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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 28, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/28\hfil
Doubly-weighted pseudo almost periodic solutions]
{Existence of doubly-weighted pseudo almost periodic
solutions to non-autonomous differential equations}

\author[T. Diagana\hfil EJDE-2011/28\hfilneg]
{Toka Diagana}

\address{Toka Diagana \newline
Department of Mathematics, Howard University,
2441 6th Street NW, Washington, DC 20059, USA}
\email{tdiagana@howard.edu}


\thanks{Submitted December 22, 2010. Published February 15, 2011.}
\subjclass[2000]{35B15, 34D09, 58D25, 42A75, 37L05}
\keywords{Weighted pseudo-almost periodic; doubly-weighted
Bohr spectrum; \hfill\break\indent
almost periodic; doubly-weighted pseudo-almost periodic}

\begin{abstract}
 First we show that if the doubly-weighted Bohr spectrum
 of an almost periodic function exists, then it is either
 empty or coincides with the Bohr spectrum of that function.
 Next, we investigate the existence of doubly-weighted
 pseudo-almost periodic solutions to some non-autonomous
 abstract differential equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

Motivated by the functional structure of the so-called weighted
Morrey spaces \cite{KA}, in Diagana \cite{DW}, a new concept
called doubly-weighted pseudo-almost periodicity, which
generalizes in a natural fashion the notion of weighted
pseudo-almost  periodicity is introduced and studied. Among other
things, in \cite{DW},  properties of these new functions have been
studied including the stability of the convolution operator, the
translation-invariance, the existence of a doubly-weighted mean
for almost periodic functions under some reasonable assumptions,
the uniqueness of the decomposition involving these new functions
as well as some results on their composition.

The main objective of this paper is twofold. We first show that if
the doubly-weighted Bohr spectrum of an almost periodic function
exists, then it is either empty or coincides with the Bohr
spectrum of that function. Next, we investigate the problem which
consists of the existence of doubly-weighted pseudo-almost
periodic mild solutions to the non-autonomous abstract
differential equations
\begin{equation} \label{2}
u'(t) = A(t) u(t) + g(t, u(t)),\quad t \in \mathbb{R},
\end{equation}
where $A(t)$ for $t\in \mathbb{R}$ is a family of closed linear operators
on $D(A(t))$ satisfying the well-known Acquistapace-Terreni
conditions, and $g: \mathbb{R} \times \mathbb{X} \to \mathbb{X}$ is doubly-weighted
pseudo-almost periodic in $t \in \mathbb{R}$ uniformly in the
second variable.

It is well-known that in this case there exists
an evolution family $\mathcal{U} =\{U(t,s)\}_{t \geq s}$
associated with the family of linear operators $A(t)$. Assuming that the
evolution family $\mathcal{U} =\{U(t,s)\}_{t \geq s}$ is exponentially
dichotomic and under some additional assumptions it
will be shown that  \eqref{2} has a unique doubly-weighted
pseudo-almost periodic solution.


The existence of weighted pseudo-almost periodic, weighted pseudo-almost automorphic,
and pseudo-almost periodic solutions to differential equations
constitutes one of the most attractive topics in qualitative theory of
differential equations due to possible applications. Some
contributions on weighted pseudo-almost periodic functions,
their extensions, and their applications to differential equations
have recently been made, among them are for instance
\cite{AG1,BB,BE,BE2,DMN,TO,TOK,DD,liang,L,LS,z4,LI}
and the references therein.
However, the problem which consists of the
existence of doubly-weighted pseudo-almost periodic(mild) solutions to evolution
equations in the form \eqref{2} is quite new and
untreated and thus constitutes one of the main
motivations of the present paper.

The paper is organized as follows: Section 2 is devoted to
preliminaries results related to the existence of an evolution
family, intermediate spaces, properties of weights, and basic definitions 
and results on the concept of doubly-weighted pseudo-almost periodic functions.
Section 3 is devoted to the existence of a doubly-weighted Bohr spectral 
theory for almost periodic functions while Section 4 is devoted to the 
existence of doubly-weighted pseudo-almost periodic solutions to \eqref{2}.

\section{Preliminaries}

Let $(\mathbb{X}, \|\cdot\|)$ be a Banach space. If $C$ is a linear
operator on $\mathbb{X}$, then $D(C)$, $\rho(C)$, and $\sigma(C)$  stand
respectively for the domain, resolvent, and spectrum of $C$.
Similarly, one sets $R(\lambda, C) := (\lambda I - C)^{-1}$ for
all $\lambda \in \rho(C)$ where $I$ is the identity operator for $\mathbb{X}$.
Furthermore, we set $Q=I-P$ for a projection $P$. We denote the
Banach algebra of bounded linear operators on $\mathbb{X}$ equipped with
its natural norm by $B(\mathbb{X})$.


If $\mathbb{Y}$ is another Banach space, we then let $BC(\mathbb{R} , \mathbb{X})$
(respectively, $BC(\mathbb{R} \times \mathbb{Y}, \mathbb{X})$) denote the collection of
all $\mathbb{X}$-valued bounded continuous functions and equip it with the
sup norm (respectively, the space of jointly bounded continuous
functions $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$).

The space $BC(\mathbb{R}, \mathbb{X})$
equipped with the sup norm is a
Banach space. Furthermore, $C(\mathbb{R}, \mathbb{Y})$ (respectively, $C(\mathbb{R} \times
\mathbb{Y}, \mathbb{X})$) denotes the class of continuous functions from $\mathbb{R}$ into
$\mathbb{Y}$ (respectively, the class of jointly continuous functions $F:
\mathbb{R} \times \mathbb{Y} \to \mathbb{X}$).

\subsection{Evolution Families}\label{EF}

The setting of this subsection follows that of Baroun {\it et al.}
\cite{W} and Diagana \cite{TOK}. Fix once and for all a Banach
space $(\mathbb{X}, \|\cdot\|)$.

\begin{definition}\label{DEF} \rm
A family of closed linear operators
$A(t)$ for $t\in \mathbb{R}$ on $\mathbb{X}$ with domain $D(A(t))$ (possibly not
densely defined) satisfy the so-called Acquistapace and Terreni
conditions, if there exist constants $\omega\in \mathbb{R}$, $\theta
\in (\pi/2,\pi)$, $L > 0$ and $\mu, \nu \in (0,1]$ with
$\mu + \nu > 1$ such that
\begin{equation}\label{AT1}
\Sigma_\theta \cup \{0\} \subset \rho(A(t)-\omega) \ni \lambda,\quad
  \|R(\lambda,A(t)-\omega)\|\le \frac{K}{1+|\lambda|} \quad \text{for all }
   t \in \mathbb{R}, \end{equation}
   and
\begin{equation}\label{AT2}
\|(A(t)-\omega)R(\lambda,A(t)-\omega)\,[R(\omega,A(t))-R(\omega,A(s))]\|
  \le L\, \frac{|t-s|^\mu}{|\lambda|^{\nu}}
  \end{equation}
for $t,s\in\mathbb{R}$, $ \lambda \in\Sigma_\theta:=
\{\lambda\in\mathbb{C}\setminus\{0\}: |\arg \lambda|\le\theta\}$.
\end{definition}

For a given family of linear operators $A(t)$, the existence of
an evolution family associated with it is not always guaranteed.
However, if $A(t)$ satisfies Acquistapace-Terreni, then there
exists a unique evolution family
$$
\mathcal{U}= \{U(t,s): t, s \in \mathbb{R} \text{ such that } t \geq s\}
$$
on $\mathbb{X}$ associated with $A(t)$ such that $U(t, s)\mathbb{X} \subseteq D(A(t))$
for all $t, s \in \mathbb{R}$ with $t \geq s$,
and

\begin{enumerate}
\item[(a)]  $U(t,s)U(s,r)=U(t,r)$ for $t,s \in \mathbb{R}$ such that
 $t \geq s \geq s$;

\item[(b)] $U(t,t)=I$ for $t \in \mathbb{R}$ where $I$ is the identity
 operator of $\mathbb{X}$;

\item[(c)] $(t,s)\to U(t,s)\in B(\mathbb{X})$ is continuous for $t>s$;

\item[(d)] $U(\cdot,s)\in C^1((s,\infty),B(\mathbb{X}))$, $
\frac{\partial U}{\partial t}(t,s) =A(t)U(t,s)$ and
\begin{align*}\label{au}
  \|A(t)^k U(t,s)\|&\le K\,(t-s)^{-k}
  \end{align*}
for $0< t-s\le 1$ and $k=0,1$.
\end{enumerate}


\begin{definition} \rm
An evolution family $\mathcal{U} = \{U(t,s): t, s \in \mathbb{R}
 \text{ such that } t \geq s\}$ is said to have an {\it exponential
  dichotomy} (or is {\it hyperbolic}) if there are projections
$P(t)$ ($t\in\mathbb{R}$) that are uniformly bounded and strongly
continuous in $t$ and constants $\delta>0$  and $N\ge1$ such that
\begin{enumerate}
\item[(e)] $U(t,s)P(s) = P(t)U(t,s)$; \item[(f)] the restriction
$U_Q(t,s):Q(s)\mathbb{X}\to Q(t)\mathbb{X}$ of $U(t,s)$ is
  invertible (we then set $\widetilde{U}_Q(s,t):=U_Q(t,s)^{-1}$); and
\item[(g)] $\|U(t,s)P(s)\| \le Ne^{-\delta (t-s)}$ and
  $\|\widetilde{U}_Q(s,t)Q(t)\|\le Ne^{-\delta (t-s)}$ for $t\ge s$
and $t,s\in \mathbb{R}$.
\end{enumerate}
\end{definition}

This setting requires some estimates related to
$\mathcal{U} =\{U(t,s)\}_{t \geq s}$. For that, we
introduce the interpolation spaces for $A(t)$.

Let $A$ be a sectorial operator on $\mathbb{X}$
(in Definition \ref{DEF}, replace
$A(t)$ with $A$) and let $\alpha\in(0,1)$. Define
the real interpolation space
$$
\mathbb{X}^A_{\alpha}: = \big\{x\in \mathbb{X}: \|x\|^A_{\alpha}:=
\sup\nolimits_{r>0}
\|r^{\alpha}(A-\omega)R(r,A-\omega)x\|<\infty\big\},
$$
which, by the way, is a Banach space when endowed with the
norm $\|\cdot\|^A_{\alpha}$. For convenience we further write
$$
\mathbb{X}_0^A:=\mathbb{X}, \quad \|x\|_0^A:=\|x\|, \quad \mathbb{X}_1^A:=D(A)
$$
and $\|x\|^A_{1}:=\|(\omega-A)x\|$. Moreover, let
$\hat{\mathbb{X}}^A:=\overline{D(A)}$ of $\mathbb{X}$.

\begin{definition} \rm
Given a family of linear operators $A(t)$ for $t\in \mathbb{R}$
satisfying the Acquistapace-Terreni conditions, we set
$\mathbb{X}^t_\alpha:=\mathbb{X}_\alpha^{A(t)}$ and $
\hat{\mathbb{X}}^t:=\hat{\mathbb{X}}^{A(t)}$ for $0\le \alpha\le 1$ and
$t\in\mathbb{R}$, with the corresponding norms.
\end{definition}


\begin{proposition}[\cite{W}]\label{pes}
For $x \in \mathbb{X}$, $ 0\leq \alpha \leq 1$ and $t > s$, the following
hold:
\begin{enumerate}
\item[(i)] There is a constant $c(\alpha)$, such that %%
 \begin{equation}\label{eq1.1}
  \|U(t,s)P(s)x\|_{\alpha}^t\leq
 c(\alpha)e^{- \frac{\delta}{2}(t-s)}(t-s)^{-\alpha} \|x\|.
  \end{equation}

\item[(ii)] There is a constant $m(\alpha)$, such that
 \begin{equation}\label{eq2.1}
 \|\widetilde{U}_{Q}(s,t)Q(t)x\|_{\alpha}^s\leq
 m(\alpha)e^{-\delta (t-s)}\|x\|, \quad t \leq s.
 \end{equation}
 \end{enumerate}
\end{proposition}

\subsection{Properties of Weights}

This subsection is similar to the one given in Diagana \cite{DW}
except that most of all the proofs will be omitted.

Let $\mathbb{U}$ denote the collection of functions (weights)
$\rho: \mathbb{R} \to (0, \infty)$, which are locally integrable over
$\mathbb{R}$ such that $\rho > 0$ almost everywhere.

In the rest of the paper, if $\mu \in \mathbb{U}$,
$T > 0$, and $a \in \mathbb{R}$, we then set
$Q_T := [-T, T]$, $Q_T + a := [-T+a, T+a]$, and
$$
\mu(Q_T) := \int_{Q_T} \mu(x) dx.
$$

Here as in the particular case when $\mu(x) = 1$ for each $x \in \mathbb{R}$,
we are exclusively interested in the weights $\mu$ for which,
$$
\lim_{T \to \infty} \mu(Q_T) = \infty.
$$
Consequently, we define the space of weights $\mathbb{U}_\infty$ by
$$
\mathbb{U}_\infty : = \big\{ \mu \in \mathbb{U}:  \inf_{x\in \mathbb{R}} \mu(x)
= \mu_0 > 0  \text{ and } \lim_{T \to \infty} \mu(Q_T)
= \infty\big\}.
$$
In addition to the above, we define the set
of weights $\mathbb{U}_B$ by
$$
\mathbb{U}_B := \big\{\mu \in \mathbb{U}_\infty: \ \sup_{x \in \mathbb{R}} \mu(x)
= \mu_1 < \infty \big\}.
$$
We also need the following set of weights, which makes
the spaces of weighted pseudo-almost periodic functions
translation-invariant,
$$
\mathbb{U}_\infty^{\rm Inv} := \big\{\mu \in \mathbb{U}_\infty:  \lim_{x \to \infty}
\frac{\mu(x+\tau)}{\mu(x)} < \infty \text{ and }
 \lim_{T \to \infty} \frac{\mu(Q_{T+\tau})}{\mu(Q_T)}
< \infty  \text{ for all } \tau \in \mathbb{R}\big\}.
$$
Let $\mathbb{U}_\infty^c$ denote the collection of all continuous functions
(weights) $\mu: \mathbb{R} \to (0, \infty)$ such that $\mu > 0$ almost
everywhere.

Define
$$
\mathbb{U}_\infty^s := \big\{\mu \in \mathbb{U}_\infty^c \cap \mathbb{U}_\infty: \ \lim_{x \to \infty}
\frac{\mu(x+\tau)}{\mu(x)} < \infty \text{ for all }
 \tau \in \mathbb{R}\big\}.
$$

\begin{lemma}[\cite{DW}] \label{TL}
The inclusion $\mathbb{U}_\infty^s \subset \mathbb{U}_\infty^{\rm Inv}$ holds.
\end{lemma}

\begin{definition} \rm
Let $\mu, \nu \in \mathbb{U}_\infty$. One says that $\mu$ is equivalent
to $\nu$ and denote it $\mu \prec \nu$, if $
\frac{\mu}{\nu} \in \mathbb{U}_B$.
\end{definition}

Let $\mu, \nu, \gamma \in \mathbb{U}_\infty$. It is clear that $\mu
\prec \mu$ (reflexivity); if $\mu \prec \nu$, then
$\nu \prec \mu$ (symmetry); and if $\mu \prec \nu$ and
$\nu \prec \gamma$, then $\mu\prec \gamma$ (transitivity).
Therefore, $\prec$ is a binary equivalence relation on $\mathbb{U}_\infty$.

\begin{proposition}\label{TOD}
Let $\mu, \nu \in \mathbb{U}_\infty^{\rm Inv}$. If $\mu \prec \nu$, then
 $\sigma = \mu + \nu \in  \mathbb{U}_\infty^{\rm Inv}$.
\end{proposition}

\begin{proposition}\label{TOKA}
 Let $\mu, \nu \in \mathbb{U}_\infty^s$. Then their product
 $\pi = \mu \nu \in  \mathbb{U}_\infty^s$. Moreover, if $\mu \prec \nu$, then
$\sigma : = \mu + \nu \in \mathbb{U}_\infty^s$.
\end{proposition}

The next theorem describes all the nonconstant polynomials
belonging to the set of weights $\mathbb{U}_\infty$.

\begin{theorem}[\cite{DW}]
If $\mu \in \mathbb{U}_\infty$ is a nonconstant polynomial of degree $N$,
then $N$ is necessarily even ($N = 2n'$ for some nonnegative
integer $n'$). More precisely, $\mu$ can be written in the
form
$$
\mu(x) = a  \prod_{k=0}^{n} (x^2 + a_k x + b_k)^{m_k}
$$
where $a> 0$ is a constant, $a_k$ and $b_k$ are some
real numbers satisfying $a_k^2 - 4b_k < 0$, and $m_k$ are
nonnegative integers for $k =0, \dots, n$.
Furthermore,  the weight $\mu$ given above belongs to $\mathbb{U}_\infty^s$.
\end{theorem}

\subsection{Doubly-weighted pseudo-almost periodic functions}

\begin{definition}\label{D} \rm
A function $f \in C(\mathbb{R} , \mathbb{X})$ is called (Bohr) almost periodic if
for each $\varepsilon > 0$ there exists
$l(\varepsilon) > 0$ such that every interval of length  $l(\varepsilon)
$ contains a number $\tau$ with the property that
$$
\|f(t +\tau) - f(t) \| < \varepsilon \quad \text{for each }
t \in \mathbb{R}.
$$
\end{definition}

The collection of all almost periodic functions will be
denoted $AP(\mathbb{X})$.

\begin{definition}\label{D2} \rm
A function $F \in C(\mathbb{R} \times \mathbb{Y}, \mathbb{X})$ is called (Bohr) almost
periodic in $t \in \mathbb{R}$ uniformly in $y \in \mathbb{Y}$ if for each
$\varepsilon > 0$ and any compact $K \subset \mathbb{Y}$
there exists $l(\varepsilon)$ such that every interval of length  $l(\varepsilon)$ contains
a number $\tau$ with the property that
$$
\|F(t + \tau, y) - F(t, y)\| < \varepsilon \quad\text{for each }
 t \in \mathbb{R}, \; y \in K.
$$
The collection of those functions is denoted by $AP(\mathbb{Y}, \mathbb{X})$.
\end{definition}


If $\mu, \nu \in \mathbb{U}_\infty$, we then define
$$
PAP_0(\mathbb{X}, \mu, \nu) := \big\{ f \in BC(\mathbb{R} , \mathbb{X}):
\lim_{T \to \infty} {\frac{1}{\mu(Q_T)}} \int_{Q_T} \|
f(\sigma)\|  \nu(\sigma) \, d\sigma = 0\big\}.
$$

Similarly, we define $PAP_0(\mathbb{Y}, \mathbb{X}, \mu, \nu)$ as the collection
of jointly continuous functions $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ such
that $F(\cdot, y)$ is bounded for each $y \in \mathbb{Y}$ and
$$
\lim_{T \to \infty} {\frac{1}{\mu(Q_T)}}
\big\{\int_{Q_T} \| F(s, y)\| \, \nu(s) \, ds\big\} = 0
$$
uniformly in $y \in \mathbb{Y}$.


\begin{definition}\label{DD} \rm
Let $\mu, \nu \in \mathbb{U}_\infty$. A function $f \in C(\mathbb{R} , \mathbb{X})$ is called
doubly-weighted pseudo-almost periodic if it can be expressed
as $f = g + \phi$, where $g \in AP(\mathbb{X})$ and $\phi \in PAP_0(\mathbb{X},
\mu, \nu)$. The collection of such functions will be denoted by
$PAP({\mathbb X}, \mu, \nu)$.
\end{definition}


\begin{definition}\label{KK} \rm
Let $\mu, \nu \in \mathbb{U}_\infty$.
A function $F \in C(\mathbb{R} \times \mathbb{Y} , \mathbb{X})$ is called doubly-weighted pseudo-almost
periodic if it can be expressed
as $F= G + \Phi$, where $G \in AP(\mathbb{Y}, \mathbb{X})$ and $\Phi \in PAP_0(\mathbb{Y}, \mathbb{X},
\mu, \nu)$. The collection of such functions will be denoted by
$PAP(\mathbb{Y}, {\mathbb X}, \mu, \nu)$.
\end{definition}


\begin{proposition}[\cite{DW}]\label{P26}
Let $\mu \in \mathbb{U}_\infty$ and let $\nu \in \mathbb{U}_\infty^{\rm Inv}$ such that
\begin{equation} \label{HHH}
\sup_{T > 0}\Big[{\frac{\nu(Q_{T})}{\mu(Q_T)}}\Big]< \infty.
\end{equation}
Let $f \in PAP_0(\mathbb{R}, \mu, \nu)$ and let $g \in L^1(\mathbb{R})$.
Suppose
\begin{equation} \label{JJ}
\lim_{T \to \infty} \Big[{\frac{\mu(Q_{T+|\tau|})}{\mu(Q_T)}}\Big]
< \infty  \text{ for all } \tau \in \mathbb{R}.
\end{equation}
Then $f \ast g$, the convolution of $f$ and $g$ on $\mathbb{R}$,
belongs to $PAP_0(\mathbb{R}, \mu, \nu)$.
\end{proposition}


\begin{proof}
It is clear that if $f \in PAP_0(\mathbb{R}, \mu, \nu)$ and
$g \in L^1(\mathbb{R})$, then their convolution
$f \ast g \in BC(\mathbb{R}, \mathbb{R})$. Now setting
$$
J(T, \mu, \nu) := {\frac{1}{\mu(Q_T)}} \int_{Q_T}
\int_{-\infty}^{+\infty} |f(t-s)|\, |g(s)| \nu(t)\, \,ds\,dt
$$
it follows that
\begin{align*}
{\frac{1}{\mu(Q_T)}} \int_{Q_T} |(f
\ast g)(t)| \nu(t) dt
&\leq J(T,\mu, \nu) \\
&= \int_{-\infty}^{+\infty} |g(s)| \Big({\frac{1}{\mu(Q_T)}}
\int_{Q_T} |f(t-s)| \nu(t) dt\Big)ds  \\
&= \int_{-\infty}^{+\infty} |g(s)| \phi_{T}(s) ds,
\end{align*}
where
\begin{align*}
 \phi_T(s) &= \frac{1}{\mu(Q_T)}
\int_{Q_T} |f(t-s)| \nu(t) dt \\
&= \frac{\mu(Q_{T+|s|})}{\mu(Q_T)} \cdot \frac{1}{\mu(Q_{T+|s|})}
\int_{Q_T} |f(t-s)| \nu(t) dt\\
&\leq \frac{\mu(Q_{T+|s|})}{\mu(Q_T)}
\cdot \frac{1}{\mu(Q_{T+|s|})}
\int_{Q_{T+|s|}} |f(t)| \nu(t+s) dt.
\end{align*}
Using the fact that $\nu \in \mathbb{U}_\infty^{\rm Inv}$ and  \eqref{JJ}, one
can easily see that
$\phi_T(s) \to 0$ as $T \to \infty$ for all $s \in \mathbb{R}$.
Next, since $\phi_T$ is bounded; i.e.,
$$
|\phi_T(s)| \leq \|f\|_\infty \cdot
 \sup_{T > 0}{\frac{\nu(Q_{T})}{\mu(Q_T)}}< \infty
$$
and $g \in L^1(\mathbb{R})$, using the Lebesgue Dominated Convergence
Theorem it follows that
$$
\lim_{T \to \infty} \Big\{\int_{-\infty}^{+\infty} |g(s)| \phi_{T}(s)
ds \Big\} = 0,
$$
and hence $f \ast g \in PAP_0(\mathbb{R}, \mu, \nu)$.
\end{proof}


\begin{corollary} \label{coro2.15}
Let $\mu \in \mathbb{U}_\infty$ and let $\nu \in \mathbb{U}_\infty^{\rm Inv}$ such that
 \eqref{HHH}--\eqref{JJ} hold. If $f \in PAP(\mathbb{R}, \mu, \nu)$
and $g \in L^1(\mathbb{R})$, then $f \ast g$ belongs to $PAP (\mathbb{R}, \mu, \nu)$.
\end{corollary}

\begin{theorem}[\cite{DW}]
If $\mu, \nu \in \mathbb{U}_\infty$ are such that the space $PAP_0(\mathbb{X}, \mu, \nu)$
is translation-invariant and if
\begin{equation}\label{I}
 \inf_{T > 0} \Big[{\frac{\nu(Q_{T})}{\mu(Q_T)}}\Big] = \delta_0 > 0,
\end{equation}
then the decomposition of the doubly-weighted pseudo-almost
periodic functions is unique.
\end{theorem}


\begin{theorem}[\cite{DW}]\label{toka}
 Let $\mu, \nu \in \mathbb{U}_\infty$ and
let $f \in PAP(\mathbb{Y}, \mathbb{X}, \mu, \nu)$ satisfying the Lipschitz condition
$$
\|f(t, u) - f(t,v)\| \leq L \cdot  \|u-v\|_{\mathbb{Y}}
\text{ for all }  u,v \in \mathbb{Y}, \; t \in \mathbb{R}.
$$
If $h \in PAP(\mathbb{Y}, \mu, \nu)$,
then $f(\cdot , h(\cdot)) \in PAP(\mathbb{X}, \mu, \nu)$.
\end{theorem}

\section{Existence of a doubly-weighted mean for almost periodic
functions}
Let $\mu, \nu \in \mathbb{U}_\infty$. If $f: \mathbb{R} \to \mathbb{X}$ is a bounded continuous
function, we define its {\it doubly-weighted mean}, if the limit
exists, by
$$
\mathcal{M}(f, \mu, \nu): = \lim_{T \to \infty}
\frac{1}{\mu(Q_T)} \int_{Q_T} f(t) \nu(t) dt.
$$

It is well-known that if $f \in AP(\mathbb{X})$, then its mean defined by
$$
\mathcal{M}(f):= \lim_{T\to \infty} \frac{1}{2T} \int_{Q_T} f(t)dt
$$
exists \cite{B}. Consequently, for every $\lambda \in \mathbb{R}$,
the following limit
$$
a(f, \lambda):= \lim_{T \to \infty} \frac{1}{2T}
 \int_{Q_T} f(t) e^{-i \lambda t} dt
$$
exists and is called the Bohr transform of $f$.

It is also well-known that
$a(f, \lambda)$ is nonzero at most at countably many points \cite{B}.
The set defined by
$$
\sigma_b(f) := \big\{\lambda \in \mathbb{R}: a(f, \lambda) \not = 0\big\}
$$
is called the Bohr spectrum of $f$ \cite{M}.

\begin{theorem}[Approximation Theorem \cite{LV, M}] \label{H}
 Let $f \in AP(\mathbb{X})$. Then for every $\varepsilon > 0$ there
exists a trigonometric polynomial
$$
P_\varepsilon (t) = \sum_{k=1}^n a_k e^{i\lambda_k t}
$$
where $a_k \in \mathbb{X}$ and $\lambda_k \in \sigma_b (f)$ such that
$\|f(t) - P_\varepsilon (t)\| < \varepsilon$
for all $t \in \mathbb{R}$.
\end{theorem}

In Liang {\it et al.} \cite{liang},
the original question which consists of the existence of a
weighted mean for almost periodic functions was raised.
In particular, Liang {\it et al.} have shown through an example
that there exist weights for which a weighted mean for almost
periodic functions may not exist. In this section we investigate
the broader question, which consists of the existence of a
doubly-weighted mean for almost periodic functions. Namely,
we give some sufficient conditions, which do guarantee the
existence of a doubly-weighted mean for almost periodic functions.
Moreover, under those conditions, it will be shown that the
doubly-weighted mean and the classical (Bohr) mean are proportional
(Theorem \ref{XX}). Further, it will be shown that if the
 doubly-weighted Bohr spectrum of an almost periodic function
 exists, then it is either empty or coincides with the Bohr
spectrum of that function. We have the following result.

\begin{theorem}\label{XX}
Let $\mu, \nu \in \mathbb{U}_\infty$ and suppose
$ \lim_{T \to \infty} \frac{\nu(Q_T)}{\mu(Q_T)} = \theta_{\mu\nu}$.
If $f: \mathbb{R} \to \mathbb{X}$ is an almost periodic function such that
\begin{equation} \label{CD}
\lim_{T \to \infty} \big|\frac{1}{\mu(Q_T)} \int_{Q_T}
e^{i\lambda t} \nu(t) dt\big| = 0
\end{equation}
for all $0 \neq \lambda \in \sigma_b(f)$, then the
doubly-weighted mean of $f$,
$$
\mathcal{M}(f, \mu, \nu) = \lim_{T \to \infty} \frac{1}{\mu(Q_T)}
\int_{Q_T} f(t) \nu(t) dt
$$
exists. Furthermore,
$\mathcal{M}(f, \mu, \nu) = \theta_{\mu\nu} \mathcal{M}(f)$.
\end{theorem}

\begin{proof}
If $f$ is a trigonometric polynomial, say,
$ f(t) = \sum_{k=0}^n a_k e^{i\lambda_k t}$
where $a_k \in \mathbb{X}-\{0\}$ and $\lambda_k \in \mathbb{R}$ for
$k = 1, 2, \dots, n$, then
$\sigma_b(f) = \{\lambda_k: \ k =1, 2, \dots, n\}$. Moreover,
\begin{align*}
\frac{1}{\mu(Q_T)} \int_{Q_T} f(t) \nu(t) dt
&= a_0 \frac{\nu(Q_T)}{\mu(Q_T)} + \frac{1}{\mu(Q_T)} \int_{Q_T} \Big[\sum_{k=1}^n a_k e^{i\lambda_k t}\Big] \nu(t) dt\\
&= a_0 \frac{\nu(Q_T)}{\mu(Q_T)} + \sum_{k=1}^n a_k \Big[\frac{1}{\mu(Q_T)}\int_{Q_T} e^{i\lambda_k t} \nu(t) dt\Big]\\
\end{align*}
and hence
\[
 \|\frac{1}{\mu(Q_T)} \int_{Q_T} f(t) \nu(t) dt
- a_0 \frac{\nu(Q_T)}{\mu(Q_T)}\|
\leq \sum_{k=1}^n \|a_k\|
\big|\frac{1}{\mu(Q_T)} \int_{Q_T} e^{i\lambda_k t} \nu(t) dt\big|
\]
which by  \eqref{CD} yields
$$
\|\frac{1}{\mu(Q_T)} \int_{Q_T} f(t) \nu(t) dt - a_0 \theta_{\mu\nu}\|
 \to 0 \quad \text{as } T \to \infty
$$
and therefore $\mathcal{M}(f, \mu, \nu) = a_0 \theta_{\mu\nu}
= \theta_{\mu\nu} M(f)$.

If in the finite sequence of $\lambda_k$ there exist
$\lambda_{n_k} = 0$ for $k = 1, 2, \dots l$ with
$a_m \in \mathbb{X}-\{0\}$ for all $m \not = n_k$ ($k=1,2,\dots,l$),
it can be easily shown that
$$
\mathcal{M}(f, \mu, \nu) = \theta_{\mu\nu} \sum_{k=1}^l a_{n_k}
= \theta_{\mu\nu} M(f).
$$
Now if $f: \mathbb{R} \to \mathbb{X}$ is an arbitrary almost periodic function,
then for every $\varepsilon > 0$ there exists a trigonometric
polynomial (Theorem \ref{H}) $P_\varepsilon$ defined by
$$
P_\varepsilon (t) = \sum_{k=1}^n a_k e^{i\lambda_k t}
$$
where $a_k \in \mathbb{X}$ and $\lambda_k \in \sigma_b (f)$ such that
\begin{equation} \label{11}
\|f(t) - P_\varepsilon (t)\| < \varepsilon \quad\text{for all }t \in \mathbb{R}.
\end{equation}

Proceeding as in Bohr \cite{B} it follows that there exists
$T_0$ such that for all $T_1, T_2 > T_0$,
\begin{align*}\label{12}
&\big\|\frac{1}{\mu(Q_{T_1})} \int_{Q_{T_1}} P_\varepsilon(t) \nu(t) dt
 - \frac{1}{\mu(Q_{T_2})} \int_{Q_{T_2}} P_\varepsilon(t) \nu(t) dt
 \big\|\\
& = \theta_{\mu\nu} \big\|M(P_\varepsilon)
- M(P_\varepsilon)\big\| = 0 < \varepsilon.
\end{align*}

In view of the above it follows that for all $T_1, T_2 > T_0$,
\begin{align*}
&\big\|\frac{1}{\mu(Q_{T_1})} \int_{Q_{T_1}} f(t) \nu(t) dt
 - \frac{1}{\mu(Q_{T_2})} \int_{Q_{T_2}} f(t) \nu(t) dt \big\|\\
&\leq  \frac{1}{\mu(Q_{T_1})} \int_{Q_{T_1}} \| f(t)
 - P_\varepsilon(t)\| \nu(t) dt\\
&\quad + \big\|\frac{1}{\mu(Q_{T_1})} \int_{Q_{T_1}} P_\varepsilon(t)
 \nu(t) dt
- \frac{1}{\mu(Q_{T_2})} \int_{Q_{T_2}} P_\varepsilon(t) \nu(t) dt\big\|\\
&\quad + \frac{1}{\mu(Q_{T_2})} \int_{Q_{T_2}} \| f(t)
 - P_\varepsilon(t)\| \nu(t) dt <3\varepsilon.
\end{align*}
\end{proof}

\begin{example} \rm
Fix a natural number $N > 1$. Let $\mu (t) = e^{|t|}$ and
$\nu(t) = (1 + |t|)^N$ for all $t \in \mathbb{R}$, which yields
$\theta_{\mu\nu} = 0$. If $\varphi: \mathbb{R} \to \mathbb{X}$ is a (nonconstant)
almost periodic function, then according to the previous theorem,
its doubly-weighted mean
$\mathcal{M}(\varphi, \mu, \nu)$ exists. Moreover,
$$
\lim_{T \to \infty} \frac{1}{2(e^T - 1)} \int_{Q_T} f(t) (1 + |t|)^N dt
= 0 . \lim_{T \to \infty} \frac{1}{2T} \int_{Q_T} f(t) dt = 0.
$$
\end{example}

Consider the set of weights $\mathbb{U}_\infty^{0}$ defined by
$$
\mathbb{U}_\infty^{0} = \big\{\mu \in \mathbb{U}_\infty: D_\tau
:= \lim_{|t| \to \infty} \frac{\mu(Q_{t+\tau})}{\mu(Q_t)}
< \infty \ \ \text{for all} \ \ \tau \in \mathbb{R} \big\}.
$$
Setting
$ C_\tau =  \lim_{|t| \to \infty} \frac{\mu(Q_{t}+\tau)}{\mu(Q_t)}$,
one can easily see that $C_\tau \leq D_\tau < \infty$ for all
$\tau \in \mathbb{R}$.

\begin{corollary}\label{X}
Fix $\mu, \nu \in \mathbb{U}_\infty^{0}$ and suppose that
$ \lim_{T \to \infty} \frac{\nu(Q_T)}{\mu(Q_T)} = \theta_{\mu\nu}$.
If $f: \mathbb{R} \to \mathbb{X}$ is an almost periodic function such that
 \eqref{CD} holds, then
\begin{equation} \label{IN}
\mathcal{M} (f_a, \mu, \nu_a)
= C_{-a} \theta_{\mu\nu} \mathcal{M} (f)
= C_{-a} \mathcal{M}(f, \mu, \nu)
\end{equation}
uniformly in $a \in \mathbb{R}$, where
$$
\mathcal{M} (f_b, \mu, \nu_b)
= \lim_{T \to \infty} \frac{1}{\mu(Q_T)}
\int_{Q_T} f_b(t) \nu_b(t) dt
 = \lim_{T\to \infty} \frac{1}{\mu(Q_T)} \int_{Q_T} f(t+b) \nu(t+b) dt
$$
for each $b \in \mathbb{R}$.
\end{corollary}

\begin{proof}
Clearly, the existence of $\mathcal{M}(f, \mu, \nu)$ is guaranteed
by Theorem \ref{XX}. Without lost of generality, suppose $a > 0$.
Now since $f \in AP(\mathbb{X})$ it follows that $f_a: t \to f(t + a)$
belongs to $AP(\mathbb{X})$. Moreover, the weight $\nu_a$ defined by
$\nu_a(t) = \nu(t+a)$ for all $t \in \mathbb{R}$ belongs to $\mathbb{U}_\infty^0$.
Now
\begin{align*}
\big| \int_{Q_T} e^{i \lambda t} \nu_a (t) dt \big|
&= \big| \int_{Q_T-a} e^{i \lambda (t-a)} \nu (t) dt \big|\\
&= \big| \int_{Q_T-a} e^{i \lambda t} \nu (t) dt \big|\\
&\leq \big| \int_{Q_{T+a}} e^{i \lambda t} \nu (t) dt \big|
\end{align*}
and hence
\begin{align*}
\lim_{T \to \infty} \big| \frac{1}{\mu(Q_T)}
 \int_{Q_T} e^{i \lambda t} \nu_a (t) dt \big|
&= \lim_{T\to \infty} \big|\frac{1}{\mu(Q_{T})} \int_{Q_{T}-a}
 e^{i \lambda t} \nu (t) dt \big| \\
&\leq  \lim_{T\to \infty} \big|\frac{1}{\mu(Q_{T})} \int_{Q_{T+a}}
 e^{i \lambda t} \nu (t) dt \big| \\
&= \lim_{T\to \infty} \big|\frac{\mu(Q_{T+a})}{\mu(Q_{T})}\frac{1}{\mu(Q_{T+a})} \int_{Q_{T+a}} e^{i \lambda t} \nu (t) dt \big| \\
&= D_a\lim_{T\to \infty} \big|\frac{1}{\mu(Q_{T+a})}
\int_{Q_{T+a}} e^{i \lambda t} \nu (t) dt \big|
=0.
\end{align*}
Now
$$
\lim_{T \to \infty} \frac{\nu_a (Q_T)}{\mu(Q_T)}
= C_{-a} \theta_{\mu\nu}.
$$
Using Theorem \ref{XX} it follows that for every
$\varphi \in AP(\mathbb{X})$,
$$
\mathcal{M}(\varphi_a, \mu, \nu_a) = \lim_{T \to \infty}
\frac{1}{\mu(Q_T)} \int_{Q_T} \varphi_a(t) \nu_a(t) dt
$$
 exists. Furthermore,
$\mathcal{M}(\varphi_a, \mu, \nu_a) = C_{-a} \theta_{\mu\nu}
\mathcal{M}(\varphi_a)$ for all $a \in \mathbb{R}$. In particular,
$\mathcal{M}(f_a, \mu, \nu_a) = C_{-a}  \theta_{\mu\nu}
\mathcal{M}(f_a)$ uniformly in $a \in \mathbb{R}$. Now from Bohr \cite{B},
$\mathcal{M}(f_a) =  \mathcal{M}(f)$ uniformly in $a \in \mathbb{R}$,
which completes the proof.
\end{proof}

\begin{definition} \rm
Fix $\mu, \nu \in \mathbb{U}_\infty$ and suppose that
$ \lim_{T \to \infty} \frac{\nu(Q_T)}{\mu(Q_T)} = \theta_{\mu\nu}$.
If $f: \mathbb{R} \to \mathbb{X}$ is an almost periodic function such that
 \eqref{CD} holds, we then define its doubly-weighted Bohr
transform as
$$
\widehat{a}_{\mu\nu}(f)(\lambda):= \lim_{T \to \infty}
\frac{1}{\mu(Q_T)} \int_{Q_T} f(t) e^{-i \lambda t} \nu(t) dt
 \quad \text{for all } \lambda \in \mathbb{R}.
$$
\end{definition}

Now since $t \to g_\lambda(t):= f(t) e^{-i \lambda t} \in AP(\mathbb{X})$
it follows that
$$
\widehat{a}_{\mu\nu}(f)(\lambda) = \theta_{\mu\nu}
\mathcal{M} (f(\cdot) e^{-i \lambda \cdot})
= \theta_{\mu\nu} a(f, \lambda).
$$
That is, under  \eqref{CD},
\begin{align*}
\widehat{a}_{\mu \nu}(f)(\lambda)
&:= \lim_{T \to \infty}
\frac{1}{\mu(Q_T)} \int_{Q_T} f(t) e^{-i \lambda t} \nu(t) dt\\
&= \theta_{\mu\nu} \lim_{T \to \infty} \frac{1}{2T}
\int_{Q_T} f(t) e^{-i \omega t} dt
 = \theta_{\mu\nu} a(f, \lambda)
\end{align*}
for all $\lambda \in \mathbb{R}$.

In summary, there are two possibilities for the doubly-weighted
Bohr spectrum of an almost periodic function. Indeed,
\begin{itemize}
\item[(1)] If $ \lim_{T \to \infty} \frac{\nu(Q_T)}{\mu(Q_T)}
 = \theta_{\mu\nu} = 0$, then
$\widehat{a}_{\mu\nu}(f)(\lambda) = \theta_{\mu\nu}a(f, \lambda) = 0$
for all $\lambda \in \mathbb{R}$. In that event, the doubly-weighted Bohr
spectrum of $f$ is
$$
\sigma_b^{\mu\nu}(f) := \big\{\lambda \in \mathbb{R}: \widehat{a}_{\mu\nu}(f)
(\lambda) \not = 0\big\} = \emptyset.
$$

\item[(2)]
If $ \lim_{T \to \infty} \frac{\nu(Q_T)}{\mu(Q_T)}
= \theta_{\mu\nu} \not = 0$, then
$\widehat{a}_{\mu\nu}(f)(\lambda) = \theta_{\mu\nu}a(f, \lambda)$
exists for all $\lambda \in \mathbb{R}$ and is nonzero at most at countably
many points.
In that event, the doubly-weighted Bohr spectrum of $f$ is
$$
\sigma_b^{\mu\nu}(f) := \big\{\lambda \in \mathbb{R}:
\widehat{a}_{\mu\nu}(f)(\lambda) \not = 0\big\}
= \big\{\lambda \in \mathbb{R}: a(f, \lambda) \not = 0\big\};
$$
that is, $\sigma_b^{\mu\nu}(f)= \sigma_b(f)$.
In particular, $\sigma_b^{\mu\mu}(f)= \sigma_b(f)$.
\end{itemize}

\section{Doubly-weighted pseudo-almost periodic solutions
to differential equations}

In this Section, we fix the two weights $\mu, \nu \in \mathbb{U}_\infty$ such that
$PAP(\mathbb{X}, \mu, \nu)$ is translation-invariant and  \eqref{I} holds.
Under these assumptions, it can be easily shown that
$PAP(\mathbb{X}, \mu, \nu)$ is a Banach space when equipped with the sup norm.

In what follows, we denote by $\Gamma_1$ and $\Gamma_2$,
the nonlinear integral operators defined by
\begin{gather*}
(\Gamma_1 u)(t) :=\int_{-\infty}^{t}U(t,s)P(s) g(s,
u(s))ds,\\
(\Gamma_2 u)(t) :=\int_{t}^{\infty}U_Q(t,s)Q(s) g(s, u(s))ds .
\end{gather*}

To study the existence of doubly-weighted pseudo-almost periodic
solutions to  \eqref{2} we will assume that the following
assumptions:

\begin{itemize}
  \item [(H1)] The family of closed linear operators
$A(t)$ for $t\in \mathbb{R}$ on $\mathbb{X}$ with domain $D(A(t))$ (possibly not
densely defined) satisfy Acquistapace and Terreni
conditions, that is, there exist constants $\omega\in \mathbb{R}$, $\theta
\in \Big(\pi/2,\pi\Big)$, $L > 0$ and $\mu, \nu \in (0,
1]$ with $\mu + \nu > 1$ such that
\[ %\label{AT1}
  \Sigma_\theta \cup \{0\} \subset \rho\big(A(t)-\omega\big)
\ni \lambda,\quad
\|R(\lambda,A(t)-\omega)\|\le \frac{K}{1+|\lambda|} \quad
 \text{for all } t \in \mathbb{R},
\]
   and
\[ %\label{AT2}
\|(A(t)-\omega)R(\lambda,A(t)-\omega)\,[R(\omega,A(t))-R(\omega,A(s))]
\|  \leq L\, \frac{|t-s|^\mu}{|\lambda|^{\nu}}
\]
for $t,s\in\mathbb{R}$, $ \lambda \in\Sigma_\theta:=
\{\lambda\in\mathbb{C} \setminus\{0\}: |\arg \lambda|\le\theta\}$.


\item[(H2)] The evolution family
$\mathcal{U} =\{U(t,s)\}_{t \geq s}$ generated by
$A(\cdot)$ has an exponential dichotomy with constants
$N,\delta>0$ and dichotomy projections $P(t)$ for $t\in\mathbb{R}$.

\item[(H3)] There exists  $0\leq \alpha<1$ such
that
$$\mathbb{X}_\alpha^t=\mathbb{X}_\alpha$$
for all $t\in \mathbb{R}$, with uniform equivalent norms.

\item[(H4)] $R(\omega, A(\cdot))  \in AP(B(\mathbb{X}_\alpha))$.

\item[(H5)] The function $g: \mathbb{R} \times \mathbb{X} \to \mathbb{X}$ belongs
to $PAP(\mathbb{X}, \mathbb{X}, \mu, \nu)$. Moreover, the
functions $g$ are uniformly  Lipschitz with respect to the second
argument in the following  sense: there exists $K > 0$ such that
$$
\|g(t,u)-g(t,v)\|\leq K \|u-v\|
$$
for all $u,v\in \mathbb{X}$ and $t\in \mathbb{R}$.
\end{itemize}

If $0 < \alpha<1$, then the nonnegative constant $k$ will denote the
bounds of the embedding  $\mathbb{X}_\alpha \hookrightarrow \mathbb{X}$; that is,
$$
\|x\| \leq k \|x\|_\alpha
\quad\text{for all }x \in \mathbb{X}_\alpha.
$$
To study the existence and uniqueness of doubly-weighted pseudo-almost
periodic solutions to  \eqref{2} we first introduce the notion of mild
solution.


\begin{definition} \rm
A continuous function $u: \mathbb{R} \to \mathbb{X}_\alpha$ is said to be a mild
solution to  \eqref{2} if
\[
u(t)=U(t,s)u(s) + \int_{s}^{t}U(t,s)P(s) g(s, u(s)) ds -
\int_{t}^{s}U(t,s)Q(s) g(s, u(s))ds
\]
for $t \geq s$ and for all $t, s \in \mathbb{R}$.
\end{definition}

Under previous assumptions (H.1)-(H.5), it can be easily shown
\eqref{2} has a unique mild solution given by
\[
u(t)=\int_{-\infty}^{t}U(t,s)P(s) g(s, u(s))ds
- \int_{t}^{\infty}U_Q(t,s)Q(s) g(s, u(s))ds
\]
for each $t \in \mathbb{R}$.

\begin{lemma}\label{ll2}
Under assumptions {\rm (H1)--(H5)}, the integral operators
 $\Gamma_1$ and $\Gamma_2$ defined
above map $PAP(\mathbb{X}_\alpha, \mu, \nu)$ into itself.
\end{lemma}

\begin{proof}
Let $u \in PAP(\mathbb{X}_\alpha, \mu, \nu)$. Setting $h(t) = g(t, u(t))$
and using the theorem of composition of doubly-weighted pseudo-almost
periodic functions (Theorem \ref{toka}) it follows that
$h \in PAP(\mathbb{X}, \mu, \nu)$. Now write $h = \phi + \zeta$ where
$\phi \in AP(\mathbb{X})$ and $\zeta \in PAP_0(\mathbb{X}, \mu, \nu)$.
The nonlinear integral operator $\Gamma_1 u$ can be rewritten as
\[
(\Gamma_1u)(t) =\int_{-\infty }^t U(t, s)P(s)\phi(s)ds +
\int_{-\infty }^tU(t,s)P(s)\zeta(s)ds.
\]
Set
\[
 \Phi(t) = \int_{-\infty }^t U(t,s)P(s)\phi(s)ds, \quad
\Psi(t)= \int_{-\infty }^t U(t,s)P(s)\zeta(s)ds
\]
for each $t \in \mathbb{R}$.

The next step consists of showing that $\Phi \in AP(\mathbb{X}_\alpha)$
and $\Psi \in PAP_0(\mathbb{X}_\alpha, \mu, \nu)$. Obviously, $\Phi \in
AP(\mathbb{X}_\alpha)$. Indeed, since $\phi \in AP(\mathbb{X})$, for every
$\varepsilon > 0$ there exists $l(\varepsilon)
> 0$ such that for every interval of length $l(\varepsilon)$ contains a $\tau$
with the property
$$
\|\phi(t +\tau) - \phi(t) \| < \varepsilon C\quad\text{for each }
 t \in \mathbb{R},
$$
where $ C = \frac{\delta^{1-\alpha}}{c(\alpha)
2^{1-\alpha} \Gamma(1 - \alpha)}$ with $\Gamma$ being the
classical Gamma function.
Now
\begin{align*}
&\Phi(t + \tau) - \Phi(t)\\
&= \int_{-\infty}^{t+\tau} U(t +\tau, s)
P(s) \phi(s)ds -  \int_{-\infty}^{t} U(t, s) P(s) \phi(s) ds\\
&= \int_{-\infty}^{t} U(t +\tau, s+\tau) P(s + \tau) \phi(s
+\tau)ds -  \int_{-\infty}^{t} U(t, s) P(s) \phi(s)
ds\\
&= \int_{-\infty}^{t} U(t +\tau, s+\tau) P(s + \tau) \phi(s
+\tau)ds \\
&\quad - \int_{-\infty}^{t} U(t+\tau, s+\tau) P(s+\tau) \phi(s)
ds\\
&\quad + \int_{-\infty}^{t} U(t +\tau, s+\tau) P(s+\tau) \phi(s)ds -
\int_{-\infty}^{t} U(t, s) P(s) \phi(s) ds\\
&= \int_{-\infty}^{t} U(t +\tau, s+\tau) P(s + \tau) \Big(\phi(s
+\tau) - \phi(s)\Big)ds\\
&\quad + \int_{-\infty}^{t} \Big(U(t +\tau, s+\tau)P(s+\tau) -
U(t,s)P(s)\Big)  \phi(s)ds.
\end{align*}
Using \cite{bar, Man-Schn} it follows that
$$
\|\int_{-\infty}^{t} \Big[U(t +\tau, s+\tau) P(s+\tau)
- U(t,s)P(s)\Big]
\phi(s)ds\|_\alpha \leq \frac{2\|\phi\|_\infty}{\delta}
\varepsilon.
$$
Similarly, using \eqref{eq1.1}, it follows that
$$
\| \int_{-\infty}^{t} U(t +\tau, s+\tau) P(s+\tau) (\phi(s
+\tau) - \phi(s))ds\|_\alpha \leq \varepsilon.
$$
Therefore,
$$
\| \Phi(t + \tau) - \Phi(t) \|_\alpha
< \Big(1 + \frac{2\|\phi\|_\infty}{\delta}\Big) \varepsilon \quad
\text{for each }  t \in \mathbb{R},
$$
and hence, $\Phi \in AP(\mathbb{X}_\alpha)$.


To complete the proof for $\Gamma_1$, we have to show that
$\Psi \in PAP_0(\mathbb{X}_\alpha, \mu, \nu)$. First, note that
$s \to \Psi(s)$ is a bounded continuous function. It remains
to show that
$$
\lim_{T\to \infty} {\frac{1}{\mu(Q_T)}} \
\int_{Q_T} \| \Psi(t) \|_\alpha \nu(t)dt = 0.
$$
Again using \eqref{eq1.1} it follows that
\begin{align*}
&\lim_{T \to \infty} {\frac{1}{\mu(Q_T)}}
\int_{Q_T} \| \Psi(t) \|_\alpha \nu(t)dt\\
&\leq \lim_{T \to \infty} {\frac{c(\alpha)}{\mu(Q_T)}}
 \int_{Q_T} \int_{0}^{+\infty} s^{-\alpha} e^{-\frac{\delta}{2}s}\|
\zeta(t -s) \| \nu(t) \,ds\,dt\\
&\leq \lim_{T \to \infty}  c(\alpha)
\int_{0}^{+\infty} s^{-\alpha} e^{-\frac{\delta}{2} s}
\frac{1}{\mu(Q_T)} \int_{Q_T}\| \zeta(t-s)\| \nu(t) dt ds.
\end{align*}
Set
$$ \Gamma_s(T) = \frac{1}{\mu(Q_T)} \int_{Q_T} \|
\zeta(t-s) \| \nu(t)dt.
$$
Since $PAP_0(\mathbb{X}, \mu, \nu)$ is assumed to be translation invariant
and that  \eqref{I} holds,
it follows that $t \to \zeta(t-s)$ belongs to $PAP_0(\mathbb{X}, \mu, \nu)$ for
each $s \in \mathbb{R}$, and hence
$$
\lim_{T \to \infty} \frac{1}{\mu(Q_T)} \int_{Q_T}
\| \zeta(t-s) \| \nu(t) dt = 0
$$
for each $s \in \mathbb{R}$.

One completes the proof by using the well-known Lebesgue Dominated
Convergence Theorem and the fact $\Gamma_s(T) \to 0$ as $T \to
\infty$ for each $s \in \mathbb{R}$.
The proof for $\Gamma_2u(\cdot)$ is similar to that of $\Gamma_1
u(\cdot)$. However one makes use of  \eqref{eq2.1} rather than
 \eqref{eq1.1}.
\end{proof}

\begin{theorem}\label{theo}
Under assumptions {\rm (H1)--(H5)}, Equation \eqref{2} has
a unique doubly-weighted pseudo-almost periodic mild solution
whenever $K$ is small enough.
\end{theorem}

\begin{proof}
Consider the nonlinear operator $\mathcal{A}$ defined on
$PAP(\mathbb{X}_\alpha, \mu, \nu)$ by
\begin{align*}
\mathcal{A} u(t)=
\int_{-\infty}^{t}U(t,s)P(s) g(s, u(s))ds
-\int_{t}^{\infty}U_Q(t,s)Q(s) g(s,  u(s))ds
\end{align*}
for each $t \in \mathbb{R}$.

In view of Lemma \ref{ll2},
it follows that $\mathcal{A}$ maps $PAP(\mathbb{X}_\alpha, \mu, \nu)$
into itself.
To complete the proof one has to show that $\mathcal{A}$ has a
unique fixed-point.
If $v,w\in PAP(\mathbb{X}_\alpha, \mu, \nu)$, then
\begin{align*}
\| \Gamma_1 (v)(t) - \Gamma_1(w)(t)\|_\alpha
&\leq \int_{-\infty}^t \|U(t,s)P(s)[g(s,v(s))-g(s,w(s))]\|_\alpha\,ds
 \\
&\leq \int_{-\infty}^t c(\alpha) (t-s)^{-\alpha}
e^{-\frac{\delta}{2} (t-s)}\|g(s,v(s))-g(s,w(s))\|\,ds
 \\
&\leq K c(\alpha) \int_{-\infty}^t (t-s)^{-\alpha}
e^{-\frac{\delta}{2} (t-s)}\|v(s)-w(s)\|\, ds
 \\
&\leq k K c(\alpha) \int_{-\infty}^t (t-s)^{-\alpha}
e^{-\frac{\delta}{2} (t-s)}\|v(s)-w(s)\|_\alpha\, ds
 \\
&\leq k K c(\alpha) 2^{1-\alpha}\,\Gamma(1-\alpha)
\delta^{\alpha-1} \|v - w\|_{\alpha, \infty},
\end{align*}
and
\begin{align*}
\| \Gamma_2 (v)(t) - \Gamma_2(w)(t)\|_\alpha
&\leq \int_{t}^{\infty} \|U_Q(t,s)Q(s)
[g(s,v(s))-g(s,w(s))]\|_\alpha\, ds
 \\
&\leq \int_{t}^{\infty} m(\alpha)
e^{\delta(t-s)}\|g(s,v(s))-g(s,w(s))\|\, ds
 \\
&\leq \int_{t}^{\infty} m(\alpha) K
e^{\delta(t-s)}\|v(s)-w(s)\| \,ds
 \\
&\leq  k m(\alpha) K \int_{t}^{\infty}
e^{\delta(t-s)}\|v(s)-w(s)\|_\alpha\, ds
 \\
&\leq K k m(\alpha) \|v - w\|_{\alpha, \infty}
  \int_{t}^{+\infty} e^{\delta(t-s)}\, ds \\
&= K  k m(\alpha) \delta^{-1}\|v -w\|_{\alpha, \infty},
\end{align*}
where $ \|u\|_{\alpha, \infty} := \sup_{t\in \mathbb{R}} \|u(t)\|_\alpha$.

Combining the previous approximations it follows that
$$
\|\mathbb{M}v-\mathbb{M}w\|_{\infty, \alpha}\leq K C(\alpha, \delta)
\cdot \|v -    w\|_{\alpha, \infty},
$$
where $C(\alpha, \delta) = k m(\alpha) \delta^{-1}
+ k c(\alpha) 2^{1-\alpha}\,\Gamma(1-\alpha)
\delta^{\alpha-1} >0$
is  constant,
and hence if the Lipschitz $K$ is small enough. Then  \eqref{2}
has a unique solution, which obviously is its
only doubly-weighted pseudo-almost periodic mild solution.
\end{proof}

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\end{document}