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\markboth{\hfil Periodic solutions of a piecewise linear beam
equation 
\hfil EJDE--2002/81} 
{EJDE--2002/81\hfil Yukun An \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 81, pp. 1--12. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Periodic solutions of a piecewise linear beam equation with
  damping and nonconstant load
 %
\thanks{ {\em Mathematics Subject Classifications:} 35L30, 35B34.
\hfil\break\indent
{\em Key words:} Keywords Piecewise linear beam equation, Damping,
   Periodic solution,     \hfil\break\indent  Lyapunov-Schmidt reduction.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted October 02, 2001. Published September 29, 2002. \hfil\break\indent
Partially supported by grant NWNU-KJCXGC-212.} }
\date{}
%
\author{Yukun An }
\maketitle

\begin{abstract}
  Using the Lyapunov-Schmidt reduction method,
  the authors discuss the existence and multiplicity of
  periodic solutions for a piecewise linear beam equation with
  damping and nonconstant load when the nonlinearities cross the
  eigenvalues. The result answers partially an open question
  possed by Lazer and McKenna \cite{LM1}.
\end{abstract}

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


\section{Introduction}

Choi and Jung \cite{CJ2} considered the
piecewise linear one-dimensional beam equation
\begin{equation}\label{q1}
\begin{gathered}
u_{tt}+u_{xxxx}+bu^{+}-au^{-}=h(x,t), \quad\text{in }(-
{\frac \pi 2},{\frac \pi 2})\times \mathbb{R}, \\
u(\pm {\frac \pi 2},t)=u_{xx}(\pm {\frac \pi 2},t)=0, \quad
 t\in \mathbb{R}
\end{gathered}
\end{equation}
with $u$ being $\pi$-periodic in $t$ and even  in $x$
and $t$. The authors assumed that the upward and the downward
restoring coefficients in the vibrating beam are constant
and different.

Let $h=s_1\phi_{00}+s_2\phi_{01}$, where $\phi_{00}=\cos x$ and
$\phi_{01}=\cos{x}\cos{2t}$ are the eigenfunctions of
$u_{tt}+u_{xxxx}$. In \cite {CJ2}, by using of Lyapunov-Schmidt
reduction method, the authors
transformed problem (\ref{q1}) into a $2$-dimensional problem on
the subspace $V=\mathop{\rm span}\{\phi_{00},\phi_{01}\}$ and investigated the
multiplicity of solutions for (\ref{q1}) with the nonlinearity
$-(bu^{+}-au^{-})$ crossing finitely many eigenvalues.

When $u_{tt}+u_{xxxx}$ is replaced by $u_{tt}-u_{xx}$ in (\ref{q1}), the
problem is the piecewise linear wave equation, for which there is a large
body of literature concerning  the existence and multiplicity of periodic
solutions; see for example \cite{CJ3,CJ4,LM6,Wi1} and references
therein. Problem (\ref{q1}) originates
from a simple mathematical model of the suspension bridge presented by Lazer
and McKenna in \cite{LM1}:
\begin{equation}\label{q2}
\begin{gathered}
u_{tt}+u_{xxxx}+\delta u_t+ku^{+}=W(x)+\epsilon h(x,t), \quad\text{in }
(0,L)\times \mathbb{R}, \\
u(0,t)=u(L,t)=u_{xx}(0,t)=u_{xx}(L,t)=0, \quad t\in \mathbb{R},
\end{gathered}
\end{equation}
where $u(x,t)$ denotes the displacement of the road bed in downward
direction at position $x$ and $t$, $W(x)$ is the weight per unit length at
$x$, and $\epsilon h(x,t)$ is an external forcing term. The constant $k$
represents the restoring force of the cables, and $\delta $ is the
viscous damping.

For (\ref{q2}), in an earlier paper, McKenna and Walter \cite{MW1}
investigated the simplified situation
\begin{equation}
\label{q3}
\begin{gathered}
u_{tt}+u_{xxxx}+ku^{+}=1+\epsilon h(x,t), \quad\text{in } (-
{\frac \pi 2},{\frac \pi 2})\times \mathbb{R}, \\
u(\pm {\frac \pi 2},t)=u_{xx}(\pm {\frac \pi 2},t)=0, \quad
 t\in \mathbb{R},
\end{gathered}
\end{equation}
where $u$ is $\pi$-periodic in $t$ and even in $x$ and $t$.
Using degree theory, they proved that (\ref{q3}) has at least two solutions
when $3<k<15$ and the one is sign-changing.
In \cite{CJ1}, using degree theory and with critical point theory,
Choi and Jung gave the relationship between
multiplicity of solutions of (\ref{q3}) and the source term
$s_1\cos x + s_2 \cos 2t\cos x$. Some other relevant
studies can be found in \cite{AH,AZ,CJ2,CM1,Fe,H}.
In all these papers $\delta=0$; i.e., there is no damping present.
As remarked in \cite{MW1}, the methods used
in all these papers do not seem to be valid for the case $\delta\neq 0$.
Meanwhile the later case is more interesting as studied in \cite{AH,AZ,K,T}.
In \cite{LM1}, an open question was stated as
Problem 6 which is relevant to this case. The question is
\begin{quote}
 Is there a non-degeneracy condition on $h(x,t)$, which will
ensure that solutions of (\ref{q3}) persist if damping is present?
\end{quote}

In this paper, we assume that there is a damping term; i.e.,
$\delta\neq 0$ and the source term is
$$
h(x,t)=\alpha \cos x+\beta \cos{2t}\cos{x}+\gamma \sin{2t}\cos{x}.
$$
As in \cite{CJ2}, we study the equation
\begin{equation}\label{q5}
\begin{gathered}
u_{tt}+u_{xxxx}+\delta u_t+bu^+-au^-
=\alpha \cos x+\beta \cos{2t}\cos{x}+\gamma \sin{2t}\cos{x}, \\
\text{in }(-{\frac{\pi }{2}}, {\frac{\pi }{2}})\times \mathbb{R}, \\
 u(\pm {\frac{\pi }{2}}, t)=u_{xx}(\pm {\frac{\pi }{2}},t)=0, \quad
t\in \mathbb{R},
\end{gathered}
\end{equation}
where $u$ is $\pi$-periodic  in  $t$ and  even in $x$.

The main goal of this paper is to establish the relationship between
the constant $a, b, \delta $ and the source term $h$, so that
there exists a  sign-changing periodic solutions when $h(x,t)$ is of
single-sign. We also guarantee the existence of multiple periodic solutions.
The results extend the corresponding results in \cite{CJ2}
and \cite{AZ} and answer partially the open problem mentioned above.


\section{Preliminaries}

Let $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ be the set of positive
integers, integers, and reals, respectively.
Let $\Lambda = \mathbb{Z}\times \mathbb{N}$,
$\Omega =(-\frac {\pi }{2}, \frac {\pi}{2})\times (- \frac{\pi }{2},
\frac {\pi }{2})$, and $L^2(\Omega )$ be the usual space of square integrable
functions with usual inner product $(\cdot ,\cdot )$ and corresponding norm
$\| \cdot \| $.
For the Sobolev space $H^1(\Omega )$, we denote the standard
inner product by $\langle u,v \rangle_1=(u,v)+(u_x,v_x)+(u_t,v_t)$,
and norm by $\| u \|_1 $.

It is known that the eigenvalue problem
\begin{gather*}
u_{tt}+u_{xxxx}=\lambda u, \quad\text{in }
 (-{\frac{\pi }{2}}, {\frac{\pi }{2}})\times \mathbb{R}, \\
 u(\pm {\frac{\pi }{2}}, t)=u_{xx}(\pm {\frac{\pi }{2}},t)=0, \quad
t\in \mathbb{R}, \\
u(x,t)=u(-x,t)=u(x,t+\pi ), \quad\text{in }
(-{\frac{\pi }{2}}, {\frac{\pi }{2}})\times \mathbb{R}
\end{gather*}
has infinitely many eigenvalues and corresponding eigenfunctions:
\begin{gather*}
\lambda_{mn}=(2n-1)^4-4m^2, \\
\phi_{mn}=e^{2mti}\cos (2n-1)x,
\end{gather*}
where $(m,n)\in \Lambda$.
Then we define the Hilbert space
$$
H=\big\{ u\in L^2(\Omega ): u(\pm {\frac{\pi }{2}}, t)= u_{xx}(\pm {\frac{\pi
}{2}},t)=0, u(x,t)=u(-x,t) \big\},
$$
for which the set of functions $\{\phi_{mn} : (m,n)\in \Lambda \}$
forms an orthogonal base.

For $\delta \neq 0$ and  functions $u(x,t)=\sum_{\Lambda} u_{mn}\phi_{mn}$,
we define the selfadjoint linear operator
$$
Au=\sum_{\Lambda }((2n-1)^4-4m^2) u_{mn}\phi_{mn}
$$
on the domain
$$
 D(A)=\{u\in H : u(x,t)=\sum_{\Lambda} u_{mn}\phi_{mn}
 \text{ and }\sum_{\Lambda }((2n-1)^4-4m^2)^2 | u_{mn} | ^2<\infty \}
$$
where $u_{-mn}=\overline {u}_{mn}$.
We also define the closed linear operator
$$
A_{\delta }u=Au+\delta u_t
$$
on the domain
\begin{multline*}
D(A_{\delta })
=\Big\{u\in H : u(x,t)=\sum_{\Lambda} u_{mn}\phi_{mn}\\
\text{ and } \sum_{\Lambda } (((2n-1)^4-4m^2)^2+4m^2\delta ^2)
| u_{mn} | ^2<\infty \Big\}.
\end{multline*}
It is easy to check that set of eigenvalues for $A$ and $A_{\delta }$ are
\begin{gather*}
\sigma (A)=\{ \lambda_{mn}=(2n-1)^4-4m^2 : (m,n)\in \Lambda \}, \\
\sigma (A_{\delta })=\{ \overline {\lambda}_{mn} =(2n-1)^4-4m^2+2m\delta
i : (m,n)\in \Lambda \}.
\end{gather*}
We consider the weak solutions of (\ref{q5}), which are
$u\in D(A_\delta)$ satisfying
\begin{equation}
\label{q6}A_{\delta}u+bu^+-au^-=h(x,t),
\end{equation}
where $h(x,t)=\alpha \cos x+\beta \cos{2t}\cos{x}+\gamma \sin{2t}\cos{x}$.
Assuming that  $\delta $ satisfies
$$
\sqrt{9+4\delta ^2}<17,
$$
we set
\begin{equation}\label{q7}
 a_0=\min{\{17, \sqrt{225+16\delta ^2}\}}, \quad b_0=\sqrt{9+4\delta ^2} .
\end{equation}
For section 4 in this paper we assume that
\begin{equation}
\label{q8}-1<a<b_0<b<a_0.
\end{equation}
For Section 5, we assume that
\begin{equation}
\label{q9}-a_0<a<-1, \quad b_0<b<a_0.
\end{equation}
To find solutions of (\ref{q6}), we first investigate the
properties of operator $A_{\delta}$. We have the following lemma.

\begin{lemma} \label{lm1}
For  all $u\in H$, the operator $A_{\delta}$ satisfies
$$
   \| A_{\delta}^{-1}u \| \leq \| u \|, \quad
   \| A_{\delta}^{-1}u \|_1 \leq 2 \| u \|.
$$
\end{lemma}

\paragraph{Proof.} Note that
$$
A_{\delta }^{-1}u=\sum_{\Lambda} \frac {1}{(2n-1)^4-4m^2+2m\delta i}
u_{mn}\phi_{mn}%
$$
for $u=\sum_{\Lambda}u_{mn}\phi_{mn}$,  $u\in H$. For
any $(m,n)\in \Lambda $,  we have
$$
| (2n-1)^4-4m^2+2m\delta i | ^2 \geq 1,
$$
then
$$
\| A_{\delta }^{-1}u \| ^2 = \sum_{\Lambda}
 | \frac{1}{(2n-1)^4-4m^2+2m\delta i} u_{mn} |^2
 \leq  \sum_{\Lambda} | u_{mn} |^2
 = \| u \| ^2 .
$$
On the other hand,
$$
\| A_{\delta}^{-1}u \| ^2_1 = \sum_{\Lambda} \frac
{1+4m^2+(2n-1)^2}{((2n-1)^4-4m^2)^2+4m^2\delta ^2} | u_{mn} |^2 ,
$$
while
\begin{eqnarray*}
\lefteqn{\frac {1+4m^2+(2n-1)^2}{((2n-1)^4-4m^2)^2+4m^2\delta ^2} }\\
  &=& \frac {1+4m^2+(2n-1)^2}{((2n-1)^2+2m)^2((2n-1)^2-2m)^2+4m^2\delta ^2} \\
  &\leq & \frac {1+4m^2+(2n-1)^2}{((2n-1)^2+2|m|)^2+4m^2\delta ^2} \\
  &\leq & \frac {((2n-1)^2+2 | m | )^2+1}{((2n-1)^2+2 | m | )^2}
  \leq  2.
\end{eqnarray*}
Therefore,
$$
\| A_{\delta}^{-1} u \| ^2_1 \leq 2\sum_{\Lambda} | u_{mn} | ^2 =2 \|
u \| ^2
$$
which completes the proof. \hfill $\square$

By Lemma \ref{lm1}, it follows that the operator $A_\delta ^{-1}\ :\ H\rightarrow
H$ is compact since the embedding $H^1\hookrightarrow L^2$ is compact.
Let
\begin{align*}
&   \phi_1(x,t)=\cos x, \\
&   \phi_2(x,t)=\cos 2t\cos x, \\
&   \phi_3(x,t)=\sin 2t\cos x.
\end{align*}
Let $V_3$ be the three-dimensional subspace of $H$ spanned by $\phi_1$,
$\phi_2$, $\phi_3$, $W_3$ be the orthogonal complement of $V_3$ in $H$. Let
$P_3$ be the orthogonal projection $H$ onto $V_3$. Then every element $u\in
H $ is expressed by $u=v+w,$ where $v=P_3u,\ w=(I-P_3)u$. Hence, equation
(\ref{q6}) is equivalent to the system
\begin{gather}
\label{q10} A_\delta w+(I-P_3)(b(v+w)^{+}-a(v+w)^{-})=0,\\
\label{q11} A_\delta v+P_3(b(v+w)^{+}-a(v+w)^{-})=\alpha \phi_1
+\beta \phi_2+\gamma \phi_2.
\end{gather}

Using the Lyapunov-Schmidt reduction method and the contraction mapping
principle, we can easily obtain the following lemma.

\begin{lemma} \label{lm2}.
Suppose that condition (\ref{q8}) or (\ref{q9}) hold,
then, for fixed $v\in V_3$, equation (\ref{q10}) has a unique
solution $w=\theta (v)$. Furthermore, $\theta (v)$ is Lipschitz continuous
(with respect to the $L^2$-norm) in terms of $v$.
\end{lemma}

By this lemma, the study of existence of solutions for equation (\ref{q6})
is reduced to that of equation (\ref{q11}).
Namely, we need to study only the problem
\begin{equation}
\label{q12}A_{\delta}v+P_3(b(v+\theta (v))^+-a(v+\theta (v))^-) =\alpha
\phi_1+\beta \phi_2+\gamma \phi_2 .
\end{equation}
on the three dimensional subspace $V_3$ spanned by
$\{\phi_1, \phi_2, \phi_3\}$.

Now, define the mapping $\Phi :V_3\rightarrow V_3$ by
$$
\Phi (v)=A_\delta v+P_3(b(v+\theta (v))^{+}-a(v+\theta (v))^{-}),
$$
and the mapping $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$ by
$$
F(s_1,s_2,s_3)=(t_1,t_2,t_3)
$$
where $v=s_1\phi_1+s_2\phi_2+s_3\phi_3,$ and
$\Phi (v)=t_1\phi_1+t_2\phi_2+t_3\phi_3$.
By Lemma \ref{lm2}, we know that $\Phi $ and $F$ are continuous on $V_3$ and
$\mathbb{R}^3$, respectively. On the other hand, we have the following Lemma.

\begin{lemma} \label{lm3}
 $\Phi (cv)=c\Phi (v)$ for $c\geq 0$.
\end{lemma}

\paragraph{Proof.} Let $c\geq 0$. If $v$ satisfies
$$
A_{\delta}\theta (v)+(I-P_3)k(v+\theta (v))^+=0 ,
$$
then
$$
A_{\delta}(c\theta (v))+(I-P_3)k(cv+c\theta (v))^+=0 ,
$$
and hence, $\theta (cv)=c\theta (v)$. Therefore, we have
\begin{eqnarray*}
  \Phi (cv) & = & A_\delta (cv)+P_3k(cv+\theta (cv))^+  \\
            & = & A_\delta (cv)+P_3k(cv+c\theta (v))^+   \\
            & = & c\Phi (v).
 \end{eqnarray*}
Lemma \ref{lm3} implies that $\Phi $ or $F$ maps a ray to a ray and a cone with
vertex $0$ onto a cone with vertex $0$.
Set
\begin{align*}
& C_1=\{S=(s_1,s_2,s_3)\in \mathbb{R}^3 : s_1\geq 0,s_2^2+s_3^2\leq s_1^2 \}, \\
& C_2=\{S=(s_1,s_2,s_3)\in \mathbb{R}^3 : s_1\leq 0,s_2^2+s_3^2\leq s_1^2 \}, \\
& C_3=\mathbb{R}^3\backslash (C_1\cup C_2).
\end{align*}
and
\begin{align*}
&  D_1=\{v=s_1\phi_1+s_2\phi_2+s_3\phi_3 : S=(s_1,s_2,s_3)\in C_1 \},\\
&  D_2=\{v=s_1\phi_1+s_2\phi_2+s_3\phi_3 : S=(s_1,s_2,s_3)\in C_2 \},\\
&  D_3=V_3 \backslash (D_1\cup D_2).
\end{align*}

\begin{lemma} \label{lm4}
If $v=s_1\phi_1+s_2\phi_2+s_3\phi_3$, then
\begin{enumerate}
\item $v\geq 0$  if only if  $v\in D_1$,

\item  $v\leq 0$  if only if  $v\in D_2$.
\end{enumerate}
\end{lemma}
The proof of this lemma is simple and we omit it.

\begin{lemma} \label{lm5}
Suppose that $v$ and $\theta (v)$ satisfy (\ref{q10})
and (\ref{q12}). If $v$ is a sign-changing solution of (\ref{q12}), then
$u=v+\theta (v)$ is a sign-changing solution of (\ref{q6}).
\end{lemma}

\paragraph{Proof.} If $u$ is a single-sign solution, then $u^+=u$ or $u^+=0$.
By (\ref{q10}) and Lemma \ref{lm2}, we know $\theta (v)=0$, and hence
$v=u$ is a single-sign solution, which is a contradiction.
\hfill$\square$

\section{Uniqueness of the solution}

In this section, we consider the  general equation
\begin{equation}
\label{q13}A_{\delta}u+bu^+-au^-=h,
\end{equation}
where $h(x,t)\in H$. By the contraction mapping principle, we obtain the
following uniqueness result for (\ref{q13}).

\begin{theorem} \label{thm1}
 If the constant $\delta \neq 0$ and the constants $a, b$ satisfy
\begin{equation}
\label{q14}  -1<a, b<3+\delta ^2,
\end{equation}
then equation (\ref{q13}) has a unique solution.
\end{theorem}
The proof of this theorem is standard and we omit it here.

\paragraph{Remark.} Condition (\ref{q14}) should be replaced by
$-1<k<3$ when $\delta \rightarrow 0$ and $a=0, b=k$.
The condition $-1<k<3$ was used in \cite{MW1} to ensure the uniqueness
of solution to (\ref{q3}) when $\delta =0$.

\section{The nonlinearity crosses one eigenvalue}

In this section, we consider equation (\ref{q6}) under
condition (\ref{q8}).
From the discussions in section 2, to consider equation (\ref{q6})
we need firstly to investigate the image
of the cones $C_1 , C_2$ and $C_3$ under $F$
and the image of the cones $D_1,D_2$ and $D_3$ under $\Phi $.
Set
\begin{align*}
&  \Theta_1=\big\{(t_1,t_2,t_3)\in \mathbb{R}^3 : t_1\geq 0,t_2^2+t_3^2
              \leq \frac{(b-3)^2+4\delta^2}{(b+1)^2}t_1^2 \big\}, \\
&  \Theta_2=\big\{(t_1,t_2,t_3)\in \mathbb{R}^3 : t_1\leq 0,t_2^2+t_3^2
              \leq \frac{(a-3)^2+4\delta^2}{(a+1)^2}t_1^2 \big\}, \\
&  \Theta_3=\mathbb{R}^3\backslash (\Theta_1\cup \Theta_2).
\end{align*}
and
\begin{align*}
& \Omega_1=\{v=t_1\phi_1+t_2\phi_2+t_3\phi_3 :
                (t_1,t_2,t_3)\in \Theta_1 \},\\
& \Omega_2=\{v=t_1\phi_1+t_2\phi_2+t_3\phi_3 :
                (t_1,t_2,t_3)\in \Theta_2 \},\\
& \Omega_3=V_3 \backslash (\Omega_1\cup \Omega_2).
\end{align*}
Suppose that $a, b, \delta $ satisfy (\ref{q8}) and
\begin{equation}
\label{q15}b>1+\frac{1}{2}\delta ^2>a,
\end{equation}
then we have
$$
\frac{(b-3)^2+4\delta^2}{(b+1)^2}<1
\quad \text{and} \quad
\frac{(a-3)^2+4\delta^2}{(a+1)^2}>1.%
$$
Denote
\begin{gather*}
C_4=C_1 \backslash \overline {\Theta_1}, \quad  C_5=\overline {\Theta
_2} \backslash C_2, \\
D_4=D_1 \backslash \overline {\Omega_1}, \quad  D_5=\overline{\Omega_2}
\backslash D_2.
\end{gather*}
By Lemma \ref{lm4}, we have that $v\geq 0$ for all $v\in D_4$ and $v$ changes
sign for every $v\in D_5$.

\begin{lemma} \label{lm6}
\begin{gather*}
F(C_1)=\Theta_1 , \quad F(C_2)=\Theta_2 , \quad  C_4\subset \Theta_3\subset
F(C_3), \\
\Phi (D_1)=\Omega_1 , \quad  \Phi (D_2)=\Omega_2 , \quad D_4\subset \Omega
_3\subset \Phi (D_3).
\end{gather*}
\end{lemma}

\paragraph{Proof.}
 Suppose $(s_1,s_2,s_3)\in C_1$, then
 $v=s_1\phi_1+s_2\phi_2+ s_3\phi_3 \in D_1 $. By Lemmas \ref{lm2} and \ref{lm4},
 we know $v\geq 0$ and  $\theta(v)=0$. At the same time, we obtain
\begin{eqnarray*}
 A_\delta v+P_3k(v+\theta (v))^+
    &=& A_\delta (s_1\phi_1+s_2\phi_2+ s_3\phi_3)
        +k(s_1\phi_1+s_2\phi_2+ s_3\phi_3) \\
    &=& (\phi_1,\phi_2,\phi_3)
        \begin{pmatrix}
          b+1 & 0 & 0 \\
          0 & b-3 & 2\delta \\
          0 & -2\delta & b-3
        \end{pmatrix}
    \begin{pmatrix}
          s_1 \\
          s_2 \\
          s_3 \\
         \end{pmatrix};
\end{eqnarray*}
therefore, for all $(s_1,s_2,s_3)\in C_1$,
$$
F(s_1,s_2,s_3) =\begin{pmatrix}
b+1 & 0 & 0 \\
0 & b-3 & 2\delta \\
0 & -2\delta & b-3
\end{pmatrix}
\begin{pmatrix}
s_1 \\
s_2 \\
s_3
\end{pmatrix}.
$$
Assuming that $(s_1,s_2,s_3)\in C_2$, in the same
way, we obtain
\begin{eqnarray*}
 A_\delta v+P_3k(v+\theta (v))^+
    &=& A_\delta (s_1\phi_1+s_2\phi_2+ s_3\phi_3) \\
    &=& (\phi_1,\phi_2,\phi_3)
    \begin{pmatrix}
          a+1 & 0 & 0 \\
          0 & a-3 & 2\delta \\
          0 & -2\delta & a-3 \\
        \end{pmatrix}
  \begin{pmatrix}
          s_1 \\
          s_2 \\
          s_3 \\
         \end{pmatrix}.
\end{eqnarray*}
Therefore, for all $(s_1,s_2,s_3)\in C_1$,
$$
F(s_1,s_2,s_3) =
\begin{pmatrix}
a+1 & 0 & 0 \\
0 & a-3 & 2\delta \\
0 & -2\delta & a-3
\end{pmatrix}
\begin{pmatrix}
s_1 \\
s_2 \\
s_3
\end{pmatrix}.
$$
It follows that
$F(C_1)=\Theta_1,\ F(C_2)=\Theta_2$ and $\Phi
(D_1)=\Omega_1,\ \ \Phi (D_2)=\Omega_2$ . Moreover, By Lemma \ref{lm3}
 and the continuity of $F$, we  have
$$
\Theta_3\subset F(C_3),\quad \Omega_3\subset \Phi (D_3).
$$
The proof of this Lemma is complete. \hfill$\square$\smallskip

Now, combining Lemmas \ref{lm2}, \ref{lm4}, \ref{lm5}, and \ref{lm6},
 we obtain the following result.

\begin{theorem} \label{thm2}
Let $h(x,t)=\alpha \cos x+\beta \cos {2t}\cos {x}+\gamma \sin {2t}\cos {x}$.
Suppose that $\delta ,a,b$ satisfy $\delta>0$, (\ref{q8}) and (\ref{q15}), then equation (\ref{q6}), and hence problem
(\ref{q5}) has a solution satisfying:
\begin{enumerate}
\item If $\alpha >0$ and
$$
\beta ^2+\gamma ^2\leq \frac{(b-3)^2+4\delta^2}{(b+1)^2}\alpha ^2,
$$
then $h>0$ and the corresponding solution is positive.

\item  If $\alpha >0$ and
$$
\frac{(b-3)^2+4\delta^2}{(b+1)^2}\alpha ^2 <\beta ^2+\gamma ^2< \alpha ^2,
$$
then $h>0$ but the corresponding solution changes sign.

\item If $\alpha <0$ and
$$
\alpha ^2<\beta ^2+\gamma ^2 \leq \frac{(a-3)^2+4\delta^2}{(a+1)^2}\alpha^2,%
$$
then $h$ changes sign but the corresponding solution is negative.

\item If $\alpha <0$ and
$\beta ^2+\gamma ^2\leq \alpha ^2$,
then $h<0$ and the corresponding solution is negative.

\item Elsewhere, the function $h$ changes sign and the corresponding
solution changes sign.
\end{enumerate}
\end{theorem}

\paragraph{Remark.}
By Theorem \ref{thm2}, if $a=0$, $b=k\rightarrow 3$, and $\delta
\rightarrow 0$, then for almost every $h>0$, problem (\ref{q5}) has
sign-changing periodic solution. If the damping $\delta $ is
large enough such that $\delta ^2>2k-2$
and condition (\ref{q8}) is satisfied, then when
$$
\alpha ^2<\beta ^2+\gamma ^2 \leq \frac{(k-3)^2+4\delta^2}{(k+1)^2}\alpha ^2,
$$
the corresponding solution is positive though $h$ changes sign.

This result answers partially the problem mentioned in section 1.


\section{The nonlinearity crosses two eigenvalues}

In this section, we consider equation (\ref{q6}) under condition (\ref{q9}).
In addition,  we suppose
\begin{equation}
\label{q16}b>1+\frac{1}{2}\delta ^2.
\end{equation}
As in section 4, we first investigate the image of
the cones $C_1 , C_2$ and $C_3$ under $F$ and
the image of the cones $D_1,D_2$ and $D_3$ under $\Phi $.
Set
\begin{align*}
& \Theta_1=\big\{(t_1,t_2,t_3)\in \mathbb{R}^3 : t_1\geq 0,t_2^2+t_3^2
              \leq \frac{(b-3)^2+4\delta^2}{(b+1)^2}t_1^2 \big\}, \\
& \Theta_6=\big\{(t_1,t_2,t_3)\in \mathbb{R}^3 : t_1\geq 0,t_2^2+t_3^2
              \leq \frac{(a-3)^2+4\delta^2}{(a+1)^2}t_1^2 \big\}, \\
& \Theta_7=\overline{\Theta_6}\backslash \overline{\Theta_1}.
\end{align*}
and
\begin{align*}
& \Omega_1=\{v=t_1\phi_1+t_2\phi_2+t_3\phi_3 :
                (t_1,t_2,t_3)\in \Theta_1 \},\\
& \Omega_6=\{v=t_1\phi_1+t_2\phi_2+t_3\phi_3 :
                (t_1,t_2,t_3)\in \Theta_6 \},\\
& \Omega_7=\overline{\Omega_6} \backslash \overline{\Omega_1}.
\end{align*}

\begin{lemma} \label{lm7}
 For every $v=s_1\phi_1+s_2\phi_2+s_3\phi_3 \in V_3 $,
there exists a constant $d>0$ such that
$(\Phi (v),\phi_1)\geq d|s_2+s_3|$.
\end{lemma}

\paragraph{Proof.} Note that
$$
bu^+-au^-+u=(b+1)u^+-(a+1)u^-\geq \min{\{b+1, -(a+1)\}}|u|=c|u|
$$
and
$$
\phi_1\geq \frac{1}{\sqrt{2}}|\phi_2+\phi_3|.
$$
Let $v=s_1\phi_1+s_2\phi_2+s_3\phi_3 $. Then we have
\begin{align*}
(\Phi (v),\phi_1)
  = & (A_\delta (s_1\phi_1+s_2\phi_2+s_3\phi_3 )
          +P_3(b(v+\theta (v))^+-a(v+\theta (v))^-), \phi_1) \\
 = & (\phi_1, v+\theta (v)+b(v+\theta (v))^+-a(v+\theta (v))^-) \\
  \geq & (\phi_1, c|v+\theta (v)|) \\
  = & c\int_{\Omega}\phi_1|s_1\phi_1+s_2\phi_2+s_3\phi_3+\theta (v)|dtdx \\
  \geq & \frac{c}{\sqrt{2}}\int_{\Omega}
           |\phi_2+\phi_3||s_1\phi_1+s_2\phi_2+s_3\phi_3+\theta (v)|dtdx \\
  \geq & \frac{c_1}{\sqrt{2}}|s_2+s_3|.
  \end{align*}
Taking $d=c_1/\sqrt{2}$, the conclusion follows. \hfill$\square$

Note that Lemma \ref{lm7} implies  $t_1\geq 0$ for every $v=s_1\phi_1+s_2\phi_2+
s_3\phi_3 \in V_3 $
and $\Phi (v)=t_1\phi_1+t_2\phi_2+t_3\phi_3 $.

\begin{lemma} \label{lm8}
\begin{gather*}
F(C_1)=\Theta_1 , \quad  F(C_2)=\Theta_6 , \quad  \Theta_7\subset F(C_3),\\
\Phi (D_1)=\Omega_1 , \quad \Phi (D_2)=\Omega_6 , \quad \Omega_7\subset \Phi
(D_3).
\end{gather*}
\end{lemma}

The proof of this lemma follows by  similar calculations as in Lemma \ref{lm6}
with $a<-1$. Now we can obtain the main result in this section.

\begin{theorem} \label{thm3}
Let $h(x,t)=\alpha
\cos x+\beta \cos{2t}\cos{x} +\gamma \sin{2t}\cos{x}$. Suppose that
$\delta , a, b$ satisfy $\delta>0$, (\ref{q9}) and (\ref{q16}), then we have:
\begin{enumerate}

\item If $h\in \overline{\Omega_1}$, (\ref{q6}) has a positive
solution, and a negative solution.

\item If $h$ belongs to interior of $\Omega_7$, (\ref{q6}) has a
negative solution and at least one sign-changing solution.

\item If $h\in \partial \Omega_6$, (\ref{q6}) has a negative solution.
\end{enumerate}
\end{theorem}

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\noindent\textsc{Yukun An}\\
College of Mathematics and Information Science\\
Northwest Normal University\\
Lanzhou, 730070, Gansu, China \\
and \\
Department of Mathematics, Suzhou University\\
Suzhou, 215006, Jiangsu, China\\
e-mail:anyk@nwnu.edu.cn


\end{document}