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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 146, pp. 1--7.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/146\hfil Weighted pseudo almost periodicity]
{Weighted pseudo almost periodic solutions for functional
differential equations}

\author[L. Zhang, Y. Xu\hfil EJDE-2007/146\hfilneg]
{Liuwei Zhang, Yuantong Xu}  

\address{Liuwei Zhang   \newline
Department of Mathematics, Sun Yat-sen University, Guangzhou
510275, China}
\email{zhangliuwei020@yahoo.com.cn}

\address{Yuantong Xu \newline
Department of Mathematics, Sun Yat-sen University, Guangzhou,
510275,  China} 
\email{xyt@mail.sysu.edu.cn}

\thanks{Submitted October 18, 2007. Published October 30, 2007.}
\thanks{Supported by grant 10471155 from  NNSF of China}
\subjclass[2000]{34K14}
\keywords{Weighted pseudo almost periodic; exponential dichotomy; finite delay}

\begin{abstract}
 By means of the exponential dichotomy method  and the properties
 of the weighted pseudo almost periodic functions, sufficient
 conditions are obtained for the existence of weighted pseudo almost
 periodic solutions of functional differential equations with finite delay.
\end{abstract}

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

\section{Introduction}

The theory of almost periodic function was created
 during 1924-1926 mainly by Harald Bohr. who gave a strong impetus
to the development of harmonic analysis on groups. In particular,
the theory of almost periodic equations has been developed in
connection with problems of differential equations, stability
theory, dynamical systems, and so on (see, e.g., \cite{l1}). The
existence of almost periodic type solutions is amongst the most
attractive topics, which arises in qualitative theory of
differential equations due to their significance and applications in
physics, mathematical biology, control theory and others. As one of
the cases, the existence of pseudo almost periodic solutions to
various differential equations has been investigated in many
papers (see, e.g.,\cite{d1,d2,d3,d4} and references therein). In
\cite{d5},  Diagnana introduced some new classes of functions
called weighted pseudo almost periodic functions. Those new
functions implement Zhang's \cite{z1} pseudo almost periodic
functions. After that, in \cite{d4}, the author investigated the
basic properties of weighted pseudo periodic functions and obtained
conditions of existence of the weighted pseudo almost periodic
solutions for abstract differential equations.

In the present paper, motivated by \cite{d1,d3}, we are concerned with
weighted pseudo almost periodic solutions of delay differential
equations. We obtain  sufficient conditions for the existence of
weighted pseudo almost periodic solution of functional differential
equations with finite delay.

\section{preliminaries}

For the reader's convenience, we recall some concepts of almost
periodicity and weighted pseudo almost periodicity, as well as some
basic facts of functional differential equations.

Throughout the paper, $ \mathbb{R}$, $\mathbb{C}$, $\mathbb{X}$
stand for the sets of real, complex numbers, and a Banach space with a
norm $\|\cdot \|$, respectively.  Let
  $(BC(\mathbb{R},\mathbb{X}),\|\cdot\|)$ denote the
  collection of all $\mathbb{X}$-valued bounded continuous functions
  equipped with the sup norm
  $\|\phi\|_{\infty}:=\sup_{t\in\mathbb{R}}\|\phi(t)\|$
  for each $\phi\in BC(\mathbb{R},\mathbb{X})$.

{\bf Definition 1.1} (\cite{l1}). \label{def1.1} \rm A continuous
function $f:\mathbb{R}\to\mathbb{X}$ is called almost periodic if it
has a relative dense set of $\epsilon$-almost periods for each
$\epsilon>0$, that is, if there is a number $l=l(\epsilon)>0$ such
that each interval $(a,a+l)\subset\mathbb{R}$ contains at least one
number $\tau=\tau(\epsilon)$ satisfying
$$
\|f(t+\tau)-f(t)\|<\epsilon
$$
for all $t\in \mathbb{R}$.
Denote by $AP(\mathbb{R},\mathbb{X})$ the
set of all such functions.


Set
$$
PAP_{0}(\mathbb{R},\mathbb{X}):=\{\varphi\in
BC(\mathbb{R},\mathbb{X}) : \lim_{T
\to\infty}\frac{1}{2T}\int_{-T}^{T}\|\varphi(t)\|dt=0 \}.$$

{\bf Definition 1.2} (\cite{z1}). \label{def1.2} \rm
 A function
$f:\mathbb{R}\to\mathbb{X}$ is called pseudo almost periodic, if
$$
f=g+\varphi,
$$
where $g\in AP(\mathbb{R},\mathbb{X})$ and $\varphi\in
PAP_{0}(\mathbb{R},\mathbb{X})$. Denote by
$PAP(\mathbb{R},\mathbb{X})$ the set of all such functions.


Let $\mathbb{U}$ be the collection of functions (weights)
$\rho:\mathbb{R}\to(0,\infty),$ which are locally integrable over
$\mathbb{R}$ such that $\rho>0$ almost everywhere. Set
\begin{gather*}
\mu(T,\rho):=\int_{-T}^{T}\rho(t)dt,\\
\mathbb{U}_{\infty}:=\{\rho\in\mathbb{U}:\lim_{T\to\infty}\mu(T,\rho)=\infty\},
\\
\mathbb{U}_{B}:=\{\rho\in\mathbb{U}_{\infty}:\rho
\text{ is bounded  with }  \inf_{t\in\mathbb{R}}\rho(t)>0\}.
\end{gather*}
Obviously,
$\mathbb{U}_{B}\subset\mathbb{U}_{\infty}\subset\mathbb{U}$, with
strict inclusions.

For $\rho\in\mathbb{U}_{\infty}$, define
$$
PAP_{0}(\mathbb{R},\mathbb{X},\rho):=\{f\in
BC(\mathbb{R},\mathbb{X}):\lim_{T\to\infty}\frac{1}{\mu(T,\rho)}
\int_{-T}^{T}\|f(t)\|\rho(t)dt=0\}.
$$

{\bf Definition 1.3} (\cite{d4}). \label{def1.3} \rm
 Let $\rho\in\mathbb{U}_{\infty}$. A
function $f\in BC(\mathbb{R},\mathbb{X})$ \ is called weighted
pseudo almost periodic or $\rho$-pseudo almost periodic if it can be
expressed as $f=g+\varphi$, where $g\in AP(\mathbb{R},\mathbb{X})$
\text{ and }$\varphi \in PAP_{0}(\mathbb{R},\mathbb{X},\rho)$.
The collection of such functions will be denoted by
$PAP(\mathbb{R},\mathbb{X},\rho)$


Let $\rho_{1},\rho_{2}\in\mathbb{U}_{\infty}$. One says that
$\rho_{1}$\ is equivalent to $\rho_{2}$, denoting this as
$\rho_{1}\prec \rho_{2}$, if
$\frac{\rho_{1}}{\rho_{2}}\in\mathbb{U}_{B}$.

\begin{remark}\label{rmk2.1} \rm
(i) Let $\rho_{1},\rho_{2},\rho_{3}\in\mathbb{U}_{\infty}$. It's clear
that $\rho_{1}\prec\rho_{1}$ (reflexivity); if
$\rho_{1}\prec\rho_{2}$, then $\rho_{2}\prec\rho_{1}$ (symmetry); and
if $\rho_{1}\prec\rho_{2}$ and $\rho_{2}\prec\rho_{3}$, then
$\rho_{1}\prec\rho_{3}$ (transitivity). So, $\prec$ is a binary
equivalence relation on $\mathbb{U}_{\infty}$. Thus the equivalence
class of a given weighted $\rho\in\mathbb{U}_{\infty}$ will then be
denoted by
$$
cl(\rho)=\{\varpi\in\mathbb{U}_{\infty}:\rho\in\varpi\}.
$$
It is then clear that
$\mathbb{U}_{\infty}=\underset{\rho\in\mathbb{U}_{\infty}}{\bigcup}cl(\rho)$.

(ii) Let $\rho\in\mathbb{U}_{\infty}$.  If $\rho_{1},\rho_{2}\in
cl(\rho)$, then
$PAP(\mathbb{R},\mathbb{X},\rho_{1})=PAP(\mathbb{R},\mathbb{X},\rho_{2})$.
In particular, if $\rho\in\mathbb{U}_{B}$, then
$PAP(\mathbb{R},\mathbb{X},\rho)=PAP(\mathbb{R},\mathbb{X},cl(1))
=PAP(\mathbb{R},\mathbb{X})$.
\end{remark}


\section{Autonomous Linear Equation}

We are concerned with the linear delay differential equations
\begin{equation}
\frac{dx(t)}{dt}=Lx_{t}+f(t)\label{e3.1}
\end{equation}
and
\begin{equation}
\frac{dx(t)}{dt}=Lx_{t}.\label{e3.2}
\end{equation}
Let $r$ be a fixed positive number,and  $C=C([-r,0],\mathbb{R}^{n})$
be the Banach space of continuous functions on $[-r,0]$ with norm
$\|\varphi\|=\sup\{|\varphi(\theta)|:-r\leq\theta\leq 0\}.$ If $x$
is a continuous map of $[a-r,b]$ into $\mathbb{R}^{n}$, then
$x_{t}\in C$ is given, for each $a\leq t\leq b,$  by
$$
x_{t}(\theta)=x(t+\theta),   -r\leq\theta\leq 0.
$$
Let L be a continuous linear operator from
$C=C([-r,0],\mathbb{R}^{n})$ into $\mathbb{R}^{n}.$ The hypothesis
on $L$ implies that there exists an $n\times n$ matrix
$\eta(\theta):-r\leq\theta\leq 0$, whose elements are of bounded
variation, and
\begin{equation}
L(\phi)=\int_{-r}^{0}d[\eta(\theta)]\phi(\theta)\label{e3.3}.
\end{equation}

If $\phi$ is any given function in $C$ and $x_{t}(\sigma,\phi)$ is
the unique solution of \eqref{e3.2} with initial data at $\sigma$,
the solution operator $T(t)$ is given by
$$
T(t)\phi=x_{t}(\sigma,\phi),  t\geq \sigma.
$$

It is well-known that the solution operator $(T(t))_{t\geq \sigma}$
is a $C_{0}$-semigroup with infinitesimal generator
$$
D(A)=\{{\phi}\in C:{\frac{d\phi}{d\theta}\in C \text{ and }
{\frac{d\phi}{d\theta}(0)}=L(\phi)} \text{ and }
A\phi=\frac{d\phi}{d\theta}\}.
$$
Furthermore the spectrum
$\sigma(A)$ of $A$ is given by
$$
\sigma(A)=\{\lambda\in \mathbb{C}:\det\Delta(\lambda)=0\},$$
where
$$
\Delta(\lambda)=\lambda
I-\int_{-r}^{0}e^{\lambda\theta}d\eta(\theta).
$$

{\bf Lemma 1} (\cite{h2}). \label{lem1}
 Let $\Lambda=\{\lambda\in \sigma(A):Re\lambda>0\}$ and suppose
 $C$ is decomposed by $\Lambda$
as
$$
C=P_{\Lambda}\oplus Q_{\Lambda},
$$
where the definition of $P_{\Lambda}$ and $Q_{\Lambda}$ one can
refer \cite[p.212]{h2}. Then there exit positive constants $k$ and
$c$ such that
\begin{gather*}
\|T(t)\phi^{P_{\Lambda}}\|\leq ke^{ct},  t\leq 0, \\
\|T(t)\phi^{Q_{\Lambda}}\|\leq ke^{-ct},  t\geq 0.
\end{gather*}


{\bf Lemma 2} (\cite{h2}). \label{lem2} Assume that
$\sigma(A)\bigcap i\mathbb{R}=\emptyset$, then for all bounded
functions $f$, \eqref{e3.1} has one and only one bounded solution.


The solution of \eqref{e3.1} with initial value $\phi$ at $\sigma$
is
$$
x_{t}=T(t-\sigma)\phi+\int_{\sigma}^{t}d[K(t,s)]f(s),
$$
where
$$
K(t,s)(\theta)=\int_{\sigma}^{s}X(t+\theta-\alpha)d\alpha,
t\geq\sigma,
$$
and
$X(\cdot)$ denotes the fundamental matrix
solution of system \eqref{e3.2}.

Assume that $C$ is decomposed by $\Lambda$ as $P\oplus Q$, then
the solution is given by
$$
x_{t} =x_{t}^{P}+x_{t}^{Q},
$$
where
\begin{gather*}
x_{t}^{P}=T(t-\sigma)\phi^{P}+\int_{\sigma}^{t}T(t-s)X_{0}^{P}f(s)ds,
t\geq\sigma,
\\
x_{t}^{Q}=T(t-\sigma)\phi^{Q}+\int_{\sigma}^{t}d[K(t,s)^{Q}]f(s)ds,
t\geq\sigma.
\end{gather*}
 We know from \cite{h2} that there exist
$\Psi$,$\Upsilon\in Q$ such that
\begin{equation}
K(t,s)^{Q}= \begin{cases}
\int_{t-s-r}^{t-r}T(\beta)\Psi d\beta & \text{if }t-s\geq r\\
\int_{0}^{t-r}T(\beta)\Psi d\beta+\Upsilon & \text{if } t-s\leq r.
\end{cases} \label{e3.4}
\end{equation}
For more details on this facts,  can be found in \cite{h1} and
\cite{h2}.
Now, we are in a position to present our main result.

\begin{theorem} \label{thm3.1}
Assume that $\sigma(A)\bigcap i\mathbb{R}=\emptyset$, if $f$ is
$\rho$-pseudo almost periodic and
$$
\sup_{T>0}\{\int_{-T}^{T}e^{-c(T+t)}\rho(t)dt\}<\infty,\eqno{(H)}\label{eH}
$$
then \eqref{e3.1} has one and only one bounded solution which is also
$cl(\rho)$-pseudo almost periodic.
\end{theorem}

\begin{proof}
By Lemma 2, \eqref{e3.1} has one and only one solution which is
$$
x_{t}=x_{t}^{P}+x_{t}^{Q}.
$$
 We will show that both
$$
\int_{-\infty}^{t}T(t-s)X_{0}^{P}f(s)ds\quad\text{and}\quad
\int_{-\infty}^{t}d[K(t,s)^{Q}]f(s)
$$
are $\rho$-pseudo almost periodic.Assume that $f=g+\phi$, where $g$
is its almost periodic component and $\phi$ satisfies
$$
\lim_{T\to\infty}\frac{1}{\mu(T,\rho)}\int_{-T}^{T}\|\phi(t)\|\rho(t)dt=0.
$$
We will  show that
$$
\int_{-\infty}^{t}T(t-s)X_{0}^{P}g(s)ds\quad\text{and}\quad
\int_{-\infty}^{t}d[K(t,s)^{Q}]g(s)ds$$
are almost periodic.
Let $\alpha=(\alpha_{n})$ be a real sequence. By
almost periodicity of $f$, there exists a subsequence of $\alpha$
noted by $\alpha{'}$ and a continuous function $h(t)$ such that
$$
h(t)=\lim_{n\to\infty}g(t+\alpha{'}) \quad \text{uniformly  in }  \mathbb{R}.
$$
So,
$$
\int_{-\infty}^{t}T(t-s)X_{0}^{P}g(s+\alpha_{n}')ds\to
\int_{-\infty}^{t}T(t-s)X_{0}^{P}h(s)ds
$$
uniformly in\ $\mathbb{R}$, and
$$
\int_{-\infty}^{t}d[K(t,s+\alpha_{n}')^{Q}]g(s+\alpha_{n}')
\to\int_{-\infty}^{t}d[K(t,s)^{Q}]h(s)
$$
uniformly in $\mathbb{R}$.
Thus
$$
\int_{-\infty}^{t}T(t-s)X_{0}^{P}g(s)ds
$$
and
$$
\int_{-\infty}^{t}d[K(t,s)^{Q}]g(s)
$$
are almost periodic. It remains to show that
\begin{equation}
\lim_{T\to\infty}\frac{1}{\mu(T,\rho)}\int_{-T}^{T}\|
\int_{-\infty}^{t}T(t-s)X_{0}^{P}\phi(s)ds\|\rho(t)dt=0,
\label{e3.5}
\end{equation}
and
\begin{equation}
\lim_{T\to\infty}\frac{1}{\mu(T,\rho)}\int_{-T}^{T}\|\int_{-\infty}^{t}d[K(t,s)^{Q}]\phi(s)\|\rho(t)dt=0.
\label{e3.6}
\end{equation}
Let us put
$$
I(t)=\int_{-\infty}^{t}T(t-s)X_{0}^{P}\phi(s)ds$$
and
$$
J(t)=\int_{-\infty}^{t}d[K(t,s)^{Q}]\phi(s).
$$
Because of \eqref{e3.4}, there exists a positive constant $M$ such
that
\begin{equation}
\begin{split}
&\lim_{T\to\infty}\frac{1}{\mu(T,\rho)}\int_{-T}^{T}\|J(t)\|\rho(t)dt\\
&\leq\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\int_{-\infty}^{t}\exp(-c(t-s))\|\phi(s)\|\rho(t)dsdt
\\ &=\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)\int_{-\infty}^{-T}\exp(-c(t-s))\|\phi(s)\|dsdt \\
&\quad +\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)\int_{-T}^{t}\exp(-c(t-s))\|\phi(s)\|dsdt\label{e3.7}
\end{split}
\end{equation}
Note that
\begin{equation}
\begin{split}
J_{1}
&=\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)
 \int_{-T}^{t}\exp(-c(t-s))\|\phi(s)\|dsdt
\\
&=\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)dt
 \int_{-T}^{t}\exp(-c(t-s))\|\phi(s)\|ds
\\
&=\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)\|\phi(t)\|dt
 \int_{-T}^{t}\exp(-c(t-s))ds\\
&=\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)\|\phi(t)\|dt
\{\frac{1}{c}[1-\exp(-c(t+T))]\}.
\label{e1.8}
\end{split}
\end{equation}
Since $-T\leq t\leq T$ and $c>0$, it follows that
$\frac{1}{c}[1-\exp(-c(t+T))]$ is bounded. Note that
$\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\|\phi(t)\|\rho(t)dt=0$.
Then it follows that $J_{1}=0$. Also, by $(H)$ we have
\begin{equation}
\begin{split}
J_{2}=
&\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)\int_{-\infty}^{-T}\exp(-c(t-s))\|\phi(s)\|dsdt
\\
&=\lim_{T\to\infty}\frac{M}{\mu(T,\rho)}\int_{-T}^{T}\rho(t)e^{-ct}dt\int_{-\infty}^{-T}e^{cs}\|\phi(s)\|ds
\\
&
\leq\lim_{T\to\infty}\{\frac{M\sup_{s\in\mathbb{R}}\|\phi(s)\|\int_{-T}^{T}\rho(t)e^{-c(c+T)}dt}{c\mu(T,\rho)}\}\\
& =0.\label{e1.9}
\end{split}
\end{equation}
 The proof of Theorem 1 is complete.

\end{proof}

\section{Nonautonomous Linear Equation}

 In this section we
consider the linear delay differential equation
\begin{equation}
\frac{dx(t)}{dt}=L(t)x_{t}.\label{e4.1}
\end{equation}

We say that linear system \eqref{e4.1} admits an exponential
dichotomy on $\mathbb{R}$, if the solution operators $T(t,s)$
satisfies the following property:

there exist positive constants $k>1,c>0$, and a projection operator
$P(s):X\to X,(P(s)=P^{2}(s)),s\in\mathbb{R}$, such that if
$Q(s)=I-P(s)$, then

(i) $T(t,s)P(s)=P(t)T(t,s), t\geq s;$

(ii) The restriction $T(t,s)R(Q(s)),t\geq s$, is an isomorphism of
$R(Q(s))$ onto $R(Q(t))$ and we define $T(s,t)$ as the inverse
mapping;

(iii) $\|T(t,s)P(s)\|\leq\exp(-c(t-s))$, for $s\leq t$;

(iv) $\|T(t,s)Q(s)\|\leq\exp(-c(t-s))$, for $s\geq t.$

We will apply the exponential dichotomy theory to prove the
existence of $\rho$-pseudo almost periodic solutions for the
following delay differential equations \eqref{e4.2} with
$\rho$-pseudo almost periodic coefficients.

For $(\sigma,\phi)\in\mathbb{R}\times C$, consider the linear system
\begin{equation}
\frac{dx(t)}{dt}=L(t)x_{t}+f(t),  t\geq\sigma\label{e4.2}
\end{equation}
$$ x_{\sigma}=\phi.$$

\begin{theorem}
Assume that $L(t)$ is almost periodic in $t$ and \eqref{e4.1} has an
exponential dichotomy. Then, for any $f\in
BC(\mathbb{R},\mathbb{R})$, it follows that \eqref{e4.2} has an
unique solution $Sf\in BC(\mathbb{R},C)$. Moreover, if $f$ is
$\rho$-pseudo almost periodic and condition $(H)$ is satisfied, then
$Sf\in PAP(\mathbb{R},\mathbb{X},cl(\rho))$.
\end{theorem}

\begin{proof}
As we know in \cite{h2} for all bounded functions $f$, the unique
bounded solution of \eqref{e4.2} is given by
$$
Sf(t)=\int_{-\infty}^{t}d[P(s)K(t,s)]f(s)-\int_{t}^{+\infty}d[Q(s)K(t,s)]f(s).
$$
where the function$K(t,s)$ is defined by
$$
K(t,s)=\int_{\sigma}^{s}X(t+\theta,\alpha)d\alpha,
$$
and $K(t,s)$ is the fundamental matrix solution to system
\eqref{e4.1}.

If $f$ is $\rho$-pseudo almost periodic, then $f(t)=g(t)+\phi(t)$,
where $g \in AP(\mathbb{R},\mathbb{R}^{n})$ ,and $\rho\in
PAP(\mathbb{R},\mathbb{R}^{n},\rho)$. By the exponential dichotomy
property, we can show similarly to the one given in Theorem 1 that
$Sg\in AP(\mathbb{R},\mathbb{R}^{n})$, $S\phi\in PAP(\mathbb{R},
\mathbb{R}^{n},cl(\rho))$, respectively.
\end{proof}

\begin{remark} \label{rmk4.2} \rm
In the particular case when $\rho=1$; that
  is, $PAP(\mathbb{R},\mathbb{R}^{n},cl(\rho))=PAP(\mathbb{R},\mathbb{R}^{n})$
by Remark 2, we retrieve the pseudo almost periodic situation,sine
condition $(H)$ is always achieved in that event. This means that
Theorem 1 and Theorem 2 we obtained here are good generalizations of
Theorem 3.4 and Proposition 4.2 in \cite{d1}.
\end{remark}

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\end{document}