\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 82, 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 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/82\hfil Existence of non-real eigenvalues]
{A priori bounds and existence of non-real eigenvalues of
 fourth-order boundary value problem with indefinite weight function}

\author[X. Han, T. Gao \hfil EJDE-2016/82\hfilneg]
{Xiaoling Han, Ting Gao}

\address{Xiaoling Han \newline
College of Mathematics and Statistics, 
Northwest Normal University,
Lanzhou, Gansu 730070, China}
\email{hanxiaoling@nwnu.edu.cn}

\address{Ting Gao \newline
College of Mathematics and Statistics, 
Northwest Normal University,
Lanzhou, Gansu 730070, China}
\email{939832645@qq.com}

\thanks{Submitted January 1, 2016. Published March 23, 2016.}
\subjclass[2010]{34B05, 34L15, 47B50}
\keywords{a priori bound; non-real eigenvalue; indefinite weight function}

\begin{abstract}
 In this article, we give a priori bounds on the possible non-real eigenvalue
 of regular fourth-order boundary value problem with indefinite weight function
 and obtain a sufficient conditions for such problem to admit non-real eigenvalue.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

In this article we  study non-real eigenvalues of differential equations
with indefinite weights.
The Sturm-Liouville problem with weighted functions is called
right-definite if the weighted function do not change signs.
Otherwise, the problem is called indefinite problem.
The spectral theory of the right-definite problem with self-adjoint boundary
conditions has been accomplished, but the spectral structure of indefinite
problems, especially both right and left indefinite problem, i.e.,
indefinite problem, is quite different from and more complicated than
that of right-definite problems. For example, there is neither upper
nor lower bound for real eigenvalues of indefinite Sturm-Liouville boundary problems.
What is more, the indefinite problem may have non-real eigenvalues.
Such problems occur in certain physical models, particularly in transport
theory and statistical physics. The indefinite nature of the problem was
noticed by Haupt \cite{h1} and Richardson \cite{r1} at the
 beginning of the previous century.
For a review of the early work in this direction, see \cite{m2}.

In  \cite{q1}, the author considered the indefinite spectral problem
$$
-y''+qy=\lambda wy, \quad y(-1)=y(1)=0,\quad  y\in L_{|w|}^2[-1,1]
$$
combined with conditions that $q$ and $w$ are real-valued functions satisfying
$$
w(x)\neq0 \text{ a.e. on } [-1,1], \quad q, w\in L^1[-1,1],
$$
and $w(x)$ changes sign on $[-1,1]$. Here, $L_{|w|}^2[-1,1]$ is a Krein space,
equipped with the indefinite inner product
$$
[f,g]=\int _{-1}^1f(x)\overline{g(x)}w(x)dx,  f,g\in L^2_{|w|}[-1,1].
$$
The indefinite problems has discrete, real eigenvalues, unbounded from both
below and above, and may also admit non-real eigenvalues.

But most  articles consider second-order differential equations;
 in this article, we consider the indefinite spectral problem
\begin{equation}
\begin{gathered}
\tau y:=y^{(4)}+qy=\lambda wy, \\
 y(-1)=y(1)=y''(-1)=y''(1)=0,
\quad  y\in L_{|w|}^2[-1,1]
\end{gathered}    \label{e1.1}
\end{equation}
combined with conditions that $q$ and $w$ are real-valued functions satisfying
\begin{equation}
w(x)\neq0\text{ a.e. on } [-1,1],\quad  q, w\in L^1[-1,1],     \label{e1.2}
\end{equation}

and $w(x)$ changes sign on $[-1,1]$. We will first obtain a priori
 bounds for possible non-real eigenvalues and then find sufficient conditions
for the existence of non-real eigenvalues of \eqref{e1.1}.



\section{A priori bounds of non-real eigenvalues}

For the indefinite problem \eqref{e1.1}, let
\begin{equation}
\begin{gathered}
\tau y:=y^{(4)}+qy=\lambda |w|y, \\
y(-1)=y(1)=y''(-1)=y''(1)=0, \quad y\in L_{|w|}^2[-1,1]
\end{gathered}   \label{e2.1}
\end{equation}
be the corresponding right-definite problem.

Firstly, we consider the fourth-order differential equation
\begin{equation}
y^{4}+qy=\lambda wy,  \quad x\in[a,b]     \label{e2.2}
\end{equation}
combined with the boundary conditions
\begin{equation}
\begin{gathered}
B_1y:=y(a)\cos(\theta_1)-y'''(a)\sin(\theta_1)=0, \\
B_2y:=y(b)\cos(\theta_2)-y'''(b)\sin(\theta_2)=0, \\
B_3y:=y'(a)\cos(\theta_3)-y''(a)\sin(\theta_3)=0, \\
B_4y:=y'(b)\cos(\theta_4)-y''(b)\sin(\theta_4)=0,
\end{gathered}    \label{e2.3}
\end{equation}
where $q$ and $w$ satisfies \eqref{e1.2} and
$\theta_1, \theta_2, \theta_3, \theta_4 \in \mathbb{R}$.
The corresponding right-definite problem is
\begin{equation}
y^{4}+qy=\lambda |w|y     \label{e2.4}
\end{equation}
combined with \eqref{e2.3}.

Assumption: $\lambda=0$ is not an eigenvalue of the boundary problems in questions.

\begin{proposition} \label{prop2.1}
 If problem \eqref{e2.2}-\eqref{e2.3} has non-real eigenvalues,
then  problem \eqref{e2.4}-\eqref{e2.3} has at least one negative eigenvalue.
\end{proposition}

\begin{proof}
Let $y=y(t,\lambda)$ be the corresponding non-real eigenfunction of the
non-real eigenvalue $\lambda$. Multiplying \eqref{e2.2} by $ \overline{y}$
integrating over $[a,b]$, we find
\begin{align*}
&\cot( \theta_2)|y(b)|^2-\cot( \theta_1)|y(a)|^2
+\cot( \theta_3)|y'(a)|^2\\
&-\cot( \theta_4)|y'(b)|^2
+\int_{a}^{b}[|y''|^2+q|y|^2]dx\\
&=\lambda \int_{a}^{b}w|y|^2dx
\end{align*}
Then the smallest eigenvalue $v$ of \eqref{e2.4} is given by the minimum
of the left side of the equation. Let $y\in S_0$, where
\begin{align*}
S_0=\big\{&y\in L_{|w|}^2[a,b]| y,y',y''\in AC_{\rm loc}[a,b],y^{(4)}
+qy \in L_{|w|}^2[a,b],\\
& B_1y=B_2y=B_3y=B_4y=0\big\}
\end{align*}
Now the non-real eigenfunction $y$ makes the left side of equation vanish.
Moreover $y\in S_0$. Hence $v<0$ since $v = 0$ is not an eigenvalue
of \eqref{e2.4}, i.e., the problem has at least one negative eigenvalue.
\end{proof}

\begin{proposition}[\cite{m1}] \label{prop2.2}
If problem \eqref{e2.4}-\eqref{e2.3} has $n$ negative eigenvalues,
then  problem \eqref{e2.2}-\eqref{e2.3} has at most $2n$ non-real eigenvalues.
\end{proposition}

Denote by $\|\cdot\|_{p}$ the norm of the space $L^{p}[-1,1]$ and by
 $\|\cdot\|_{C}$ the maximum norm of $C[-1,1]$.
If $xw(x) > 0$ a.e. on $[-1,1]$, we set
\begin{equation}
S_1(\varepsilon_1) = \{x\in [-1,1]: xw(x) <\varepsilon_1\},\quad
m_1(\varepsilon_1)= \operatorname{meas}S_1(\varepsilon_{_1}).     \label{e2.5}
\end{equation}
If $w \in AC_{\rm loc}[-1, 1]$, $w'\in L^2[-1,1]$, $w''\in L^2[-1,1]$, we set
\begin{equation}
S_2(\varepsilon_2) = \{x\in [-1,1]: w^2(x) <\varepsilon_2\},\quad
m_2(\varepsilon_2) = \operatorname{meas}S_2(\varepsilon_2).     \label{e2.6}
\end{equation}

A value of $x$ about which $w(x)$ changes its sign will be called a turning point.
If $w(x)$ has only one turning point, we will obtain the
following a priori bounds for possible non-real eigenvalues.

\begin{theorem} \label{thm2.3}
 Suppose that $\lambda$ is, if it exists, a non-real eigenvalue of \eqref{e1.1}.
If $xw(x) > 0$ a.e. on $[-1,1]$, then
\begin{gather*}
|\operatorname{Re}\lambda|
\leq \frac{2\sqrt{2}\|\phi\|_{C}(1+\sqrt{\|q_{-}\|_1}\|\phi\|_{C}
+\sqrt{2}\|q_{-}\|_1\|\phi\|_{C})}{\varepsilon_1}, \\
|\operatorname{Im} \lambda|
\leq \frac{2\sqrt{2}\|\phi\|_{C}(1+\sqrt{\|q_{-}\|_1}\|\phi\|_{C})}{\varepsilon_1},
\end{gather*}
where $\varepsilon_1>0$ satisfies
 $(1-m_1(\varepsilon_1)\|\phi\|^2_{C})\geq \frac{1}{2}$ and
$q_{-}(x) = -\min\{0, q(x)\}$.
\end{theorem}

\begin{proof}
Let $\lambda$ be a non-real eigenvalue of \eqref{e1.1} and $\phi$ is the
corresponding eigenfunction with
$\|\phi\|_2=1$, $\|\phi\|_{C}=\max\{|\phi|, |\phi'|, |\phi'''|\}$. Multiplying
both sides of $\phi^{(4)}+q\phi=\lambda w\phi$ by $\overline{\phi}$ and
integrating over the interval $[x,1]$ we have
\begin{equation}
-(\phi'''\overline{\phi})(x)+(\phi''\overline{\phi'})(x)
+\int_{x}^1|\phi''|^2dx+\int_{x}^1q|\phi|^2dx
=\lambda\int_{x}^1w|\phi|^2dx.     \label{e2.7}
\end{equation}
Separating the real and imaginary parts of both sides of  \eqref{e2.4} yields
\begin{gather}
\operatorname{Re}\lambda\int_{x}^1w|\phi|^2dx
=\operatorname{Re}(-\phi'''\overline{\phi})(x)
+\operatorname{Re}(\phi''\overline{\phi'})(x)
+\int_{x}^1|\phi''|^2dx+\int_{x}^1q|\phi|^2dx,     \label{e2.8}\\
\operatorname{Im}\lambda\int_{x}^1w|\phi|^2dx
=\operatorname{Im}(-\phi'''\overline{\phi})(x)
+\operatorname{Im}(\phi''\overline{\phi'})(x).     \label{e2.9}
\end{gather}
We will use \eqref{e2.8} and \eqref{e2.9} to estimate
 $\operatorname{Re} \lambda$ and $\operatorname{Im} \lambda$.
To do this, let $x=-1$ in \eqref{e2.9}.
From $\operatorname{Im} \lambda\neq0$ and $\phi(-1)=0$ and
$\phi''(-1)=0$, we have $\int_{-1}^1w|\phi|^2dx=0$ and hence, by \eqref{e2.8},
\begin{equation}
\int_{-1}^1|\phi''|^2dx+\int_{-1}^1q|\phi|^2dx=0.     \label{e2.10}
\end{equation}
Let $\|q_{-}\|_1=\int_{-1}^1q_{-}dx$, then
$$
\int_{-1}^1|\phi''|^2dx
=-\int_{-1}^1q|\phi|^2dx\leq\int_{-1}^1q_{-}|\phi|^2dx
\leq\|\phi\|^2_{C}\|q_{-}\|_1
$$
and $\int_{-1}^1q_{-}|\phi|^2dx\leq\|\phi\|^2_{C}\|q_{-}\|_1$, hence
\begin{equation}
\|\phi''\|^2_2\leq\|\phi\|^2_{C}\|q_{-}\|_1, \quad
\int_{-1}^1q_{-}|\phi|^2dx\leq\|\phi\|^2_{C}\|q_{-}\|_1.\label{e2.11}
\end{equation}
Since $xw(x) > 0$ a.e. on $[-1,1]$, one can find $\varepsilon_1>0$
such that $(1-m_1(\varepsilon_1)\|\phi\|^2_{C})\geq \frac{1}{2}$,
where $m_1(\varepsilon_1)$ is defined in \eqref{e2.5}.
Using $\int_{-1}^1w|\phi|^2dx=0$, from \eqref{e2.11}, we have
\begin{equation}
\begin{aligned}
\int_{-1}^1\int_{x}^1w(t)|\phi(t)|^2\,dt\,dx
&=\int_{-1}^1xw(x)|\phi(x)|^2dx  \\
&\geq \varepsilon_1\Big(\int_{-1}^1|\phi(x)|^2dx
 -\int_{S_1(\varepsilon_1)}|\phi(x)|^2dx\Big) \\
&\geq \varepsilon_1(1-m_1(\varepsilon_1)\|\phi\|^2_{C})
 \geq\frac{\varepsilon_1}{2}.
\end{aligned} \label{e2.12}
\end{equation}
Set $q_{+}(x)=\max\{0,q(x)\}$, then $q=q_{+}-q_{-}$ and
$|q|=q_{+}+q_{-}=q+2q_{-}$. Repeatedly using \eqref{e2.10}, we have
\begin{align*}
|\int_{-1}^1\int_{x}^1(|\phi''|^2+q|\phi|^2)\,dt\,dx|
&=|\int_{-1}^1x(|\phi''|^2+q|\phi|^2)dx| \\
&\leq\int_{-1}^1(|\phi''|^2+q|\phi|^2+2q_{-}|\phi|^2)dx \\
&\leq2\int_{-1}^1q_{-}|\phi|^2dx\leq2\|\phi\|^2_{C}\|q_{-}\|_1.
\end{align*}
Now, by \eqref{e2.11} integrating  \eqref{e2.8} gives
\begin{align*}
&|\operatorname{Re}\lambda|\int_{-1}^1\int_{x}^1w(t)|\phi(t)|^2\,dt\,dx\\
&=|\int_{-1}^1\operatorname{Re}(-\phi'''\overline{\phi})(x)dx
 +\int_{-1}^1\operatorname{Re}(\phi''\overline{\phi'})(x)dx
 +\int_{-1}^1\int_{x}^1(|\phi''|^2+q|\phi|^2)\,dt\,dx| \\
&\leq\sqrt{2}\|\phi\|_{C}(1+\sqrt{\|q_{-}\|_1}\|\phi\|_{C}
 +\sqrt{2}\|q_{-}\|_1\|\phi\|_{C}).
\end{align*}
Therefore, in view of \eqref{e2.11}, we conclude that
\begin{equation}
|\operatorname{Re}\lambda|
\leq \frac{2\sqrt{2}\|\phi\|_{C}(1+\sqrt{\|q_{-}\|_1}\|\phi\|_{C}
+\sqrt{2}\|q_{-}\|_1\|\phi\|_{C})}{\varepsilon_1}.   \label{e2.13}
\end{equation}
Moreover, integrating \eqref{e2.9} and using \eqref{e2.12} and \eqref{e2.11},
we have
\begin{equation}
\begin{aligned}
\frac{\varepsilon_1}{2}|\operatorname{Im} \lambda|
&\leq |\operatorname{Im}\lambda|\int_{-1}^1\int_{x}^1w|\phi|^2\,dt\,dx\\
&=|\int_{-1}^1(\operatorname{Im}(-\phi'''\overline{\phi})(x)
+\operatorname{Im}(\phi''\overline{\phi'})(x))|   \\
&\leq\sqrt{2}\|\phi\|_{C}(1+\sqrt{\|q_{-}\|_1}\|\phi\|_{C}).
\end{aligned}  \label{e2.14}
\end{equation}
This completes the proof.
\end{proof}

When $w(x)$ is allowed to have more turning points, we have the following result.

\begin{theorem} \label{thm2.4}
 Suppose that $\lambda$ is, if it exists, a non-real eigenvalue of \eqref{e1.1}.
If $w \in AC_{\rm loc}[-1, 1]$, $w', w''\in L^2[-1,1]$, then
\begin{gather*}
|\operatorname{Re}\lambda|
\leq \frac{2\|\phi\|_{C}[2\|w\|_{C}\|q_{-}\|_1\|\phi\|_{C}
+2\sqrt{2}\|w'\|_2\sqrt{\|q_{-}\|_1}\|\phi\|_{C}
+\sqrt{\|q_{-}\|_1}\|w''\|_2]}{\varepsilon_2},
\\
|\operatorname{Im} \lambda|
\leq \frac{2\|\phi\|_{C}\sqrt{\|q_{-}\|_1}[2\sqrt{2}\|w'\|_2\|\phi\|_{C}
+\|w''\|_2]}{\varepsilon_2},
\end{gather*}
where $\varepsilon_2>0$ satisfies
$(1-m_2(\varepsilon_2)\|\phi\|^2_{C})\geq 1/2$.
\end{theorem}

\begin{proof}
 Let $\lambda$ be a non-real eigenvalue of \eqref{e1.1} and $\phi$ the
corresponding eigenfunction with
$\|\phi\|_2=1$, $\|\phi\|_{C}=\max\{|\phi|, |\phi'|, |\phi'''|\}$.
In this case we still can make use of \eqref{e2.7}, \eqref{e2.8}
and \eqref{e2.9}.
From \eqref{e2.9}, since  $\operatorname{Im} \lambda\neq0$,
we have $\int_{-1}^1w|\phi|^2dx=0$. Thus, \eqref{e2.10} and \eqref{e2.11} holds,
and particularly,
\begin{equation}
\begin{gathered}
|\phi|^2\leq\|\phi\|^2_{C},  \quad  \|\phi'\|^2_2\leq2\|\phi\|^2_{C},\\
\|\phi''\|^2_2\leq\|\phi\|^2_{C}\|q_{-}\|_1, \quad
  \int_{-1}^1q_{-}|\phi|^2dx\leq\|\phi\|^2_{C}\|q_{-}\|_1.
\end{gathered} \label{e2.15}
\end{equation}
Multiplying both sides of $\phi^{(4)}+q\phi=\lambda w\phi$ by
$w\overline{\phi}$ and integrating over the interval $[-1,1]$ we have
\begin{equation}
\begin{aligned}
&\int_{-1}^1w|\phi''|^2dx+2\int_{-1}^1\phi''w'\overline{\phi'}dx
+\int_{-1}^1\phi''w''\overline{\phi}dx
+\int_{-1}^1wq|\phi|^2dx\\
&=\lambda\int_{-1}^1w^2|\phi|^2dx.
\end{aligned}    \label{e2.16}
\end{equation}
Separating the real and imaginary parts of both sides of  \eqref{e2.17} yields
\begin{gather}
\begin{aligned}
\operatorname{Re}\lambda\int_{-1}^1w^2|\phi|^2dx
&=\operatorname{Re}\Big(2\int_{-1}^1\phi''w'\overline{\phi'}dx\Big)
 +\operatorname{Re}\Big(\int_{-1}^1\phi''w''\overline{\phi}dx\Big) \\
&\quad +\int_{-1}^1w|\phi''|^2dx+\int_{-1}^1wq|\phi|^2dx,
\end{aligned}  \label{e2.17} \\
\operatorname{Im}\lambda\int_{-1}^1w^2|\phi|^2dx
=\operatorname{Im}\Big(2\int_{-1}^1\phi''w'\overline{\phi'}dx\Big)
+\operatorname{Im}\Big(\int_{-1}^1\phi''w''\overline{\phi}dx\Big).  \label{e2.18}
\end{gather}
Now, using \eqref{e2.15} and  $|q|=q_{+}+q_{-}=q+2q_{-}$ and
$\int_{-1}^1q|\phi|^2dx=-\int_{-1}^1|\phi''|^2dx$, we obtain
\begin{equation}  \label{e2.19}
\begin{gathered}
\big|\int_{-1}^1w|\phi''|^2dx\big|\leq \|w\|_{C}\|\phi\|^2_{C}\|q_{-}\|_1,\\
\big|\int_{-1}^1wq|\phi|^2dx\big| \leq \|w\|_{C}\|\phi\|^2_{C}\|q_{-}\|_1, \\
\begin{aligned}
\big|\int_{-1}^1\phi''w'\overline{\phi'}dx\big|
&\leq \Big(\int_{-1}^1|\phi''|^2dx\Big)^{1/2}
\Big(\int_{-1}^1|w'|^2dx\Big)^{1/2}
\Big(\int_{-1}^1|\phi'|^2dx\Big)^{1/2} \\
&\leq \sqrt{2\|q_{-}\|_1}\|w'\|_2\|\phi\|^2_{C},
\end{aligned} \\
\big|\int_{-1}^1\phi''w''\overline{\phi}dx\big|
\leq \sqrt{\|q_{-}\|_1}\|w''\|_2\|\phi\|_{C}.
\end{gathered}
\end{equation}
Recall that $m_2(\varepsilon_2) = \operatorname{meas}  S_2(\varepsilon_2)$
defined by \eqref{e2.6} and $w^2(x)\geq\varepsilon_2$ on the set
$\Omega(\varepsilon_2):=[-1,1]\backslash S_2(\varepsilon_2)$.
Then $(1-m_2(\varepsilon_2)\|\phi\|^2_{C})\geq \frac{1}{2}$ yields
\begin{equation}
\begin{aligned}
\int_{-1}^1w^2(x)|\phi(x)|^2dx
&\geq\varepsilon_2\int_{\Omega(\varepsilon_2)}|\phi(x)|^2dx  \\ 
&=\varepsilon_2\Big(\int_{-1}^1|\phi(x)|^2dx
-\int_{S_2(\varepsilon_2)}|\phi(x)|^2dx\Big)\\
&\geq \varepsilon_2(1-m_2(\varepsilon_2)\|\phi\|^2_{C})
\geq\frac{\varepsilon_2}{2},
\end{aligned}\label{e2.20}
\end{equation}
which, together with \eqref{e2.17}, \eqref{e2.18} and \eqref{e2.19},
completes the proof.
\end{proof}

In the particular case when $q\geq0$,  by Theorems \ref{thm2.3}
 and \ref{thm2.4} we see that \eqref{e1.1} has no any non-real eigenvalues,
 which is in accordance  with the conclusion in Proposition \ref{prop2.1} 
since \eqref{e2.1} does not  have any negative eigenvalues.

In what follows, we impose the symmetry conditions on $q$ and $w$, namely,
\begin{equation}
q(x)=q(-x), \quad w(-x)=-w(x).\label{e2.21}
\end{equation}
Under the conditions \eqref{e1.2} and \eqref{e2.21}, it is easy to see that
if $\lambda\in \mathbb{C}$ be a eigenvalue of \eqref{e1.1} and $\phi$ the
corresponding eigenfunction,  then $-\overline{\lambda}$ is an
eigenvalue of \eqref{e1.1} with the eigenfunction $\overline{\phi(-x)}$.
Thus, if $\lambda=i\alpha$ with $\alpha\in \mathbb{R}$, then
 $\overline{\phi(-x)}=c\phi(x)$ for some $c\neq0$ since the geometric
multiplicity is one. Then it follows that $|c|=1$ from
$\overline{\phi(0)}=c\phi(0)$,
 $\overline{\phi'(0)}=c\phi'(0)$, and $|\phi(0)|+|\phi'(0)|\neq0$.
To sum up, we have a lemma.

\begin{lemma} \label{lem2.5}
 Let \eqref{e1.2} and \eqref{e2.21} hold. If $\lambda\in \mathbb{C}$ is
 an eigenvalue of \eqref{e1.1} with an eigenfunction $\phi$, 
then $-\overline{\lambda}$ is an eigenvalue of \eqref{e1.1} with the 
eigenfunction $\overline{\phi(-x)}$. Particularly, if $\lambda=i\alpha$ 
with $\alpha\in \mathbb{R}$ and $\alpha\neq0$, then
 $\overline{\phi(-x)}=c\phi$ for some $c\in \mathbb{C}$ with $|c|=1$.
\end{lemma}

In this case, more accurate a priori bounds on imaginary eigenvalues can 
be found if $q$ is bounded below and $w$ keeps away from zero.

\begin{theorem} \label{thm2.6}
Suppose that \eqref{e2.21} holds and $xw(x)>0$  a.e. on
$[-1,1]$. If, for some $q_0<0$ and $w_0>0$,
\begin{equation}
q(x)\geq q_0,\quad |w(x)|\geq w_0, \quad \text{a.e. }  x\in [-1,1],\label{e2.22}
\end{equation}
then for any possible pure imaginary eigenvalue $\lambda$ of \eqref{e1.1},
we have
\begin{equation}
|\operatorname{Im}\lambda|\leq \frac{8\sqrt{2}\|\phi\|^3_{C}(1+\sqrt{-q_0})}{w_0}.
\label{e2.23}
\end{equation}
\end{theorem}

\begin{proof}
Let $\phi$ be an eigenfunction corresponding to $\lambda=i\alpha$ with  
$\|\phi\|_2=1$, $\|\phi\|_{C}=\max\{|\phi|,|\phi'|,|\phi'''|\}$. 
It follows from Lemma \ref{lem2.5} that there exists an $\omega\in [0,2\pi)$ 
such that $\overline{\phi(-x)}=e^{i\omega}\phi(x)$ and
 $-\overline{\phi'(-x)}=e^{i\omega}\phi'(x)$. So, $|\phi(x)|$ 
and $|\phi'(x)|$ are even functions. We see that \eqref{e2.7}-\eqref{e2.11} 
hold for this  $\phi$. And
\begin{equation}
|\phi(x)|^2\leq (x+1)\int _{-1}^{x}|\phi'(t)|^2dt
\leq\int _{-1}^{0}|\phi'(t)|^2dt=\frac{1}{2}\|\phi'\|_2^2, \quad
x\in [-1,0]\label{e2.24}
\end{equation}
since $|\phi'(x)|$ is even. Actually, \eqref{e2.24} is true for $x\in [-1,1]$
since $|\phi(x)|$ is even.

Since $q(x)\geq q_0$, on $[-1,1]$, it follows from \eqref{e2.10} and 
$\|\phi\|_2=1$, that
$\|\phi''\|_2^2=-\int_{-1}^1q|\phi|^2dx\leq-q_0$.
Then  integrating  \eqref{e2.9} produces
\begin{equation}   \label{e2.25}
\begin{aligned}
|\operatorname{Im}\lambda\|\int_{-1}^1\int_{x}^1w|\phi|^2\,dt\,dx|
&=|\int_{-1}^1(\operatorname{Im}(-\phi'''\overline{\phi})(x)
+\operatorname{Im}(\phi''\overline{\phi'})(x))dx| \\
&\leq\sqrt{2}\|\phi\|_{C}(1+\sqrt{-q_0}).
\end{aligned}
\end{equation}
Let $\delta= 1/(4\|\phi\|^2_{C})$. By \eqref{e2.24}, we have
\begin{equation}
\begin{aligned}
&\big|\int_{-1}^1\int_{x}^1w(t)|\phi(t)|^2\,dt\,dx\big|\\
&=\int_{-1}^1xw(x)|\phi(x)|^2dx
 \geq w_0\int_{-1}^1|x\|\phi(x)|^2dx  \\
&\geq w_0\delta\int_{|x|\geq\delta}|\phi(x)|^2dx \\
& \geq w_0\delta\Big(\int_{-1}^1|\phi(x)|^2dx-\int_{-\delta}^{\delta}|\phi(x)|^2dx\Big)\\
&\geq w_0\delta(1-2\delta\|\phi\|^2_{C})
 \geq\frac{w_0}{8\|\phi\|^2_{C}}.
\end{aligned}\label{e2.26}
\end{equation}
Now, \eqref{e2.23} follows from \eqref{e2.25} and \eqref{e2.26}.
\end{proof}

\section{Existence of non-real eigenvalues }

Although a priori estimate can be given in section 2 and the exact number of non-real 
eigenvalues are still difficult; there are recent studies by means of the operator 
theory in Krein spaces \cite{c1}. 
In this section we  prove the existence of non-real eigenvalues.

\begin{lemma}[\cite{c1}] \label{lem3.1}
 If $w_{j}\in L^1[-1,1]$ and $w_{j}(x)>0$ a.e. on $[-1,1]$ for
$j=1,2$, then the two eigenvalue problems
\begin{equation}
y^{(4)}+qy=\lambda w_{j}(x)y, \quad y(-1)=y(1)=y''(-1)=y''(1)=0, \quad
j=1,2     \label{e3.1}
\end{equation}
have the same number of negative eigenvalues.
\end{lemma}

Let $K$ be the Krein space $L^2_{|w|}[-1,1]$, equipped with the indefinite 
inner product
\begin{equation}
[f,g]=\int _{-1}^1f(x)\overline{g(x)}w(x)dx, \quad f,g\in L^2_{|w|}[-1,1],
   \label{e3.2}
\end{equation}
and $T$ be a self-adjoint operator in $K$ with domain
$D(T)=\{y\in L_{|w|}^2[-1,1]|y,y',y''\in AC_{\rm loc}[-1,1]$,
$T\in L_{|w|}^2[-1,1]\}$. See \cite{b1,b3,b5}.
We say that the operator $T$ has $k$ negative squares, $k\in N_0$, if there exists
a $k$-dimensional subspace $X$ of $K$ in $D(T)$ such that $[Tf,f]<0$ if
$f\in X $and $f\neq0$, but no $(k + 1)$-dimensional subspace with this
property.

\begin{theorem} \label{thm3.2}
 Let \eqref{e2.21} hold. If the eigenvalue problem
\begin{equation}
y^{(4)}+qy=\lambda |w|y, \quad y(-1)=y(1)=y''(-1)=y''(1)=0     \label{e3.3}
\end{equation}
has one negative eigenvalue and the rest eigenvalues are all positive,
then \eqref{e1.1} has exactly two purely imaginary eigenvalues.
\end{theorem}

\begin{proof}
 Let $A=\frac{1}{w}\tau$ and $B=\frac{1}{|w|}\tau$ be the operators 
associated with $y^{(4)}+qy=\lambda wy$ and $y^{(4)}+qy=\lambda |w|y$ with
boundary conditions, respectively. Then $B$ is self-adjoint with respect 
to the definite inner product
$$
(f,g)=\int _{-1}^1f(x)\overline{g(x)}|w(x)|dx,  f,g\in L^2_{|w|}[-1,1]
$$
and $A$ is self-adjoint with respect to the indefinite inner product \eqref{e3.2}.

It follows from Lemma \ref{lem3.1} and the assumption in Theorem \ref{thm3.2} that $B$ 
has one negative eigenvalue and the rest are positive, and hence,
$A$ has exactly one negative square since $[Af,f]=(Bf, f)$ and $0$ 
is a resolvent point of $A$. It is well known (see, \cite{b5,c1}) 
that this implies the existence of exactly one eigenvalue 
$\lambda$ of \eqref{e1.1} in $\mathbb{R}$ or the upper half-plane 
$\mathbb{C}^{+}$ and that if
$\lambda \in \mathbb{R}$ with eigenfunction $\phi$ then 
$[Af,f]=\lambda(f, f)\leq0$. Let $\lambda$ be such
an eigenvalue with eigenfunction $\phi$. If $\lambda$ is real, 
then $-\lambda=-\overline{\lambda}$ is also
an eigenvalue with the eigenfunction $\overline{\phi(-x)}$ by Lemma \ref{lem2.5} and
$$
-\lambda[\overline{\phi(-x)},\overline{\phi(-x)}]=\lambda[\phi,\phi]\leq 0
$$
by the odd symmetry of $w$. Thus, we get that $\lambda$ and $-\lambda$ 
are two such eigenvalues, which is a contradiction. 
Since $\lambda\in \mathbb{C}^{+}$ implies
$-\overline{\lambda}\in \mathbb{C}^{+}$,
we see that $\lambda=-\overline{\lambda}$, i.e., $\lambda$ is purely imaginary. 
The proof is complete.
\end{proof}

 For more details about non-real eigenvalue of second-order boundary 
value problems, please see \cite{b1,b2,b3,b4,q2,x1}.

\subsection*{Acknowledgments}
This research was supported by NNSF  of China (No.  1156\-1063).


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\end{document}