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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 232, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/232\hfil Anti-periodic solutions]
{Anti-periodic solutions to Rayleigh-type equations with
two deviating arguments}

\author[M. Feng, X. Zhang\hfil EJDE-2012/232\hfilneg]
{Meiqiang Feng, Xuemei Zhang}  % in alphabetical order

\address{Meiqiang Feng \newline
School of Applied Science, 
Beijing Information Science and Technology University\\
 Beijing, 100192,  China}
\email{meiqiangfeng@sina.com}

\address{Xuemei Zhang \newline
Department of Mathematics and Physics,
North China Electric Power University \\
Beijing, 102206,  China}
\email{zxm74@sina.com}

\thanks{Submitted September 20, 2012. Published December 21, 2012.}
\subjclass[2000]{34K13, 34K15, 34C25}
\keywords{Rayleigh equation; anti-periodic solution; deviating argument}

\begin{abstract}
 In this article, the Rayleigh equation with two deviating arguments
 $$
 x''(t)+f(x'(t))+g_1(t,x(t-\tau_1(t)))+g_2(t,x(t-\tau_2(t)))=e(t)
 $$
 is studied. By using Leray-Schauder fixed point theorem, we obtain
 the existence of anti-periodic solutions to this equation.
 The results are illustrated with an example, which can not be
 handled using previous results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Consider the Rayleigh equation with two deviating arguments
\begin{equation}
 x''(t)+f(x'(t))+g_1(t,x(t-\tau_1(t)))+g_2(t,x(t-\tau_2(t)))=e(t),
\label{e1.1}
\end{equation}
 where $f\in C(\mathbb{R},\mathbb{R})$, $g_i\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})$,
$i=1,2$, $e,\tau_i\in C(\mathbb{R},\mathbb{R})$, $i=1,2$,
$g_i(t+T,x)=g_i(t,x)$,
$g_i(t+\frac{T}{2},-x)=-g_i(t,x)$,
$\tau_i(t+T)=\tau_i(t)$, $\tau_i(t+\frac{T}{2})=-\tau_i(t)$, $i=1,2$, and
$e(t+T)=e(t)$, $e(t+\frac{T}{2})=-e(t)$.

 The dynamic behavior of Rayleigh equation have been widely investigated
due to their applications in many fields such as physics, mechanics and the
engineering technique fields. For example, an excess voltage of
ferro-resonance known as some kind of nonlinear resonance having
long duration arises from the magnetic saturation of inductance in
an oscillating circuit of a power system, and a boosted excess
voltage can give rise to some problems in relay protection. To probe
this mechanism, a mathematical model was proposed in \cite{f1,j1,w1},
which is a special case of the Rayleigh equation with two delays.
This implies that \eqref{e1.1} can represent analog voltage
transmission. In a mechanical problem, $f$ usually represents a
damping or friction term, $g_i$ represents the restoring force,
$e$ is an externally applied force and $\tau_i$ is the time lag of
the restoring force (see \cite{b1}). Some other examples in practical
problems concerning physics and engineering technique fields can be
found in \cite{h1,k2,y1}.

Arising from problems in applied sciences, it is well-known that
anti-periodic problems of nonlinear differential equations have been
extensively studied by many authors during the past twenty years,
see \cite{a3,c4,l2,l3,l4,y2}
and references therein. For example,
anti-periodic trigonometric polynomials are important in the study
of interpolation problems \cite{d1,d4},
 and anti-periodic wavelets are discussed in \cite{c2}.
 Recently, anti-periodic boundary conditions have
been considered for the Schr\"odinger and Hill differential
operator \cite{d2,d3}. Also anti-periodic boundary conditions appear in
the study of difference equations \cite{c1,w2}. Moreover, anti-periodic
boundary conditions appear in physics in a variety of situations
\cite{a1,a2,k1}. There exist only few results for the existence of
anti-periodic solutions for Rayleigh equation and Rayleigh type
equations with and without deviating arguments in the literature.
The main difficulty lies in the middle term $f(x'(t))$ of
\eqref{e1.1}, the existence of which obstructs the usual method of
finding a priori bounds for delay Duffing or Li\'enard
equations from working. Thus, it is worthwhile to continue to
investigate the anti-periodic solutions of Rayleigh equation in this
case.


At the same time, the periodic solutions for Rayleigh equations
with two deviating arguments  have been studied
by authors \cite{l1,h2,p1}. But all the results of \cite{l1,h2,p1}
 are periodic solutions, not anti-periodic solutions. Thus, it is worth discussing
the existence of the anti-periodic solutions of Rayleigh equations
with two deviating arguments in this case.

The main purpose of this paper is to establish sufficient conditions
for the existence of anti-periodic solution of  \eqref{e1.1} by using
the Leray-Schauder fixed theorem. We remark that our methods are
different from those used in \cite{l1,h2,p1} to some degree. In particular,
one example is also given to illustrate the effectiveness of our
results.

For ease of exposition, we assume that $T>0$, and define the
following assumptions to be used in this article.
\begin{itemize}
\item[(H1)] $ f\in C(\mathbb{R},\mathbb{R})$,
$g_i\in C(\mathbb{R}^2,\mathbb{R})$,
$\tau_i\in C(\mathbb{R},\mathbb{R})$,
$i=1,2$, $e\in C(\mathbb{R},\mathbb{R})$,
$g_i(t+T,x)=g_i(t,x)$, $\tau_i(t+T)=\tau_i(t)$,
$g_i(t+\frac{T}{2},-x)=-g_i(t,x)$,
$\tau_i(t+\frac{T}{2})=-\tau_i(t)$, $i=1,2$, and
$e(t+T)=e(t)$, $e(t+\frac{T}{2})=-e(t)$.

\item[(H2)] $f(0)=0$, and there exists $\gamma>0$ such that
$xf(x)\geq \gamma |x|^2$, for all $x\in \mathbb{R}$
(or $xf(x)\leq -\gamma |x|^2$, for all $x\in \mathbb{R}$).

\item[(H3)] $g_i$ is differentiable with respect to $t$,
and there exist $a_i>0$, $b_i>0$, $i=1,2$, such that
$$
|g'_{it}(t,x)|\leq a_i+b_i|x|,\quad \forall (t,x)\in \mathbb{R}^2,\; i=1,2.
$$

\item[(H4)] There exist $l_i>0$ such that
$|g_i(t,x_1)-g_i(t,x_2)|\leq l_i|x_1-x_2|,\quad \forall t\in
\mathbb{R}, x_1,x_2\in \mathbb{R}$, $i=1,2$.

\item[(H5)] There exist  integers  $n_i$ such that
$\delta_i :=
\max_{t\in [0,T]}|\tau_i(t)-n_iT|\leq T$, $i=1,2$.

\end{itemize}

The main result in this article is the following theorem, which
will be proved in Section 3.

\begin{theorem} \label{thm1.1}
 If {\rm (H1)--(H5)} hold, and
$(b_1+b_2)\gamma^{-1}T^2+8\sqrt{2}(l_1\delta_1+l_2\delta_2)
\pi^2\gamma^{-1} <8\pi^2$, then \eqref{e1.1} has at least one
anti-periodic solution.
\end{theorem}

\section{Preliminaries}

In this section, to establish the existence of anti-periodic solutions for
\eqref{e1.1}, we provide some background definitions and some
 well-known results, which are crucial in our arguments.

Let $X$  be a real Banach space, and $A:X \to X$ be a
completely continuous operator.

\begin{definition} \label{def2.1}\rm
 Let $u:\mathbb{R}\to \mathbb{R}$ be continuous. $u(t)$
is said to be anti-periodic on $\mathbb{R}$ if
$$
u(t+T)=u(t),\quad
u(t+\frac{T}{2})=-u(t), \quad \forall t\in \mathbb{R}.
$$
\end{definition}

\begin{lemma}[Leray-Schauder Fixed point theorem \cite{g2,z1}] \label{lem2.1}
 Let $X$ be a real Banach space, and $A:X\to X$ be a completely
  continuous operator. If
$$\big\{x\in X: x=\lambda Ax,\; 0<\lambda  <1\big\}
$$
is bounded, then $A$ has a fixed point $x^{*}\in\Omega$, where
$$
\Omega=\big\{x\in X: \|x\|\leq l\big\},\quad
l=\sup\big\{ x\in X: x=\lambda Ax,\; 0<\lambda  <1\big\}.
$$
\end{lemma}

\begin{lemma}[Wirtinger inequality \cite{m1}] \label{lem2.2}
Suppose that $x(t)\in C^1(\mathbb{R},\mathbb{R}), x $ is $T$-periodic and
 $\int_0^T x(t)dt=0$.
Then $\int_0^T |x(t)|^2dt\leq
\frac{T^2}{4\pi^2}\int_0^T |x'(t)|^2dt$.
\end{lemma}

\begin{lemma}[\cite{g1}] \label{lem2.3}
Let $0\leq \alpha\leq T$ be constant,
 $s\in C(\mathbb{R},\mathbb{R})$ be periodic with period $T$, and
 $\max_{t\in [0,T]}|s(t)|\leq \alpha$. Then for any
$u\in C^1(\mathbb{R},\mathbb{R})$ which is periodic with period $T$, we have
$$
\int_0^T |u(t)-u(t-s(t))|^2dt
\leq 2\alpha^2\int_0^T |u'(t)|^2dt.
$$
\end{lemma}

\section{Proof of Theorem \ref{thm1.1}}

In this section, we will use  Lemma \ref{lem2.1} to prove Theorem \ref{thm1.1}.
Let
\begin{gather*}
X=\big\{x\in C(\mathbb{R},\mathbb{R}): x(t+T)=x(t),\;
x(t+\frac{T}{2})=-x(t)\big\},\\
Y=\big\{x\in C^1(\mathbb{R},\mathbb{R}): x(t+T)=x(t),\;
 x(t+\frac{T}{2})=-x(t)\big\}.
\end{gather*}
Then $X$ and $Y$ are real Banach space endowed with the norms
$$
\|x\|_{\infty}=\max_{t\in[0,T]}|x(t)|\quad\text{and}\quad
\|x\|=\|x\|_{\infty}+\|x'\|_{\infty},
$$
respectively.

Choosing $m>0$ with
 $m\neq(\frac{2k\pi}{T})^2$ $(k=1,2,\dots)$, then equation
$$
x''(t)+mx(t)=0
$$
has only the trivial solution in $Y$.
 In fact, it is easy to see
the general solution of $x''(t)+mx(t)=0$ is
$$
x(t)=c_1\sin\sqrt{m}t+c_2\cos\sqrt{m}t.
$$
By the periodic properties we obtain that $x=0$ is its unique
solution in $Y$.
Then for $ h\in X$,
$$
-x''(t)-mx(t)=h(t)
$$
has unique solution $x\in Y$. Writing $x=Kh$, then $K:X\to
Y$ is a completely continuous operator.

 Define an operator $G:Y\to X$ by
$$
(Gx)(t)=f(x'(t))+g_1(t,x(t-\tau_1(t)))+g_2(t,x(t-\tau_2(t)))-mx(t)-e(t),x\in Y.
$$
Then $G:Y\to X$ is continuous and bounded. Let $A=KG:Y\to Y$.
Then $A$ is also a completely continuous operator.
By Lemma \ref{lem2.1}, if
$$
\big\{x\in Y: x=\lambda Ax,\; 0<\lambda <1 \big\}
$$
is bounded in $Y$, then $A$ has a
fixed point in $Y$. Thus \eqref{e1.1} has anti-periodic solution.

Now suppose that $x\in Y$, $0<\lambda<1$ satisfying $x=\lambda Ax$.
Then $x(t)$ is a solution of
\begin{equation}
x''(t)+\lambda f(x'(t))+\lambda g_1(t,x(t-\tau_1(t))) +\lambda
g_2(t,x(t-\tau_2(t)))+(1-\lambda)mx(t)=\lambda e(t),
\label{e3.1}
\end{equation}
and $x(t)$ satisfies
$$
\int_0^T x(t)dt=\int_0^{T/2}x(t)dt
 +\int_{\frac{T}{2}}^T x(t)dt=\int_0^{T/2}x(t)dt
+\int_0^{T/2}x(t+\frac{T}{2})dt=0.
$$
Thus, there exists $\xi\in [0,T]$ such that $x(\xi)=0$. So we have
$$
|x(t)|=|x(\xi)+\int_{\xi}^{t}x'(s)ds|\leq \sqrt{T}\|x'\|_{L^2}.
$$
  Then
$$
\|x\|_{\infty}\leq \sqrt{T}\|x'\|_{L^2},
$$
where $\|\cdot\|_{L^2}$ is the norm of $L^2[0,T]$.

Multiplying \eqref{e3.1} by $x'(t)$ and integrating from $0$ to $T$,  we have
\begin{equation}
\begin{aligned}
\lambda\int_0^T f(x'(t))x'(t)dt
& =- \lambda\int_0^T g_1(t,x(t-\tau_1(t)))x'(t)dt\\
&\quad -\lambda\int_0^T g_2(t,x(t-\tau_2(t)))x'(t)dt+
\lambda\int_0^T e(t)x'(t)dt.
\end{aligned} \label{e3.2}
\end{equation}
By (H2), we know that
\begin{equation}
\int_0^T f(x'(t))x'(t)dt\geq
\gamma\int_0^T |x'(t)|^2dt.\label{e3.3}
\end{equation}
By H\"older's inequality, from \eqref{e3.2} and \eqref{e3.3}, we have
\begin{equation}
\begin{aligned}
&\gamma\int_0^T x'^2(t)dt\\
&\leq |\int_0^T g_1(t,x(t-\tau_1(t)))x'(t)dt|
+|\int_0^T g_2(t,x(t-\tau_2(t)))x'(t)dt|+\|e\|_{L^2}\|x'\|_{L^2}
\\
& \leq \int_0^T |g_1(t,x(t-\tau_1(t)))-g_1(t,x(t))\|x'(t)|dt
 +|\int_0^T g_1(t,x(t))x'(t)dt|
\\
&\quad +\int_0^T |g_2(t,x(t-\tau_2(t)))-g_2(t,x(t))\|x'(t)|dt
\\
&\quad +|\int_0^T g_2(t,x(t))x'(t)dt|+\|e\|_{L^2}\|x'\|_{L^2}.
\end{aligned}\label{e3.4}
\end{equation}
Since the functions $\int_0^{x(t)}g_i(t,v)dv$, $i=1,2$ are
$T$-periodic , differentiable and
$$
\frac{d}{dt}\int_0^{x(t)}g_i(t,v)dv
=g_i(t,x(t))x'(t) +\int_0^{x(t)}g'_{it}(t,v)dv,\quad i=1,2,
$$
we have
\begin{equation}
\int_0^T g_i(t,x(t))x'(t)dt=-\int_0^T dt\int_0^{x(t)}g'_{it}(t,v)dv,\quad
 i=1,2.\label{e3.5}
\end{equation}
Combining \eqref{e3.4} and \eqref{e3.5} with (H3)  and (H4) we obtain
\begin{equation}
\begin{aligned}
&\gamma\int_0^T x'^2(t)dt\\
&\leq l_1\int_0^T |x(t)-x(t-\tau_1(t))\|x'(t)|dt
 +l_2\int_0^T |x(t)-x(t-\tau_2(t))\|x'(t)|dt\\
&\quad +\int_0^T dt\int_0^{|x(t)|}
 (a_1+b_1|v|)dv+\int_0^T dt\int_0^{|x(t)|}
 (a_2+b_2|v|)dv+\|e\|_{L^2}\|x'\|_{L^2}\\
&\leq l_1\|x'\|_{L^2}\big(\int_0^T |x(t)- x(t-\tau_1(t)-n_1T)|
 ^2dt\big)^{1/2}\\
&\quad +l_2\|x'\|_{L^2}\big(\int_0^T |x(t)-x(t-\tau_2(t)-n_2T)|
 ^2dt\big)^{1/2}\\
&\quad +(a_1+a_2)\int_0^T |x(t)|dt
 +\frac{b_1+b_2}{2}\int_0^T |x(t)|^2dt+\|e\|_{L^2}\|x'\|_{L^2}.
\end{aligned} \label{e3.6}
\end{equation}
By Lemma \ref{lem2.1} we have
\begin{equation}
\int_0^T |x(t)|^2dt\leq \frac{T^2}{4\pi^2}\|x'\|_{L^2}^2.\label{e3.7}
\end{equation}
By (H5) and  Lemma \ref{lem2.2}, we have
\begin{equation}
\Big(\int_0^T |x(t)-x(t-\tau_i(t)-n_iT)| ^2dt\Big)^{1/2}
\leq \sqrt{2}\delta_i
 \Big(\int_0^T |x'(t)|^2dt\Big)^{1/2},\quad i=1,2.
\label{e3.8}
\end{equation}
By H\"older's inequality and \eqref{e3.7}, we have
\begin{equation}
\int_0^T |x(t)|dt\leq
\sqrt{T}(\int_0^T |x(t)|^2dt)^{1/2}\leq
\sqrt{T}\frac{T}{2\pi}(\int_0^T |x'(t)|^2dt)^{1/2}
=\frac{T^{3/2}}{2\pi}\|x'\|_{L^2}.\label{e3.9}
\end{equation}
Thus, it follows from \eqref{e3.6},\eqref{e3.7}  \eqref{e3.8} and \eqref{e3.9}
that
\begin{align*}
\gamma \|x'\|^2_{L^2}
&\leq \sqrt{2}(l_1\delta_1+l_2\delta_2) \|x'\|^2_{L^2}+
\frac{(a_1+a_2)T^{3/2}}{2\pi} \|x'\|_{L^2}\\
&\quad +\frac{(b_1+b_2)T^{2}}{8\pi^2}\|x'\|^2_{L^2}
+\|e\|_{L^2}\|x'\|_{L^2}.
\end{align*}
Combining this with
$(b_1+b_2)\gamma^{-1}T^2+8\sqrt{2}(l_1\delta_1+l_2\delta_2)
\pi^2\gamma^{-1} <8\pi^2$, we know that there exists $c_1$ such
that $\|x'\|_{L^2}\leq c_1$. Then
\begin{equation}
\|x\|_{\infty}\leq \sqrt{T}c_1:= M_1.\label{e3.10}
\end{equation}
Multiplying \eqref{e3.1} by $x''(t) $ and integrating from $0$ to $T$,
we have
\begin{align*}
\|x''\|^2_{L^2}
&\leq|-\lambda
\int_0^T g_1(t,x(t-\tau_1(t)))x''(t)dt-\lambda
\int_0^T g_2(t,x(t-\tau_2(t)))x''(t)dt\\
&\quad -(1-\lambda)m\int_0^T x(t)x''(t)dt+\lambda
\int_0^T e(t)x''(t)dt|\\
&\leq (g_{1M_1}+g_{2M_1})\sqrt{T}\|x''\|_{L^2}
 +mM_1\sqrt{T}\|x''\|_{L^2}+\|e\|_{L^2}\|x''\|_{L^2},
\end{align*}
where
$$
g_{1M_1}=\max_{t\in[0,T],\|x\|_{\infty}\leq
M_1}|g_1(t,x(t))|,\quad
g_{2M_1}=\max_{t\in[0,T],\|x\|_{\infty}\leq
M_1}|g_2(t,x(t))|.
$$
Thus
$$
\|x''\|_{L^2}
\leq (g_{1M_1}+g_{2M_1})\sqrt{T}+mM_1\sqrt{T}+\|e\|_{L^2}
 := M_2.
$$
Selecting $\eta\in[0,T]$ such that $x'(\eta)=0$, we
have
\begin{equation}
|x'(t)|\leq \int_0^T |x''(t)|dt\leq \sqrt{T}M_2.\label{e3.11}
\end{equation}
Thus from \eqref{e3.10} and \eqref{e3.11}, we know that
$\|x\|\leq M_1+ \sqrt{T}M_2:= M$.
It is following that
$$
\big\{x\in Y: x=\lambda Ax,\; 0<\lambda <1\big\}
$$
is bounded.
Therefore, by Lemma \ref{lem2.1}, we obtain that $A$ has a fixed point
$x^{*}\in\Omega$, where
$\Omega=\{x\in Y: \|x\|\leq M\}$.
Therefore,  \eqref{e1.1} has an anti-periodic solution.


 \section*{An example}

In this section, we give one example to demonstrate the results
obtained in previous sections.
Consider the forced Rayleigh-type  equation with period $2\pi$,
\begin{equation}
x''(t)+f(x'(t))+g_1(t,x(t-\tau_1(t)))+g_2(t,x(t-\tau_2(t)))=e(t),\label{e4.1}
\end{equation}
 where
\begin{equation}
f(x)= \begin{cases} e^{x}-1,\quad  x\geq 0,\\
1-e^{-x},\quad x\leq 0,
 \end{cases}  \label{e4.2}
\end{equation}
and
\begin{equation}
\begin{gathered}
g_1(t,x)=\frac{1}{9}\sin^2 (t) x(t-\theta\cos t)+\cos t,\\
g_2(t,x)=\frac{1}{9}\cos^2 (t) x(t-\theta\sin t)+\sin t,\\
e(t)=\sin t,\quad \tau_1(t)=\theta\cos t,\quad \tau_2(t)=\theta\sin t,\quad
\theta\in(0,1).
\end{gathered} \label{e4.3}
\end{equation}
Then \eqref{e4.1} has at least one anti-periodic solution with period $2\pi$.

 By \eqref{e4.2} and \eqref{e4.3}, it is not difficult to see that
condition (H1) holds, $T=2\pi$, $|f(0)|=0$, 
\begin{gather*}
|g'_{1t}(t,x)|=|\frac{1}{9}x\sin (2 t)-\sin t|\leq \frac{1}{9}|x|+1,\quad
\forall (t,x)\in \mathbb{R}^2,\\
|g'_{2t}(t,x)|=|-\frac{1}{9}x\sin (2 t)+\cos
t|\leq \frac{1}{9}|x|+1,\quad \forall (t,x)\in \mathbb{R}^2.
\end{gather*}
On the other hand, let $\gamma=1$, $\delta_i=\theta$,
$l_i=\frac{1}{9}$, $b_i=\frac{1}{9}$, $i=1,2$. If 
$\theta\in (0,\frac{\sqrt{2}}{2})$, then
$xf(x)\geq |x|^2$, for all $x\in \mathbb{R}$,
$$
(b_1+b_2)\gamma^{-1}T^2+8\sqrt{2}(l_1\delta_1+l_2\delta_2)
\pi^2\gamma^{-1} <8\pi^2.
$$
Hence, (H1)--(H5) are satisfied. Thus, by Theorem \ref{thm1.1}, Equation
\eqref{e4.1} has at least one anti-periodic
solution with period $2\pi$.


\subsection*{Acknowledgements} 
We would like to express our gratitude to the
anonymous referees for their very valuable observations that have
greatly improved this article.

This work is sponsored by the project NSFC (11161022, 11171032)
 and by the Fundamental Research Funds for the Central Universities (11ML30).

\begin{thebibliography}{99}

\bibitem{a1} A. Abdurahman, F. Anton, J. Bordes;
\emph{Half-string oscillator approach to string field theory
 (Ghost sector: I)}, Nuclear Phys. B 397 (1993) 260-282.

\bibitem{a2} C. Ahn, C. Rim;
\emph{Boundary flows in general coset theories},
 J. Phys. A 32 (1999) 2509-2525.

\bibitem{a3} S. Aizicovici, M. McKibben, S. Reich;
\emph{Anti-periodic solutions to nonmonotone evolution equations
 with discontinuous nonlinearities},
  Nonlinear Anal. 43 (2001) 233-251.

 \bibitem{b1} T. A. Burton;
\emph{Stability and Periodic Solutions  of Ordinary and Functional Differential
 Equations}, Academic Press, Orland, FL, 1985.

\bibitem{c1} A. Cabada, D. R. Vivero;
\emph{Existence and uniqueness of solutions of
 higher-order antiperiodic dynamic equations}, 
Adv. Difference Equ. 4 (2004) 291-310.

\bibitem{c2} H. L. Chen;
\emph{Antiperiodic wavelets}, J. Comput. Math. 14 (1996) 32-39.

\bibitem{c4} Y. Chen, J. J. Nieto, D. O'Regan;
\emph{Anti-periodic solutions for fully nonlinear first-order
differential equations}, Math. Comput. Modelling
 46 (2007) 1183-1190.

\bibitem{d1} F. J. Delvos, L. Knoche;
\emph{Lacunary interpolation by antiperiodic
 trigonometric polynomials}, BIT 39 (1999) 439-450.

\bibitem{d2} P. Djiakov, B. Mityagin;
\emph{Spectral gaps of  the periodic Schrodinger operator when its
 potential is an entire function},
 Adv. Appl. Math. 31 (2003) 562-596.

\bibitem{d3} P. Djiakov, B. Mityagin;
\emph{Simple and double eigenvalues of the Hill operator
 with a two-term potential}, J. Approx. Theory 135 (2005) 70-104.

\bibitem{d4} J. Y. Du, H. L. Han, G. X. Jin;
\emph{On trigonometric and paratrigonometric  Hermite interpolation},
J. Approx. Theory 131 (2004) 74-99.

\bibitem{f1} C. H. Feng;
\emph{On the existence and uniqueness of almost periodic
 solutions for some delay differential equation appearing in a power system},
  Acta. Math. Sin. 46 (2003) 932-936 (in Chinese).

\bibitem{g1} S. Lu, W. Ge;
\emph{Periodic solutions for a kind of
second order differential equations with multiple with deviating
arguments}, Appl. Math. Comput. 146 (2003) 195-209.

\bibitem{g2} D. Guo;
\emph{Nonlinear functional analysis}, Shandong
 Science and Technology Press, Ji Nan, 1985 (in Chinese).

\bibitem{h1} J. K. Hale;
\emph{Theory of Functional Differential Equations},
  Springer-Verlag, New York, 1977.

\bibitem{h2} C. Huang, Y. He, L. Huang, W. Tan;
\emph{New results on the periodic solutions for a kind of Rayleigh
equation with two deviating arguments},
Math. Comput. Modelling 46 (2007) 604-611.

\bibitem{j1} G. J. Ji, Z. X. Wang, D. W. Lai;
\emph{On the existence of periodic
 solutions of overvoltage model in power system},
 Acta Math. Sci. 16 (1996) 99-104 (in Chinese).

\bibitem{k1} H. Kleinert, A. Chervyakov;
\emph{Functional determinants from Wronski Green function},
J. Math. Phys. 40 (1999) 6044-6051.

 \bibitem{k2} Y. Kuang;
\emph{Delay Differential Equations with Applications
 in Population Dynamics}, Academic Press, New York, 1993.

\bibitem{l1} B. Liu;
\emph{Existence and uniqueness of periodic solutions for a kind of
Rayleigh equation with two deviating arguments},
 Comput. Math. Appl. 55 (2008) 2108-2117.

\bibitem{l2} B. Liu;
\emph{Anti-periodic solutions for forced Rayleigh-type
equations}, Nonlinear Anal. RWA. 10 (2009) 2850-2856.

\bibitem{l3} Y. Li, T. Zhang;
\emph{Existence and uniqueness of anti-periodic  solution for a kind of forced
Rayleigh equation with state dependent delay and impulses}, Commun
Nonlinear Sci Numer Simulat 15 (2010) 4076-4083.

\bibitem{l4} X. Lv, P. Yan, D. Liu;
\emph{Anti-periodic solutions for a class of nonlinear second-order Rayleigh
equations with delays}, Commun Nonlinear Sci Numer Simulat
 15 (2010) 3593-3598.

\bibitem{m1} J. Mawhin, M. Willem;
\emph{Critical point theory and Hamiltonian systems}, 
Application of mathematical science, vol.74.
Springer-Verlag, New York, 1989.

\bibitem{p1} L. Peng, B. Liu, Q. Zhou, L. Huang;
\emph{Periodic solutions for a kind of Rayleigh equation with two
  deviating arguments}, J. Franklin Inst. 7 (2006) 676-687.

\bibitem{w1} Z. X. Wang, D. W. Lai;
\emph{A delay differential equation appeared in the study of
 overvoltage}, Report to the Italian Symposium, 1984, 12.

\bibitem{w2} Y. Wang, Y. M. Shi;
\emph{Eigenvalues of second-order  difference equations with periodic
and antiperiodic boundary conditions},
  J. Math. Anal. Appl. 309 (2005) 56-69.

\bibitem{y1} T. Yashizaw;
\emph{Asymptotic behavior of solutions of  differential equations}, 
in: Differential Equation: Qualitative Theory (Szeged, 1984),
in: Colloq. Math. Soc. J\'anos Bolyai, vol. 47, North-Holland, Amsterdam,
1987, pp. 1141-1172.

\bibitem{y2} Y. Yue, J. Shao, G. Yue;
\emph{Existence and uniqueness of anti-periodic solutions for
a kind of Rayleigh equation with two deviating arguments}, Nonlinear
Anal. 71 (2009) 4689-4695.

\bibitem{z1} E. Zeidler;
\emph{Nonlinear functional analysis and  its applications
 I: fixed-point theorems},
 Springer-Verla Inc., New York, 1986.

\end{thebibliography}

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