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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 100, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2010/100\hfil Periodic solutions]
{Periodic solutions for a second-order neutral differential
equation with variable parameter and multiple deviating arguments}

\author[B. Du, X. Wang\hfil EJDE-2010/100\hfilneg]
{Bo Du, Xiaojing Wang}  % in alphabetical order

\address{Department of Mathematics, Huaiyin Normal University\\
 Huaian Jiangsu, 223300, China}
\email[Bo Du]{dubo7307@163.com} 
\email[Xiaojing Wang]{wwxxjj@126.com}

\thanks{Submitted April 12, 2010. Published July 21, 2010.}
\subjclass[2000]{34B15, 34B13}
\keywords{Mawhin's continuation theorem; periodic
solution; neutral; \hfil\break\indent variable parameter}

\begin{abstract}
 By employing the continuation theorem of coincidence
 degree theory developed by Mawhin, we obtain periodic solution
 for a class of neutral differential equation with variable
 parameter and multiple deviating arguments.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}{Example}[section]
\allowdisplaybreaks

\section{Introduction}

Neutral functional differential equations (in short NFDEs) are an
important research subject of functional differential
equations and provide good models in many fields including
physics, mechanics, biology and economics (see
\cite{cor,bab,dix,kuang,peng}). With
 such clear indications of the importance of
NFDEs in the applications, it is not surprising that the subject has
undergone a rapid development in the previous twenty years.
Particularly, in recent years the problems of periodic solution for
second-order NFDEs have been studied by many authors. In \cite{lu1},
by employing the continuation theorem of coincidence degree theory,
Lu and Ge studied the following second-order NFDE:
$$
(x(t)+cx(t-r))''+f(x'(t))+g(x(t-\tau(t)))=p(t).
$$
After that, Lu and Gui \cite{lu2} went still one step further to
study the above equation in the critical case and obtained more
profound results. Furthermore, in \cite{lu3} Lu and Ren investigated
the second-order NFDE with multiple deviating arguments as follows:
$$
\frac{d^2}{dt^2}(u(t)-ku(t-\tau))=f(u(t))u'(t)+\alpha(t)g(u(t))
+\Sigma_{j=1}^{n}\beta_j(t)g(u(t-\gamma_j(t)))+p(t).
$$
The authors used new techniques and methods for multiple deviating
arguments and obtained some new results. In very recent years,
$p$-Laplacian NFDEs were studied by some researchers. In
\cite{zhu1}-\cite{zhu2}, Zhu and Lu studied the following
$p$-Laplacian NFDEs:
$$
\left(\varphi_p[(x(t)-cx(t-\sigma))']\right)'+g(t,x(t-\tau(t)))=e(t)
$$
and
$$
\left(\varphi_p[(x(t)-cx(t-\sigma))']\right)'
=f(x(t))x'(t)+\Sigma_{j=1}^{n}\beta_j(t)g(x(t-\gamma_j(t)))+p(t).
$$
However, for all the above papers they obtained the existence of
periodic solution to NFDEs based on the properties of neutral
operator $A$. In 1995, Zhang \cite{zhang} obtained the following
results. Define $A$ on $C_T$
$$
A:C_T\to C_T, [Ax](t)=x(t)-cx(t-\tau),\forall t\in \mathbb{R},
$$
where $C_T=\{x:x\in C(\mathbb{R},\mathbb{R}),x(t+T)\equiv x(t)\}$,
$c$ is constant. When $|c|\neq 1$, then $A$ has a unique continuous
bounded inverse $A^{-1}$ satisfying
$$
[A^{-1}f](t)= \begin{cases}
\sum _{j\geq 0}c^{j}f(t-j\tau),&\text{if }|c|< 1,\;\forall
f\in C_T, \\
 -\sum _{j\geq 1}c^{-j}f(t+j\tau),&\text{if }|c|>1,\;\forall f\in C_T.
 \end{cases}
$$
Obviously, we have
\begin{itemize}
\item[(1)] $\|A^{-1}\|\le \frac{1}{|1-|c\|}$;
\item[(2)] $\int_0^T|[A^{-1}f](t)|dt\le\frac{1}{|1-|c\|}
\int_0^T|f(t)|dt,\forall f\in C_T$;
\item[(3)]
$\int_0^T|[A^{-1}f](t)|^2dt\le\frac{1}{|1-|c\|}
 \int_0^T|f(t)|^2dt,\forall f\in
C_T$.
\end{itemize}
When $c$ is a variable $c(t)$, we have obtained the properties
of the neutral operator $A:C_{T}\to C_{T}$,
$[Ax](t)=x(t)-c(t)x(t-\tau)$ in \cite{du}. We note that there
are few results on the existence of periodic solutions to
second-order neutral equations for the cases of a variable $c(t)$.
The purpose of this article is to investigate the existence of
periodic solution for the second-order NFDE with variable parameter
and multiple deviating arguments by using the properties of the
operator $A$ in \cite{du} and Mawhin's continuation theorem. Here we
use the same technique, but our results extend and complement the
existing ones. We will study the following NFDE:
\begin{equation}
(x(t)-c(t)x(t-\tau))''+\sum_{j=1}^n\beta_{j}(t)g(x(t-\gamma_j(t)))
=e(t),\label{e1}
\end{equation}
where $g\in C(\mathbb{R},\mathbb{R})$;
$c\in C^2(\mathbb{R},\mathbb{R})$ with $c(t)=c(t+T)$ and
$|c(t)|\neq1$; $e(t),\beta_{j}(t),\gamma_j(t)$
are $T-$periodic functions on $\mathbb{R}$ ($j=1,2,\dots,n$);
$\tau,T>0$ are given constants.

 In this article, we assume that $e(t)$ is not a constant
function on $\mathbb{R}$. Furthermore, we suppose that
$\gamma_j\in C^1(\mathbb{R},\mathbb{R})$ with
$\gamma_j'(t)<1, \forall t\in \mathbb{R}, (j=1,2,\dots,n)$.
It is obvious that the function
$t-\gamma_j(t)$ has a unique inverse denoted by
$\mu_j(t)$, ($j=1,2,\dots,n$). Let
$$
\Gamma(t)=\Sigma_{j=1}^n\frac{\beta_{j}(\mu_j(t)}{1-\gamma_j'(\mu_j(t))},
\quad
 \bar{h}=\frac{1}{T}\int_0^Th(s)ds.
$$

\section {Preliminary}

In this section, we give some  lemmas which will be used in this
paper.

\begin{lemma}[\cite{du}]\label{lem1}
If $|c(t)|\neq1$, then operator $A$  has continuous inverse
 $A^{-1}$ on $C_T$, satisfying:
(1)
$$
[A^{-1}f](t)= \begin{cases}
f(t)+\sum _{j= 1}^{\infty}\prod_{i=1}^{j}c(t-(i-1)\tau)f(t-j\tau),
&c_{0}< 1, \forall f\in C_{T} , \\
 -\frac{f(t+\tau)}{c(t+\tau)}-\sum _{j=
1}^{\infty}\prod_{i=1}^{j+1}\frac{1}{c(t+i\tau)}f(t+j\tau+\tau),&
 \sigma> 1, \forall f\in C_{T},
 \end{cases}
$$
(2)
$$
\int_{0}^{T}|[A^{-1}f](t)|dt
\leq \begin{cases}
 \frac{1}{1-c_{0} }\int_{0}^{T}|f(t)|dt, &c_{0}< 1 , \forall f\in C_{T},\\
 \frac{1}{\sigma-1}\int_{0}^{T}|f(t)|dt, &\sigma> 1 , \forall f\in C_{T},
  \end{cases}
$$
where
$$
c_{0}=\max_{t\in[0,T]}|c(t)|, \quad
\sigma=\min_{t\in[0,T]}|c(t)|, \quad
 c_{1}=\max_{t\in[0,T]}|c'(t)|.
$$
\end{lemma}

 Let $X$ and $Y$ be two Banach spaces and let $L:D(L)\subset
X\to Y$ be a a linear operator, Fredholm operator with index
zero (meaning that $\operatorname{Im}L$ is closed in $Y$ and
$dim\ker L=codim\operatorname{Im}L<+\infty$. If $L$ is  a Fredholm operator with
index zero, then there exist continuous projectors $P:X\to
X, Q:Y\to Y$ such that
$\operatorname{Im}P=\ker L, \operatorname{Im}L
=\ker Q=\operatorname{Im}(I-Q)$ and
$L_{D(L)\cap \ker P}:(I-P)X\to \operatorname{Im}L$ is invertible.
Denote by
$K_p$ the inverse of $L_P$.

 Let $\Omega$ be an open bounded
subset of $X$, a map $N :\bar{\Omega}\to Y$ is said to be
$L$-compact in $\bar{\Omega}$ if $QN(\bar{\Omega})$ is bounded and
the operator $K_p(I-Q)N(\bar{\Omega})$ is relatively compact. We
first give the famous Mawhin's continuation theorem.

\begin{lemma}[\cite{ma}]\label{lem2}
Suppose that $X$ and $Y$ are Banach spaces, and $L:D(L)\subset
X\to Y$, is a Fredholm operator with index zero.
Furthermore, $\Omega\subset X$ is an open bounded set and
$N:\bar{\Omega}\to Y$ is $L$-compact on $\bar{\Omega}$. if
all the following conditions hold:
\begin{itemize}
    \item[(1)] $Lx\neq\lambda Nx$, for all $x\in\partial\Omega\cap D(L)$,
     and all $\lambda\in(0,1)$,
    \item[(2)] $Nx\notin \operatorname{Im}L$, for all $x\in\partial\Omega\cap \ker L$,
    \item[(3)]$ \deg\{QN,\Omega\cap \ker L,0\}\neq 0$,
\end{itemize}
Then the equation $Lx=Nx$ has a solution on $\bar{\Omega}\cap D(L)$.
\end{lemma}

Define the linear operator
$L:D(L)\subset C_{T}\to C_{T}$ as $Lx=(Ax)''$,
and a nonlinear operator
$N:C_{T}\to C_{T}$,
$$
Nx=-\sum_{j=1}^n\beta_{j}(t)g(x(t-\gamma_j(t)))+e(t),
$$
where $D(L)=\{x|x\in C_{T}^1\}$. For $x\in \ker L$, we have
$(x(t)-c(t)x(t-\tau))''=0$.
Then
$$
x(t)-c(t)x(t-\tau)=\tilde{c_1}t+\tilde{c_2},
$$
where $\tilde{c_1},  \tilde{c_2}\in \mathbb{R}$.
Since $x(t)-c(t)x(t-\tau)\in C_T$, then $\tilde{c_1}=0$. Let
$\varphi(t)$ be a solution of $x(t)-c(t)x(t-\tau)=1$ and
$\int_0^T\varphi^2(t)dt\neq0$. We get
$$
\ker L=\{a_0\varphi(t), a_0\in \mathbb{R}\},
\operatorname{Im}L=\{y|y\in C_{T},
\int_{0}^{T}y(s)ds=0\}.
$$
Obviously, $\operatorname{Im}L$
is a closed in $C_{T}$ and $\dim\ker L=\operatorname{codim\,Im}L=1$,
So $L$ is a Fredholm operator with index zero. Define continuous
projectors $P, Q$
\begin{gather*}
 P:C_{T}\to \ker L, \quad
 (Px)(t)=\frac{\int_0^Tx(t)\varphi(t)dt}{\int_0^T\varphi^2(t)dt}
 \varphi(t),\\
Q:C_{T}\to C_{T}/\operatorname{Im}L, \quad
Qy=\frac{1}{T} \int_{0}^{T}y(s)ds.
\end{gather*}
Let
$$
L_{P}=L|_{D(L)\cap \ker P}:D(L)\cap \ker P\to \operatorname{Im}L,
$$
then
 $$
L_{P}^{-1}=K_p:\operatorname{Im}L\to D(L)\cap \ker P.
$$
Since $\operatorname{Im}L\subset C_T$ and $D(L)\cap \ker P\subset
C_T^1$, so $K_p$ is an embedding operator. Hence  $K_p$ is a
completely operator in $\operatorname{Im}L$. By the definitions
of $Q$ and $N$, it follows that $QN(\bar{\Omega})$ is bounded
on $\bar{\Omega}$.
Hence nonlinear operator $N$ is $L$-compact on $\overline{\Omega}$.

\section{Existence of periodic solution for \eqref{e1}}

For convenience when applying Lemma 2.1 and Lemma 2.2, we
introduce some notation and sate some assumptions:
\begin{gather*}
C_{T}=\{x\in C(\mathbb{R},\mathbb{R}): x(t+T)= x(t), \;
 \forall t\in \mathbb{R}\},\\
|\varphi|_{0}=\max_{t\in[0,T]}|\varphi(t)|,  \quad \forall
\varphi \in C_{T}, \\
C_{T}^{1}=\{x\in C^{1}(\mathbb{R},\mathbb{R}): x(t+T)= x(t),\;
  \forall t\in \mathbb{R}\},\\
\|\varphi\|=\max_{t\in[0,T]}{\{|\varphi|_{0}, |\varphi'|_{0}}\}, \quad
 \forall\varphi \in C_{T}^1,
\end{gather*}
where $|\cdot|_{0}$ and $\|\cdot\|$ are the norms of
$C_{T}$ and $C_{T}^{1}$ respectively. Obviously, $C_T,C_T^1$ are
both Banach space.
\begin{itemize}
\item[(H1)] $\Gamma(t)>0$, for all $t\in \mathbb{R}$;
\item[(H2)] $\lim_{|x|\to+\infty}\frac{|g(x)|}{|x|}\leq r\in[0,\infty)$;
\item[(H3)] There exists a positive constant $d$ such that $xg(x)>0$,
whenever $|x|>d$.
\end{itemize}

\begin{theorem}\label{thm1}
Suppose that $\int_0^Te(s)ds=0$, $\int_0^T\varphi^2(s)ds\neq0 $,
$|c(t)|\neq1$ for all $t\in \mathbb{R}$, and assumptions
{\rm(H1)--(H3)} hold, where
$\varphi(t)$ is a solution of $x(t)-c(t)x(t-\tau)=1$.
Then \eqref{e1} has at least one $T$-periodic solution, if
$$
\frac{T^{1/2}}{1-c_0}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)r}
+\frac{c_1T}{1-c_0}<1  \quad\text{for } c_0<\frac{1}{2},
$$
or if
$$
\frac{T^{1/2}}{\sigma-1}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)r}
+\frac{c_1T}{\sigma-1}<1  \quad\text{for }\sigma>1.
$$
\end{theorem}

\begin{proof} Take $\Omega_1=\{x\in D(L): Lx=\lambda
Nx,\lambda\in(0,1)\}$. For $x\in \Omega_1 $, we have
\begin{equation} \label{e2}
(x(t)-c(t)x(t-\tau))''+\lambda\sum_{j=1}^n\beta_{j}(t)
g(x(t-\gamma_j(t)))=\lambda e(t).
\end{equation}
We claim that there exists a point $\xi\in \mathbb{R}$ such that
\begin{equation}
|x(\xi)|\leq d,\label{e3}
\end{equation}
where $d$ is a constant which is independent with $\lambda$.
Integrating two sides of \eqref{e2} over the interval $[0,T]$,
$$
\sum_{j=1}^n\int_0^T\beta_{j}(t)g(x(t-\gamma_j(t)))dt=0;
$$
i.e.,
$$
\int_0^T\Gamma(t)g(x(t))dt=0.
$$
By  mean value theorem for integrals, there exists a point
$\xi_1\in [0,T]$ such that
$$
g(x(\xi_1))\bar{\Gamma}T=0.
$$
By $\bar{\Gamma}\neq0$, then $g(x(\xi_1))=0$.  From the assumption
(H3), the inequality \eqref{e3} holds. Furthermore we have
\begin{equation}
|x(t)|\leq d+\int_0^T|x'(t)|dt. \label{e4}
\end{equation}
 From the conditions
$$
\frac{T^{1/2}}{1-c_0}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)r}
+\frac{c_1T}{1-c_0}<1,
$$
and
$$
\frac{T^{1/2}}{\sigma-1}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)r}
+\frac{c_1T}{\sigma-1}<1,
$$
there exists a constant $\varepsilon_1>0$ such that
\begin{equation}
\frac{T^{1/2}}{1-c_0}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)
(r+\varepsilon_1)}+\frac{c_1T}{1-c_0}<1, \label{e5}
\end{equation}
or
\begin{equation}
\frac{T^{1/2}}{\sigma-1}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)
(r+\varepsilon_1)}+\frac{c_1T}{\sigma-1}<1.\label{e6}
\end{equation}
For such a constant $\varepsilon_1$, by (H2), there exists a
constant $\rho>0$  such that
\begin{equation}
|g(u)|\leq (r +\varepsilon_1)|u|, \quad  |u|>\rho>d.\label{e7}
\end{equation}
Let
$$
E_{1j}=\{t|t\in[0,T],|x(t-\gamma_j(t))|\leq\rho\},\quad
E_{2j}=\{t|t\in[0,T],|x(t-\gamma_j(t))|>\rho\},
$$
for $j=1,2\dots,n$.
Multiplying both sides of \eqref{e2} by $(Ax)(t)$ and integrating
over $[0,T]$, from \eqref{e4} and \eqref{e7}, we obtain
\begin{equation}
\begin{aligned}
&\int_0^T|(Ax)'(t)|^2dt\\
&=\lambda\int_0^T\sum_{j=1}^n\beta_{j}(t)g(x(t-\gamma_j(t)))(Ax)(t)dt-\lambda\int_0^Te(t)(Ax)(t)dt\\
&\leq |Ax|_0\int_{E_{1j}}\sum_{j=1}^n|\beta_{j}(t)\|g(x(t-\gamma_j(t)))|dt\\
&\quad+|Ax|_0\int_{E_{2j}}\sum_{j=1}^n|\beta_{j}(t)\|g(x(t-\gamma_j(t)))|dt+T|Ax|_0|e|_0 \\
&\leq T\sum_{j=1}^n|\beta_{j}|_0g_{\rho}|Ax|_0+T\sum_{j=1}^n|\beta_{j}|_0|Ax|_0(r+\varepsilon_1)|x|_0+T|Ax|_0|e|_0\\
&\leq\Big(T\sum_{j=1}^n|\beta_{j}|_0g_{\rho}(1+c_0)
 +T|e|_0(1+c_0)\Big)|x|_0+T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)(r+\varepsilon_1)|x|_0^2\\
&\leq k_1\int_0^T|x'(t)|dt+k_2\Big(\int_0^T|x'(t)|dt\Big)^2+k_3,
\end{aligned}\label{e8}
\end{equation}
where
\begin{gather*}
g_\rho=\max_{|x(t-\gamma_j(t))|\leq\rho}|g(x(t-\gamma_j(t)))|,\\
k_1=T\sum_{j=1}^n|\beta_{j}|_0g_{\rho}(1+c_0)+T|e|_0(1+c_0)
 +2T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)(r+\varepsilon_1)d,\\
k_2=T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)(r+\varepsilon_1),\\
k_3=T\sum_{j=1}^n|\beta_{j}|_0g_{\rho}(1+c_0)d+T|e|_0(1+c_0)d
 +T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)(r+\varepsilon_1)d^2.
\end{gather*}
 From $(Ax')(t)=(Ax)'(t)+c'(t)x(t-\tau)$, \eqref{e8} and
Lemma 2.1, if $c_0<\frac{1}{2}$, we have
\begin{align*}
\int_0^T|x'(t)|dt
&=\int_0^T|(A^{-1}Ax')(t)|dt\\
&\leq\frac{1}{1-c_0}\int_0^T|(Ax')(t)|dt\\
&\leq\frac{1}{1-c_0}\int_0^T|(Ax)'(t)|dt+\frac{c_1T}{1-c_0}|x|_0\\
&\leq\frac{T^{1/2}}{1-c_0}\Big(\int_0^T|(Ax)'(t)|^2dt
 \Big)^{1/2}+\frac{c_1T}{1-c_0}\int_0^T|x'(t)|dt+\frac{c_1Td}{1-c_0}\\
&\leq\frac{T^{1/2}}{1-c_0}\Big[k_1\int_0^T|x'(t)|dt+k_2
\Big(\int_0^T|x'(t)|dt\Big)^2+k_3\Big]^{1/2}\\
&\quad +\frac{c_1T}{1-c_0}\int_0^T|x'(t)|dt+\frac{c_1Td}{1-c_0}.
\end{align*}
By \eqref{e5}, there exists a constant $M_1>0$ which
is independent with $\lambda$ such that
$$
\int_0^T|x'(t)|dt\leq M_1.
$$
Similarly, for $\sigma> 1$, by \eqref{e6}, there exists
a  constant$M_1'>0$ which is independent
with $\lambda$ such that
$$
\int_0^T|x'(t)|dt\leq M_1'.
$$
Combining \eqref{e4} with the above two inequalities, we obtain
$$
|x|_0\leq d+\max\{M_1,M_1'\}:=M_2.
$$
 From
$$
(Ax'')(t)=(Ax)''(t)+2c'(t)x'(t-\tau)+c''(t)x(t-\tau),
$$
if
$c_0<1/2$, we have
\begin{align*}
&\int_0^T|x''(t)|dt\\
&=\int_0^T|[A^{-1}Ax''](t)|dt\\
&\leq\int_0^T\frac{|(Ax'')(t)|}{1-c_0}dt\\
&=\int_0^T\frac{|(Ax)''(t)+2c'(t)x'(t-\tau)+c''(t)x(t-\tau)|}{1-c_0}dt\\
&\leq\frac{1}{1-c_0}\Big(\int_0^T\sum_{j=1}^n|\beta_{j}(t)
\|g(x(t-\gamma_j(t)))|dt+\int_0^T|e(t)|dt+2c_1M_1+c_2M_2T\Big)\\
&\leq\frac{1}{1-c_0}(\sum_{j=1}^n|\beta_{j}|_0Tg_{M_2}
+T|e|_0+2c_1M_1+c_2M_2T):=M_3;
\end{align*}
if $\sigma>1$, we have
$$
\int_0^T|x''(t)|dt
\leq\frac{1}{\sigma-1}(\sum_{j=1}^n|\beta_{j}|_0Tg_{M_2}+T|e|_0
+2c_1M_1+c_2M_2T):=M_3',
$$
where $g_{M_2}=\max_{|x|\leq M_2}|g(x)|, c_{2}
=\max_{t\in[0,T]}|c''(t)|$. Since $x\in \Omega_1$,
so $x(0)=x(T)$ and there exists a point $\eta\in [0,T]$ such that
$x'(\eta)=0$. Then
\begin{gather*}
x'(t)=x'(\eta)+\int_{\eta}^{t}x''(s)\mathrm{d}s,\\
|x'|_0\leq\int_{0}^{T}|x''(t)|dt\leq \max\{M_3,M_3'\}:=M_4.
\end{gather*}
Then
$$
\|x\|=\max_{t\in[0,T]}{\{|x|_{0}, |x'|_{0}}\}\leq\max\{M_2,M_4\}.
$$
Hence $\Omega_1$ is bounded.

Take $\Omega_2=\{x\in \ker L\cap C_T^1: Nx\in \operatorname{Im}L\}$,
for all $x\in \Omega_2$, then $x(t)=a_0\varphi(t)$,
$a_0\in \mathbb{R}$ satisfying
\begin{equation}
\int_0^T\Gamma(t)g(a_0\varphi(t))dt=0. \label{e9}
\end{equation}
When $c_0<1/2$, we have
\begin{align*}
\varphi(t)&=A^{-1}(1)=1+\sum _{j=
1}^{\infty}\prod_{i=1}^{j}c(t-(i-1)\tau)\\
&\geq1-\sum _{j= 1}^{\infty}\prod_{i=1}^{j}c_0\\
&=1-\frac{c_0}{1-c_0} \\
&=\frac{1-2c_0}{1-c_0}:=\delta_1>0.
\end{align*}
Then we have
$a_0\leq d/\delta_1$. Otherwise, for all
$t\in[0,T]$, $a_0\varphi(t)>d$, from assumption (H3), we have
$$
\int_0^T\Gamma(t)g(a_0\varphi(t))dt>0
$$
which is contradiction to \eqref{e9}.
When $\sigma>1$, we have
\begin{align*}
\varphi(t)&=A^{-1}(1)= -\frac{1}{c(t+\tau)}-\sum _{j=
1}^{\infty}\prod_{i=1}^{j+1}\frac{1}{c(t+i\tau)}\\
&\leq-\frac{1}{\sigma}-\sum _{j=
1}^{\infty}\prod_{i=1}^{j+1}\frac{1}{\sigma}\\
&=-\frac{1}{\sigma-1}:=\delta_2<0.
\end{align*}
Then we have
$a_0\leq-d/\delta_2$. Otherwise, for all
$t\in[0,T]$, $a_0\varphi(t)<-d$, from assumption (H3), we have
$$
\int_0^T\Gamma(t)g(a_0\varphi(t))dt<0
$$
which is contradiction to \eqref{e9}. Then we have
$$
|x|=|a_0\varphi(t)|\leq\max\{\frac{d}{\delta_1},
-\frac{d}{\delta_2}\}|\varphi|_0.
$$
 Hence $\Omega_2$ is a bounded set.

Let $\Omega \supset \Omega_1 \cup \Omega_2 $ be a bounded set. For
$x\in \partial\Omega\cup D(L)$, $\forall\lambda\in(0,1)$,
we have $Lx\neq\lambda Nx$. For all
$x\in\partial\Omega\cap \ker L$, we have $Nx\notin \operatorname{Im}L$.
Hence the conditions (1) and (2) of Lemma 2.2 hold.
It remains to verify conditions (3)  of Lemma 2.2.
 Now, for $x\in \partial \Omega \cap \ker L$,
take the homotopy
$$
H(x,\mu)=\begin{cases}
 -\mu x-\frac{1}{T} (1-\mu)\int_0^T\sum_{j=1}^n\beta_{j}(t) g(x)dt ,
 &\text{if }(\sum_{j=1}^n\bar{\beta_j})xg(x)>0; \\
\mu x-\frac{1}{T} (1-\mu)\int_0^T\sum_{j=1}^n\beta_{j}(t) g(x)dt ,
 &\text{if } (\sum_{j=1}^n\bar{\beta_j})xg(x)<0.
\end{cases}
$$
Clearly,
$$
H(x,\mu)=\begin{cases}
 -\mu x-(1-\mu)g(x)\sum_{j=1}^n\bar{\beta_{j}}, &\text{if }
  (\sum_{j=1}^n\bar{\beta_j})xg(x)>0; \\
 \mu x-(1-\mu)g(x)\sum_{j=1}^n\bar{\beta_{j}}, & \text{if }
  (\sum_{j=1}^n\bar{\beta_j})xg(x)<0.
\end{cases}
$$
For $x\in \partial \Omega \cap \ker L$ and $\mu\in[0,1]$,
$xH(x,\mu)\neq 0$. So we have
\begin{align*}
\deg\{ QN,\Omega \cap \ker L,0\}
&= \deg\big\{-\frac{1}{T} \int_0^T\sum_{j=1}^n\beta_{j}(t) g(x) dt ,
 \Omega \cap \ker L,0\big\}\\
&=\deg\{ -x,\Omega \cap \ker L,0\}\neq 0.
\end{align*}
Applying Lemma 2.2, we reach the conclusion.
\end{proof}



As an application, we consider the following example.

\begin{example}\label{exa3.1} \rm Consider the equation
\begin{equation}
\begin{aligned}
&(x(t)-\frac{1}{10}(2-\sin t)x(t-\tau))''
+(1+\frac{1}{2}\sin t)\frac{u(t-\frac{1}{2}\cos t)}{80000}\\
&+(1-\frac{1}{2}\sin t)\frac{u(t-\frac{1}{2}\sin t)}{80000}=\sin t,
\end{aligned}\label{e10}
\end{equation}
where
\begin{gather*}
c(t)=\frac{1}{10}(2-\sin t), \quad
\beta_1(t)=1+\frac{1}{2}\sin t, \quad
\beta_2(t)=1-\frac{1}{2}\sin t,\\
\gamma_1(t)=\frac{1}{2}\cos t,\quad
\gamma_2(t)=\frac{1}{2}\sin t, \quad
e(t)=\sin t, \quad T=2\pi.
\end{gather*}
 From simple calculations, we have
$$
c_0=\frac{3}{10},\quad c_1=\frac{1}{10}, \quad
|\beta_1|_0=\frac{3}{2}, \quad |\beta_2|_0=\frac{3}{2}, \quad
r=\frac{1}{80000}.
$$
 Let $\mu_1(t)$ and
$\mu_2(t)$ be the inverses of $t-\frac{1}{2}\cos t$ and
$t-\frac{1}{2}\sin t$ respectively.
 We have
\begin{align*}
\Gamma(t)&=\frac{\beta_1(\mu_1(t))}{1-\gamma_1'(\mu_1(t))}
 +\frac{\beta_2(\mu_2(t))}{1-\gamma_2'(\mu_2(t))}\\
&=\frac{1+\frac{1}{2}\sin\mu_1(t)}{1+\frac{1}{2}\sin\mu_1(t)}
 +\frac{1-\frac{1}{2}\sin\mu_2(t)}{1-\frac{1}{2}\cos\mu_2(t)}\\
&=1+\frac{1-\frac{1}{2}\sin\mu_2(t)}{1-\frac{1}{2}\cos\mu_2(t)}>0
\end{align*}
and
$$
\frac{T^{1/2}}{1-c_0}\sqrt{T\sum_{j=1}^n|\beta_{j}|_0(1+c_0)r}
+\frac{c_1T}{1-c_0}\approx0.96<1.
$$
Applying Theorem 3.1, Equation \eqref{e10} has at least
one 2$\pi$-periodic solution.
\end{example}


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\end{document}