$ are in fact
more appropriate than $S_k^{\nu}$ when the multi-point boundary
condition \eqref{eq11} is considered.
\end{remark}

Let $\mathbb{E}=\mathbb{R}\times Y$ under the product topology. As
in \cite{Rabinowitz1973}, we add the points
$\{(\lambda,\infty)|\lambda\in \mathbb{R}\}$ to the space
$\mathbb{E}$. Let $\Phi_k^{+}=\mathbb{R}\times T_k^{+},
\Phi_k^{-}=\mathbb{R}\times T_k^{-}$, and
$\Phi_k=\mathbb{R}\times T_k$.

We first convert  \eqref{eq11} into another form. Notice that
\begin{gather*}
  u''(t)+v(t)=0,\quad t\in(0,1),\\
  u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i).
\end{gather*}
 Thus $u(t)$ can be written as
\begin{eqnarray}\label{eq25}
u(t)=Lv(t),
\end{eqnarray}
where the operator $L$ is defined by
\begin{equation} \label{eq26}
L v(t)=\int_0^{1}G(t,s)v(s)\,{\rm d}s, \forall v\in Y,
\end{equation}
where
\begin{gather} \label{eq27}
G(t,s)=g(t,s)+\frac{t}{1-\sum_{i=1}^{m-2} a_i\eta_i}\sum_{i=1}^{m-2}
 a_ig(\eta_i,s),\\
\label{eq28}
g(t,s)=\begin{cases}
s(1-t), &0\leq s\leq t\leq 1,\\
t(1-s), &0\leq t\leq s\leq 1.
\end{cases}
\end{gather}
Let $v(t)=-u^{(4)}(t)$. Then we obtain the following equivalent
form of \eqref{eq11}
\begin{equation} \label{eq29}
    \begin{gathered}
     v''(t)+f((-L^2v)(t), (Lv)(t),-v(t))=0,\quad t\in(0,1),\\
     v(0)=0,\quad v(1)=\sum_{i=1}^{m-2}a_iv(\eta_i).
     \end{gathered}
\end{equation}

For the rest of this paper we assume that the initial value
problem
\begin{equation}\label{eq210}
    \begin{gathered}
     v''(t)+f((-L^2v)(t), (Lv)(t),-v(t))=0,\quad t\in(0,1),\\
     v(t_0)=0, \quad v'(t_0)=0,
     \end{gathered}
\end{equation}
has the unique trivial solution $v\equiv0$ on $[0,1]$ for any
$t_0\in[0,1]$; in fact some suitable conditions such as a
Lipschitz assumption or $f\in C^{1}$ guarantee this.

Define two operators on $Y$ by
\begin{gather}\label{eq211}
(Av)(t):=(LFv)(t),\\ \label{eq212}
(Fv)(t):=f((-L^2v)(t), (Lv)(t),-v(t)),\quad t\in[0, 1], \quad v\in Y.
\end{gather}
Then it is easy to see the following lemma holds.

\begin{lemma}\label{lem23} 
The linear operator $L$ and operator $A$ are
both completely continuous from $Y$ to $Y$ and
\[
\|Lv\|_1\leq M\|v\|\leq M\|v\|_1,\quad  \forall v\in Y,
\]
where
\[
M=\max\Big\{1,\,
\frac{1}{2}+\frac{\sum_{i=1}^{m-2}a_i}{6(1-\sum_{i=1}^{m-2}a_i
\eta_i)}\Big\}.
\]
\end{lemma}

Moreover, $u\in C^{6}[0, 1]$ is a solution of \eqref{eq11} if
and only if $v=-u^{(4)}$ is a solution of the operator equation
$v=Av$. In fact, if $u$ is a solution of \eqref{eq11}, then
$v=-u^{(4)}$ is a solution of the operator equation $v=Av$.
Conversely, if $v$ is a solution of the operator equation $v=Av$,
then $u=-L^2v$ is a solution of \eqref{eq11}

Let the function $\Gamma(s)$ be defined by
\[
\Gamma(s)=\sin s- \sum_{i=1}^{m-2}a_i\sin \eta_is,~~ s\in
\mathbb{R}^{+}.
\]
The following lemma can be found in \cite{Rynne}.

\begin{lemma}\label{lem24} 
(i) For each $k\geq1$, $\Gamma(s)$ has exactly one
zero $s_k\in I_k:=((k-\frac{1}{2})\pi,
(k+\frac{1}{2})\pi)$, so
\[
s_1<s_2<\dots<s_k\to\infty(k\to +\infty);
\]

(ii) the characteristic value of $L$ is exactly given by
$\mu_k=s_k^2, k=1,2,\dots$, and the eigenfunction $\phi_k$
corresponding to $\mu_k$ is $\phi_k(t)=\sin s_kt$;

(iii) the algebraic multiplicity of each characteristic value
$\mu_k$ of $L$ is 1;

(iv) $\phi_k\in T_k^{+}$ for $k=1,2,3,\dots$, and $\phi_1$ is
strictly positive on $(0,1)$.
\end{lemma}

\begin{lemma}\label{lem25}
 For $d=(d_1, d_2, d_3)\in \mathbb{R}^{+}\times
\mathbb{R}^{+}\times \mathbb{R}^{+}\backslash \{(0,0,0)\}$, define a
linear operator
\begin{equation}\label{eq216}
L_dv(t)=(d_1L^{3}+d_2L^2+d_3L)v(t), \quad \forall t\in [0,1], v\in Y,
\end{equation}
where $L$ is defined as in \eqref{eq26}. Then the generalized
eigenvalues of $L_d$ are simple and are given by
\[
0<\lambda_1(L_d)<\lambda_2(L_d)<\dots<\lambda_k(L_d)\to\infty
\quad \text{as }k\to +\infty,
\]
where
\[
\lambda_k(L_d)=\frac{\mu_k^{3}}{d_1+d_2\mu_k+d_3\mu_k^2}.
\]
The generalized eigenfunction corresponding to $\lambda_k(L_d)$is
\[
\phi_k(t)=\sin s_kt,
\]
where $\mu_k, s_k,\phi_k$ are as in Lemma \ref{lem24}.
\end{lemma}

\begin{proof} 
Suppose there exist $\lambda$ and $u\neq0$ such that
 $u=\lambda L_du$. By \eqref{eq25}-\eqref{eq216}, we have
\begin{equation} \label{eq217}
    \begin{gathered}
     u^{(6)}(t)=\lambda(-d_1u(t)+d_2u''(t)-d_3u^{(4)}(t)),\quad t\in(0,1),\\
     u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i),\\
     u''(0)=0,\quad u''(1)=\sum_{i=1}^{m-2}a_iu''(\eta_i),\\
     u^{(4)}(0)=0,\quad u^{(4)}(1)=\sum_{i=1}^{m-2}a_iu^{(4)}(\eta_i).
     \end{gathered}
\end{equation}
Denote $D=\frac{d}{dt}$, Then there exist three complex numbers
$r_1, r_2$ and $r_3$ such that
\[
(D^{6}+\lambda d_1-\lambda d_2D^2+\lambda
d_3D^{4})u(t)=(D^2+r_1)(D^2+r_2)(D^2+r_3)u(t)
\]
By the properties of differential operators, if \eqref{eq217} has a
nonzero solution, then there exists $r_i(1\leq i\leq 3)$ such that
$r_i =\mu_k=s_k^2, k\in \mathbb{N}$. In this case,
$\sin s_kt$ is a nonzero solution of \eqref{eq217}. On substituting this
solution into \eqref{eq217}, we have
\[
-\mu_k^{3}=\lambda(-d_1-d_2\mu_k-
d_3\mu_k^2).
\]
Hence,
$\{\lambda_k=\frac{\mu_k^{3}}{d_1+d_2\mu_k+
d_3\mu_k^2}, k=1,2,\dots\}$ is the sequence of all
eigenvalues of the operator $L_d$. Then $\lambda$ is one of the
values $\lambda_1<\lambda_2<\dots<\lambda_n<\dots, $ and the
eigenfunction corresponding to the eigenvalue $\lambda_n$ is
\[
u_n(t)=C\sin s_nt,
\]
where $C$ is a nonzero constant. By the ordinary method, we can show
that any two eigenfunctions corresponding to the same eigenvalue
$\lambda_n$ are merely nonzero constant multiples of each other.
Consequently,
\[
\dim \ker(I-\lambda_nL_d)=1.
\]
Now we show that
\begin{equation} \label{eq219}
\ker(I-\lambda_nL_d)=\ker(I-\lambda_nL_d)^2.
\end{equation}
Obviously, we need to show only that
\begin{equation} \label{eq220}
\ker(I-\lambda_nL_d)^2\subset\ker(I-\lambda_nL_d).
\end{equation}
For any $v\in\ker(I-\lambda_nL_d)^2$, $(I-\lambda_nL_d)v$
is an eigenfunction of the linear operator $L_d$ corresponding to
the eigenvalue $\lambda_n$ if $(I-\lambda_nL_d)v\neq\theta$.
Then there exists nonzero constant $\gamma$ such that
\[
(I-\lambda_nL_d)v=\gamma \sin s_nt, \quad t\in[0,1].
\]
By direct computation, we have
\begin{equation} \label{eq221}
    \begin{gathered}
     v^{(6)}(t)=\lambda_n(-d_1v(t)+d_2v''(t)-d_3v^{(4)}(t))-\gamma \mu_n^{3}\sin s_nt,\\
     v(0)=0,\quad v(1)=\sum_{i=1}^{m-2}a_iv(\eta_i),\\
     v''(0)=0,\quad v''(1)=\sum_{i=1}^{m-2}a_iv''(\eta_i),\\
     v^{(4)}(0)=0,\quad v^{(4)}(1)=\sum_{i=1}^{m-2}a_iv^{(4)}(\eta_i).
     \end{gathered}
\end{equation}
The characteristic equation associated with \eqref{eq221} is
\[
\lambda^{6}-\frac{\mu_n^{3}}{d_1+d_2\mu_n+
d_3\mu_n^2}(-d_1+d_2\lambda^2-d_3\lambda^{4})=0.
\]

Case (i): If there exists two real number $a>0, b>0$ and $a\neq b$
such that
\begin{align*}
&(\lambda^2+\mu_n)(\lambda^2-a)(\lambda^2-b)\\
&=\lambda^{6}-\frac{\mu_n^{3}}{d_1+d_2\mu_n+
d_3\mu_n^2}(-d_1+d_2\lambda^2-d_3\lambda^{4})=0.
\end{align*}
 It is easy to see that
the general solution of \eqref{eq221} is of the form
\[
v(t)=C_1e^{\sqrt{a}t}+C_2e^{-\sqrt{a}t}+C_3e^{\sqrt{b}t}
+C_4e^{-\sqrt{b}t}+C_5
\sin s_nt+C_6 \cos s_nt+K t\cos s_nt,
\]
for  $t\in[0,1]$, where $C_1,C_2,C_3,C_4,C_5,C_6$ are six nonzero
constants, and 
\[
 K=\frac{\gamma s_n(d_1+d_2\mu_n+
d_3\mu_n^2)}{6d_1+4d_2\mu_n+2d_3\mu_n^2}.
\]
 Applying the conditions $v(0)=0, v''(0)=0, v^{(4)}(0)=0$, we obtain 
$C_1+C_2=0, C_3+C_4=0, C_6=0$, Then
\begin{gather*}
v(t)=C_1(e^{\sqrt{a}t}-e^{-\sqrt{a}t})
 +C_3(e^{\sqrt{b}t}-e^{-\sqrt{b}t})+C_5 \sin s_nt+K t\cos s_nt,
\\
\begin{aligned}
v''(t)&=C_1a(e^{\sqrt{a}t}-e^{-\sqrt{a}t})
 +C_3b(e^{\sqrt{b}t}-e^{-\sqrt{b}t})-C_5 s_n^2\sin s_nt\\
&\quad +K (-s_n^2t\cos s_nt-2s_n\sin s_nt),
\end{aligned}
\\
\begin{aligned}
v^{(4)}(t)&=C_1a^2(e^{\sqrt{a}t}-e^{-\sqrt{a}t})
 +C_3b^2(e^{\sqrt{b}t}-e^{-\sqrt{b}t}) +C_5 s_n^{4}\sin s_nt\\
&\quad +K (s_n^{4}t\cos s_nt+4s_n^{3}\sin s_nt).
\end{aligned}
\end{gather*}
Applying the conditions
\begin{gather*}
v(1)=\sum_{i=1}^{m-2}a_iv(\eta_i), \quad
v''(1)=\sum_{i=1}^{m-2}a_iv''(\eta_i), \\
v^{(4)}(1)=\sum_{i=1}^{m-2}a_iv^{(4)}(\eta_i), \quad
\sin s_n= \sum_{i=1}^{m-2}a_i\sin \eta_is_n,
\end{gather*}
we have
\begin{equation} \label{eq222}
\begin{gathered}
     C_1F+C_3G+KH=0,\\
     C_1aF+C_3b G-s_n^2K H=0,\\
     C_1a^2F+C_3b^2 G+s_n^{4}K H=0,
     \end{gathered}
\end{equation}
where
\begin{gather*}
     F=e^{\sqrt{a}}-e^{-\sqrt{a}}-\sum_{i=1}^{m-2}a_i(e^{\sqrt{a}\eta_i}
-e^{-\sqrt{a}\eta_i}),\\
     G=e^{\sqrt{b}}-e^{-\sqrt{b}}-\sum_{i=1}^{m-2}a_i(e^{\sqrt{b}\eta_i}
-e^{-\sqrt{b}\eta_i}),\\
     H=\cos s_n-\sum_{i=1}^{m-2}a_i\eta_i\cos \eta_is_n.
     \end{gather*}

If $H\neq0$, then the solution of \eqref{eq222} is
$C_1=C_3=K=0$, which is a contradiction to $\gamma\neq0$, and
\[
v(t)=C_5\sin s_nt\in \ker(I-\lambda_nL_d).
\]
So, \eqref{eq220}  holds. Hence, \eqref{eq219} holds.
If $H = 0$, then
\[
\cos s_n=\sum_{i=1}^{m-2}a_i\eta_i\cos \eta_is_n.
\]
By the Schwarz inequality, we obtain
\begin{align*}
1-\sin^2 s_n
&=\Big(\sum_{i=1}^{m-2}a_i\eta_i\cos \eta_is_n\Big)^2\\
&\leq\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\big(\sum_{i=1}^{m-2}a_i^2\cos^2
\eta_is_n\Big)\\
&=\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\Big)-
\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\sin^2
\eta_is_n\Big).
\end{align*}
Applying the condition $\sin s_n= \sum_{i=1}^{m-2}a_i\sin \eta_is_n$, we obtain
\begin{align*}
1&\leq
\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\Big)
+\Big(\sum_{i=1}^{m-2}a_i\sin \eta_is_n\Big)^2-
\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\sin^2
\eta_is_n\Big)\\
&=\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\Big)
+\Big(1-\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\sin^2
\eta_is_n\Big)\\
&\quad +\sum_{i\neq j}a_ia_{j}\sin \eta_is_n\sin \eta_{j}s_n
\\
&\leq\Big(\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\Big)
+\Big(1-\sum_{i=1}^{m-2}\eta_i^2\Big)\Big(\sum_{i=1}^{m-2}a_i^2\Big)+\sum_{i\neq j}a_ia_{j}\\
&=\Big(\sum_{i=1}^{m-2}a_i\Big)^2.
\end{align*}
which is a contradiction to
$\sum_{i=1}^{m-2}a_i<1$. Thus, \eqref{eq219}
holds. It follows from \eqref{eq219} and \eqref{eq220} that the
algebraic multiplicity of the eigenvalue $\mu_n$ is 1.
\smallskip


Case (ii): There exists a real number $a>0$ such that
\[
(\lambda^2+\mu_n)(\lambda^2-a)^2=\lambda^{6}-\frac{\mu_n^{3}}{d_1+d_2\mu_n+
d_3\mu_n^2}(-d_1+d_2\lambda^2-d_3\lambda^{4})=0.
\]
It is easy to see that the general solution of \eqref{eq221} is of
the form
\[
v(t)=(C_1+C_2t)e^{\sqrt{a}t}+(C_3+C_4t)e^{-\sqrt{a}t}+C_5
\sin s_nt+C_6 \cos s_nt+K t\cos s_nt,\]
for $t\in[0,1]$,
where $C_1,C_2,C_3,C_4,C_5,C_6$ are six nonzero
constants, and $ K=\frac{\gamma s_n(d_1+d_2\mu_n+
d_3\mu_n^2)}{6d_1+4d_2\mu_n+2d_3\mu_n^2}$.

 Applying the conditions $v(0)=0$, $v''(0)=0$, $v^{(4)}(0)=0$, we obtain 
$C_1+C_3=0$, $C_2-C_4=0$, $C_6=0$, Then
\begin{gather*}
v(t)=C_1(e^{\sqrt{a}t}-e^{-\sqrt{a}t})+C_2t(e^{\sqrt{a}t}+e^{-\sqrt{a}t})
+C_5 \sin s_nt+K t\cos s_nt,\\
\begin{aligned}
v''(t)&= C_1a(e^{\sqrt{a}t}-e^{-\sqrt{a}t})+2C_2\sqrt{a}(e^{\sqrt{a}t}
-e^{-\sqrt{a}t}) +C_2at(e^{\sqrt{a}t}+e^{-\sqrt{a}t})\\
&\quad -C_5 s_n^2\sin s_nt+K (-s_n^2t\cos s_nt-2s_n\sin s_nt),
\end{aligned}\\
\begin{aligned}
v^{(4)}(t)&=C_1a^2(e^{\sqrt{a}t}-e^{-\sqrt{a}t})
 +4C_2a\sqrt{a}(e^{\sqrt{a}t}-e^{-\sqrt{a}t})
 +C_2a^2t(e^{\sqrt{a}t}+e^{-\sqrt{a}t})\\
&\quad +C_5 s_n^{4}\sin s_nt+K (s_n^{4}t\cos s_nt+4s_n^{3}\sin s_nt).
\end{aligned}
\end{gather*}
Applying the conditions
\begin{gather*}
v(1)=\sum_{i=1}^{m-2}a_iv(\eta_i), \quad
v''(1)=\sum_{i=1}^{m-2}a_iv''(\eta_i),\\
v^{(4)}(1)=\sum_{i=1}^{m-2}a_iv^{(4)}(\eta_i), \quad
\sin s_n= \sum_{i=1}^{m-2}a_i\sin \eta_is_n,
\end{gather*}
we have
\begin{equation} \label{eq223}
    \begin{gathered}
     C_1F+C_2G+KH=0,\\
C_1aF+C_2(2\sqrt{a}F+a G)-s_n^2K H=0,\\
C_1a^2F+C_2(4a\sqrt{a}F+a^2 G)+s_n^{4}K H=0,
     \end{gathered}
\end{equation}
where
\begin{gather*}
F=e^{\sqrt{a}}-e^{-\sqrt{a}}-\sum_{i=1}^{m-2}a_i(e^{\sqrt{a}\eta_i}
 -e^{-\sqrt{a}\eta_i}),\\
G=e^{\sqrt{a}}+e^{-\sqrt{a}}-\sum_{i=1}^{m-2}a_i\eta_i(e^{\sqrt{a}\eta_i}
 +e^{-\sqrt{a}\eta_i}),\\
H=\cos s_n-\sum_{i=1}^{m-2}a_i\eta_i\cos \eta_is_n.
\end{gather*}
If $H\neq0$, then the solution of \eqref{eq223} is
$C_1=C_2=K=0$, which is a contradiction to $\gamma\neq0$, and
\[
v(t)=C_5\sin s_nt\in \ker(I-\lambda_nL_d).
\]
So, \eqref{eq220} holds. Hence, \eqref{eq219} holds.

If $H = 0$, in this case, the proof is similar to Case (i), we omit it.
\smallskip

Case (iii): There exists a real number $a, b>0$ such that
\begin{align*}
&(\lambda^2+\mu_n)[\lambda^2-(a^2-b^2+2abi)]
[\lambda^2-(a^2-b^2-2abi)]\\
&=\lambda^{6}-\frac{\mu_n^{3}}{d_1+d_2\mu_n+
d_3\mu_n^2}(-d_1+d_2\lambda^2-d_3\lambda^{4})=0.
\end{align*}
It is easy to see that the general solution of \eqref{eq221} is of
the form
\begin{align*}
v(t)&=(C_1\cos bt+C_2\sin bt)e^{at}+(C_3\cos bt+C_4\sin
bt)e^{-at}\\
&\quad +C_5 \sin s_nt+C_6 \cos s_nt+K t\cos s_nt, \quad t\in[0,1].
\end{align*}
where $C_1,C_2,C_3,C_4,C_5,C_6$ are six nonzero
constants, and 
\[
K=\frac{\gamma s_n(d_1+d_2\mu_n+
d_3\mu_n^2)}{6d_1+4d_2\mu_n+2d_3\mu_n^2}.
\]
 Applying the conditions $v(0)=0$, $v''(0)=0$, $v^{(4)}(0)=0$, we obtain that
$C_1+C_3=0$, $C_2-C_4=0$, $C_6=0$, Then
\begin{gather*}
v(t)=C_1\cos bt(e^{at}-e^{-at})+C_2\sin
bt(e^{at}+e^{-at})+C_5
\sin s_nt+K t\cos s_nt,
\\
\begin{aligned}
v''(t)&=C_1(a^2-b^2)\cos bt(e^{at}-e^{-at})-2C_1ab \sin  bt(e^{at}+e^{-at})\\
&\quad +2C_2ab \cos bt(e^{at}-e^{-at})+C_2(a^2-b^2)\sin bt(e^{at}+e^{-at})\\
&\quad -C_5 s_n^2\sin s_nt+K (-s_n^2t\cos s_nt-2s_n\sin s_nt),
\end{aligned}\\
\begin{aligned}
v^{(4)}(t)
&=C_1(a^{4}+b^{4}-6a^2b^2)\cos bt(e^{at}-e^{-at})\\
&\quad +4C_1(ab^{3}-a^{3}b)\sin bt(e^{at}+e^{-at})
  +4C_2(a^{3}b-ab^{3})\cos bt(e^{at}-e^{-at})\\
&\quad +C_2(a^{4}
 +b^{4}-6a^2b^2)\sin bt(e^{at}+e^{-at})
 +C_5 s_n^{4}\sin s_nt\\
&\quad +K (s_n^{4}t\cos s_nt+4s_n^{3}\sin s_nt).
\end{aligned}
\end{gather*}
Applying the conditions
\begin{gather*}
v(1)=\sum_{i=1}^{m-2}a_iv(\eta_i), \quad
v''(1)=\sum_{i=1}^{m-2}a_iv''(\eta_i), \\
v^{(4)}(1)=\sum_{i=1}^{m-2}a_iv^{(4)}(\eta_i), \quad 
\sin s_n= \sum_{i=1}^{m-2}a_i\sin \eta_is_n,
\end{gather*}
we have
\begin{equation} \label{eq224}
\begin{gathered}
     C_1F+C_2G+KH=0,\\
 C_1[(a^2-b^2)F-2abG]+C_2[2abF+(a^2-b^2)G]-s_n^2K H=0,\\
\begin{aligned}
&C_1[(a^{4}+b^{4}-6a^2b^2)F+4(ab^{3}-a^{3}b)G]
+C_2\big[4(a^{3}b-ab^{3})F \\
&+ (a^{4}+b^{4}-6a^2b^2)G\big]+s_n^{4}KH=0,
\end{aligned}
\end{gathered}
\end{equation}
where
\begin{gather*}
      F=\cos b(e^{a}-e^{-a})-\sum_{i=1}^{m-2}a_i\cos
b\eta_i(e^{a\eta_i}-e^{-a\eta_i}),\\
      G=\sin b(e^{a}+e^{-a})-\sum_{i=1}^{m-2}a_i\sin
b\eta_i(e^{a\eta_i}+e^{-a\eta_i}),\\
      H=\cos s_n-\sum_{i=1}^{m-2}a_i\eta_i\cos
\eta_is_n.
     \end{gather*}
If $H\neq0$, then the solution of \eqref{eq224} is
$C_1=C_2=K=0$, which is a contradiction to $\gamma\neq0$, and
\[
v(t)=C_5\sin s_nt\in \ker(I-\lambda_nL_d).
\]
So, \eqref{eq220} holds. Hence, \eqref{eq219} holds.

If $H = 0$,  the proof is similar to Case (i), we omit it.

To sum up, the generalized eigenvalues of $L_d$ are simple, and
the proof of this lemma is complete.
\end{proof}


\section{Main Results}

We now list the following hypotheses for convenience.
\begin{itemize}
\item[(H1)] There exists $a=(a_1, a_2, a_3)\in \mathbb{R}^{+}\times
\mathbb{R}^{+}\times \mathbb{R}^{+}\backslash \{(0,0,0)\}$ such that
\[
f(x,y,z)=-a_1x+a_2y-a_3z+o(|(x,y,z)|), \quad \text{as }
|(x,y,z)|\to0,
\]
where $(x,y,z)\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}$, and
$|(x,y,z)|:=\max\{|x|, |y|, |z|\}$.

\item[(H2)] There exists $b=(b_1, b_2, b_3)\in \mathbb{R}^{+}\times
\mathbb{R}^{+}\times \mathbb{R}^{+}\backslash \{(0,0,0)\}$ such that
\[
f(x,y,z)=-b_1x+b_2y-b_3z+o(|(x,y,z)|), ~\text{as}~
|(x,y,z)|\to\infty.
\]

\item[(H3)] There exists $R>0$ such that
\[
|f(x,y,z)|<\frac{R}{M}, \quad \text{for }(x,y,z)\in\{(x,y,z):|x|\leq
M^2R, |y|\leq MR, |z|\leq R\},
\]
where $M$ is defined as in Lemma \ref{lem23}.

\item[(H4)] There exist two constants $r_1<0<r_2$ such that $f(x, y,
-r_1)\geq0$ and $f(x, y, -r_2)\leq0$ for 
$(x, y)\in[-Mr^2, Mr^2]\times[-Mr, Mr]$, and $f(x, y, -z)$ satisfies
 a Lipschitz condition in $z$ for $(x,y,z)\in[-Mr^2, Mr^2]\times[-Mr,
Mr]\times[r_1, r_2]$, where $r=\max\{|r_1|,r_2\}$.
\end{itemize}
Now we are ready to give our main results.
To set it up we first consider global results for the equation
\begin{equation}
v=\lambda A v, \label{e3.1lamb}
\end{equation}
on $Y$, where $\lambda\in \mathbb{R}$, and the operator $A$ is
defined as in \eqref{eq211}. Under the condition (H1), Equation
\eqref{e3.1lamb} can be rewritten as
\begin{equation} \label{eq34}
v=\lambda L_{a}v+H_{a}(\lambda, v),
\end{equation}
here $H_{a}(\lambda, v)=\lambda A v-\lambda L_{a}v$, $ L_{a}$ is
defined as in \eqref{eq216} (replacing $d$ with $a$). Obviously, by
(H1) and Lemma \ref{lem23}-\ref{lem25}, it can be seen that
$H_{a}(\lambda, v)$ is $o(\|v\|_1)$ for $v$ near 0 uniformly on
bounded $\lambda$ intervals and $L_{a}$ is a compact linear map on
$Y$. A solution of \eqref{e3.1lamb}) is a pair $(\lambda, v)\in E$.
By (H1), the known curve of solutions $\{(\lambda, 0)|\lambda\in
\mathbb{R}\}$ will henceforth be referred to as the trivial
solutions. The closure of the set on nontrivial solutions of
$\eqref{e3.1lamb})$ will be denoted by $\mathscr{S}$ as in Lemma
\ref{lem13}.

If $H_{a}(\lambda, v)\equiv0$, then \eqref{eq34} becomes a linear
system
\begin{equation} \label{eq35}
v=\lambda L_{a} v,
\end{equation}
By Lemma \ref{lem25}, \eqref{eq35} possesses an increasing sequence
of simple eigenvalues
\[
0<\lambda_1<\lambda_2<\dots<\lambda_k\to\infty,\quad \text{as }k\to
+\infty,
\]
where
\begin{equation} \label{eq36}
\lambda_k=\frac{\mu_k^{3}}{a_1+a_2\mu_k+a_3\mu_k^2}.
\end{equation}
Any eigenfunction $\phi_k(t)=\sin s_kt$ corresponding to
$\lambda_k$ is in $T_k^{+}$.

A similar analysis as in \cite[Lemma 3.4, 3.5]{Liu2009} and
\cite[Proposition 4.1]{Rynne} yield the following results.

\begin{lemma}\label{lem31} 
Suppose that $(\lambda, v)$ is a solution of \eqref{e3.1lamb} and $v\neq0$. Then
$v\in\cup_{i=1}^{\infty}T_i$.
\end{lemma}

\begin{lemma}\label{lem32} 
Assume that {\rm (H1)} holds and $\lambda_k$ is
defined by \eqref{eq36}. Then for each integer $k>0$ and each
$\nu=+$, or $-$, there exists a continua $\mathscr{C}_k^{\nu}$ of
solutions of $\eqref{e3.1lamb})$ in
$\Phi_k^{\nu}\cup\{(\lambda_k, 0)\}$, which meets
$\{(\lambda_k, 0)\}$ and $\infty$ in $\mathscr{S}$.
\end{lemma}

Under  condition (H2), \eqref{e3.1lamb} can be rewritten as
\begin{equation} \label{eq37}
v=\lambda L_{b}v+K_{b}(\lambda, v),
\end{equation}
here $K_{b}(\lambda, v)=\lambda A v-\lambda L_{b}v$, $ L_{b}$ is
defined as in \eqref{eq216} (replacing $d$ with $b$).
Here $h(x, y, z)=f(x, y, z)+b_1x-b_2y+b_3z$. Then from (H2) it
follows that 
\[
\lim_{|(x,y,z)|\to\infty}\frac{h(x, y,z)}{|(x,y,z)|}=0.
\]
 Define a function
\[
\widehat{h}(r):=\max\{|h(x, y, z)|:|(x,y,z)|\leq r\}.
\]
Then $\widehat{h}(r)$ is nondecreasing and
\begin{equation} \label{eq39}
\lim_{r\to\infty}\frac{\widehat{h}(r)}{r}=0.
\end{equation}
Obviously, by \eqref{eq39} and Lemma \ref{lem23}, it can be seen
that $K_{b}(\lambda, v)$ is $o(\|v\|_1)$ for $v$ near $\infty$
uniformly on bounded $\lambda$ intervals and $L_{b}$ is a compact
linear map on $Y$.

Similar to \eqref{eq35}, by Lemma \ref{lem25}, $L_{b}$ possesses an
increasing sequence of simple eigenvalues
\[
0<\overline{\lambda}_1<\overline{\lambda}_2<\dots
<\overline{\lambda}_k\to\infty,\quad \text{as }k\to +\infty,
\]
where
\begin{equation} \label{eq310}
\overline{\lambda}_k=\frac{\mu_k^{3}}{b_1+b_2\mu_k+b_3\mu_k^2}.
\end{equation}
Note $\phi_k(t)=\sin s_kt$ is an eigenfunction corresponding to
$\overline{\lambda}_k$. Obviously, it is in $T_k^{+}$.


\begin{lemma}\label{lem33} 
Assume that {\rm (H1)-(H2)} hold.
 Then for each integer $k>0$ and each $\nu=+, \text{or} -$,
there exists a continua $\mathscr{D}_k^{\nu}$ of $\mathscr{T}$ in
$\Phi_k^{\nu}\cup\{(\overline{\lambda}_k, \infty)\}$ coming
from $\{(\overline{\lambda}_k, \infty)\}$, which meets
$\{(\overline{\lambda}_k, 0)\}$ or has an unbounded projection on
$\mathbb{R}$.
\end{lemma}


 \begin{theorem}\label{the34} 
Assume that {\rm (H1)--(H2)} hold. Suppose there exists two integers
 $i_0\geq0$ and $k>0$ such that either
\begin{equation} \label{eq311}
\frac{\mu_{i_0+k}^{3}}{a_1+a_2\mu_{i_0+k}+a_3\mu_{i_0+k}^2}
<1<\frac{\mu_{i_0+1}^{3}}{b_1+b_2\mu_{i_0+1}+b_3\mu_{i_0+1}^2}
\end{equation}
or
\begin{equation} \label{eq312}
\frac{\mu_{i_0+k}^{3}}{b_1+b_2\mu_{i_0+k}+b_3\mu_{i_0+k}^2}
<1<\frac{\mu_{i_0+1}^{3}}{a_1+a_2\mu_{i_0+1}+a_3\mu_{i_0+1}^2}
\end{equation}
holds. Then  \eqref{eq11} has at least $2k$ nontrivial solutions.
\end{theorem}

\begin{proof} 
First suppose that \eqref{eq311} holds.
Using the notation of \eqref{eq36} and \eqref{eq310}, this means
$\lambda_{i_0+k}<1<\overline{\lambda}_{i_0+1}$ and so from Lemma
\ref{lem25} we know that
\[
\lambda_{i_0+1}<\lambda_{i_0+2}<\dots<\lambda_{i_0+k}<1<
\overline{\lambda}_{i_0+1}<\overline{\lambda}_{i_0+2}<\dots
<\overline{\lambda}_{i_0+k}.
\]
Consider \eqref{eq34} as a bifurcation problem from the trivial
solution. We need only show that
\begin{equation} \label{eq315}
\mathscr{C}_{i_0+j}^{\nu}\cap(\{1\}\times Y)\neq\emptyset,
\quad j=1,2,\dots, k;\; \nu=+,\text{or}-.
\end{equation}
Suppose, on the contrary and without loss of generality, that
\begin{equation} \label{eq316}
\mathscr{C}_{i_0+i}^{+}\cap(\{1\}\times Y)=\emptyset,
\quad \text{for some }i,\; 1\leq i\leq k.
\end{equation}
By Lemma \ref{lem32} we know that $\mathscr{C}_{i_0+i}^{+}$ joins
$(\lambda_{i_0+i}, 0)$ to infinity in $\mathscr{S}$ and
$(\lambda, v)=(0,0)$ is the unique solution of \eqref{e3.1lamb}
 (in which $\lambda=0$) in $\mathbb{E}$. This together with
$\lambda_{i_0+i}<1$ guarantee that there exists a sequence
$\{(\xi_{m},y_{m})\}\subset \mathscr{C}_{i_0+i}^{+}$ such that
$\xi_{m}\in(0, 1)$ and $\|y_{m}\|_1\to\infty$ as $m\to\infty$. We
may assume that $\xi_{m}\to\overline{\xi}\in[0,1]$ as $m\to\infty$.
Let $x_{m}:=\frac{y_{m}}{\|y_{m}\|_1}, m\geq1$. From the fact that
\[
y_{m}=\xi_{m}L_{b}y_{m}+K_{b}(\xi_{m},y_{m}),
\]
it follows that
\begin{equation} \label{eq318}
x_{m}=\xi_{m}L_{b}x_{m}+\frac{K_{b}(\xi_{m},y_{m})}{\|y_{m}\|_1}.
\end{equation}
Notice that $L_{b}: Y \to Y$ is completely continuous. We may assume
that there exists $\omega\in Y$ with $\|\omega\|_1=1$ such that
$\|x_{m}-\omega\|_1\to 0$ as $m\to\infty$.

Letting $m\to\infty$ in \eqref{eq318} and noticing
$\frac{K_{b}(\xi_{m},y_{m})}{\|y_{m}\|_1}\to 0$ as $m\to\infty$
one obtains
\[
\omega=\overline{\xi}L_{b}\omega.
\]
Since $\omega\neq0$, then $\overline{\xi}\neq0$ is an eigenvalue of
 $L_{b}$; that is,
$\overline{\xi}=\overline{\lambda}_{i_0+i}$, which contradicts
$\overline{\lambda}_{i_0+i}>1$. Thus \eqref{eq316} is not true,
which means \eqref{eq315} holds.

Next suppose that \eqref{eq312} holds. This means
\[
\overline{\lambda}_{i_0+1}<\overline{\lambda}_{i_0+2}<\dots
<\overline{\lambda}_{i_0+k}<1<
\lambda_{i_0+1}<\lambda_{i_0+2}<\dots<\lambda_{i_0+k}.
\]
Consider \eqref{eq37} as a bifurcation problem from infinity. As
above we need only to prove that
\begin{equation} \label{eq322}
\mathscr{D}_{i_0+j}^{\nu}\cap(\{1\}\times
Y)\neq\emptyset, \quad j=1,2,\dots, k;\; \nu=+,\text{or}-.
\end{equation}
From Lemma \ref{lem33}, we know that $\mathscr{D}_{i_0+j}^{\nu}$
comes from $\{(\overline{\lambda}_{i_0+j},\infty)\}$, meets
$\{(\lambda_{i_0+j},0)\}$ or has an unbounded projection on
$\mathbb{R}$. If it meets $\{(\lambda_{i_0+j},0)\}$, then the
connectedness of $\mathscr{D}_{i_0+j}^{\nu}$ and
$\lambda_{i_0+j}>1$ guarantees that \eqref{eq322} is satisfied. On
the other hand, if $\mathscr{D}_{i_0+j}^{\nu}$ has an unbounded
projection on $\mathbb{R}$, notice that $(\lambda, v)=(0,0)$ is the
unique solution of \eqref{e3.1lamb} (in which $\lambda=0$) in $E$,
so \eqref{eq322} also holds.
 \end{proof}

\begin{theorem}\label{the35} 
Assume that {\rm (H1), (H2)} hold and one of 
{\rm (H3)} or {\rm (H4)} hold. Suppose there exists two integers
 $i_0$ and $j_0$ such that
\begin{equation} \label{eq323}
\frac{\mu_{i_0}^{3}}{a_1+a_2\mu_{i_0}+a_3\mu_{i_0}^2}<1,
\frac{\mu_{j_0}^{3}}{b_1+b_2\mu_{j_0}+b_3\mu_{j_0}^2}<1.
\end{equation}
Then \eqref{eq11} has at least $2(i_0+j_0)$ nontrivial
solutions.
\end{theorem}

\begin{proof} 
First suppose that (H3) holds. Then there exists
$\varepsilon>0$ such that
\begin{equation} \label{eq324}
(1+\varepsilon)|f(x,y,z)|<\frac{R}{M}, (x,y,z)\in\{(x,y,z):|x|\leq
M^2R, |y|\leq MR, |z|\leq R\},
\end{equation}
Let $(\lambda, v)$ be a solution of \eqref{e3.1lamb} such that
$0\leq\lambda<1+\varepsilon$ and $\|v\|_1\leq R$. Then by
\eqref{eq211}, \eqref{eq212}, \eqref{e3.1lamb}, \eqref{eq324} and
Lemma \ref{lem23} it is easy to see
\begin{equation} \label{eq325}
\begin{aligned}
\|v\|_1&=\lambda\|A v\|_1=\lambda\|LF v\|\\
&\leq\lambda M\|Fv\|
=M\max_{t\in[0,1]}|\lambda f((-L^2v)(t), (Lv)(t),-v(t))|\\
&<M\frac{R}{M}=R,
\end{aligned}
\end{equation}
Therefore,
\begin{equation} \label{eq326}
\mathscr{S}\cap([0,1+\varepsilon]\times\partial\overline{B}_{R})=\emptyset.
\end{equation}
This together with \eqref{eq325} and Lemma \ref{lem32} and Lemma
\ref{lem33} implies that
\begin{gather}
\mathscr{C}_k^{\nu}\cap([0,1+\varepsilon]\times\overline{B}_{R})\subset[0,
1+\varepsilon]\times B_{R},\quad k=1,2,\dots,i_0;\label{eq327}
\\
\mathscr{D}_{j}^{\nu}\cap([0,1+\varepsilon]\times\partial\overline{B}_{R})
=\emptyset, \quad j=1,2,\dots,j_0;\label{eq328}
\end{gather}
where $B_{R}=\{v\in Y| \|v\|_1<R\}$ and
$\overline{B}_{R}=\{v\in Y| \|v\|_1\leq R\}$,
$\mathscr{C}_k^{\nu}$ and $\mathscr{D}_{j}^{\nu}$ are obtained
from Lemma \ref{lem32} and Lemma \ref{lem33}, respectively.

Since $\mathscr{C}_k^{\nu}$ is a unbounded component of solutions
of \eqref{e3.1lamb} joining $(\lambda_k,0)$ in $\mathbb{E}$, it
follows from \eqref{eq326} and \eqref{eq327} that
$\mathscr{C}_k^{\nu}$ crosses the hyperplane $\{1\}\times Y$ with
$(1,v^{\nu})$ such that $\|v^{\nu}\|_1<R,
(\nu=+~\text{or}~-$, $k=1,2,\dots,i_0)$. This implies that \eqref{eq29}
has $2i_0$ nontrivial solutions $\{v_i^{\nu}\}_{i=1}^{i_0}$ in
which $\{v_i^{+}\}$ and $\{v_i^{-}\}$ are positive and negative
solutions, respectively.

On the other hand, by \eqref{eq326}, \eqref{eq328}, and Lemma
\ref{lem33} one can obtain
\[
\mathscr{D}_{j}^{\nu}\cap(\{
1\}\times(Y\backslash\overline{B}_{R}))\neq\emptyset,
\quad j=1,2,\dots,j_0.
\]
This implies that \eqref{eq29} has $2j_0$ nontrivial solutions
$\{\omega_i^{\nu}\}_{i=1}^{j_0}$ in which $\{\omega_i^{+}\}$
and $\{\omega_i^{-}\}$ are positive and negative solutions,
respectively.

Now it remains to show this theorem holds when the condition (H4) is
satisfied.
From the above we need only to prove that
(i) for $(\lambda, v)\in \mathscr{C}_k^{\nu}(\nu=+~\text{or}~-$,
$k=1,2,\dots,i_0)$,
\[
r_1<v(t)<r_2, \quad t\in[0,1].
\]
(ii) for $(\lambda, v)\in \mathscr{D}_{j}^{\nu}(\nu=+~\text{or}~-$,
$j=1,2,\dots,j_0)$, we have that either
\[
\max_{t\in[0,1]}v(t)>r_2, \quad t\in[0,1]
\]
or
\[
\min_{t\in[0,1]}v(t)<r_1, \quad t\in[0,1].
\]
In fact, like in \cite{An2008}, suppose on the contrary that there exists
$(\lambda, v)\in \mathscr{C}_k^{\nu}\cap \mathscr{D}_{j}^{\nu}$
such that either
\[
\max\{v(t): t\in[0,1]\}=r_2
\]
or
\[
\min\{v(t): t\in[0,1]\}=r_1
\]
for some $i, j$.

First consider the case $\max\{v(t): t\in[0,1]\}=r_2$.
Then there exists $\overline{t}\in[0,1]$ such that $v(\overline{t})=r_2$. Let
\begin{gather*} 
\tau_0:=\inf\{t\in[0,\overline{t}]:
v(s)\geq0 \text{ for } s\in[t, \overline{t}]\},\\
\tau_1:=\sup\{t\in[\overline{t}, 1]: v(s)\geq0 \text{ for }
s\in[ \overline{t}, t]\}.
\end{gather*}
Then
\begin{gather}
\max\{v(t): t\in[\tau_0,\tau_1]\}=r_2,\label{eq336}\\
0\leq v(t)\leq r_2,\quad t\in[\tau_0,\tau_1].
\end{gather}
Therefore, $v(t)$ is a solution of the following equation
\[
-v''(t)=\lambda f((-L^2v)(t), (Lv)(t),-v(t)), \quad
 t\in(\tau_0,\tau_1)
\]
with $v(\tau_0)=v(\tau_1)=0$ if $0\leq\tau_0<\tau_1<1$.

By (H4), there exists $\overline{M}\geq0$ such that $f(x, y,
-z)+\overline{M}z$ is strictly increasing in $z$ for
$(x,y,z)\in[-Mr^2, Mr^2]\times[-Mr, Mr]\times[r_1, r_2]$,
where $r=\max\{|r_1|, r_2\}$. Then
\[
-v''(t)+\lambda\overline{M}v=\lambda(f((-L^2v)(t),
(Lv)(t),-v(t))+\overline{M}v),\quad t\in(\tau_0,\tau_1)
\]
Using (H4) and Lemma \ref{lem23} again, we can obtain
\begin{equation}
\begin{split}
&-(r_2-v(t))''+\lambda\overline{M}(r_2-v(t)) \\
&=-\lambda[f((-L^2v)(t),
(Lv)(t),-v(t))+\overline{M}v(t)-\overline{M}r_2] \\
&=-\lambda[f((-L^2v)(t),
(Lv)(t),-v(t))+\overline{M}v(t)-(f((-L^2v)(t), (Lv)(t),-r_2)
+\overline{M}r_2)] \\
&\quad -\lambda f((-L^2v)(t),
(Lv)(t),-r_2) \\
&\geq 0, \quad t\in(\tau_0,\tau_1).
\end{split}\label{eq340}
\end{equation}
and if $\tau_1=1$, by \eqref{eq12} we know $v(1)<r_2$.
Therefore,
\begin{gather*}
r_2-v(\tau_0)>0, \quad r_2-v(\tau_1)>0 \quad \text{if }
0\leq\tau_0<\tau_1<1; \\
r_2-v(\tau_1)>0 \quad \text{if } \tau_1=1.
\end{gather*}
This together with \eqref{eq340} and the maximum principle
\cite{Protter1984} imply that $r_2-v(t)>0$ in
$[\tau_0,\tau_1]$, which contradicts \eqref{eq336}.

The proof in the case $\min\{v(t): t\in[0,1]\}=r_1$ is similar, so
we omit it.
\end{proof}

\subsection*{Acknowledgments}
Supported by grants 10971045 from the National Natural Science
 Foundation of China, A2011208012  from the Natural Science
Foundation of Hebei Province, and  XL201245 from the Foundation
of Hebei University of Science and Technology.


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\end{document}