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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2020 (2020), No. 43, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2020 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2020/43\hfil $\mu$ pseudo rotating-periodic solutions]
{$\mu$ pseudo rotating-periodic solutions for differential equations}

\author[D. Li, J. Du \hfil EJDE-2020/43\hfilneg]
{Dandan Li, Jiayin Du}

\address{Dandan Li \newline
Department of Mathematics,
Northeast Normal University,
Changchun 130024, China}
\email{lidandan0104@163.com}

\address{Jiayin Du (corresponding author) \newline
Department of Mathematics,
Jilin University,
Changchun, 130012, China}
\email{dujiayinmath@163.com}

\thanks{Submitted November 24, 2019. Published May 16, 2020.}
\subjclass[2010]{34D09, 34K14}
\keywords{$\mu$ pseudo rotating periodic functions; exponential dichotomy}

\begin{abstract}
 In this article, we combine rotating periodic functions with $\mu$ ergodic functions
 to obtain a new class of functions called $\mu$ pseudo rotating periodic functions.
 Then we study the existence and uniqueness of $\mu$ pseudo rotating periodic
 solutions for linear systems, semi-linear systems, and non-linear systems by
 exponential dichotomy.
\end{abstract}

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

\section{Introduction}

Researchers pay a lot of attention to the existence and uniqueness of all kinds
of periodic solutions. Newton \cite{newton} was the first person who mentioned
the existence of periodic solutions. Then, Bohl and Esclangon gave the concept of
quasi-periodic functions. In 1931, Bohr \cite{H.Bohr2} obtained the concept of almost
periodic functions.  Hale \cite{J.K.Hale} studied their properties.
In 1955, Bochner \cite{S.Bochner1,S.Bochner2} gave the concept of almost autonomous
periodic functions. His student Veech \cite{Veech} researched the group properties
of this kind of functions. Shen and Yi \cite{Yi.Yingfei}, Liang and Xiao
\cite{Liang.Jin2,Liang.Jin3, Liang.Jin1,Liang.Jin4}, Campos and Tarallo \cite{Juan},
Wang and Li \cite{wang yan} also achieved a series of results in the field.
Recently, many researchers have studied rotating periodic functions and obtained a
series of results; see \cite{Li.Yong3, Liu.G, Wang.H, Li.Yong4,Yang1, Yang.X,Li.Yong2}.

In the past, because of the development of ergodic perturbation theory,
the study of systems with perturbations gradually became one of the focus in
the field of dynamics. Soon after, Frech\'et gave the definition of asymptotically
almost periodic function.

\begin{definition} \rm
A function $f\in\mathcal{C}(\mathbb{R}^{+})$ is called asymptotically almost periodic if
\begin{equation}
f=g+\phi,
\end{equation}
where $g\in \mathcal{AP}(\mathbb{R}^{+})$, and
$\phi\in C_0(\mathbb{R}^{+})
=\{h\in C(\mathbb{R}^{+}): \lim_{t\to+\infty}h(t)=0\}$.
\end{definition}

However, there were some shortcomings about asymptotically almost periodic functions.
For example, the indefinite integral of asymptotically almost periodic function may not
be asymptotically almost periodic function. Hence, Zhang \cite{Zhang.ChuangYi}
put forward the concept of pseudo almost periodic functions.

 \begin{definition} \rm
A function $f\in\mathcal{C}(\mathbb{R})(\mathcal{C}(\Omega\times\mathbb{R}))$
is called pseudo almost periodic if
\begin{equation}
f=g+\phi,
\end{equation}
where $g\in\mathcal{AP}(\mathbb{R})(\mathcal{AP}(\Omega\times\mathbb{R}))$,
and $\phi\in \mathcal{C}_1(\mathbb{R})(\mathcal{C}_1(\Omega\times\mathbb{R}))$.
Here,
\begin{equation}\label{S2.1}
\begin{gathered}
\mathcal{C}_1(\mathbb{R})=\{h\in \mathcal{C}(\mathbb{R}),\;
\lim_{r\to\infty}\frac{1}{2r}\int^{r}_{-r}\|h(t)\|dt=0\},\\
\mathcal{C}_1(\Omega\times\mathbb{R})=\{h\in\mathcal{C}(\Omega\times\mathbb{R}):
\lim_{r\to\infty}\frac{1}{2r}\int^{r}_{-r}\|h(t,x)\|dt=0\},
\end{gathered}
\end{equation}
where $g$ and $\phi$ are called the almost periodic component and the ergodic perturbation,
respectively, of the function of $f$.
\end{definition}

Moreover, the completeness and translation invariance of pseudo almost periodic
function were also proved in \cite{Zhang.ChuangYi}.
Then, Ait \cite{E.Ait.Dads} gave the concept of generalized pseudo almost periodic
functions by extending the ergodic perturbation part of the function $f$ to an
unbounded function. And he proved the existence and uniqueness of generalized pseudo
almost periodic solutions for differential equations with exponential dichotomy.

In 2006, Diagana \cite{Diagana} introduced a new class of functions called weighted
pseudo almost periodic functions, which generalize the classical pseudo almost
periodic functions.

\begin{definition} \rm
A function $f\in\mathcal{BC}(\mathbb{R},\mathbb{X})$ is called weighted pseudo almost
periodic if it can be expressed as $f=g+\phi$,
where $g\in\mathcal{AP}(\mathbb{X})$ and $\phi\in\mathcal{C}_2(\mathbb{R},\rho)$.
Here,
\begin{equation}\label{S2.2}
\mathcal{C}_2(\mathbb{R},\rho)=\{h\in\mathcal{BC}(\mathbb{R},\mathbb{X}):
\lim_{r\to\infty}\frac{1}{m(r,\rho)}\int^{r}_{-r}\|h(t)\|\rho(t)dt=0\}.
\end{equation}
\end{definition}

Similarly, he obtained the completeness and translation invariance of weighted pseudo
almost periodic function.

In 2012, Blot \cite{J.Blot} gave the definition of $\mu$ pseudo almost periodic
functions, which further expanded the weighted pseudo almost periodic functions.

\begin{definition} \rm
A function $f\in\mathcal{BC}(\mathbb{R},\mathbb{X})$ is called $\mu$ pseudo almost periodic
if it can be expressed as $f=g+\phi$, where
$g\in\mathcal{AP}(\mathbb{X})$, and $\phi\in\mathcal{C}_{3}(\mathbb{R},\mu)$. Here,
\begin{equation}
\mathcal{C}_{3}(\mathbb{R},\mu)=\{h\in \mathcal{BC}(\mathbb{R},\mathbb{X},\mu):
\lim_{r\to\infty}\frac{1}{\mu[-r,r]}\int^{r}_{-r}\|h(t)\|d\mu(t)=0\},
\end{equation}
where $\mu$ represents a measure function defined on $\mathbb{R}$.
\end{definition}

 Blot et al, transformed the density function $\rho$ of weighted pseudo
almost periodic function into measure function by measure theory.
They also proved the completeness and translation invariance of $\mu$ pseudo almost
periodic function by introducing measure equivalence.
In this article, we introduce a kind of pseudo rotating-periodic functions and
prove the existence uniqueness of pseudo rotating periodic solutions.

This article is organized as follows:
in section \ref{sec2}, we show the concept of $\mu$ pseudo rotating periodic
function and exponential dichotomy.
The basic properties of the space of $\mu$ pseudo rotating periodic function
are shown in section \ref{sec3}.
In section \ref{sec4}, we prove the existence and uniqueness of $\mu$ pseudo
rotating periodic solution for linear systems, semi-linear systems,
 and non-linear systems. Finally, section \ref{sec5} verifies the flexibility
 of our conditions by an example.


\section{$\mu$ pseudo rotating periodic function}\label{sec2}

Firstly, we give some notation. $\mathcal{B}$ denotes the Lebesgue $\sigma$-field of
$\mathbb{R}$. $\mathcal{M}$ is the set that consist of all positive measures $\mu$ on
$\mathcal{B}$ satisfying $\mu(\mathbb{R})=+\infty$ and  $\mu([a,b])<+\infty$,
for all $a,b\in\mathbb{R}$  $(a\leq b)$. We denote the space of all $n$-order
orthogonal matrices by $\mathbb{O}(n)$.
$\mathcal{BC}(\mathbb{R},\mathbb{R}^n )$ denotes the Banach space of all bounded
continuous functions from $\mathbb{R}$ to $\mathbb{R}^n $, equipped with the
supremum norm $\|f\|=\sup_{t\in\mathbb{R}}\|f(t)\|$.

\begin{definition}\label{def2.1} \rm
A function $g\in\mathcal{BC}(\mathbb{R},\mathbb{R}^n )$ is said to be  $(T,Q)$
rotating periodic if there exist  $T>0$ and $Q\in\mathbb{O}(n)$ such that
\[
g(t+T)=Q g(t)  \quad  \forall t\in\mathbb{R}.
\]
\end{definition}

\begin{definition}\label{def2.2} \rm
A function $g\in\mathcal{BC}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n )$
is said to be  $(T,Q)$ rotating periodic if the exist
$T>0$ and $Q\in\mathbb{O}(n)$ such that
\[
g(t+T,x)=Q g(t,Q^{-1}x)  \quad  \forall t\in\mathbb{R}.
\]
We denote the space of rotating periodic functions by
$\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$
 $(\mathcal{RP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ))$,
equipped with the supremum norm.
\end{definition}


\begin{remark} \rm
If $Q=\operatorname{id}$ or $Q=-\operatorname{id}$, then a $(T,Q)$ rotating periodic
function is the usual $T$-periodic function or $T$-anti-periodic one, respectively.
If for some positive integer $k_0$ such that $Q^{k_0}=id$, a $(T,Q)$ rotating
periodic function is a subharmonic one. If for all positive integer $k_1$ such that
$Q^{k_1}\neq \operatorname{id}$, then a $(T,Q)$ rotating periodic function is a
quasi-periodic function.
\end{remark}

\begin{definition} \label{def2.3} \rm
Let $\mu\in\mathcal{M}$. A function $\phi\in\mathcal{BC}(\mathbb{R},\mathbb{R}^n )$
is called $\mu$-ergodic if
\[
\lim_{r\to\infty}\frac{1}{\mu([-r,r])}\int^{r}_{-r}\|\phi(t)\|d\mu(t)=0.
\]
\end{definition}

\begin{definition}\label{def2.4} \rm
Let $\mu\in\mathcal{M}$. A function
$\phi\in\mathcal{BC}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n )$ is called $\mu$-ergodic if
\begin{equation}
\lim_{r\to\infty}\frac{1}{\mu([-r,r])}\int^{r}_{-r}\|\phi(t,x)\|d\mu(t)=0.
\end{equation}
We denote the space of all such functions by $\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$,
$\mathcal{E}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$, equipped with
the supremum norm.
\end{definition}

\begin{remark} \rm
If $\mu$ is the Lebesgue measure, then the $\mu$ ergodic function $\phi$ is the same
as the function given by Zhang \cite{Zhang.ChuangYi}.
If the Radon-Nikodym derivative of $\mu$ with respect to the Lebesgue measure on
 $\mathbb{R}$ is $\rho$, the $\mu$-ergodic function is a Weighted ergodic one by
Diagana \cite{Diagana}.
\end{remark}

\begin{definition} \rm
Let  $\mu\in\mathcal{M}$. A function $f\in\mathcal{BC}(\mathbb{R},\mathbb{R}^n )$
is said to be $\mu$ pseudo rotating periodic  if $f$ is written in the form
\[
f=g+\phi,
\]
where $g\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$, and
$\phi\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$.
\end{definition}

\begin{definition} \rm
Let $\mu\in\mathcal{M}$. A function
$f\in\mathcal{BC}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n )$ is said to be
$\mu$ pseudo rotating periodic  if $f$ is written in the form
\[
f=g+\phi,
\]
where $g\in\mathcal{RP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n )$, and
$\phi\in\mathcal{E}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$.

We denote the space of such functions by $\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$
$(\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu))$,
 equipped with the new norm $\|f\|_1=\|g\|+\|\phi\|$.
\end{definition}

Consider the system
\begin{equation}\label{27}
\dot{x}=A(t)x,
\end{equation}
where $A(t)$ is a continuous matrix on $\mathbb{R}$ and $X(t)$ is a fundamental
matrix of system \eqref{27} satisfying $X(0)=\operatorname{id}$.

\begin{definition}[\cite{Coppel}] \label{def2.4b}
The system of differential equations \eqref{27} is said to possess exponential dichotomy
on $\mathbb{R}$, if there exist a projection matrix P and constants $K>1, \alpha>0$,
such that
\begin{gather*}
\|X(t)PX^{-1}(s)\| \leq  K\exp({-\alpha(t-s)}),\quad   t\geq s,\\
\|X(t)(I-P)X^{-1}(s)\| \leq  K\exp({-\alpha(s-t)}),\quad   t\leq s.
\end{gather*}
\end{definition}

We define the Green's function as follows
\begin{equation}\label{40}
U(t,s)= \begin{cases}
  X(t)PX^{-1}(s)  &  t\geq s,\\
 -X(t)(I-P)X^{-1}(s) &   t\leq s.\\
 \end{cases}
\end{equation}

\section{Basic properties}\label{sec3}

For $\mu\in\mathcal{M}$, and $\tau\in\mathbb{R}$, we define
$\mu_{\tau}(A)=\mu(\{a+\tau:a\in A\})$ for $A\in\mathcal{B}$.
Then we make the following hypothesis.
\begin{itemize}
\item[(H1)]  Let $\mu\in\mathcal{M}$, there exist $\beta>0$ and a bounded interval
$I$ such that $\mu_{\tau}(A)\leq \beta\mu(A)$,
when $A\in\mathcal{B}$ satisfies $A\cap I=\emptyset$.
\end{itemize}

\begin{lemma}\label{lemma3.1}
If $\mu$ satisfies {\rm (H1)}, then
$(\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu),\|\cdot\|_1)$ is a Banach space.
\end{lemma}

\begin{proof}
Let $\{f_{n}\}$ be a Cauchy sequence in $\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$.
For all $\varepsilon>0$, there exist $N>0$ such that
\[
\|f_{n}(t)-f_{m}(t)\|_1<\varepsilon \quad( m,n>N,\;   \forall t\in\mathbb{R}),
\]
and
\begin{gather*}
f_{n}(t)=g_{n}(t)+\phi_{n}(t)  \quad         \forall t\in\mathbb{R},\\
f_{m}(t)=g_{m}(t)+\phi_{m}(t)  \quad         \forall t\in\mathbb{R}.
\end{gather*}
where $g_{n},g_{m}\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$, and
$\phi_{n},\phi_{m}\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$.
So, we have
\begin{align*}
\|f_{n}(t)-f_{m}(t)\|_1
&= \|g_{n}(t)+\phi_{n}(t)-g_{m}(t)-\phi_{m}(t)\|_1\\
&= \|g_{n}(t)-g_{m}(t)\|+\|\phi_{n}(t)-\phi_{m}(t)\|\\
&= \sup_{t\in\mathbb{R}}|g_{n}(t)-g_{m}(t)|+\sup_{t\in\mathbb{R}}|\phi_{n}(t)-\phi_{m}(t)|
< \varepsilon.
\end{align*}
We deduce that
\begin{gather*}
\sup_{t\in\mathbb{R}}|g_{n}(t)-g_{m}(t)| \leq \varepsilon \quad     \forall t\in\mathbb{R},\\
\sup_{t\in\mathbb{R}}|\phi_{n}(t)-\phi_{m}(t)| \leq \varepsilon \quad  \forall t\in\mathbb{R}.
\end{gather*}
According to the definition of Cauchy sequence, we can conclude that
$\{g_{n}\}$ and $\{\phi_{n}\}$ are the Cauchy sequence of
$\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$ and $\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$,
respectively. Then we use that
$\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$ and $\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$
are Banach space \cite{chengcheng,J.Blot}. Hence there exist
$g_0(t)\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$ and
$\phi_0(t)\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$ such that
\[
\lim_{t\to\infty}\|g_{n}(t)-g_0(t)\|=0,\quad
\lim_{t\to\infty}\|\phi_{n}(t)-\phi_0(t)\|=0.
\]
Let $f_0(t)=g_0(t)+\phi_0(t)$. We deduce that
$f_0\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$. So we can prove that
$(\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu),\|\cdot\|_1)$ is a Banach space.
\end{proof}

\begin{lemma}
If $\mu$ satisfies {\rm (H1)}, then
$(\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu),\|\cdot\|_1)$
is a Banach space.
\end{lemma}

The proof of the above lemma is similar to that of Lemma \ref{lemma3.1}.

\begin{lemma}\label{lemma3.3}
Let $\mu$ satisfies {\rm (H1)}.
If $y\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$,
$f\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$, and
$f$ satisfies the Lipschitz condition in $x$, then the convolution product
\begin{equation}
(f*U)(t)=\int^{+\infty}_{-\infty}U(t,s)f(s,y(s))ds
\end{equation}
is also a $\mu$ pseudo rotating periodic function.
\end{lemma}

\begin{proof}
Since $y\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$ and
$f\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$, we have
\begin{align*}
f(t,y(t))
&=f_1(t,y_1(t))+f(t,y(t))-f(t,y_1(t))+f_2(t,y_1(t))\\
&=: f_1(t,y_1(t))+\widetilde{f}(t,y(t),y_1(t)),
\end{align*}
where $y_1\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$,
$y_2\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$,
$f_1\in\mathcal{RP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n )$,
$f_2\in\mathcal{E}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$. Then
\begin{align*}
(f*U)(t)
&= \int^{+\infty}_{-\infty}U(t,s)f_1(s,y_1(s))ds
 +\int^{+\infty}_{-\infty}U(t,s)\widetilde{f}(s,y(s),y_1(s))ds\\
&=: (f*U)_1(t)+(f*U)_2(t).
\end{align*}
For $f_1\in\mathcal{RP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n )$,
it is easy to prove that $(f*U)_1(t)\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$.

Next we prove that $(f*U)_2(t)$ is a $\mu$ ergodic function.
i.e.
\[ % \label{28}
\lim_{r\to\infty}\frac{1}{\mu([-r,r])}\int^{r}_{-r}\|(f*U)_2(t)\| d\mu(t)=0.
\]
Since $f$ satisfies the Lipschitz condition, we have a constant $N$ such that
\begin{align*}
\|f(s,y(s))-f(s,y_1(s))\|
&\leq \|f(s,y(s))-f(s,y_1(s))\|_1\\
&\leq N\|y(s)-y_1(s)\|_1\\
&= N\|y_2(t)\|.
\end{align*}
Hence, we deduce that
\begin{align*}
&\int^{r}_{-r}\|\int^{+\infty}_{-\infty}U(t,s)\widetilde{f}(s,y(s),y_1(s))ds\| d\mu(t)\\
&\leq N\int^{r}_{-r}\int^{+\infty}_{-\infty}\|U(t,s)\| \|y_2(s)\| \,ds\,d\mu(t)\\
&\quad +\int^{r}_{-r}\int^{+\infty}_{-\infty}\|U(t,s)\| \|f_2(s,y_1(s))\| \,ds\,d\mu(t)\\
&=: Z_1(t)+Z_2(t).
\end{align*}
According to \eqref{40} and definition \ref{def2.4}, we conclude that
\begin{align*}
Z_1(t)
&\leq \int^{r}_{-r}\int^{t}_{-\infty}KN\exp(-\alpha(t-s))\|y_2(s)\| \,ds\,d\mu(t)\\
&\quad +\int^{r}_{-r}\int^{+\infty}_{t}KN\exp(\alpha(t-s))\|y_2(s)\| \,ds\,d\mu(t)\\
&\leq \int^{0}_{-\infty}\int^{r}_{-r}KN\|y_2(s+t)\| d\mu(t)ds\\
&\quad +\int^{+\infty}_0\int^{r}_{-r}KN\|y_2(s+t)\| d\mu(t)ds\\
&= KN\int^{+\infty}_{-\infty}\int^{r}_{-r}\|y_2(s+t)\| d\mu(t)ds.
\end{align*}
For $y_2\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$, Jo\"el Blot \cite{J.Blot} proved
 $y_2(t+s)\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$.
Using Lebesgue's dominated convergence theorem, we deduce that
\[
\lim_{r\to\infty}\frac{Z_1(t)}{\mu([-r,r])}
\leq \int^{+\infty}_{-\infty}\lim_{r\to\infty}\frac{KN}{\mu([-r,r])}
\int^{r}_{-r}\|y_2(s+t)\| d\mu(t)ds
=0.
\]
Similarly, $\lim_{r\to\infty}\frac{Z_2(t)}{\mu([-r,r])}=0$. Therefore,
we conclude that $(f*U)_2(t)$ is a $\mu$ ergodic function. Finally, we obtain that
$(f*U)(t)\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$.
\end{proof}

\section{Existence and uniqueness}\label{sec4}

\begin{theorem}\label{T.4.1}
Consider the linear system
\begin{equation}\label{30}
\dot{x}=A(t)x+f(t),
\end{equation}
where $A(t)\in\mathcal{PR}(\mathbb{R},\mathbb{R}^n )$,
$f(t)\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$.
If the linear homogeneous system $\dot{x}=A(t)x$ satisfies exponential dichotomy,
then  \eqref{30} has a unique $\mu$ pseudo rotating periodic solution.
\end{theorem}

\begin{proof}
Since $f(t)\in \mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$, we have a bounded solution
 (see \cite{Coppel})
\[
x(t)=\int^{+\infty}_{-\infty}U(t,s)g(s)ds+\int^{+\infty}_{-\infty}U(t,s)\phi(s)ds
=: x_1(t)+x_2(t),
\]
where $g\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$,
$\phi\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$.
It follows from $g(s)\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$ that
$x_1(t)\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$.

Then we will prove that $x_2(t)$ is a $\mu$ ergodic function.
By \eqref{40} and definition \ref{def2.4}, we have
\begin{align*}
&\int^{r}_{-r}\|x_2(t)\|d\mu(t)\\
&\leq \int^{r}_{-r}\int^{t}_{-\infty}K\exp(-\alpha(t-s))\|\phi(s)\|\,ds\,d\mu(t)\\
&\quad +\int^{r}_{-r}\int^{+\infty}_{t}K\exp(-\alpha(s-t))\|\phi(s)\|\,ds\,d\mu(t)\\
&\leq K\int^{r}_{-r}\int^{0}_{-\infty} \|\phi(s+t)\|\,ds\,d\mu(t)
 +K\int^{r}_{-r}\int^{+\infty}_0 \|\phi(s+t)\|\,ds\,d\mu(t)\\
&= K\int^{0}_{-\infty}\int^{r}_{-r}\|\phi(s+t)\|d\mu(t)ds
 +K\int^{+\infty}_0\int^{r}_{-r}\|\phi(s+t)\|d\mu(t)ds,
\end{align*}
by Fubini's theorem.

The translation invariance of $\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$ was
proved in \cite{J.Blot}, i.e.\ $\phi(s+t)\in\mathcal{E}(\mathbb{R},\mathbb{R}^n ,\mu)$.
So we can obtain that $\lim_{r\to+\infty}\frac{x_2(t)}{\mu([-r,r])}=0$
by Lebesgue's dominated convergence theorem. Then system \eqref{30} has a unique
 $\mu$ pseudo rotating periodic solution.
\end{proof}

\begin{theorem}\label{41}
Consider the semi-linear system
\begin{equation}\label{31}
\dot{x}=A(t)x+f(t,x),
\end{equation}
where $A(t)\in\mathcal{RP}(\mathbb{R},\mathbb{R}^n )$,
$f(t,x)\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$,
and it satisfies the Lipschitz condition in $x$.
If the linear homogeneous system $\dot{x}=A(t)x$ satisfies exponential dichotomy
and there is a Lipschitz constant $N>0$, such that $\frac{2KN}{\alpha}<1$,
then system \eqref{31} has a unique $\mu$ pseudo rotating periodic solution.
\end{theorem}

\begin{proof}
Firstly, for all $y(t)\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$, we consider
the system
\[
\dot{x}=A(t)x+f(t,y(t)),
\]
which has a bounded solution
\[
x(t)=\int^{+\infty}_{-\infty}U(t,s)f(s,y(s))ds
\]
and $x(t)\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$
by lemma \ref{lemma3.3}. Then we define the mapping
$H(y)(t):\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)
\to\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$ by
\begin{equation}
H(y)(t)=\int^{+\infty}_{-\infty}U(t,s)f(s,y(s))ds.
\end{equation}
For all $y_1(t), y_2(t)\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$, we deduce that
\begin{align*}
\|H(y_1)(t)-H(y_2)(t)\|_1
&\leq \int^{+\infty}_{-\infty}\|U(t,s)\|_1 N \|y_1(s)-y_2(s)\|_1ds\\
&\leq KN\int^{t}_{-\infty}\exp(-\alpha(t-s)) \|y_1(s)-y_2(s)\|_1ds\\
&\quad +KN\int^{+\infty}_{t}\exp(\alpha(t-s)) \|y_2(s)-y_1(s)\|_1ds\\
&\leq \frac{2KN}{\alpha}\|y_1(s)-y_2(s)\|_1.
\end{align*}
Since $\frac{2KN}{\alpha}<1$, the mapping $H(y)(t)$ is a strict contraction.
Using Banach fixed point theorem, there exists a unique
$y(t)\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$ such that $H(y)(t)=y(t)$.
So system \eqref{31} has a unique $\mu$ pseudo rotating periodic solution.
\end{proof}

In the following, we replace the Lipschitz condition by the linear growth condition.
Then the existence of $\mu$ pseudo rotating wave solutions for semi-linear systems
is also going to be proved.

\begin{corollary}
Consider system \eqref{31}, where $A(t)$ is an rotating periodic matrix,
$f(t,x)\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$, and
$f$ is uniformly continuous function in $x$. Suppose that the homogeneous linear
system $\dot{x}=A(t)x$ satisfies an exponential dichotomy,
and the function $f$ satisfies linear growth condition, i. e.
\begin{equation}
\|f(t,x)\|_1\leq a\|x\|_1+b   \quad (t\in\mathbb{R}, x\in\mathbb{R}^n ),
\end{equation}
where $a>0,\frac{2Ka}{\alpha}<1$. Then system \eqref{31} has a $\mu$ pseudo rotating
periodic solution.
\end{corollary}

\begin{proof}
Let $G:=\{y\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu) , \|y\|_1\leq N_1\}$,
where $N_1=\frac{2Kb}{\alpha-2Ka}$. We can deduce that $G$ is a closed convex subset
of the space of Banach. For all $y\in G$, we have
\begin{align*}
\|H(y)(t)\|_1
&=\|\int^{+\infty}_{-\infty}U(t,s)f(s,y(s)) ds\|_1\\
&\leq \int^{+\infty}_{-\infty}\|U(t,s)\|_1\|f(s,y(s))\|_1ds\\
&\leq (a\|y\|_1+b)(\int^{t}_{-\infty}K\exp(-\alpha(t-s))ds+\int^{+\infty}_{t}K\exp(\alpha(t-s))ds)\\
&\leq (a N_1+b)\frac{2K}{\alpha}\\
&\leq N_1.
\end{align*}
By using the uniform continuity of $f$, for all $\varepsilon>0$,
there exist $\delta>0$ such that $\|y_1-y_2\|_1<\delta$, and we have
\[
\|f(t,y_1)-f(t,y_2)\|_1<\varepsilon.
\]
From the inequality
\begin{align*}
& \|H(y_1)(t)-H(y_2)(t)\|_1\\
&\leq \|\int^{+\infty}_{-\infty}U(t,s) f(s,y_1(s)) ds-\int^{+\infty}_{-\infty}U(t,s) f(s,y_2(s)) ds\|_1\\
&\leq \int^{t}_{-\infty}\|X(t)PX^{-1}(s)\|_1\|f(s,y_1(s))-f(s,y_2(s))\|_1ds\\
&\quad +\int^{+\infty}_{t}\|X(t)(I-P)X^{-1}(s)\|_1\|f(s,y_2(s))-f(s,y_1(s))\|_1ds\\
&\leq \int^{t}_{-\infty}K\exp(-\alpha(t-s))\varepsilon ds+\int^{+\infty}_{t}K\exp(\alpha(t-s))\varepsilon ds\\
&\leq \frac{2K\varepsilon}{\alpha} ,
\end{align*}
we deduce that $H(y)(t)$ is uniformly continuous in $x$ and satisfies \eqref{31}.
Then we have
\[
\|\dot{H}(y)(t)\|_1
=\|A(t)\|_1\|H(y)(t)\|_1+\|f(t,x)\|_1
\leq (a+\sup_{t\in\mathbb{R}}|A(t)|) M_1+b.
\]
Therefore, the mapping $H(y)(t)$ is equicontinuous.
By using Arzela-Ascoli theorem, there exist a subsequence of
$\{f_{k}(t)\}\subset\{f_{n}(t)\}\in G$ such that
\[
\lim_{k\to\infty}f_{k}(t)=y_0(t),
\]
where $y_0(t)\in G$. We can prove that $H(y)(t)$ has fixed point in the space $G$
by Schauder fixed point theorem. Then the system \eqref{31} has a $\mu$ pseudo
rotating periodic solution.
\end{proof}


\begin{theorem}\label{T.4.4}
Consider the nonlinear system
\begin{equation}\label{32}
\dot{x}=f(t,x),
\end{equation}
where $f\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$,
and $f$ is a continuous differentiable function in $x$. Suppose
\begin{equation}\label{37}
\dot{x}=\int^{1}_0\frac{\partial f}{\partial x}(\theta y)d\theta \cdot x, \quad
  y\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)
\end{equation}
satisfies the exponential dichotomy, i.e.
\begin{gather*}
\|\Phi(t,y)P_{y}\Phi^{-1}(s,y)\| \leq  K\exp(-\alpha(t-s))  \quad    t\geq s,\\
\|\Phi(t,y)(I-P_{y})\Phi^{-1}(s,y)\| \leq  K\exp(-\alpha(s-t)) \quad     t\leq s,
\end{gather*}
where $\Phi(t,y)$ is a fundamental matrix of system \eqref{37}, and satisfies
$\Phi(0,x)=\operatorname{id}$. Then  \eqref{32} has a unique $\mu$ pseudo rotating
periodic solution.
\end{theorem}

\begin{proof}
Firstly, we prove uniqueness.
Suppose $x,y$ are $\mu$ pseudo rotating wave solutions of the system \eqref{32}.
Let $u=x-y$. Then we have
\begin{align*}
\dot{u}&= \dot{x}-\dot{y}=f(t,x)-f(t,y)\\
&= \int^{1}_0\frac{\partial f}{\partial x}(\theta x+(1-\theta)y)d\theta\cdot(x-y)\\
&= \int^{1}_0\frac{\partial f}{\partial x}(y+\theta u)d\theta \cdot u.
\end{align*}
Because \eqref{37} satisfies the exponential dichotomy, we get a unique solution $u=0$,
 i.e.\ $x=y$.

Next we prove existence. It is easy to obtain
\[
 \dot{x}=f(t,x)=\int^{1}_0\frac{\partial f}{\partial x}(\theta x)d\theta\cdot x+f(t,0).
\]
For all $y\in\mathcal{PRP}(\mathbb{R},\mathbb{R}^n ,\mu)$, we consider the equation
\begin{equation}\label{38}
\dot{x}=\int^{1}_0\frac{\partial f}{\partial x}(\theta y)d\theta \cdot x+f(t,0),
\end{equation}
which has a solution
\begin{align*}
x_{y}(t)
&= \int^{t}_{-\infty}\Phi(t,y)P_{y}\Phi^{-1}(s,y)f(s,0)ds\\
&\quad -\int^{+\infty}_{t}\Phi(t,y)(I-P_{y})\Phi^{-1}(s,y)f(s,0)ds.
\end{align*}
Because $f\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$,
we easily prove that $x_{y}\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu)$.
For all $s\in\mathbb{R}$, there exists a constant $M_1$ such that
$\|f(s,0)\|_1\leq M_1$.
Then we consider the set
\[
\mathcal{D}=\{h\in\mathcal{PRP}(\mathbb{R}\times\mathbb{R}^n ,\mathbb{R}^n ,\mu),
\|h\|_1\leq M\},
\]
where $M=\frac{2KM_1}{\alpha}+1$. For all $y\in\mathcal{D}$, we
define the mapping $T_{[y]}(t)= x_{y}(t)$. We have
\begin{align*}
\|T_{[y]}(t)\|_1
&\leq \int^{t}_{-\infty}\|\Phi(t,y)P_{y}\Phi^{-1}(s,y)\|_1\|f(s,0)\|_1ds\\
&\quad +\int^{\infty}_{t}\|\Phi(t,y)(I-P_{y})\Phi^{-1}(s,y)\|_1\|f(s,0)\|_1ds\\
&\leq \int^{t}_{-\infty}K\exp(-\alpha(t-s))M_1 ds+\int^{\infty}_{t}K\exp(\alpha(t-s))M_1 ds\\
&\leq \frac{2KM_1}{\alpha}\\
&\leq  M.
\end{align*}
Then $T_{[y]}:\mathcal{D}\to\mathcal{D}$ and uniformly bounded with respect to $y$.
Because $T_{[y]}$ satisfies the system \eqref{38}, we have
\begin{align*}
\|\dot{T}_{[y]}(t)\|_1
&\leq \|\int^{1}_0\frac{\partial f}{\partial x}(\theta y)d\theta\|_1\| T_{[y]}(t)\|_1+\|f(t,0)\|_1\\
&\leq \int^{1}_0\|\frac{\partial f}{\partial x}(\theta y)\|_1d\theta M+M_1.
\end{align*}
It follows from $f$ is a continuous differentiable function in $x$ that
$T_{[y]}$ is equi-continuous.
Hence,  \eqref{32} has $\mu$ pseudo rotating periodic solution by Schauder's
fixed point theorem.
\end{proof}

\section{Example}\label{sec5}

Consider the equation
\begin{equation}\label{55}
\dot{x}=-3x+\exp(-t)+x^{2}\exp(-t),
\end{equation}
for all $t\in\mathbb{R}^{+}, x\in\Omega$, where
 $\Omega(\subseteq\mathbb{R})$ is a bounded set.

We will prove that \eqref{55} has a $\mu$ pseudo rotating periodic solution.
Let
\[
f(t,x)= \left(-3x+\exp(-t)\right)+x^{2}\exp(-t)
=: g(t,x)+\phi(t,x).
\]
 For a given $\tau\in\mathbb{R}$, let
\begin{align*}
g(t+\tau,x)
&= -3x+\exp(-(t+\tau))\\
&= \exp(-\tau)(-3\exp(\tau)x+\exp(-t))\\
&= \exp(-\tau)g(t,\exp(\tau)x).
\end{align*}
Therefore, $g(t,x)\in\mathcal{PR}(\mathbb{R}^{+}\times\Omega,\Omega)$.

If the Radon-Nikodym derivative $\rho$ is defined by $\rho(t)=1$, then
\begin{align*}
\lim_{r\to+\infty}\frac{\int^{r}_0x^2\exp{(-t)}d\mu(t)}{\mu([0,r])}
&= \lim_{r\to+\infty}\frac{\int^{r}_0x^2\exp{(-t)}dt}{\int^{r}_01dt}\\
&= \lim_{r\to+\infty}\frac{x^2(1-\exp{(-r)})}{r}
= 0.
\end{align*}
Hence, $f\in\mathcal{PRP}(\mathbb{R}^{+}\times\Omega,\Omega,\mu)$.
By theorem \ref{T.4.4}, equation \eqref{55} has a unique $\mu$ pseudo rotating
periodic solution.


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\end{document}
