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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 170, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/170\hfil Li\'enard-type systems with delays]
{Existence and uniqueness of pseudo almost periodic solutions
for Li\'enard-type systems \\ with delays}

\author[C. Xu, M. Liao \hfil EJDE-2016/170\hfilneg]
{Changjin Xu, Maoxin Liao}

\address{Changjin Xu \newline
Guizhou Key Laboratory of Economics System Simulation,
Guizhou University of Finance and Economics,
Guiyang 550004, China}
\email{xcj403@126.com}

\address{Maoxin Liao \newline
School of Mathematics and Physics,
University of South China,
Hengyang 421001, China}
\email{maoxinliao@163.com}

\thanks{Submitted February 2, 2015. Published July 4, 2016.}
\subjclass[2010]{34C25, 34K13, 34K25}
\keywords{Li\'enard-type system; pseudo almost periodic solution;
\hfill\break\indent exponential dichotomy; time-varying delay; contraction mapping principle}

\begin{abstract}
 This article concerns Li\'enard-type systems with time-varying delays.
 By applying the theory of exponential dichotomies, the properties of
 pseudo almost periodic function, inequality analysis techniques, and
 contraction mapping principle, new criteria for the existence
 and uniqueness of pseudo almost periodic solutions are established.
 An example is given to illustrate the theoretical findings.
\end{abstract}

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

\section{Introduction}

It is well known that the Li\'enard-type systems have
potential applications in many fields such as mechanics, physics and
engineering and so on \cite{f1,g1,h1,j1,k1,l1}.
 Thus the problem on almost periodic
solutions of the Li\'enard-type systems has attracted
numerous scholars. In recent years, Gao and Liu \cite{g1} investigated the
following Li\'enard-type equation with time-varying delays
\begin{equation} \label{e1.1}
\ddot{x}+g(x(t))\dot{x}(t)+h_0(x(t))+\sum_{l=1}^mh_l(x(t-\sigma_l(t)))=\rho(t),
\end{equation}
where $g$ and $h_l$ $(l=0,1,2,\dots,m)$ are continuous functions on $\mathbb{R}$,
$\sigma_l(t)\geq0$ $(l=1,2,\dots,m)$ and
$\rho(t)$ are almost periodic functions on $\mathbb{R}$. Form the viewpoint
of mechanics, $g$ usually stands for a damping or friction term,
$h_l$ denotes the restoring force, and $\rho_l(l=1,2,\dots,m)$
represents an externally force, $\sigma_i$ denotes a time delay of
the restoring force. For more details, we refer the reader to \cite{b1,g1}.
Letting $c^*$ be a positive constant and defining
$$
f(x)=\int_0^x[g(u)-c^*]du, y=\frac{dx}{dt}+f(x),
$$
Gao and Liu \cite{g1} obtained the following equivalent form of system
\eqref{e1.1},
\begin{equation} \label{e1.2}
\begin{gathered}
 \dot{x}(t)=-f(x(t))+y(t),\\
 \dot{y}(t)=-c^*y(t)-[h_0(x(t))-c^*f(x(t))]
 -\sum_{l=1}^mh_l(x(t-\sigma_l(t)))+\rho(t).
\end{gathered}
\end{equation}
By applying some analysis techniques and constructing a suitable
Lyapunov function, they establish some sufficient conditions which
guarantee the existence and exponential stability of the almost
periodic solutions for system \eqref{e1.2}.

As we know, since the existence of pseudo almost periodic solutions
has wide applications in various fields, especially in the economic,
physics and biology \cite{l2,z1},
it is a hot topic in qualitative theory
of differential equations \cite{l2,z1}.
 In many cases, the nature of the
almost periodic functions do not always hold in the set of pseudo
almost periodic functions (see, e.g. \cite{l2,z1}).
Based on the analysis
above, we think that it is worthwhile to investigate the pseudo
almost periodic solutions of Li\'enard-type systems.
To the best of our knowledge, there is no paper that deal with this
aspect for Li\'enard-type systems.
Let
\begin{equation}
\begin{gathered}
 y=\dot{x}+ax-\theta_1(t),\\
 \theta_2(t)=\rho(t)+a\theta_1(t)-\dot{\theta}_1(t).
\end{gathered}\label{e1.3}
\end{equation}
Then \eqref{e1.2} becomes
\begin{equation}
\begin{gathered}
 \dot{x}(t)=-ax(t)+y(t)+\theta_1(t),\\
\begin{aligned}
 \dot{y}(t)
&=ay(t)-a^2x(t)-g(x(t))[y-ax+\theta_1(t)] -h_0(x(t))\\
&\quad -\sum_{l=1}^mh_l(x(t-\sigma_l(t)))+\theta_2(t).
\end{aligned}
\end{gathered} \label{e1.4}
\end{equation}
Considering that the more reasonable models require the inclusion of
the effect of changing environment, we modify system \eqref{e1.4} as
follows
\begin{equation}
\begin{gathered}
 \dot{x}(t)=-a(t)x(t)+y(t)+\theta_1(t),\\
\begin{aligned}
 \dot{y}(t)&=a(t)y(t)-a^2(t)x(t)-g(x(t))[y-a(t)x+\theta_1(t)] -h_0(x(t))\\
&\quad -\sum_{l=1}^mh_l(x(t-\sigma_l(t)))+\theta_2(t).
\end{aligned}
\end{gathered} \label{e1.5}
\end{equation}
The main aim of this article is to establish some sufficient
conditions for the existence and uniqueness of pseudo almost
periodic solutions of \eqref{e1.5}. The obtained results of this article
are completely new and complement some previous studies.

The remainder of the paper is organized as follows. In Section 2, we
introduce some notations, lemmas and definitions. In Section 3, we
present some new sufficient conditions for the existence and
uniqueness of pseudo almost periodic solutions for the
Li\'enard-type system with delays. An example is
given to illustrate the effectiveness of the obtained results in
Section 4. In Section 5, we give a brief conclusion.

\section{Preliminary results}

In this section, we would like to recall some notation, basic
definitions and lemmas which are used in what follows. For
convenience, we denote by
${\mathbb{R}}^q({\mathbb{R}}={\mathbb{R}}^1)$ the set of all
$q$-dimensional real vectors (real numbers). Let
$$
\{x_i(t)\}=(x_1(t),x_2(t),\dots,x_n(t))\in{\mathbb{R}}^n.
$$
For any $x(t)=\{x_i(t)\}\in{\mathbb{R}}^n$, we let $|x|$ denote the
absolute value vector given by $|x|=\{|x_i(t)|\}$ and define
$\|x(t)\|=\max_{1\leq i\leq n}\{|x_i(t)|\}$.
A matrix or vector
$U\geq 0$ means that all entries of $U$ are greater than or equal to
zero. $U>0$ can be defined similarly. For matrices or vectors $U$
and $V$, $U\geq V$ (resp. $U>V$) means that $U-V\geq0$
(resp. $U-V>0$). Throughout this paper, we use the notation
$$
v^{+}=\sup_{t\in\mathbb{R}}|v(t)|, \quad
v^{-}=\inf_{t\in\mathbb{R}}|v(t)|,
$$
where $v(t)$ is a bounded continuous function.
Let $\mathrm{BC}(\mathbb{R},\mathbb{R})$ denote the
set of bounded continued functions from $\mathbb{R}$ to
$\mathbb{R}$, and $\mathrm{BUC}(\mathbb{R}, \mathbb{R})$ be the set of all
bounded and uniformly continuous functions from $\mathbb{R}$ to
$\mathbb{R}$. Obviously, $(\mathrm{BC}(\mathbb{R},\mathbb{R}), \|\cdot\|)$ is a
Banach space where $\|\cdot\|$ denotes the sup norm
$\|v\|_\infty:= \sup_{t\in \mathbb{R}}\|v(t)\|$.


\begin{definition}[\cite{f2,z1}] \label{def2.1} \rm
 Let $v(t)\in \mathrm{BC}(\mathbb{R},\mathbb{R}),v(t)$ is said to be almost
periodic on $\mathbb{R}$ if, for any $\varepsilon>0$, the set
$T(u,\varepsilon)=\{\varrho:\|v(t+\varrho)-v(t)\|< \varepsilon$
for all $t\in \mathbb{R}\}$ is relatively dense; that is, for
any $\varepsilon> 0$, it is possible to find a real number
$l=l(\varepsilon) >0$; for any interval with length
$l(\varepsilon)$, there exists a number
$\varrho=\varrho(\varepsilon)$ in this interval such that
$\|v(t+\varrho)-v(t)\|< \varepsilon$, for all $t\in\mathbb{R}$.
\end{definition}


 We denote by
$\mathrm{AP}(\mathbb{R},\mathbb{R})$ the set of the almost periodic
functions from $\mathbb{R}$ to $\mathbb{R}$. Besides, the concept of
pseudo almost periodicity (PAP) was introduced by Zhang \cite{z1} in
the early nineties. It is a natural generalization of the classical
almost periodicity. Precisely, define the class of functions
$\mathrm{PAP}_0(\mathbb{R},\mathbb{R})$ as follows:
$$
\big\{v\in \mathrm{BC}(\mathbb{R},\mathbb{R}):
 \lim _{T\to +\infty}\frac{1}{2T}
\int_{-T}^T|v(t)|dt=0 \big\}.
$$
A function $v\in \mathrm{BC}(\mathbb{R},\mathbb{R})$ is called pseudo
almost periodic if it can be expressed as $v=v_1+v_2$, where
$v_1\in\mathrm{AP}(\mathbb{R},\mathbb{R})$ and
$v_2\in \mathrm{PAP}_0(\mathbb{R},\mathbb{R})$. The collection of such
functions will be denoted by $ \mathrm{PAP}(\mathbb{R},\mathbb{R})$. The
functions $v_1$ and $v_2$ in the above definition are, respectively,
called the almost periodic component and the ergodic perturbation of
the pseudo almost periodic function $v$. The decomposition given in
definition above is unique. Observe that
$(\mathrm{PAP}(\mathbb{R},\mathbb{R}),\|\cdot\|)$ is a Banach space and
$\mathrm{AP}(\mathbb{R},\mathbb{R})$ is a proper subspace of
$(\mathrm{PAP}(\mathbb{R},\mathbb{R})$ since the function
$v_2(t)=\sin^2t+\sin^4\sqrt{11}t + \exp(-t^6\sin^4t)$ is pseudo almost
periodic function but not almost periodic.

\begin{definition}[\cite{d1,z2}] \label{def2.2}
 Let $x\in \mathbb{R}^n$ and
$Q(t)$ be a $n\times n$ continuous matrix defined on $\mathbb{R}$.
The linear system
\begin{equation} \label{e2.1}
\dot{x}(t)=Q(t)x(t)
\end{equation}
 is said to admit an exponential dichotomy on
$\mathbb{R}$ if there exist positive constants $k,\alpha$ and
projection $P$ and the fundamental solution matrix $X(t)$ of \eqref{e2.1}
satisfying
\begin{gather*}
\|X(t)PX^{-1}(s)\|\leq k e^{-\alpha(t-s)}, \quad \text{for } t\geq s,\\
\|X(t)(I-P)X^{-1}(s)\|\leq k e^{-\alpha(s-t)}, \quad \text{for } t\leq s,
\end{gather*}
where $I$ is the identity matrix.
\end{definition}

\begin{lemma}[\cite{h1}] \label{lem2.1}
Suppose that $Q(t)$ is an almost
periodic matrix function and $g(t)\in \mathrm{PAP}(\mathbb{R},\mathbb{R}^p)$.
If the linear system \eqref{e2.1}
admits an exponential dichotomy, then pseudo almost periodic system
\begin{equation} \label{e2.2}
\dot{x}(t)=Q(t)x(t)+g(t)
\end{equation}
has a unique pseudo almost
periodic solution $x(t)$, and
\begin{equation} \label{e2.3}
x(t)=\int_{-\infty}^tX(t)PX^{-1}(s)g(s)ds
-\int_t^{+\infty}X(t)(I-P)X^{-1}(s)g(s)ds.
 \end{equation}
\end{lemma}

\begin{lemma}[\cite{h1}] \label{lem2.2}
Let $Q(t)=(q_{ij})_{n\times n}$ be
an almost matrix defined on $\mathbb{R}$, and there exists a
positive constant $\varsigma$ such that
 $$
|q_{ii}|-\sum_{j=1,j\neq i}|q_{ij}|\geq \varsigma,\quad
\forall t\in \mathbb{R}, i=1,2,\dots,n.
$$
Then the linear system \eqref{e2.1} admits an exponential dichotomy
 on $\mathbb{R}$.
\end{lemma}

Let
\begin{equation} \label{e2.4}
\begin{gathered}
\varrho=\max\big\{\frac{\sup_{t\in\mathbb{R}}|\theta_1(t)|}{a^{-}},
\frac{\sup_{t\in\mathbb{R}}|\theta_2(t)|}{a^{-}}\big\},
\\
 \kappa=\max\big\{\frac{1}{a^{-}},
\frac{\sup_{t\in\mathbb{R}}[a^2(t)+G(1+a(t)+\theta_1(t))+\sum_{l=0}^mH_l]}
{a^{-}}\big\},
\\
 \delta=\max\Big\{\frac{1}{a^{-}}, \frac{\sup_{t\in
\mathbb{R}}\big[a^2(s)+G|a(t)|+G|\theta_1(t)|+2G+\sum_{l=0}^mH_l\big]}{a^{-}}\Big\}
\end{gathered}
\end{equation}
and
$$
\Omega^*=\Big\{\varphi\||\varphi-\varphi_0\|_\infty\leq\frac{\kappa
\varrho}{1-\kappa}, \varphi=(\varphi_1,\varphi_2)^T\in
\mathrm{PAP}(\mathbb{R},\mathbb{R}^2)\cap
\mathrm{BUC}(\mathbb{R},\mathbb{R}^2)\Big\},
$$
where
$$
\varphi_0=\Big(\int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}\theta_1(s)ds,
\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}|\theta_2(s)|ds
\Big)^T.
$$

\begin{lemma} \label{lem2.3} $\Omega^*$ is a closed subset
$\mathrm{PAP}(\mathbb{R}, \mathbb{R}^2)$.
\end{lemma}

The proof of the above lemma is similar to that of
\cite[Lemma 2.1]{l2}. Here we omit it.


Throughout this paper, we make the following assumptions for system
\eqref{e1.5}:
\begin{itemize}
\item[(H1)] For $l=0,1,2,\dots,m$, there exist nonnegative constants
$H_0,H_1,H_2,\dots,H_m$ such that
$$
|h_l(x)-h_l(y)|\leq H_l|x-y| \quad \text{for all }
x,y\in\mathbb{R}, h_l(0)=0.
$$

\item[(H2)] There exists a positive constant $G$ such that
$g(x)\leq G$ for all $x\in\mathbb{R}$.

\item[(H3)] $a(t),\theta_1(t),\theta_2(t),\sigma_l(t)\in
\mathrm{PAP}(\mathbb{R},\mathbb{R}), a(t)>0$, for all
$t\in \mathbb{R}$,
where $ i=1,2,\dots,m$.


\item[(H4)] $\kappa<1, \frac{\varrho}{1-\kappa}<1,\delta<1$.
\end{itemize}

 \section{Existence and uniqueness of pseudo almost periodic solutions}

 In this section, we establish sufficient conditions on the
existence and uniqueness of pseudo almost periodic solutions of
\eqref{e1.5}.

\begin{theorem} \label{thm3.1}
 Suppose that {\rm (H1)--(H4)} hold. Then
there exists a unique pseudo almost periodic solution of system
\eqref{e1.5} in the region $\Omega^*$.
\end{theorem}

\begin{proof}
Denote $x(t)=x(t;t_0,\varphi)$.
Let $\varphi=(\varphi_1,\varphi_2)^T\in \Omega$ and
$f(t,\mu)=\varphi_j(t-\mu)$. In view of
\cite[Theorem 5.3 p.58 ]{z1},
and \cite[Definition 5.7 p.59]{l1}, we can conclude that
the uniform continuity of $\varphi_2$ implies that
$f\in \mathrm{PAP}(\mathbb{R}\times\Omega)$ and $f$ is continuous
in $\mu\in L$ and uniformly in $t\in \mathbb{R}$ for all compact
subset $L$ of $\Omega\subset \mathbb{R}$.
According to
$\sigma_l\in \mathrm{PAP}(\mathbb{R},\mathbb{R})(l=1,2,\dots,m)$ and
\cite[Theorem 5.11]{z1}, we know that
 $\varphi_1(t-\sigma_l(t))\in \mathrm{PAP}(\mathbb{R},\mathbb{R})$.
It follows from \cite[Corollary 5.4]{z1} and the composition theorem
of pseudo almost periodic functions that
\begin{gather*}
\varphi_2(t)+\theta_1(t)\in \mathrm{PAP}(\mathbb{R},\mathbb{R}),\\
\begin{aligned}
&-a^2(t)\varphi_1(t)-g(\varphi_1(t))[\varphi_2(t)-a(t)\varphi_1(t)+\theta_1(t)]\\
&-h_0(\varphi_1(t))-\sum_{l=1}^mh_l(\varphi_1(t-\sigma_l(t))) \in
\mathrm{PAP}(\mathbb{R},\mathbb{R}).
\end{aligned}
\end{gather*}
Now we consider the auxiliary system
\begin{equation} \label{e3.1}
\begin{bmatrix}
 \dot{x}(t) \\
 \dot{y}(t)
 \end{bmatrix}
=\begin{bmatrix}
 -a(t) & 0 \\
 0 & a(t)
 \end{bmatrix}
\begin{bmatrix}
 x(t) \\
 y(t)
 \end{bmatrix}
+ \begin{bmatrix}
 \gamma_1(t) \\
 \gamma_2(t)
 \end{bmatrix},
\end{equation}
where
\begin{gather*}
\gamma_1(t)=\varphi_2(t)+\theta_1(t)\\
\begin{aligned}
\gamma_2(t)&=-a^2(t)\varphi_1(t)-g(\varphi_1(t))[\varphi_2(t)-a(t)
 \varphi_1(t)+\theta_1(t)]\\
&\quad -h_0(\varphi_1(t))-\sum_{l=1}^mh_l(\varphi_1(t-\sigma_l(t)))+\theta_2(t).
\end{aligned}
\end{gather*}
Then it follows from Lemma \ref{lem2.2} that the following system
\begin{equation} \label{e3.2}
\begin{bmatrix}
 \dot{x}(t) \\
 \dot{y}(t)
 \end{bmatrix}
=\begin{bmatrix}
 -a(t) & 0 \\
 0 & a(t)
 \end{bmatrix}
  \begin{bmatrix}
 x(t) \\
 y(t)
 \end{bmatrix}
\end{equation}
admits an exponential dichotomy on $\mathbb{R}$. Define a projection
$P$ as follows
$$P= \begin{bmatrix}
 1 & 0 \\
 0 & 0
 \end{bmatrix}.
$$
Applying Lemma \ref{lem2.1}, we can conclude that system \eqref{e3.1} has exactly
one pseudo almost periodic solution which takes the form
$$
 \begin{bmatrix}
 x_\varphi(t) \\
 y_\varphi(t)
 \end{bmatrix}
= \begin{bmatrix}
 \int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}\gamma_1(s)ds \\
 -\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}\gamma_2(s)ds
 \end{bmatrix},
$$
where
\begin{gather*}
\gamma_1(s)=\varphi_2(s)+\theta_1(s),\\
\begin{aligned}
\gamma_2(s)&=-a^2(s)\varphi_1(s)-g(\varphi_1(s))[\varphi_2(s)
-a(s)\varphi_1(s)+\theta_1(s)]\\
&\quad -h_0(\varphi_1(s))-\sum_{l=1}^mh_l(\varphi_1(s-\sigma_l(s)))+\theta_2(s).
\end{aligned}
\end{gather*}
Define a mapping
$\Gamma: \Omega^*\to \mathrm{PAP}(\mathbb{R},\mathbb{R}^2)$ as follows
$$
(\Gamma\varphi)(t)= \begin{bmatrix}
 x_\varphi(t) \\
 y_\varphi(t)
 \end{bmatrix}, \quad \forall \varphi\in \Omega^*.
$$
In view of the definition of the norm in Banach space
$\mathrm{PAP}(\mathbb{R},\mathbb{R}^2)$, we have
\begin{equation} \label{e3.3}
\begin{aligned}
\|\varphi_0\|
&\leq \sup_{t\in
\mathbb{R}}\max\Big\{\int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}|\theta_1(s)|ds,
\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}|\theta_2(s)|ds\Big\} \\
&\leq \max\big\{\frac{\sup_{t\in\mathbb{R}}|\theta_1(t)|}{a^{-}},
\frac{\sup_{t\in\mathbb{R}}|\theta_2(t)|}{a^{-}}\big\}=\varrho.
\end{aligned}
\end{equation}
Thus
\begin{equation} \label{e3.4}
\|\varphi\|_\infty\leq\|\varphi-\varphi_0\|+\|\varphi_0\|_\infty\leq\frac{\kappa
\varrho}{1-\kappa}+\varrho=\frac{\varrho}{1-\kappa}<1.
\end{equation}
By \eqref{e3.4}, we have
\begin{equation} \label{e3.5}
\begin{aligned}
&\|\Gamma\varphi-\varphi_0\|_\infty \\
&=\sup_{t\in \mathbb{R}}\max
\Big\{\big|\int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}|\varphi_2(s)|ds\big|,
\big|\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}|\gamma_2^*(s)|ds\big|\Big\} \\
&\leq\max\Big\{\frac{1}{a^{-}},\frac{\sup_{t\in\mathbb{R}}[a^2(t)
 +G(1+a(t)+\theta_1(t))+\sum_{l=0}^mH_l]}{a^{-}}\Big\}\|\varphi\|_\infty \\
&=\kappa\|\varphi\|_\infty
\leq\frac{\kappa \varrho}{1-\kappa},
\end{aligned}
\end{equation}
where
\begin{align*}
\gamma_2^*(s)
&=-a^2(s)\varphi_1(s)-g(\varphi_1(s))[\varphi_2(s)-a(s)\varphi_1(s)+\theta_1(s)]\\
&\quad -h_0(\varphi_1(s))-\sum_{l=1}^mh_l(\varphi_1(s-\sigma_l(s))).
\end{align*}
Then
\begin{equation} \label{e3.6}
\|\Gamma\varphi\|_\infty\leq\|\Gamma\varphi-\varphi_0\|
+\|\varphi_0\|_\infty\leq\frac{\kappa
\varrho}{1-\kappa}+\varrho=\frac{\varrho}{1-\kappa}<1.
\end{equation}
In view of \eqref{e3.1}, we know that
$(\Gamma(\varphi)(t))'$ is
bounded on $\mathbb{R}$. Thus we can conclude that
$\Gamma(\varphi)(t)$ is uniformly continuous on $\mathbb{R}$, and
$\Gamma\varphi\in\Omega^*$ which implies that $T$ is a self-mapping
from $\Omega^*$ to $\Omega^*$.

Now we prove that $\Gamma$ is a contraction mapping. According
to \eqref{e2.4}, we obtain
\begin{equation}
\begin{aligned}
&\|(\Gamma\varphi)(t)-(\Gamma\psi)(t)\|\\
&=(|((\Gamma\varphi)(t)-(\Gamma\psi)(t))_1|,((\Gamma\varphi)(t)
 -(\Gamma\psi)(t))_2|)^T \\
&\leq\Big(\big|\int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}|\varphi_2(s)
 -\psi_2(s)|ds\big|,
\big|\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}|
 \gamma_2^*(s)-\bar{\gamma}_2^*(s)|ds\big|\Big)^T
\\
&\leq\Big(\int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}ds
\sup_{t\in \mathbb{R}}|\varphi_2(s)-\psi_2(s)|,
\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}ds \\
&\quad \times \Big\{a^2(s)\sup_{t\in \mathbb{R}}|\varphi_1(s)-\psi_1(s)|
 +G|a(t)|\sup_{t\in \mathbb{R}}|\varphi_1(s)-\psi_1(s)| \\
&\quad +G|\theta_1(t)|\sup_{t\in \mathbb{R}}|\varphi_1(s)-\psi_1(s)|
  +G\sup_{t\in \mathbb{R}}|\varphi_1(s)-\psi_1(s)| 
  +G\sup_{t\in \mathbb{R}}|\varphi_2(s)-\psi_2(s)| \\
&\quad +H_0\sup_{t\in \mathbb{R}}|\varphi_1(s)-\psi_1(s)|
+\sum_{l=1}^mH_l\sup_{t\in \mathbb{R}}|\varphi_1(s-\sigma_l(s))
 -\psi_1(s-\sigma_l(s))|\Big\}\Big)^T \\
&\leq\Big(\int_{-\infty}^te^{-\int_s^ta(\theta)d\theta}ds
\|\varphi(s)-\psi(s)\|_\infty,
\int_{t}^{+\infty}e^{-\int_t^sa(\theta)d\theta}ds \\
&\quad \sup_{t\in \mathbb{R}}\Big[a^2(s)+G|a(t)|+G|\theta_1(t)|
 +2G+\sum_{l=0}^mH_l\Big]\|\varphi(s)-\psi(s)\|_\infty\Big)^T,
\end{aligned}  \label{e3.7}
\end{equation}
where
\begin{align*}
\bar{\gamma}_2^*(s)
&=-a^2(s)\psi_1(s)-g(\psi_1(s))[\psi_2(s)-a(s)\psi_1(s)+\theta_1(s)]\\
&\quad -h_0(\psi_1(s))-\sum_{l=1}^mh_l(\psi_1(s-\sigma_l(s))).
\end{align*}
It follows from \eqref{e3.7} that
\begin{align*}
&\|\Gamma\varphi-\Gamma\psi\|_\infty \\
&\leq\max\Big\{\frac{1}{a^{-}},
\frac{\sup_{t\in
\mathbb{R}}\big[a^2(s)+G|a(t)|+G|\theta_1(t)|+2G
+\sum_{l=0}^mH_l\big]}{a^{-}}\Big\}\|\varphi(s)-\psi(s)\|_\infty \\
&=\delta\|\varphi(s)-\psi(s)\|_\infty.
\end{align*}% {e3.8}
By (H4), we can conclude that the mapping $\Gamma$ is a contraction.
It follows from Lemma \ref{lem2.3} that the mapping $\Gamma$ has a unique
fixed point $\bar{z}=(\bar{x},\bar{y})^T\in \Omega^*, \Gamma
\bar{z}=\bar{z}$. Thus system \eqref{e1.5} has a pseudo almost periodic
solution in $\Omega^*$. The proof is complete.
\end{proof}


\section{Example}

In this section, we will give an example to support our theoretical predictions.
Considering the following Li\'enard-type system with
time-varying delays
\begin{equation} \label{e4.1}
\begin{gathered}
 \dot{x}(t)=-a(t)x(t)+y(t)+\theta_1(t),\\
\begin{aligned}
 \dot{y}(t)
&=a(t)y(t)-a^2(t)x(t)-g(x(t))[y-a(t)x+\theta_1(t)]\\
&\quad -h_0(x(t))-\sum_{l=1}^mh_l(x(t-\sigma_l(t)))+\theta_2(t).
\end{aligned}
\end{gathered}
\end{equation}
where
\begin{gather*}
a(t)=24+2\sin t, \beta_1(t)=-2-20\cos t,\theta_2(t)
=\cos \sqrt{5}t+\sin \sqrt{7}t,\\
h_l(x)=\frac{1}{2}(|x+1|-|x-1|), g(x)
=\arctan x, \sigma_l(t)=\frac{l}{2}\cos^2t,
\end{gather*}
where $l=0,1,2$. Then
$H=G=1,a^{-}=22$, $\kappa=\frac{1}{22}<1$,
$\delta=\frac{1}{22}<1$, $\varrho=\frac{11}{12}<1$.
Then (H1)--(H4) in Theorem \ref{thm3.1} hold, thus system \eqref{e4.1} has an
unique positive pseudo almost periodic solution.

\subsection*{Conclusions}
In this article, a Li\'enard-type system with
time-varying delays is studied. With the aid of the theory of
exponential dichotomies, the properties of pseudo almost periodic
function, inequality analysis technique and contraction mapping
principle, some new sufficient conditions to ensure the existence
and uniqueness of pseudo almost periodic solutions of the model are
derived. These conditions are expressed in simple algebraic formulae
which are very easily checked in practice. We give an example to
support the theoretical predictions. The obtained results are
essentially new and complement the previously known studies.

\subsection*{Acknowledgments}
This work was supported by the National Natural
Science Foundation of China
 (No.11261010), Natural Science and
Technology Foundation of Guizhou Province(J[2015]2025),
125 Special Major Science and Technology of Department of Education of Guizhou
Province ([2012]011).


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\end{document}