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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 117, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2006/117\hfil
Existence and uniqueness of periodic solutions]
{Existence and uniqueness of periodic solutions
for first-order neutral functional differential equations
with two deviating arguments}

\author[J. Xiao,B. Liu \hfil EJDE-2006/117\hfilneg]
{Jinsong Xiao, Bingwen Liu}  

\address{Jinsong Xiao \newline
Department of Mathematics and Computer Science,
 Hunan City University,
Yiyang, Hunan  413000, China} 
\email{bingxiao209@yahoo.com.cn}

\address{Bingwen Liu \newline
 College of Mathematics and Information Science,
Jiaxing University,
Jiaxing, Zhejiang 314001, China}
\email{liubw007@yahoo.com.cn}

\date{}
\thanks{Submitted March 16, 2006. Published September 26, 2006.}
\thanks{Supported by grants 10371034 from the NNSF of China
and 05JJ40009 from the \hfill\break\indent
 Hunan Provincial Natural Science Foundation of China}
\subjclass[2000]{34C25, 34D40}
\keywords{First order; neutral; functional differential equations;
\hfill\break\indent deviating argument; periodic solutions; coincidence degree}

\begin{abstract}
 In this paper, we use the coincidence degree  theory to establish
 the existence and uniqueness of  $T$-periodic solutions  for
 the first-order neutral functional differential equation, with two
 deviating arguments,
$$
(x(t)+Bx(t-\delta))'=  g_{1}(t,x(t-\tau_{1}(t)))
                         +g_{2}(t,x(t-\tau_{2}(t))) +p(t).
$$
\end{abstract}

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

\section{Introduction}

    Consider  the first-order neutral functional differential
         equation (NFDE), with two deviating arguments,
\begin{equation}
   (x(t)+Bx(t-\delta))'=  g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t)))
    +p(t),
\label{e1.1}
\end{equation}
where    $\tau_{1}$, $\tau_{2}$, $p:\mathbb{R}\to \mathbb{R}$
   and $g_{1},g_{2}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous
    functions, $B  $ and $\delta$ are constants,  $\tau_{1},  \tau_{2}$ and $p$
     are $T$-periodic,
    $g_{1}$ and  $g_{2}$ are $T$-periodic in the first argument,
   $|B|\neq 1$ and $T>0$.

The above equation has been used for the study of distributed
networks containing lossless transmission lines \cite{k1,k2}.
Hence, in recent years, the  problem of the existence
of periodic solutions for \eqref{e1.1}  has been extensively studied.
For more details, we refer the reader to  \cite{b1,g1,h2,h3,k1,k2,l2,z1}
and the references cited therein.
    However, to the best of our knowledge, there exist no results for
the existence  and uniqueness
    of periodic solutions of \eqref{e1.1}.

    The main purpose of this paper is to establish sufficient conditions for
    the existence and uniqueness of $T$-periodic solutions of \eqref{e1.1}.
The results  of this paper are new and they complement previously  known
    results. An illustrative   example is given in Section 4.

     For ease of exposition, throughout this paper we will adopt the following
     notation:
$$
    |x|_k=\Big(\int^T_0|x(t)|^kdt\Big)^{1/k}, \quad
    |x|_\infty=\max_{t\in [0,T]}|x(t)|.
$$
Let $ X=\{x|x\in C(\mathbb{R},  \mathbb{R}),  x(t+T)=x(t), \mbox{ for all }
 t\in \mathbb{R}\} $ be a Banach space with the norm
$ \|x\|_{X}=|x|_\infty$.
Define  the two linear operators
\begin{equation}
\begin{gathered}
A:X\to X, \quad  (Ax)(t)=x(t)+Bx(t-\delta); \\
L: D(L)\subset X\to X, \quad  Lx=(Ax)',  \label{e1.2}
\end{gathered}
\end{equation}
where $D(L)=\{x|x\in X,x'\in C(\mathbb{R},  \mathbb{R})\}$.

   We also define the nonlinear operator $N: X\to X$ by
$$
Nx= g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t)))
    +p(t).\label{e1.3}
$$
By Hale's terminology \cite{h2}, a solution $u(t)$ of \eqref{e1.1} is that
 $u\in C(\mathbb{R},  \mathbb{R})$ such that
$Au\in C^{1}(\mathbb{R},  \mathbb{R})$ and \eqref{e1.1} is
  satisfied on $\mathbb{R}$. In
general, $u\not\in C^{1}(\mathbb{R},  \mathbb{R})$.
But  from \cite[Lemma 1]{l2}, in view of $|B|\neq 1$, it is easy to
see that $(Ax)'=Ax'$.
So a $T$-periodic solution $u(t)$ of \eqref{e1.1} must be such that
$u\in C^{1}(\mathbb{R},  \mathbb{R})$. Meanwhile, according
to \cite[Lemma 1]{l2}, we can easily get that
$ \ker L=\mathbb{R}$,   and
$\mathop{\rm Im} L=\{x\in X : \int^{T}_{0}x(s)ds=0 \}$.
Therefore,  the operator $L$ is a Fredholm operator with index zero.
Define the continuous projectors
$P: X \to \ker L$  and  $ Q:X \to X/ImL$ by setting
\begin{gather*}
 Px(t)=\frac{1}{T}\int^{T}_{0}x(s)ds,
\\
 Qx(t)=\frac{1}{T}\int^{T}_{0}x(s)ds.
\end{gather*}
Hence, $\mathop{\rm Im}P=\ker L$ and $ \ker Q=\mathop{\rm Im}L$.
Set $L_{P}=L|_{D(L)\cap \rm{Ker} P}$, then $L_{P}$ has continuous
inverse $L^{-1}_{P}$ defined by
\begin{equation}
L^{-1}_{P}y(t)=A^{-1}\Big(\frac{1}{T}\int^{T}_{0}sy(s)ds
+\int^{t}_{0}y(s)ds\Big).
\label{e1.4}
\end{equation}
Therefore, it is easy to see from \eqref{e1.3}  and \eqref{e1.4} that $N$ is
$L$-compact on $\overline{\Omega}$, where $\Omega$ is an open
bounded set in $X$.

\section{Preliminary Results}

In view of  \eqref{e1.2} and \eqref{e1.3}, the operator equation
$$
Lx=\lambda Nx
$$
is equivalent to the  equation
$$
    x'(t)+Bx'(t-\delta)=\lambda [ g_{1}(t,x(t-\tau_{1}(t)))
+g_{2}(t,x(t-\tau_{2}(t))) +p(t)],
    \label{e2.1}
$$
    where $ \lambda\in (0,  1) $.

    For convenience of use, we introduce the Continuation Theorem
    \cite{g1} as follows.

\begin{lemma} \label{lem2.1}
 Let $X$ be a Banach space.  Suppose that $L:D(L)\subset X\to X$
is a Fredholm  operator with index zero and $N: \overline{\Omega}\to X$
is $L$-compact on $\overline{\Omega}$, where $\Omega$ is an open
bounded subset of $X$. Moreover, assume that all the following
conditions  are  satisfied:
\begin{enumerate}
\item  $Lx\neq \lambda Nx$, for all $x\in \partial \Omega\cap D(L)$,
$\lambda\in (0,1)$;

\item  $Nx \not \in ImL$, for all $x\in \partial \Omega\cap \ker L$;

\item  For the Brower  degree,
$\deg\{QN,  \Omega\cap \ker L ,  0     \}\neq 0 $.

\end{enumerate}
 Then the equation $Lx= Nx$ has at  least one
    solution on $\overline{\Omega}\cap D(L)$.
\end{lemma}

By using a similar argument of the proof of \cite[Lemma 2.5]{l1},
from  \cite[Theorem 225]{h1}, we can obtain the following Lemma.

\begin{lemma} \label{lem2.2}
 Let $x(t)\in X\cap C^{1}(\mathbb{R},\mathbb{R})$ .
    Suppose that there exists a constant $D\geq 0$ such that
$$
    |x(\tau_0)|\leq D,  \tau_0\in [0,T].
    \label{e2.2}
$$
    Then
\begin{equation}
    |x|_2\le \frac{T}{\pi}|x'|_2+\sqrt{T} D.
    \label{e2.3}
\end{equation}
\end{lemma}

\begin{lemma}[\cite{l3}] \label{lem2.3}
  Let $\mu \in [0,  T]$ be a constant,
$\overline{\delta}\in  C (\mathbb{R},\mathbb{R})$  be periodic with
period $T$,  and $\sup_{t \in [0,  T]}|\overline{\delta}(t)|\leq \mu$.
Then for any     $  h\in C^{1} (\mathbb{R},\mathbb{R})$ which is periodic
with period $T$, we have
\begin{equation}
\int^T_0|h(s)-h(s-\overline{\delta}(s))|^{2}ds\leq
2\mu^{2}\int^T_0|h'(s)|^{2}ds. \label{e2.4}
\end{equation}
For the next lemma we need the following conditions
\begin{itemize}
\item[(H)]   For $i=1,2$, there exist a constants $\mu_{i}$  and an
integers $K_{i}$  such that
 $$
\mu_{i}=\sup_{t\in [0,T]}|\tau_{i}(t)-K_{i}T|\leq T.
$$

\item[(A0)] One of the following conditions holds:\\
  (1)  $(g_{i}(t,u_{1})-g_{i}(t,u_{2}))(u_{1}-u_{2})>0$,
     for $i=1,2, u_{i}\in \mathbb{R}$, for all $t\in \mathbb{R}$
     and $u_{1}\neq u_{2}$;\\
  (2)  $(g_{i}(t,u_{1})-g_{i}(t,u_{2}))(u_{1}-u_{2})<0$,
     for $i=1,2, u_{i}\in \mathbb{R}$, for all $t\in \mathbb{R}$ and
     $u_{1}\neq u_{2}$;

\item[(A0')] One of the following conditions holds:\\
 (1) there exists  constants $b_{1}$ and $ b_{2}$ such that
$ b_{1}  (\sqrt{2}\mu_{1}+ \frac{T}{\pi}
)+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi})< 1-|B|$, and
$$
    |g_{i}(t,u_{1})-g_{i}(t,u_{2})|
   \leq b_{i}|u_{1}-u_{2}|,\mbox{  for  } i=1,2,  u_{i}\in \mathbb{R}, \forall t\in \mathbb{R}
   ,
$$
(2) There exists  constants $ b_{1}$
    and $ b_{2}$ such that $b_{1}  (\sqrt{2}\mu_{1}+ \frac{T}{\pi}
)+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi})<|B|-1$, and
$$
    |g_{i}(t,u_{1})-g_{i}(t,u_{2})|
   \leq b_{i}|u_{1}-u_{2}|,\quad \mbox{for  } i=1,2,  u_{i}\in \mathbb{R}, \forall t\in \mathbb{R}.
$$
\end{itemize}
\end{lemma}


\begin{lemma} \label{lem2.4}
Under assumptions (A0) and (A0'),
Equation \eqref{e1.1} has at most one $T$-periodic  solution.
\end{lemma}

\begin{proof} Suppose that $x_{1}(t)$ and $x_{2}(t)$ are two
$T$-periodic solutions of \eqref{e1.1}. Then
$$
(x_{1}(t)+Bx^{}_{1}(t-\delta))'
   -g_{1}(t,x_{1}(t-\tau_{1}(t))) -g_{2}(t,x_{1}(t-\tau_{2}(t)))=p(t)
$$
and
$$
(x_{2}(t)+Bx_{2}(t-\delta))'
   -g_{1}(t,x_{2}(t-\tau_{1}(t)))-g_{2}(t,x_{2}(t-\tau_{2}(t)))=p(t).
$$
This implies
\begin{equation}
\begin{aligned}
&[(x_{1}(t)-x_{2}(t))+B(x_{1}(t-\delta)-x_{2}(t-\delta))]'
-(g_{1}(t,x_{1}(t-\tau_{1}(t)))\\
&-g_{1}(t,x_{2}(t-\tau_{1}(t))))
-(g_{2}(t,x_{1}(t-\tau_{2}(t)))-g_{2}(t,x_{2}(t-\tau_{2}(t))))=0.
\end{aligned}\label{e2.5}
\end{equation}
Set $Z(t)=x_{1}(t)-x_{2}(t)$. Then, from \eqref{e2.5}, we obtain
\begin{equation} \label{e2.6}
\begin{aligned}
&Z'(t)+BZ'(t-\delta)-(g_{1}(t,x_{1}(t-\tau_{1}(t)))\\
&-g_{1}(t,x_{2}(t-\tau_{1}(t))))
-(g_{2}(t,x_{1}(t-\tau_{2}(t)))-g_{2}(t,x_{2}(t-\tau_{2}(t))))=0.
\end{aligned}
\end{equation}
Thus, integrating \eqref{e2.6} from $0$ to $T$, we have
\begin{align*}
&\int^T_0[(g_{1}(t,x_{1}(t-\tau_{1}(t)))
 -g_{1}(t,x_{2}(t-\tau_{1}(t))))\\
& +(g_{2}(t,x_{1}(t-\tau_{2}(t)))
 -g_{2}(t,x_{2}(t-\tau_{2}(t))))]dt=0.
\end{align*}
Therefore, in view of integral mean value theorem, it follows that there
exists a constant $\gamma \in [0,  T]$ such that
\begin{equation}
\begin{aligned}
& (g_{1}(\gamma,x_{1}(\gamma-\tau_{1}(\gamma)))-g_{1}(\gamma,x_{2}
 (\gamma-\tau_{1}(\gamma))))\\
&+(g_{2}(\gamma,x_{1}(\gamma-\tau_{2}(\gamma)))
 -g_{2}(\gamma,x_{2}(\gamma-\tau_{2}(\gamma))))=0.
\end{aligned}\label{e2.7}
\end{equation}
 From (A0), \eqref{e2.7} implies
 $$
(x_{1}(\gamma-\tau_{1}(\gamma))-x_{2}(\gamma-\tau_{1}(\gamma)))(x_{1}(\gamma-\tau_{2}(\gamma))
-x_{2}(\gamma-\tau_{2}(\gamma)))\leq
0.
 $$
 Since $Z(t)=x_{1}(t)-x_{2}(t)$ is a continuous function on $\mathbb{R}$,
 it follows that there
    exists a constant $\xi \in \mathbb{R}$ such that
\begin{equation}
    Z(\xi)=0. \label{e2.8}
\end{equation}
Let  $  \xi=n T+\widetilde{\gamma}$,  where
$\widetilde{\gamma}\in [0,  T]$     and $n$ is an integer.
Then,  \eqref{e2.8} implies that there
    exists a constant $\widetilde{\gamma} \in [0,  T]$ such that
\begin{equation}
Z(\widetilde{\gamma})=Z(\xi)=0. \label{e2.9}
\end{equation}
 Then, from Lemma \ref{lem2.2}, using Schwarz inequality and
the inequality
$$
    |Z(t)|=|Z(\widetilde{\gamma})+\int^t_{\widetilde{\gamma}}Z'(s)ds|\le
    \int^T_0|Z'(s)|ds,
    \mbox{  for  all  } t\in [0,  T],
$$
 we obtain
\begin{equation}
    |Z|_\infty\le \sqrt{T}
    |Z'|_2, \quad\mbox{and}\quad  |Z|_2\leq \frac{T}{\pi}
    |Z'|_2.
\label{e2.10}
\end{equation}
Now, we  consider two cases.

\subsection*{Case (i)}
  If (A0')(1) holds, multiplying both sides of \eqref{e2.6}
   by $Z'(t)$ and then integrating them from
    $0$ to $T$,  using (H), \eqref{e2.4}, \eqref{e2.10}
 and Schwarz inequality, we have
\begin{align*}
&|Z'|_2^2\\
& =   \int^T_0|Z'(t)|^2dt \\
& = -B\int^T_0Z'(t)Z'(t-\delta)dt
 +\int^T_0(g_{1}(t,x_{1}(t-\tau_{1}(t)))
-g_{1}(t,x_{2}(t-\tau_{1}(t))))Z'(t)dt\\
& \quad +\int^T_0(g_{2}(t,x_{1}(t-\tau_{2}(t)))
-g_{2}(t,x_{2}(t-\tau_{2}(t))))Z'(t)dt \\
&\le   |B||Z'|_2^2+b_{1} \int^T_0|x_{1}(t-\tau_{1}(t))-x_{2}(t-\tau_{1}(t))||Z'(t)| dt\\
& \quad +b_{2} \int^T_0|x_{1}(t-\tau_{2}(t))-x_{2}(t-\tau_{2}(t))||Z'(t)| dt\\
&\le   |B||Z'|_2^2+b_{1} \int^T_0|Z(t-\tau_{1}(t))-Z(t )||Z'(t)| dt+b_{1} \int^T_0|Z(t )||Z'(t)| dt\\
& \quad +b_{2}\int^T_0|Z(t-\tau_{2}(t))-Z(t )||Z'(t)| dt+b_{2} \int^T_0|Z(t )||Z'(t)| dt \\
&\le   |B||Z'|_2^2+b_{1} (\int^T_0|Z(t-\tau_{1}(t))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2}+b_{1}  |Z |_{2}|Z' |_{2}\\
& \quad +b_{2} (\int^T_0|Z(t-\tau_{2}(t))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2} +b_{2}  |Z |_{2}|Z' |_{2}\\
&=   |B||Z'|_2^2+b_{1} (\int^T_0|Z(t-(\tau_{1}(t)-K_{1}T))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2}+b_{1}  |Z |_{2}|Z' |_{2}\\
& \quad +b_{2} (\int^T_0|Z(t-(\tau_{2}(t)-K_{2}T))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2} +b_{2}  |Z |_{2}|Z' |_{2}\\
&\le    [|B| +b_{1}  (\sqrt{2}\mu_{1}+ \frac{T}{\pi}
)+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi}  )]|Z'|_2^2. %\eqref{e2.11}
\end{align*}
  From \eqref{e2.10} and (A0')(1), the above inequalit
implies
$$
Z(t)\equiv Z'(t) \equiv 0, \quad \mbox{for all  } t\in \mathbb{R}.
$$
  Hence, $x_{1}(t)\equiv x_{2}(t)$,
  for all $ t\in \mathbb{R}$. Therefore, \eqref{e1.1} has at most
one $T$-periodic     solution.

\subsection*{Case (ii)}
 If (A0')(2) holds,
  multiplying both sides of \eqref{e2.6}
   by $Z'(t-\delta)$ and then integrating them from
    $0$ to $T$, using $(H)$, \eqref{e2.4}, \eqref{e2.10} and Schwarz inequality, we have
\begin{align*}
&|B||Z'|_2^2 \\
& =    |\int^T_0B|Z'(t-\delta)|^2dt |\\
& = |-\int^T_0Z'(t)Z'(t-\delta)dt \\
&\quad +\int^T_0(g_{1}(t,x_{1}(t-\tau_{1}(t)))
-g_{2}(t,x_{2}(t-\tau_{1}(t))))Z'(t-\delta)dt\\
& \quad +\int^T_0(g_{2}(t,x_{1}(t-\tau_{2}(t)))
-g_{2}(t,x_{2}(t-\tau_{2}(t))))Z'(t-\delta)dt |\\
&\le   |Z'|_2^2 +b_{1} \int^T_0|x_{1}(t-\tau_{1}(t))-x_{2}(t-\tau_{1}(t))||Z'(t-\delta)| dt\\
& \quad +b_{2} \int^T_0|x_{1}(t-\tau_{2}(t))-x_{2}(t-\tau_{2}(t))||Z'(t-\delta)| dt\\
&\le    |Z'|_2^2+b_{1} \int^T_0|Z(t-\tau_{1}(t))-Z(t )||Z'(t-\delta)| dt+b_{1} \int^T_0|Z(t )||Z'(t-\delta)| dt\\
& \quad +b_{2}\int^T_0|Z(t-\tau_{2}(t))-Z(t )||Z'(t-\delta)| dt+b_{2} \int^T_0|Z(t )||Z'(t-\delta)| dt \\
&\le    |Z'|_2^2+b_{1} (\int^T_0|Z(t-\tau_{1}(t))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2}+b_{1}  |Z |_{2}|Z' |_{2}\\
& \quad +b_{2} (\int^T_0|Z(t-\tau_{2}(t))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2} +b_{2}  |Z |_{2}|Z' |_{2}\\
&=    |Z'|_2^2+b_{1} (\int^T_0|Z(t-(\tau_{1}(t)-K_{1}T))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2}+b_{1}  |Z |_{2}|Z' |_{2}\\
& \quad +b_{2} (\int^T_0|Z(t-(\tau_{2}(t)-K_{2}T))-Z(t )| ^{2} dt)^{\frac{1}{2}}|Z' |_{2} +b_{2}  |Z |_{2}|Z' |_{2}\\
&\le    [1 +b_{1}  (\sqrt{2}\mu_{1}+ \frac{T}{\pi}
)+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi}  )]|Z'|_2^2 %  \eqref{e2.12}
\end{align*}
Then using the methods similar to those used in Case (i), from the above
inequality, \eqref{e2.10}, and (A0')(2), we can conclude  that
 \eqref{e1.1} has at most one $T$-periodic
    solution. The proof of Lemma \ref{lem2.4} is now  complete.
\end{proof}

For the next lemma we use the following assumptions:
\begin{itemize}
\item[(A1)]   $x(g_{1}(t,x)+g_{2}(t,x)+p(t))>0$, for all
$t\in \mathbb{R}, |x|\ge d$;
\item[(A2)]  $x(g_{1}(t,x)+g_{2}(t,x)+p(t))<0$, for all
$t\in \mathbb{R}, |x|\ge d$.
\end{itemize}

\begin{lemma} \label{lem2.5}
 Assume (A0) and that there exists a positive constant $d$
 such that one of the two conditions (A1) or (A2) holds.
If $x(t)$ is a $T$-periodic solution of \eqref{e2.1}, then
\begin{equation}
    |x|_\infty\le d+\sqrt{T}|x'|_2.
    \label{e2.13}
\end{equation}
\end{lemma}

\begin{proof}
Let $x(t)$ be a $T$-periodic solution of \eqref{e2.1}.
    Then, integrating \eqref{e2.1} from
    $0$ to $T$,  we have
$$
\int^T_0[g_{1}(t, x(t-\tau_{1}(t)))+g_{2}(t,
x(t-\tau_{2}(t)))+p(t)] dt=0.
$$
This implies that there exists a constant $t_{1}\in \mathbb{R}$ such that
\begin{equation}
g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{2}(t_{1})))+p(t_{1})=0. \label{e2.14}
\end{equation}
We show next the following
Claim:  If $x(t)$ is a $T$-periodic solution of \eqref{e2.1}, then
    there exists a constant $t_{2}\in \mathbb{R}$ such that
\begin{equation}
    |x(t_{2})|\leq d .  \label{e2.15}
\end{equation}
 Assume, by way of contradiction, that \eqref{e2.15} does not hold. Then
$$
|x(t)|> d,   \quad\text{for  all }  t\in \mathbb{R},
 \label{e2.16}
$$
which, together with (A1), (A2) and \eqref{e2.14},  implies that
one of the following relations holds:
\begin{gather}
x(t_{1}-\tau_{1}(t_{1}))>
x(t_{1}-\tau_{2}(t_{1}))> d;\label{e2.17} \\
x(t_{1}-\tau _{2}(t_{1}))> x(t_{1}-\tau_{1}(t_{1}))> d;\label{e2.18}\\
x(t_{1}-\tau_{1}(t_{1}))< x(t_{1}-\tau_{2}(t_{1}))<- d;\label{e2.19}\\
x(t_{1}-\tau_{2}(t_{1}))< x(t_{1}-\tau_{1}(t_{1}))<- d.\label{e2.20}
\end{gather}
If \eqref{e2.17} holds, in view of (A0)(1),
(A0)(2),  (A1) and (A2), we shall consider four
cases as follows.

\noindent  Case (i).  If (A1) and (A0)(1) hold,  according to
    \eqref{e2.17}, we obtain
\begin{align*}
0&<g_{1}(t_{1}, x(t_{1}-\tau_{2}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{2}(t_{1})))+p(t_{1}) \\
&<g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{2}(t_{1})))+p(t_{1}),
\end{align*}
which contradicts  \eqref{e2.14}.
This contradiction implies that \eqref{e2.15} holds.

\noindent  Case (ii).  If (A1) and (A0)(2) hold,  according to
    \eqref{e2.17}, we obtain
\begin{align*}
0&<g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{1}(t_{1})))+p(t_{1})\\
&<g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{2}(t_{1})))+p(t_{1}),
\end{align*}
which contradicts  \eqref{e2.14}.
This contradiction implies that \eqref{e2.15} holds.

\noindent Case (iii).  If (A2) and (A0)(1) hold,  according to
 \eqref{e2.17}, we obtain
\begin{align*}
&g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{2}(t_{1})))+p(t_{1}) \\
&<g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{1}(t_{1})))+p(t_{1})<0,
\end{align*}
which contradicts \eqref{e2.14}. This contradiction implies that \eqref{e2.15}
holds.

 \noindent Case (iv).  If (A2) and (A0)(2) hold,  according to
 \eqref{e2.17}, we obtain
\begin{align*}
&g_{1}(t_{1}, x(t_{1}-\tau_{1}(t_{1})))+g_{2}(t_{1}, x(t_{1}-\tau_{2}(t_{1})))
+p(t_{1})\\
&<g_{1}(t_{1}, x(t_{1}-\tau_{2}(t_{1})))+g_{2}(t_{1},
x(t_{1}-\tau_{2}(t_{1})))+p(t_{1})<0,
\end{align*}
which contradicts \eqref{e2.14}. This contradiction implies that
 \eqref{e2.15} holds.

If \eqref{e2.18} (or \eqref{e2.19}, or \eqref{e2.20}) holds,
using the methods similar to those used in Case (i) - Case (iv), we can
show that \eqref{e2.15} holds. This completes the proof of the Claim.

 Let $t_{2}=m T+t_0$,  where  $ t_0\in [0,T]$ and $m$ is an integer.
Then, using Schwarz inequality and the inequality
$$
    |x(t)|=|x(t_0)+\int^t_{t_0}x'(s)ds|\le d+\int^T_0|x'(s)|ds,
     \quad \mbox{for  all  }  t\in [0,  T],
$$
we obtain
$$
    |x|_\infty=\max_{t\in[0,T]}|x(t)|\le d+\sqrt{T}
    |x'|_2.
$$
This completes the proof.
\end{proof}

\section{Main Results}


\begin{theorem} \label{thm3.1}
Assume that (H), (A0), (A0') and either  (A1) or (A2).
 Then \eqref{e1.1} has a unique $T$-periodic solution.
\end{theorem}

\begin{proof}
From Lemma \ref{lem2.4}, together with (H),  (A0) and (A0'), it i
s easy to see that  \eqref{e1.1} has at most one
$T$-periodic     solution. Thus, to prove Theorem \ref{thm3.1}, it suffices to
show that \eqref{e1.1} has at least one $T$-periodic
solution. To do this, we shall apply Lemma \ref{lem2.1}.
 Firstly, we will claim that the set of all possible
    $T$-periodic solutions of  \eqref{e2.1} is bounded.

    Let $x(t)$ be a $T$-periodic solution of equation
    \eqref{e2.1}.
 In view of (A0')(1) and (A0')(2), we shall consider two
 cases as follows.

\noindent Case (i).  If (A0')(1) holds, multiplying both
sides of \eqref{e2.1}   by $x'(t)$ and then integrating them from
    $0$ to $T$, from \eqref{e2.3}, \eqref{e2.4},   \eqref{e2.15},
(H),  (A0')(1)  and the  Schwarz inequality,
we have
\begin{equation}
\begin{aligned}
&|x'|_2^2\\
& =    \int^T_0|x'(t)|^2dt \\
& =   -\int^T_0Bx'(t-\delta)x'(t)dt+\lambda\int^T_0
g_{1}(t,x(t-\tau_{1}(t))) x'(t)dt \\
& \quad +\lambda\int^T_0g_{2}(t,x(t-\tau_{2}(t)))x'(t)dt  +\lambda
        \int^T_0p(t)x'(t)dt \\
&\le  |B||x'|_2^2+|p|_{2}|x'|_2+\lambda\int^T_0(
g_{1}(t,x(t-\tau_{1}(t)))-g_{1}(t,x(t))+g_{1}(t,x(t)) \\
 & \quad  -g_{1}(t,0))
x'(t)dt
+\lambda\int^T_0(g_{2}(t,x(t-\tau_{2}(t)))-g_{2}(t,x(t))+g_{2}(t,x(t))\\
& \quad  -g_{2}(t,0))
        x'(t)dt+\lambda\int^T_0 g_{1}(t,0)  x'(t)dt +\lambda\int^T_0
g_{2}(t,0)
x'(t)dt \\
&\quad |B||x'|_2^2+|p|_{2}|x'|_2+b_{1}(\int^T_0|
x(t-\tau_{1}(t))-x(t)|^{2}dt) ^{\frac{1}{2}}| x'|_{2}+b_{1}|
x|_{2}| x'|_{2}\\
 & \quad  +b_{2}(\int^T_0| x(t-\tau_{2}(t))-x(t)|^{2}dt)
^{\frac{1}{2}}| x'|_{2}+b_{2}| x|_{2}| x'|_{2}\\
 & \quad  +(\max_{t\in [0,
T]}|g_{1}(t,0)| +\max_{t\in [0,  T]}|g_{2}(t,0)|)\sqrt{T}| x'|_{2}  \\
&=  |B||x'|_2^2+|p|_{2}|x'|_2+b_{1}(\int^T_0|
x(t-(\tau_{1}(t)-K_{1}T))-x(t)|^{2}dt) ^{\frac{1}{2}}|
x'|_{2}+b_{1}|
x|_{2}| x'|_{2}\\
 & \quad  +b_{2}(\int^T_0| x(t-(\tau_{2}(t)-K_{2}T))-x(t)|^{2}dt)
^{\frac{1}{2}}| x'|_{2}+b_{2}| x|_{2}| x'|_{2}\\
 & \quad  +(\max_{t\in [0,
T]}|g_{1}(t,0)| +\max_{t\in [0,  T]}|g_{2}(t,0)|)\sqrt{T}| x'|_{2}  \\
&\le  [|B| +b_{1}  (\sqrt{2}\mu_{1}+ \frac{T}{\pi}
)+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi}
)]|x'|_2^2+|p|_{2}|x'|_2\\
 & \quad  +(b_{1}d+b_{2}d+\max_{t\in [0,
T]}|g_{1}(t,0)| +\max_{t\in [0,  T]}|g_{2}(t,0)|)\sqrt{T}|
x'|_{2}.
\end{aligned}  \label{e3.1}
\end{equation}
Now, let
$$
D_{1}=\frac{|p|_{2}+(b_{1}d+b_{2}d+\max_{t\in
[0,  T]}|g_{1}(t,0)|+\max_{t\in [0,
T]}|g_{2}(t,0)|)\sqrt{T}}{1-|B|-(b_{1}  (\sqrt{2}\mu_{1}+
\frac{T}{\pi} )+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi} ))}.
$$
In view of \eqref{e2.13} and \eqref{e3.1}, we obtain
\begin{equation}
|x'|_2\leq D_{1}, |x|_\infty\le d+\sqrt{T}D_{1} .\label{e3.2}
\end{equation}

\noindent Case (ii).  If (A0')(2) holds,  multiplying both
sides of \eqref{e2.1}  by $x'(t-\delta)$  and then integrating them from
    $0$ to $T$,  from \eqref{e2.3}, \eqref{e2.4}, \eqref{e2.13},
\eqref{e2.15}, (A0')(2) and the inequality of Schwarz, we have
\begin{equation}
\begin{aligned}
|B||x'|_2^2
& =   |\int^T_0B|x'(t-\delta)|^2dt |\\
& =   | -\int^T_0x'(t-\delta)x'(t)dt+\lambda\int^T_0
g_{1}(t,x(t-\tau_{1}(t)))
x'(t-\delta)dt  \\
& \quad +\lambda\int^T_0g_{2}(t,x(t-\tau_{2}(t)))x'(t-\delta)dt+\lambda
        \int^T_0p(t)x'(t-\delta)dt |\\
&\le  |x'|_2^2+|p|_{2}|x'|_2+|\lambda\int^T_0(
g_{1}(t,x(t-\tau_{1}(t)))-g_{1}(t,x(t))+g_{1}(t,x(t)) \\
 & \quad  -g_{1}(t,0)) x'(t-\delta)dt+\lambda\int^T_0(g_{2}(t,x(t-\tau_{2}(t)))-g_{2}(t,x(t))+g_{2}(t,x(t))\\
& \quad  -g_{2}(t,0)) x'(t-\delta)dt+ \lambda\int^T_0 g_{1}(t,0)
x'(t)dt +\lambda\int^T_0 g_{2}(t,0)
x'(t-\delta)dt | \\
&\le  |x'|_2^2+|p|_{2}|x'|_2+b_{1}(\int^T_0|
x(t-\tau_{1}(t))-x(t)|^{2}dt) ^{\frac{1}{2}}| x'|_{2}+b_{1}|
x|_{2}| x'|_{2}\\
 & \quad  +b_{2}(\int^T_0| x(t-\tau_{2}(t))-x(t)|^{2}dt)
^{\frac{1}{2}}| x'|_{2}+b_{2}| x|_{2}| x'|_{2}\\
 & \quad  +(\max_{t\in [0,
T]}|g_{1}(t,0)| +\max_{t\in [0,  T]}|g_{2}(t,0)|)\sqrt{T}| x'|_{2}  \\
&\le  [1 +b_{1}  (\sqrt{2}\mu_{1}+ \frac{T}{\pi}
)+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi}
)]|x'|_2^2+|p|_{2}|x'|_2\\
 & \quad  +(b_{1}d+b_{2}d+\max_{t\in [0,
T]}|g_{1}(t,0)| +\max_{t\in [0,  T]}|g_{2}(t,0)|)\sqrt{T}|
x'|_{2}.
\end{aligned}        \label{e3.3}
\end{equation}
Now, let
$$
\overline{D}_{1}=\frac{|p|_{2}+(b_{1}d+b_{2}d+\max_{t\in [0,
T]}|g_{1}(t,0)|+\max_{t\in [0,
T]}|g_{2}(t,0)|)\sqrt{T}}{|B|-1-(b_{1}  (\sqrt{2}\mu_{1}+
\frac{T}{\pi} )+b_{2}(\sqrt{2}\mu_{2}+ \frac{T}{\pi} ))}.
$$
In view of \eqref{e2.13} and \eqref{e3.3}, we obtain
\begin{equation}
|x'|_2\leq \overline{D}_{1}, |x|_\infty\le
 d+\sqrt{T}\overline{D}_{1} . \label{e3.4}
\end{equation}
If $x\in \Omega_{1}=\{x\in \ker L \cap X : Nx\in ImL\}$,
then there exists a constant $M_{1}$ such that
\begin{equation}
x(t)\equiv M _{1} \quad\mbox{and}\quad
  \int^{T}_{0}[g_{1}( t,M_{1})+g_{2}(t, M_{1})+p(t)]dt=0.    \label{e3.5}
\end{equation}
Thus,
\begin{equation}
|x(t)|\equiv |M_{1}|<d,  \quad \mbox{for  all } x(t)\in \Omega_{1}.
    \label{e3.6}
\end{equation}
Let $M=(D_{1}+\overline{D}_{1})\sqrt{T}+d+1$. Set
$$
\Omega=\{x|x\in X,  |x|_\infty<M \}.
$$
It is easy to see from \eqref{e1.3}  and \eqref{e1.4} that $N$ is $L$-compact
on $\overline{\Omega}$. We have from \eqref{e3.5}, \eqref{e3.6} and the fact
$M>\max\{D_{1}\sqrt{T}+d , \overline{D}_{1}\sqrt{T}+d , d \}$
that  the conditions (1) and (2) in Lemma \ref{lem2.1} hold.

  Furthermore, define the continuous
    functions
\begin{gather*}
   H_{1}(x,\mu)=(1-\mu)x+\mu\cdot
   \frac{1}{T}\int^{T}_{0}[g_{1}(t,x)+g_{2}(t,x)+p(t)]dt;  \mu\in [0 \, 1],
\\
H_{2}(x,\mu)=-(1-\mu)x+\mu\cdot
   \frac{1}{T}\int^{T}_{0}[g_{1}(t,x)+g_{2}(t,x)+p(t)]dt;  \mu\in [0 \, 1].
\end{gather*}
 If (A1) holds, then
$$
  xH_{1}(x,\mu)\neq 0  \quad \mbox{for  all}   x\in \partial \Omega\cap \ker L.
$$
Hence, using the  homotopy invariance theorem,
    we have
\begin{align*}
 \deg\{QN,  \Omega\cap \ker L ,  0  \}
& =\deg\{\frac{1}{T}
  \int^{T}_{0}[g_{1}(t,x)+g_{2}(t,x)+p(t)]dt,  \Omega\cap \ker L ,  0 \}\\
& =deg\{x,  \Omega\cap \ker L ,  0   \}\neq 0.
\end{align*}
If (A2) holds, then
$ xH_{2}(x,\mu)\neq  0 $ for  all  $x\in \partial \Omega\cap \ker L$.
Hence, using the  homotopy invariance  theorem, we obtain
\begin{align*}
 \deg\{QN,  \Omega\cap \ker L ,  0     \}
& =\deg\{\frac{1}{T}\int^{T}_{0}[g_{1}(t,x)+g_{2}(t,x)+p(t)]dt,
 \Omega\cap \ker L ,  0   \} \\
&=\deg\{-x,  \Omega\cap \ker L ,  0   \}\neq 0.
\end{align*}
In view of  all the discussions above and Lemma \ref{lem2.1},
Theorem \ref{thm3.1} is proved.
\end{proof}

\section{Concluding remarks}

 \begin{example} \label{examp4.1} \rm
The  first-order neutral functional differential
\begin{equation}
   (x(t)+\frac{1}{8}x(t-\delta))'
=- \frac{3}{8 } x(t-\frac{\sqrt{2}}{64}\sin ^{2} t)+ \frac{1
    }{32}[1-x(t-\frac{\sqrt{2}}{64}\cos ^{2}   t)]+e^{\cos  t}
    \label{e4.1}
\end{equation}
    has a unique $2\pi$-periodic solution.

  From \eqref{e4.1}, we have $B=\frac{1}{8}$,
$g_{1}(x)=-\frac{3}{8 } x$,
    $g_{2}(x )=\frac{1}{32}[1-x ]$ and $p(t)=e^{\cos  t}$. Then,
$ \mu_{1}=\sup_{t\in [0,T]}|\frac{\sqrt{2}}{64}\sin ^{2} t|
=\frac{\sqrt{2}}{64}< 2\pi$,
$\mu_{2}=\sup_{t\in [0,T]}|\frac{\sqrt{2}}{64}\cos ^{2}
t|=\frac{\sqrt{2}}{64}< 2\pi$,
 $b_{1}=\frac{3}{8 }$, $b_{2}=\frac{1}{32}$. It is straight
forward to check that all the conditions needed in Theorem \ref{thm3.1} are
satisfied. Therefore, \eqref{e4.1} has a unique $2\pi$-periodic
solution.
\end{example}

\begin{remark} \label{rmk4.1} \rm
Equation \eqref{e4.1} is a very simple version of first order NFDE.
Since $B\neq 0$, all the results in the references and their references
 can not be  applicable to \eqref{e4.1} to obtain the existence
and uniqueness of     $2\pi$-periodic solutions. This
    implies that the results of this paper are essentially new.
\end{remark}

\begin{remark} \label{rmk4.2} \rm
By using the methods similarly to those used
    for \eqref{e1.1}, we can deal with the NFDE with multiple
    deviating arguments, for example
\begin{equation}
    (x(t)+Bx(t-\delta))'=\sum_{i=1}^{n}g_{i}(t,x(t-\tau_{i}(t)))
    +p(t), \label{e4.2}
\end{equation}
where  $\tau_{i}(i=1,2,\dots, n)$,  $p:\mathbb{R}\to \mathbb{R}$
   and  $g_{i}(i=1,2,\dots, n) :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$
are continuous  functions,  $\tau_{i}(i=1,2,\dots, n)$ and $p$ are
$T$-periodic, $g_{i}, i=1,2,\dots, n, $ are $T$-periodic in the first
argument,  and $T>0$. One may also establish the results similarly to
 those in Theorem \ref{thm3.1}  under some minor additional assumptions on
$g_{i}(t,x)(i=1,2,\dots, n)$.
\end{remark}

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\end{document}