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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 210, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/210\hfil
 $\mathcal{S}$-asymptotically $\omega$-periodic solutions]
{$\mathcal{S}$-asymptotically $\omega$-periodic  solutions
 for abstract neutral differential equations}

\author[M. Pierri, D. O'Regan \hfil EJDE-2015/210\hfilneg]
{Michelle Pierri, Donal O'Regan}

\address{Michelle Pierri \newline
Departamento de Computa\c{c}\~{a}o e Matem\'atica,
 Faculdade de Filosofia Ci\^encias e Letras de Ribeir\~{a}o Preto \\
Universidade de S\~ao Paulo \\
 CEP 14040-901 Ribeir\~{a}o Preto, SP, Brazil}
\email{michellepierri@ffclrp.usp.br}

\address{Donal O'Regan \newline
School of Mathematics, Statistics and Applied Mathematics,
National University of Ireland, Galway, Ireland}
\email{donal.oregan@nuigalway.ie}

\thanks{Submitted January 30, 2014. Published August 10, 2015.}
\subjclass[2010]{34K30, 34K40, 34K14, 35B15, 47D06}
\keywords{Neutral equation; $\mathcal{S}$-asymptotically $\omega$-periodic 
function; \hfill\break\indent  analytic semigroup; strict solutions;
partial delayed differential equation}

\begin{abstract}
 In this article we study the existence of $ \mathcal{S}$-asymptotically
 $\omega$-pe\-riodic solutions for abstract neutral functional differential
 equations recently, introduced in the literature. An application involving
 a partial neutral differential equation is presented.
\end{abstract}

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

\section{introduction}\label{introduction}

In this work we study the existence of $\mathcal{S}$-asymptotically
$\omega$-periodic solutions for a class of abstract neutral differential equations
 of the form
 \begin{gather}\label{1}
 u' (t)=Au(t)+ f(t,u_{t}, u _{t}'), \quad t\in [0,\infty), \\
 u_0=\varphi\in\mathcal{B}, \label{2}
 \end{gather}
where $A: D(A) \subset X\to X $ is the infinitesimal generator of an analytic
semigroup of bounded linear operators $ (T(t))_{t\geq 0} $ defined on a
Banach space $(X,\|\cdot\|)$, the history $u_{t}$ belongs to an abstract
Banach space $(\mathcal{B},\| \cdot\|_{\mathcal{B}} )$ defined axiomatically,
 $ u _{t}'$ denotes the derivative at $t$ of the function $s\to u_{s} $
and $f(\cdot)$ is a suitable function.

The neutral system \eqref{1}--\eqref{2} was introduced recently by
Hern\'andez and O' Regan \cite{Hernandez1}.
 As pointed out in \cite{Hernandez1}, the study of the existence of solutions
for this class of problems via semigroup methods and fixed point
techniques is highly nontrivial since the temporal derivative of the
solution appears in the integral equation used to define the concept of
mild solution of \eqref{1}--\eqref{2}, see Definition \ref{def1}.
As a consequence, it is necessary to work on spaces of differentiable
functions which is a complex problem under the semigroup framework.

To the best of our knowledge, the paper \cite{Hernandez1} is the first and only
 work treating neutral problems described in the abstract form
\eqref{1}--\eqref{2}. On the current state of the theory of abstract
neutral differential equations we cite
 \cite{adimy1,Datko1,Haad1,Hale1,H1,H2,H3,H4,H5,H6,H10,H11,wu3}
and the references therein.

The concept of $\mathcal{S}$-asymptotically $\omega$-periodic was
introduced recently in the literature, see \cite{Pierri2,Pierri1}.
We note that a continuous function $u(\cdot)$ defined on
$[0, \infty)$ is said to be $\mathcal{S}$-asymptotically periodic
if there exists $\omega\in \mathbb{R}$ such that
$\lim_{t \to \infty} [f(t+\omega )-f(t)]=0$.

 For qualitative properties of $\mathcal{S}$-asymptotically $\omega$-periodic
functions, we cite \cite{Pierri2,Pierri3,Pierri1}. Concerning
the problem of the existence of $\mathcal{S}$-asymptotically $\omega$-periodic
solutions for differential equations, we cite
 \cite{grimmer1,Xiaohu,waltman1,wong1,zho1} for ordinary differential
equations on finite dimensional spaces and \cite{C3,C1,C2,Pierri2,Pierri1}
for ordinary differential equations defined on abstract Banach spaces.

The abstract problem \eqref{1}--\eqref{2} arises, for example, in the
theory of heat conduction in fading memory material. In the
classical theory of heat conduction, it is assumed that the internal
energy and the heat flux depends linearly on the temperature
$u(\cdot)$ and on its gradient
$\nabla u(\cdot)$. Under these conditions, the classical heat equation
describes sufficiently well the evolution of the temperature in different types of
materials.
 However, this description is not satisfactory in
 materials with fading memory. In the theory developed in
\cite{Gurtin1,Nunziato1}, the internal energy and the heat flux
are described as functionals of $u$ and $u_{x}$. The next
 system, see \cite{Sforza2,Clement3,Alessandra3}, has been frequently
used to describe these phenomena,
\begin{equation}
\begin{gathered}
\frac{d }{dt} ( u(t,x)+
\int_{-\infty}^{t}k_1(t-s)u(s,x)ds )
= c \Delta u(t,x)+\int_{-\infty}^{t}k_{2}(t-s)\Delta u(s,x)ds, \\
 u(t,x) = 0,\quad x\in \partial\Omega.
\end{gathered}
\end{equation}
In this system, $\Omega\subset\mathbb{R}^{n}$ is open,
bounded and has smooth boundary, $(t,x)\in
[0,\infty)\times\Omega$, $u(t,x)$ denotes the temperature in $x$ at
time $t$, $c $ is a physical constant and $k_{i}:\mathbb{R}\to
\mathbb{R}$, $i=1,2,$ are the internal energy and the heat flux
relaxation respectively. If we assume that
$k_1=\gamma_1+\gamma_{2}$ and the solution $u(\cdot)$ is known
on $(-\infty,0]$, we can study the above system via the
initial-value problem
\begin{equation}
\begin{gathered}
\begin{aligned}
&\frac{d }{dt} ( u(t,x)+\int_{-\infty}^{t}\gamma_1(t-s)u(s,x)ds ) \\
&= c \Delta u(t,x)+\int_{-\infty}^{t}k_{2}(t-s)\Delta u(s,x)ds
 - \int_{-\infty}^{t}\gamma_{2}(t-s)u'(s,x)ds ,
\end{aligned} \\
 u(s,x) = \varphi(s,x),\quad s\geq 0, x\in \Omega,
\end{gathered}
\end{equation}
which can be represented in the abstract form \eqref{1}--\eqref{2}.
For additional applications and examples on neutral differential equations
we cite our recent papers \cite{H4,Hernandez1} and the references therein.

Next, we include some notations, definitions and technicalities.
Let $(Z,\| \cdot\| _{Z})$ and $(W,\| \cdot \| _{W})$ be Banach spaces.
In this paper, $\mathcal{L}(Z,W)$ denotes the space of bounded linear
operators from $Z$ into $W$ endowed with the norm of operators denoted
$\| \cdot\|_{\mathcal{L}(Z,W)} $ and we write $\mathcal{L}(Z)$ and
$\| \cdot\|_{\mathcal{L}(Z)} $ when $Z = W$. We use the notation
$Z\hookrightarrow W$ to indicate that $Z$ is continuously included in $W$.


Let $I\subset \mathbb{R}$. As usual, $C(I;Z)$ is the space formed
 by all the bounded continuous functions from $I$ into $Z$ endowed
with the sup-norm denoted by $\|\cdot\|_{C(I;Z)}$ and $C^{1}(I;Z)$ the
space formed by all the functions $u\in C(I;Z)$ such that $u'\in C(I;Z)$
endowed with the norm $\| u\|_{C^{1}(I;Z)}=\| u\|_{C(I;Z)}+\| u'\|_{C(I;Z)}$.
 In addition, $C^{ \gamma}([0,\infty);Z)$ (with $\gamma\in (0,1)$)
is the space formed by all the functions $ \xi \in C([0,\infty);Z)$
such that \[
 [ \xi ]_{C^{\gamma}([0,\infty); Z)}
= \sup_{t,s\in [0,\infty) , t\neq s }
\frac{\| \xi(s)-\xi(t)\|_{Z}}{\mid t-s\mid^{\gamma}}
\]
is finite, provided with the norm
$ \| \xi\|_{C^{ \gamma}([0,\infty); Z)}
=\| \xi\|_{_{C([0,\infty); Z)}}+[ \xi ]_{C^{ \gamma}([0,\infty); Z)} $.
 The notation $C^{1+ \gamma}([0,\infty); Z)$ is used for the space
of all the differentiable functions $\xi \in C^{\gamma} ([0,\infty); Z)$
such that $ \xi'\in C^{\gamma}([0,\infty); Z)$ endowed with the norm
$\| \xi \|_{C^{1+\gamma}([0,\infty);Z)}
= \| \xi \|_{C^{ \gamma}([0,\infty);Z)} +\| \xi' \|_{C^{\gamma}([0,\infty);Z)}$.

From \cite{Pierri2} we note the following concept.

\begin{definition} \label{def1.1} \rm
 A function $f\in C([0, \infty),Z) $ is said to be $\mathcal{S}$-asymptotically
periodic if there exists $\omega\in \mathbb{R}$ such that
$\lim_{t \to \infty} [f(t+\omega )-f(t)]=0$.
In this case, we say that $\omega$ is an
 asymptotic period of $f(\cdot)$ and that $f(\cdot)$ is
$\mathcal{S}$-asymptotically $\omega$-periodic.

Here $SAP_{\omega}(Z)$ denotes the space formed by the
$Z$-valued $\mathcal{S}$-asymptotically $\omega$-periodic functions provided with
 the norm $ \|\cdot \|_{C([0,\infty);Z)}$.
\end{definition}


In this article, $A: D(A) \subset X\to X$ is the generator of an
uniformly stable analytic semigroup of bounded linear operators
$(T(t))_{t\geq 0} $ on $X$ and $\gamma $, $C_{i}$, $i \in \mathbb{N}$,
are positive constants such that
\[
\| A^{i}T(t)\|_{\mathcal{L}(X)}\leq \frac{ C_{i} e^{-\gamma t}}{t^{ i}}
\]
 for all $t>0$ and each $i\in \mathbb{N}$. For $\beta>0 $, we represent by
$X_{\beta}$ the domain of the fractional power $(-A)^{\beta}$ of $-A$
endowed with the norm $\| x\|_{\beta}= \| (-A)^{\beta}x\| $.
In addition, for $\beta >0$ we assume that $C_{\beta}>0$ is such that
\[
\| AT(t)\|_{\mathcal{L}(X_{\beta},X)}
\leq \frac{ C_{\beta}e^{-\gamma t} }{t^{ \beta}},\quad \forall t>0.
\]
The notation $D_{A}(\eta,\infty)$, $\eta\in (0,1)$, stands for the space
$$
D_{A}(\eta,\infty)=\{x\in X:[x]_{\eta,\infty}
=\sup_{t\in (0,1)}\| t^{1-\eta}AT(t)x\| <\infty\},
$$
 with the norm
$\| x\|_{ \eta,\infty }= [x]_{\eta,\infty}+\| x\|$ and we
assume that for all $k\in \mathbb{N}\cup \{0\}$ there exits a constant
 $C_{k,\eta}$ such that
\[
\| A^{k}T(t) \|_{\mathcal{L}(D_{A}(\eta,\infty),X)}
\leq \frac{C_{k,\eta} e^{-\gamma t}}{t^{1-\eta}}
\]
for all $t>0$. For additional details on analytic semigroups
and interpolation spaces we cite \cite{Alessandra2}.


In this article, $(\mathcal{B},\| \cdot\|_{\mathcal{B}} )$ is a
Banach space formed by functions defined from a connected interval
$\{0\}\subset J\subset (-\infty,0]$ into $X$, satisfying
the following conditions.
\begin{itemize}
\item[(A1)] If $x: (J+\{\sigma\})\cup [\sigma, \sigma+b)\to X$,
$ b>0$, $\sigma\in \mathbb{R}$, is continuous on $[\sigma,\sigma +b)$
and $x_{\sigma}\in \mathcal{B}$, then for every $t\in [\sigma,\sigma+b)$
the following conditions hold:
\begin{itemize}
\item[(i)] the function $s\to x_{s}$ belongs to $C([\sigma,\sigma+b),\mathcal{B})$,
 \item[(ii)] $\|x(t)\| \leq H \| x_{t}\|_{\mathcal{B}}$,
\item[(iii)] $\| x_{t}\|_{\mathcal{B}} \leq K(t-\sigma)
\sup\{\| x(s)\| :\sigma\leq s\leq t\}+
 M(t-\sigma)\| x_{\sigma}\|_{\mathcal{B}}$,
 where $H>0$ is a constant; $ K,M\in C([0,\infty); \mathbb{R}^{+})$
and $H,K(\cdot),M(\cdot)$
are independent of $x(\cdot)$.
 \item[(iv)] If $(\psi^n )_{n\in\mathbb{ N}} $ is a sequence in
$C(J,X)\cup \mathcal{B}$ and $\psi^n\to \psi $ uniformly on compact
subsets of $J$, then $\psi\in \mathcal{B}$ and
$\| \psi^n - \psi\|_{\mathcal{B}}\to 0 $ as $n \to \infty $.
\end{itemize}
\end{itemize}

\begin{remark} \label{rmk1.2} \rm
 For $\beta > 0 $, we represent by $\mathcal{B}_{\beta}$ the space
$\mathcal{B}_{\beta}=\{ (-A)^{-\beta}\psi: \psi \in \mathcal{B}\}$
endowed with the norm
$\| \psi\|_{\mathcal{B}_{\beta}}=\| (-A)^{\beta} \psi\| _{\mathcal{B}_{\beta}} $.
We note that $\mathcal{B}_{\beta} $ verifies the axiom (A1) with
$X_{\beta}$ in place $X$.
\end{remark}

\begin{remark} \label{rmk1.3} \rm
In the remainder of this paper, to simplify, we assume that
$ \mathcal{K} >0 $ is a constant such that $\max\{K(t), M(t)\}\leq \mathcal{K} $
for all $t\geq 0$.
\end{remark}

 From from \cite{Hernandez1} we note the following result.
In this result,  $P_u$ is the function defined  by $P_u(t)=u_{t}$.
We will use this notation for the remainder of this article. 

\begin{lemma}\label{lemma2}
If $u\in C^{1}(J \cup [0,b];X)$, then
$P_u\in C^{1}([0,b];\mathcal{B})$ and
$\frac{d}{dt}P_u(t)=P_{\frac{d u}{dt} }(t)$
for all $t\in [0,b]$.
\end{lemma}

This article has three sections.
In the next section we study the existence of $\mathcal{S}$-asymptotically
$\omega$-periodic strict solutions for the problem \eqref{1}--\eqref{2}.
In the last section, an application involving a partial neutral
differential equation is presented.

\section{Existence results} \label{section1}

In this section we study the existence of $\mathcal{S}$-asymptotically
$\omega$-periodic strict solution for the abstract neutral problem
\eqref{1}--\eqref{2}. To prove our results, we consider the following conditions.
\begin{itemize}
\item[(H1)] There are $\alpha\in (0,1)$, $ \beta \in (0,1]$ and
 functions $f_1\in C^{\alpha}([0,\infty);\mathcal{L}
(\mathcal{B}_{\beta},X))$,
$f_{2}\in C^{\alpha}([0,\infty);\mathcal{L}(\mathcal{B},X))$
such that $f(t,\psi_1,\psi_{2})=f_1(t,\psi_1) + f_{2}(t,\psi_{2})$ for all
$t\geq 0$, $\psi_1\in \mathcal{B}_{\beta} $ and $\psi_{2} \in \mathcal{B}$.

\item[(H2)] There is a Banach space
$ (Y,\| \cdot \|_{Y})\hookrightarrow ( X ,\| \cdot \| )$, a integrable
function $H\in L^{1}([0,\infty);\mathbb{R}^{+})$, a continuous function
 $L_{f}\in C([0,\infty); \mathbb{R}^{+})$ and $\omega >0$ such that
$f\in C([0,\infty)\times \mathcal{B}_{\beta}\times\mathcal{B} ; Y)$,
$\| AT(s)\|_{\mathcal{L}(Y,X)} \leq H(s)$ for all $s>0 $, and
 \begin{equation}
 \| f(t,\psi_1, \zeta_1)-f(t,\psi_{2} ,\zeta_{2}) \|_{Y}
\leq L_{f}(t) ( \|\psi_1- \psi_{2} \|_{\mathcal{B}_{\beta}} +
 \| \zeta_1-\zeta_{2} \|_{\mathcal{B}}),
 \end{equation}
 for all $ \psi_{i} \in \mathcal{B}_{\beta} $, $ \zeta_{i} \in \mathcal{B} $,
 $i=1,2,$ and every $ t\geq 0$.

 \item[(H3)] There is a Banach space
$ (Y,\| \cdot \|_{Y})\hookrightarrow ( X ,\| \cdot \| )$, such that the
function $f(\cdot)$ belongs to $ C([0,\infty)\times
\mathcal{B}_{\beta}\times\mathcal{B} ; Y)$ and $f(\cdot)$ is uniformly
$\mathcal{S}$-asymptotically $\omega$-periodic on bounded sets;
 that is,
$$ \lim_{t\to \infty }\sup_{\| \psi\|_{\mathcal{B}_{\beta}}
\leq r,\| \xi\|_{\mathcal{B}}\leq r }\| f(t+\omega,\psi,\xi)
- f(t ,\psi,\xi) \|_{Y}= 0 .
$$
\end{itemize}

To prove our results, is convenient to include some comments on the problem
 \begin{gather}\label{4}
 u'(t)=Au(t)+ \xi(t), \quad t\in I, \\
 u(0) = x  \quad \in X, \label{5}
\end{gather}
 with $I= [0,a]$ or $I=[0,\infty)$ and $\xi\in L^{1}(I;X)$.
We note that the function $u: I\to X$ given by
$u(t)= T(t)x+ \int_0^{t}T(t-s)\xi(s) ds $ is called a mild solution of
\eqref{4}-\eqref{5} on $I$ and that a function $v\in C(I;X)$ is said to be
a strict solution \eqref{4}-\eqref{5} on $I$ if $v \in C^{1} (I;X)\cap C(I; X_1)$
and $v(\cdot)$ satisfies \eqref{4}-\eqref{5}.

 The proof of Proposition \ref{prop1} is similar to the proof of
\cite[Theorem 4.3.1]{Alessandra2}. We include some details of the proof
for completeness.

\begin{proposition}\label{prop1}
Assume $\xi\in C^{\alpha}([0,\infty);X)$, $x\in X_1$ and
$Ax+\xi(0) \in D_{A}(\alpha,\infty)$. If $u(\cdot)$ is the mild solution
of \eqref{4}-\eqref{5} on $[0,\infty)$, then $u(\cdot)$ is a strict solution,
 $ u\in C^{\alpha}([0,\infty);X_1) $ and
\begin{equation} \label{e1}
 u'(t) = AT(t)x+\int_0^{t}AT(t-s)(\xi(s)-\xi(t))ds + T(t) \xi(t) , \quad
\forall  t\geq 0.
\end{equation}
Moreover,
\begin{gather}
 [ u ]_{C^{\alpha}([0,\infty);X_1)}
 \leq \frac{C_{1,\alpha}}{\alpha} \| Ax+\xi(0)\|_{\alpha,\infty}
+ \Lambda \| \xi\|_{C^{\alpha} ([0,\infty);X)} , \label{e2}
\\
  [ u' ]_{C^{\alpha}([0,\infty);X )}
\leq \frac{C_{1,\alpha}}{\alpha} \| Ax
+\xi(0)\|_{\alpha,\infty} + ( \Lambda +1) \| \xi\|_{C^{\alpha} ([0,\infty);X)} ,
  \label{e4} \\
 \| u \|_{C([0,\infty);X_1)}
 \leq C_0\| A x\| + \Lambda_1 [ \xi ]_{C^{\alpha}([0,\infty);X)}
+ \Lambda_{2} \| \xi \|_{C([0,\infty);X)}, \label{e3} \\
 \| u' \|_{C([0,\infty);X)}
 \leq C_0\| A x\| + \Lambda_1 [ \xi ]_{C^{\alpha}([0,\infty);X)}
+ (\Lambda_{2} +1) \| \xi \|_{C([0,\infty);X)},
\label{e5}
\end{gather}
 where
$\Lambda =( \frac{2 C_1 }{\alpha }+ 3 C_0 +1 + \frac{ C_{2}}{ \alpha(1-\alpha)})$,
$\Lambda_1 = C_1(\frac{1}{\gamma}+\frac{1}{\alpha} )$ and
$\Lambda_{2} = ( C_0+1)$.
\end{proposition}

\begin{proof}
 Let $T>0$. From \cite[Theorem 4.3.1]{Alessandra2} we know that
$u_{\mid_{[0,T]}} $ is a strict solution of the problem \eqref{4}-\eqref{5} on
 $[0,T]$, $ u\in C^{\alpha}([0,T]);X_1) $ and that the representation
\eqref{e1} is valid on $[0,T]$. Moreover, a review of the proof of
\cite[Theorem 4.3.1]{Alessandra2} permit us to assert that
$$
 [ u]_{ C^{\alpha}([0,T];X_1)}
 \leq  \frac{ C_{1,\alpha}}{\alpha} \| Ax+\xi(0)\|_{\alpha,\infty}
+( \frac{2 C_1 }{\alpha }+ 3 C_0 +1
+ \frac{ C_{2}}{ \alpha(1-\alpha)}) [ \xi ]_{C^{\alpha}([0,T];X)}.
$$
 From the above, we infer that $u(\cdot)$ is a strict solution of the problem
\eqref{4}-\eqref{5} on $[0,\infty)$, the representation \eqref{e1} is satisfied on
$[0,\infty)$ and \eqref{e2} is valid with
$\Lambda = ( \frac{2 C_1 }{\alpha }+ 3 C_0 +1
+ \frac{ C_{2}}{ \alpha(1-\alpha)})$. In addition, from the representation
$$
 A u(t) = T(t) Ax+\int_0^{t}AT(t-s)(\xi(s)-\xi(t))ds + (T(t)-I) \xi(t),
 $$
it is easy to see that
$$
\| u \|_{C([0,\infty);X_1)} \leq
 C_0\| A x\| + C_1(\frac{1}{\gamma}
+\frac{1}{\alpha} ) [ \xi ]_{C^{\alpha}([0,\infty);X)}
 +( C_0+1) \| \xi \|_{C([0,\infty);X)},
$$
 which establish \eqref{e3}. Finally, from \eqref{e2} and \eqref{e3},
and the fact that $u(\cdot)$ is a strict solution we obtain \eqref{e4}
and \eqref{e5}. This completes the proof.
\end{proof}

 For completeness, from \cite{Hernandez1} we quote the followings result.

 \begin{lemma}\label{lemma3}
Assume the condition {\rm (H2)} is satisfied, $x \in X_1$ and
$\xi\in C([0,b];Y)$. Then, the mild solution $w(\cdot)$ of \eqref{4}-\eqref{5}
is a strict solution and
$$
w'(t) = T(t)Ax+ \int_0^t AT(t-s) \xi(s) ds +\xi(t), \quad  \forall  t\in I.
$$
 \end{lemma}

 \begin{remark} \label{rmk2.3} \rm
In the remainder of this paper, for $u\in C(J\cup [0,\infty);X_{\beta})$,
$v\in C(J\cup [0,\infty);X)$ with $u_0\in \mathcal{B}_{\beta}$ and
$v_0\in \mathcal{B}$, we use the notations $ \mathcal{P}_{v}$,
$ \mathcal{P}_{u}$ and $f_{u,v}$ for the functions
$\mathcal{P}_{v}:[0,\infty)\to \mathcal{B}$,
$\mathcal{P}_{u}:[0,\infty)\to \mathcal{B}_{\beta}$ and
$f_{u,v}:[0,\infty)\to X$ given by $\mathcal{P}_{v}(t)=v_{t}$,
$\mathcal{P}_{u}(t)=u_{t} $ and $f_{u,v}(t)=f(t,u_{t},v_{t})$.
\end{remark}
By considering the above remarks, we adopt the following concepts of solution.

\begin{definition}  \label{def1} \rm
A function $u: J\cup [0,\infty)\to X $ is called a mild solution of the abstract
problem \eqref{1}--\eqref{2} if $u_0=\varphi$,
$\mathcal{P}_{u}\in C^{1}([0,\infty);\mathcal{B}) \cap C([0,\infty);\mathcal{B}_{\beta})$
and
 $$
u(t)=T(t)\varphi(0)+\int_0^{t}T(t-s)f_{u,u'}(s) ds ,\quad
 \forall  t\in [0,\infty).
$$
\end{definition}

\begin{definition} \label{def2} \rm
A function $u: J\cup [0,\infty)\to X $ is said to be a strict solution of
\eqref{1}--\eqref{2} if
$\mathcal{P}_{u}\in C^{1}([0,\infty);\mathcal{B})
\cap C([0,\infty);\mathcal{B}_{\beta})$,
$ u\in C([0,\infty);X_1) $ and $u(\cdot)$ satisfies \eqref{1}--\eqref{2}.
\end{definition}

Next, we include some results on $\mathcal{S}$-asymptotically $\omega$-periodic
functions. In the next results, $(Z,\| \cdot\|_{Z}) $ is a Banach space.

\begin{lemma}\label{lemma8}
Assume $(Z,\| \cdot\|_{Z})\hookrightarrow (X ,\| \cdot\|)$,
$u\in C(J \cup [0,\infty);Z)$ and $u_0\in \mathcal{B}$. If
$u\big|_{[0,\infty)} \in SAP_{\omega}(Z)$ and $M(t)\to 0 $ as $t\to\infty$,
then $\mathcal{P}_{u}\in SAP_{\omega}(\mathcal{B})$. Similarly, if
$(Z,\| \cdot\|_{Z})\hookrightarrow (X_{\beta} ,\| \cdot\|)$,
$w\in C(J \cup [0,\infty);Z)$, $w_0\in \mathcal{B}_{\beta}$,
$w\big|_{[0,\infty)}\in SAP_{\omega}(Z)$ and $M(t)\to 0 $ as $t\to\infty$,
then $\mathcal{P}_{w}\in SAP_{\omega}(\mathcal{B}_{\beta})$.
 \end{lemma}

\begin{proof}
 We only prove the assertion involving the space $X$. Let $i_{Z,X}:Z\to X $
be the inclusion map from $Z$ into $X$. For $1>\varepsilon>0$ there exists
$L_{\varepsilon}>0$ such that $\| u(t+\omega )-u(t)\|_{Z} \leq \varepsilon$
and $M(t )\leq \varepsilon$ for all $t\geq L_{\varepsilon}$.
Then, for $t\geq 2 L_{\varepsilon}$ we obtain
 \begin{align*}
&\| \mathcal{P}_{u}(t+\omega)-\mathcal{P}_{u}(t)\|_{\mathcal{B}} \\
& =\| u_{t+\omega} - u_{t} \|_{\mathcal{B}}  \\
&\leq M(t-L_{\varepsilon}) \| u_{_{L_{\varepsilon}}} \|_{\mathcal{B}}
 +\mathcal{K} \| i_{Z,X} \|_{\mathcal{L}(Z,X)}
 \sup_{s\geq L_{\varepsilon}}\| {u}(s+\omega)-u(s)\|_{Z} \\
&\leq M(t-L_{\varepsilon}) \big(M(L_{\varepsilon}) \| u_0 \|_{\mathcal{B}}
+ \mathcal{K} \| i_{Z,X} \|_{\mathcal{L}(Z,X)} \| u \|_{C( [0,\infty);Z)}\big)
+ \mathcal{K} \| i_{Z,X} \|_{\mathcal{L}(Z,X)} \varepsilon
 \\
&\leq \varepsilon \big( \varepsilon \| u_0 \|_{\mathcal{B}}
+ \mathcal{K} \| i_{Z,X} \|_{\mathcal{L}(Z,X)} \| u \|_{C( [0,\infty);Z)}
+ \mathcal{K} \| i_{Z,X} \|_{\mathcal{L}(Z,X)} \big) ,
\end{align*}
which implies that
 $\lim_{t\to\infty } \| \mathcal{P}_{u}(s+\omega)
-\mathcal{P}_{u}(s)\|_{\mathcal{B}} = 0 $ and
$\mathcal{P}_{u}\in SAP_{\omega}(\mathcal{B})$.
 \end{proof}

\begin{lemma}\label{lemma6}
Assume  $\mathcal{Q}\in C([0,\infty); \mathcal{L}(Z,X))\cap L^{1}([0,\infty);
\mathcal{L}(Z,X)) $, $v \in SAP_{\omega}(Z)$ and let $u :[0,\infty )\to X$
be the function given by $ u(t)=\int_0^t\mathcal{Q}(t-s) v(s)ds$.
Then $u \in SAP_{\omega}(X)$.
\end{lemma}

\begin{proof}
From Bochner's criteria for integrable functions and the estimate
 \[
 \| u(t ) \| \leq \int_0^t \|\mathcal{Q}(t -s) \|_{\mathcal{L}(Z,X)}
  \| v(s) \|_{Z} ds
 \leq \| \mathcal{Q}\|_{L^{1}([0 ,\infty), \mathcal{L}(Z,X))}
\| v\|_{ C([0,\infty);Z)},
 \]
it follows that $v\in C( [0,\infty ); X)$. On the other hand, for
$\varepsilon>0$ given there is $ L_{\varepsilon} >0 $ such that
$\| v(s+\omega)-v(s)\|_{Z} \leq \varepsilon $ and
$\| \mathcal{Q}\|_{L^{1}([L_{\varepsilon} ,\infty), \mathcal{L}(Z,X))}
 \leq \varepsilon $ for all $s\geq L_{\varepsilon} $.
Then, for $t\geq 2L_{\varepsilon} $ we obtain
 \begin{align*}
 &\| u(t + \omega) - u(t) \|\\
 & \leq  \int_0^\omega \|\mathcal{Q}(t+ \omega-s) \|_{\mathcal{L}(Z,X)}
 \| v(s) \|_{Z} ds  \\
&\quad + \int_0^t \| \mathcal{Q}(t-s)\|_{\mathcal{L}(Z,X)} \|
 v(s+\omega) - v(s) \|_{Z} ds \\
& \leq  \| v\|_{ C([0,\infty);Z)} \int_{t}^{t+\omega }
 \| \mathcal{Q} ( s) \|_{\mathcal{L}(Z,X)} ds \\
&\quad + 2 \| v\|_{ C([0,\infty);Z)} \int_0^{L_{\varepsilon}}
 \| \mathcal{Q} (t-s) \|_{\mathcal{L}(Z,X)} ds
 + \varepsilon \int_{L_{\varepsilon}}^{t}
 \| \mathcal{Q} (t-s) \|_{\mathcal{L}(Z,X)} ds \\
& \leq  \| v\|_{ C([0,\infty);Z)} \| \mathcal{Q} \|_{L^{1}
 ([L_{\varepsilon},\infty); \mathcal{L}(Z,X) )}
+ 2 \| v\|_{ C([0,\infty);Z)} \| \mathcal{Q} \|_{L^{1}([L_{\varepsilon},\infty);
 \mathcal{L}(Z,X) )} \\
&\quad + \varepsilon \| \mathcal{Q} \|_{L^{1}([0,\infty); \mathcal{L}(Z,X) )} \\
& \leq  \varepsilon( 3 \| v\|_{ C([0,\infty);Z)}
 + \| \mathcal{Q} \|_{L^{1}([0,\infty); \mathcal{L}(Z,X) )}),
 \end{align*}
which implies that $ \lim_{t\to\infty } [ u(s + \omega) - u(s) ] =0$ and
$u \in SAP_{\omega}(X)$.
\end{proof}

In addition to the above, from \cite{Pierri2} we note (without proof)
the following corollary.

\begin{corollary}[{\cite[Corollary 3.2]{Pierri2}}] \label{cor1}
Let $w : [0, \infty) \to Z$ be a
$S$-asymptotically $\omega$-periodic function and assume that $w^{\prime}$ is
bounded and uniformly continuous. Then $w^{\prime}$ is $S$-asymptotically
$\omega$-periodic.
\end{corollary}

We include now some Lemmas on $\alpha$-H\"older functions. We omit the proofs.

\begin{lemma} \label{lemma4}
Assume that $u\in C^{\alpha} ( [0,b];X)$, $\psi \in \mathcal{B}\cap C^{\alpha}(J;X)$
and $u(0)=\psi(0)$. Let $v$ be the function $v: J\cup [0,b]\to X $ given by
 $v= \psi $ on $J$ and $v=u$ on $[0,b]$.
Then $ v\in C^{\alpha}(J\cup [0,b];X)$ and
$ [v]_{C^{\alpha}(J\cup [0,b];X)}
\leq [ \psi ]_{C^{\alpha}(J;X)} + [ u ]_{C^{\alpha}([0,b];X)}$.
Moreover, the assertion is valid replacing $X$ by $X_{\beta}$.
\end{lemma}

\begin{lemma}\label{lemma7}
Assume that the condition {\rm (H1)} is satisfied,
$z\in C^{\alpha}([0,\infty);\mathcal{B}_{\beta}) $ and
$w \in C^{\alpha}([0,\infty);\mathcal{B}) $. Then
$f_1(\cdot,w(\cdot))\in C^{\alpha}([0,\infty);X)$,
$ f_{2}(\cdot,w(\cdot))\in C^{\alpha}([0,\infty);X)$ and
\begin{align*}
 [f_1(\cdot,z )]_{C^{\alpha}([0,\infty);X)}
&\leq [ f_1]_{C^{\alpha}([0,\infty); \mathcal{L}(\mathcal{B}_{\beta},X))}
\| z\|_{C([0,\infty);\mathcal{B}_{\beta})} \\
&\quad  + \| f_1 \|_{C ([0,\infty); \mathcal{L}(\mathcal{B}_{\beta},X))}
 [ z]_{ C^{\alpha}([0,\infty);\mathcal{B}_{\beta})},
\\
{{[f_{2}(\cdot,w)]}_{C^{\alpha}([0,\infty);X) } }
& \leq  [ f_{2} ]_{C^{\alpha}([0,\infty); \mathcal{L}
(\mathcal{B} ,X))} \| w\|_{C([0,\infty);\mathcal{B} )} \\
&\quad  + \| f_{2} \|_{C ([0,\infty); \mathcal{L}(\mathcal{B} ,X))}
 [w]_{C^{\alpha}([0,\infty);\mathcal{B} )}.
 \end{align*}
 \end{lemma}

We can establish now our first result on the existence of an
$\mathcal{S}$-asymptotically $\omega$-periodic strict solution for
\eqref{1}--\eqref{2}. Next, $\Lambda, \Lambda_1 $ and $ \Lambda_{2}$
are the constant in Proposition \ref{prop1} and
$ \tilde{\varphi}:J \cup [0,\infty)\to X$ is the function given by
 $\tilde{\varphi}(t)=\varphi(t)$ for $t\in J$ and
$\tilde{\varphi}(t)=T(t)\varphi(0)$ for $t\geq 0$.

 \begin{theorem}\label{thm1}
Assume {\rm (H1)} is satisfied, $ C(J;X)\hookrightarrow \mathcal{B}$,
$M(t)\to 0 $ as $t\to\infty$,
$ f_1\in SAP_{\omega}( \mathcal{L} (\mathcal{B}_{\beta},X))$ and
$f_{2}\in SAP_{\omega}( \mathcal{L} (\mathcal{B},X))$.
Suppose that the function $\varphi(\cdot)$ belongs to
$C^{1+\alpha }(J;X) \cap C ^{ \alpha }(J;X_{\beta}) $, $\varphi(0) \in X_1 $,
$\frac{d^{-}\varphi}{dt} (0)= A\varphi(0) \in D_{A}(\alpha,\infty)$,
$ f(0,\varphi,\varphi')=0$,
$\mathcal{P}_{\tilde{\varphi}} \in C^{1+\alpha}([0,\infty);\mathcal{B})
\cap C^{\alpha}([0,\infty);\mathcal{B}_{\beta})$ and
\[
 \Xi = \mathcal{K}\big[ (\mathcal{K}+1)( \mathcal{A } \Lambda +1)
+ \mathcal{A }( \Lambda_1 + \Lambda_{2})+1 \big]\Theta (f_1,f_{2}) <1,
\]
where
\[
 \Theta (f_1,f_{2}) = \| f_1 \| _{ C^{\alpha}([0,\infty);
\mathcal{L}(\mathcal{B}_{\beta},X))}+ \| f_{2}\| _{C^{\alpha}( [0,\infty);
\mathcal{L}(\mathcal{B},X))}
\]
 and $\mathcal{A}= \| A^{ -1}\| +\| (-A)^{\beta-1} \| +1$.
 Then there exists a unique $\mathcal{S}$-asymptotically $\omega$-periodic
strict solution
$u\in C ^{1+\alpha}(J\cup \mathbb{R}^{+};X) \cap C(J\cup \mathbb{R}^{+} ; X_{\beta}) $
 of \eqref{1}--\eqref{2}.
\end{theorem}

\begin{proof}
Let $\mathcal{Y}= C ^{1+\alpha}(J\cup \mathbb{R}^{+};X)
\cap C ^{ \alpha}(J\cup \mathbb{R}^{+} ; X_{\beta})$ endowed with the norm
$\| \cdot\|_{\mathcal{Y}} $ given by
$\| \cdot\|_{\mathcal{Y}}= \| \cdot\|_{C ^{1+\alpha}(J\cup \mathbb{R}^{+};X)}
+ \| \cdot\|_{ C(J\cup \mathbb{R}^{+} ; X_{\beta})} $ and
${\mathfrak{S}}$ be the space
\begin{equation}\label{def3}
 {\mathfrak{S}} = \{ u\in \mathcal{Y} :
u\big|_{[0,\infty)} \in SAP_{\omega}(X) , \,
\mathcal{P}_{u}\in C^{1+\alpha }([0,\infty);\mathcal{B})
\cap C^{\alpha}( [0,\infty);\mathcal{B}_{\beta}) \},
 \end{equation}
endowed with the metric
$$
\Phi(u,v)=\| P_u- P_v\|_{C^{1+\alpha}( [0,\infty);\mathcal{B}) }
 + \| P_u- P_v\|_{C^{\alpha}( [0,\infty);\mathcal{B}_{\beta}) } .
$$

 Let $\Gamma $ be the map
 $\Gamma :\mathfrak{S} \to\mathfrak{S} $ given by $(\Gamma u)_0=\varphi$ and
\begin{equation}
\Gamma u(t)=T(t)\varphi(0)+\int_0^{t}T(t-s)f_{u,u'}(s) ds,\quad t\geq 0,
\end{equation}
where $f_{u,u'}:[0,\infty)\to X$ is the function defined 
by  $f_{u,u'}(t)=f(t,u_{t},u'_{t})$. 

In the remainder of this proof we show that $ \Gamma$ is a contraction on
$ \mathfrak{S}$. To this end, next we assume that $u,v\in \mathfrak{S}$.
\smallskip

\noindent\textbf{Step 1.}
 $\Gamma u_{ \mid_{[0,\infty)}} \in SAP_{\omega}(X)
\cap C ^{ \alpha}( \mathbb{R}^{+};X_1) $ and
$\Gamma u \in C ^{1+\alpha}(J\cup \mathbb{R}^{+};X)
\cap C ^{ \alpha}(J\cup \mathbb{R}^{+} ; X_{\beta})$.

From the definition of $\mathcal{Y}$ we have that $u'(\cdot) $ is uniformly
continuous on $[0,\infty) $, which implies via Corollary \ref{cor1} that
$u' \in SAP_{\omega}(X)$. Moreover, from Lemma \ref{lemma8} and Lemma
\ref{lemma7} we infer that $ f_1\circ P_u$ and
$f_{2}\circ P_u'$ belong to $ SAP_{\omega}(X)$ and from
Lemma \ref{lemma6} (with $\mathcal{Q}(s)=T(s)$ and $Z=X$) we obtain
that $\Gamma u_{ \mid_{[0,\infty)}} \in SAP_{\omega}(X)$.

On the other hand, from Lemma \ref{lemma7} we have that
$f_{u,u'}\in C^{\alpha}([0,\infty);X)$ which implies via Proposition \ref{prop1}
that $ \Gamma u_{ \mid_{[0,\infty)}} \in C^{ \alpha}([0,\infty);X_1)
\cap C^{ 1+\alpha}([0,\infty);X )\cap C^{\alpha}([0,\infty);X_{\beta}) $ and
\begin{gather}
{ [ \Gamma u ]_{C^{\alpha}([0,\infty);X_1)} }
 \leq \frac{C_{1,\alpha}}{\alpha} \| A\varphi(0)\|_{\alpha,\infty}
+ \Lambda [ f_{u,u'} ]_{C^{\alpha} ([0,\infty);X)} \label{des9} ,  \\
{ \| \Gamma u \|_{C([0,\infty);X_1)} }
 \leq C_0\| A\varphi(0) \| + ( \Lambda_1
+ \Lambda_{2}) \| f_{u,u'} \|_{C^{\alpha}([0,\infty);X)}, \label{des10} \\
 { [ ( \Gamma u )' ]_{C^{\alpha}([0,\infty);X)} }
 \leq \frac{C_{1,\alpha}}{\alpha} \| Ax\|_{\alpha,\infty}
+ (\Lambda+1) [ f_{u,u'} ]_{C^{\alpha} ([0,\infty);X)} , \label{des12} \\
 { \| ( \Gamma u )' \|_{C([0,\infty);X)}}
 \leq C_0\| A\varphi(0) \| + ( \Lambda_1
+ \Lambda_{2}+1) \| f_{u,u'} \|_{C^{\alpha}([0,\infty);X)} , \label{des13} \\
 { \| \Gamma u \|_{C([0,\infty);X_{\beta})} }
 \leq \| (-A)^{\beta-1}\| \Gamma u \|_{C([0,\infty);X_1)}, \label{des14}\\
 { [ \Gamma u ]_{C^{\alpha}([0,\infty);X_{\beta})} }
 \leq \| (-A)^{\beta-1}\| [ \Gamma u ]_{C^{\alpha}([0,\infty);X_1)}. \label{des15}
\end{gather}
Moreover, by noting that
$(\Gamma u )'(0)=A\varphi(0)+f_{u,u'}(0)=A\varphi(0)=\varphi'(0)$ and
$\Gamma u(0)= \varphi(0)$, from the properties of the function
$\tilde{\varphi}$ and Lemma \ref{lemma4} we infer that
$\Gamma u\in C^{1+\alpha} (J\cup [0,b];X) \cap C^{\alpha} (J\cup [0,b];X_{\beta})$,
which completes the proof of this step.
\smallskip


\noindent\textbf{Step 2.}
 $\mathcal{P}_{\Gamma u}\in C^{1+\alpha }([0,\infty);\mathcal{B}) $ and
 \begin{gather}
[ P_{\Gamma u} - P_{\Gamma v} ]_{C^{\alpha}([0,b];\mathcal{B})}
 \leq \mathcal{K} (\mathcal{K}+1) \| A^{-1}
 \| \Lambda [ f_{u,u'} - f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)} , \label{des3}
 \\
 \| P_{\Gamma u} - P_{\Gamma v} \|_{C ([0,b];\mathcal{B}) }
\leq  \mathcal{K} \| A^{-1} \| (\Lambda_1 + \Lambda_{2})
 \| f_{u,u'} -f_{v,v'}\|_{C^{\alpha} ([0,\infty);X)}, \label{des4}
\\
[ (P_{\Gamma u})' - (P_{\Gamma v})' ]_{C^{\alpha}([0,b];\mathcal{B}) }
 \leq \mathcal{K} (\mathcal{K}+1) ( \Lambda +1)
 [ f_{u,u'} -f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)} , \label{des5}
 \\
\| (P_{\Gamma u})' - (P_{\Gamma v})' \|_{C([0,b];\mathcal{B})}
 \leq \mathcal{K}( \Lambda_1 + \Lambda_{2}+ 1)
 \| f_{u,u'} -f_{v,v'}\|_{C ^{\alpha} ([0,\infty);X)}) . \label{des6}
 \end{gather}
From the properties of the space $\mathcal{B}$ it is easy to see that
$\mathcal{P}_{\Gamma u}\in C ([0,\infty);\mathcal{B}) $.
To show that $\mathcal{P}_{\Gamma u}\in C^{\alpha }([0,\infty);\mathcal{B}) $
is convenient to estimate $\| P_{ \Gamma u }(t) -\varphi \|_{\mathcal{B} } $.
By using the properties of $P_{\tilde{\varphi}}$, for $t\geq 0 $ we obtain
 \begin{align*}
 &\| P_{ \Gamma u}(t) -\varphi \|_{\mathcal{B} }\\
 &\leq  \| P_{ \Gamma u}(t) - P_{ \widetilde{\varphi} }(t)\|_{\mathcal{B} }
 + \| P_{ \widetilde{\varphi} }(t) - \varphi\|_{\mathcal{B} }
 \\
 &\leq  \mathcal{K}\sup_{s\in [0,t]}\| \Gamma u (s) - \widetilde{\varphi} (0)\|
 + \mathcal{K} \sup_{s\in [0,t]}\| \widetilde{\varphi} (0)
 - \widetilde{\varphi} (s)\| +
 t^{\alpha}[P_{ \widetilde{\varphi}}]_{C^{\alpha}([0,\infty);\mathcal{B} )}
\\
 &\leq  t^{\alpha}\mathcal{K} [ \Gamma u]_{C^{ \alpha}([0,\infty); X)}
 + t^{\alpha}(\mathcal{K}H+1)[P_{ \widetilde{\varphi}}]_{C^{\alpha}
 ([0,\infty);\mathcal{B} )} . %\label{18}
 \end{align*}
 From this estimate, for $t>0$ and $h>0$, we see that
\begin{align*}
 &\| P_{ \Gamma u} (t+h)-P_{ \Gamma u} (t)\|_{\mathcal{B} }\\
 &\leq  \mathcal{K} \| P_{ \Gamma u} ( h)-\varphi \|_{\mathcal{B} }
 + \mathcal{K} \sup_{s\in [0,t]}\| \Gamma u (s+h)-\Gamma u(s)\| \\
&\leq \mathcal{K}^{2} h^{\alpha} [ \Gamma u]_{C^{ \alpha}([0,\infty); X)}
+ h^{\alpha}\mathcal{K}(\mathcal{K}H+1)
 [P_{ \widetilde{\varphi}}]_{C^{\alpha}([0,\infty);\mathcal{B} )}
+ \mathcal{K} h^{\alpha} [\Gamma u]_{C^{ \alpha}([0,\infty); X)} ,
 \end{align*}
which implies that
 \begin{equation}\label{des2}
 [ P_{ \Gamma u}]_{C^{\alpha}([0,\infty);\mathcal{B} )}
 \leq \mathcal{K} (\mathcal{K}+1)[ \Gamma u]_{C^{ \alpha}([0,\infty); X)}
+ \mathcal{K} (\mathcal{K}H+1)[P_{ \widetilde{\varphi}}]_{C^{\alpha}
([0,\infty);\mathcal{B} )} ,
\end{equation}
and $\mathcal{P}_{\Gamma u}\in C^{ \alpha }([0,\infty);\mathcal{B}) $
since $ [ \Gamma u]_{C^{ \alpha}([0,\infty); X)}
\leq \| A^{-1}\| [ \Gamma u]_{C^{ \alpha}([0,\infty); X_1)} <\infty $.
Moreover, by noting that
 $\Gamma u- \Gamma v$ is the mild solution of \eqref{4}-\eqref{5} with $x=0$
and $\xi= f_{u,u'} - f_{v,v'} $, from the above remarks, the estimative \eqref{des2}
 and Proposition \ref{prop1} we infer
 \begin{align*}
 [ P_{ \Gamma u}- P_{ \Gamma v}]_{C^{\alpha}([0,\infty);\mathcal{B} )}
 &\leq \mathcal{K} (\mathcal{K}+1)[ \Gamma u - \Gamma v]_{C^{ \alpha}([0,\infty); X)}
\\
 &\leq \mathcal{K} (\mathcal{K}+1) \| A^{-1} \|
 [ \Gamma u - \Gamma v]_{C^{ \alpha}([0,\infty); X_1)} \\
 &\leq \mathcal{K} (\mathcal{K}+1) \| A^{-1} \|\Lambda
 [ f_{u,u'} - f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)},
 \end{align*}
 which establish the estimate \eqref{des3}.


Proceeding as above, we also can prove that
$(\mathcal{P}_{\Gamma u})'\in C^{ \alpha }([0,\infty);\mathcal{B}) $.
Since $\varphi'\in \mathcal{B}$ and $ (\Gamma u )'_0=\varphi'$
 (note that $(\Gamma u )'(0)=A\varphi(0)+f_{u,u'}(0)=A\varphi(0)=\varphi'(0)$),
from the properties of $\mathcal{B}$ and the estimate \eqref{des13} we obtain
that $(\mathcal{P}_{\Gamma u})'\in C([0,\infty);\mathcal{B} )$.
In addition, for $t\geq 0 $ it is easy to see that
 \begin{equation}
 \| P_{ (\Gamma u)'}(t) -\varphi '\|_{\mathcal{B} }
 \leq t^{\alpha}\mathcal{K} [ (\Gamma u)']_{C^{ \alpha}([0,\infty); X)} +
 t^{\alpha}(\mathcal{K}H+1)[P_{ \widetilde{\varphi}'}]_{C^{\alpha}([0,\infty);\mathcal{B} )} .
 \end{equation}
Using this estimate, we have that
\begin{align*}
 &\| P_{ (\Gamma u)'} (t+h)-P_{ (\Gamma u)'} (t)\|_{\mathcal{B} } \\
 &\leq \mathcal{K} \| P_{ (\Gamma u)'} ( h)-\varphi'\|_{\mathcal{B} }
 + \mathcal{K} \sup_{s\in [0,t]}\| (\Gamma u)' (s+h)-(\Gamma u)'(s)\| \\
 &\leq \mathcal{K}(h^{\alpha} \mathcal{K} [ (\Gamma u)']_{C^{ \alpha}([0,\infty); X)}
+  h^{\alpha}(\mathcal{K}H+1)[P_{ \widetilde{\varphi}'}]_{C^{\alpha}([0,\infty);
 \mathcal{B} )} )  \\
&\quad + \mathcal{K} h^{\alpha} [(\Gamma u)']_{C^{ \alpha}([0,\infty); X)}
 \\
 &\leq h^{\alpha}\mathcal{K} (\mathcal{K}+1)[(\Gamma u)']_{C^{ \alpha}
 ([0,\infty); X)} +  h^{\alpha}\mathcal{K}(\mathcal{K}H+1)
 [P_{ \widetilde{\varphi}'}]_{C^{\alpha}([0,\infty);\mathcal{B} )} ) ,
\end{align*}
which implies
 \begin{equation}
 [ P_{ (\Gamma u)'}]_{C^{\alpha}([0,\infty);\mathcal{B} )}
 \leq \mathcal{K} (\mathcal{K}+1)
[(\Gamma u)']_{C^{ \alpha}([0,\infty); X)} + \mathcal{K}
(\mathcal{K}H+1)[P_{ \widetilde{\varphi}'}]_{C^{\alpha}([0,\infty);\mathcal{B} )} ,
\label{des16}
 \end{equation}
and $ P_{( \Gamma u)'}\in C^{\alpha}([0,\infty);\mathcal{B} )$ since
$[(\Gamma u)']_{C^{ \alpha}([0,\infty); X)}$ is finite.
This completes the proof that
$P_{\Gamma u}\in C^{1+\alpha }([0,\infty);\mathcal{B})$.
Proceeding as above, we also note that
 \begin{align*}
 [ P_{ (\Gamma u)'}- P_{ (\Gamma v)'}]_{C^{\alpha}([0,\infty);\mathcal{B} )}
 &\leq \mathcal{K} (\mathcal{K}+1)[ (\Gamma u)' - (\Gamma v)']_{C^{ \alpha}
 ([0,\infty); X)} \\
 &\leq \mathcal{K} (\mathcal{K}+1)(\Lambda+1) [ f_{u,u'}-f_{v,v'} ]_{C^{\alpha}
([0,\infty);X)},
 \end{align*}
which establish \eqref{des5}.

 Concerning the estimate \eqref{des4}, from Proposition \ref{prop1} we have
\begin{align*}
&\| P_{\Gamma u}(t) - P_{\Gamma v}(t) \|_{\mathcal{B}}\\
&\leq \mathcal{K} \sup_{s\in [0,t]}\| \Gamma u (s)-\Gamma v(s)\| \\
&\leq \mathcal{K} \| A^{-1}\| \sup_{s\in [0,t]}\| \Gamma u (s)
- \Gamma v (s)\|_{X_1} \\
 &\leq \mathcal{K} \| A^{-1}\| ( \Lambda_1
[ f_{u,u'} -f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)}
+ \Lambda_{2} \| f_{u,u'} -f_{v,v'}\|_{C ([0,\infty);X)}) ,
 \end{align*}
which proves \eqref{des4}. Finally, from the estimate
\begin{align*}
&\| (P_{\Gamma u})' - (P_{\Gamma v})' \|_{C([0,b];\mathcal{B})} \\
&\leq \mathcal{K} \| (\Gamma u)' - (\Gamma v)' \|_{C([0,b];X)} \\
&\leq \mathcal{K} ( \| \Gamma u - \Gamma v \|_{C([0,b];X_1)}
 + \| f_{u,u'} -f_{v,v'}\|_{C ([0,\infty);X)}) \\
&\leq \mathcal{K}( \Lambda_1 [ f_{u,u'} -f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)}
+ \Lambda_{2} \| f_{u,u'} -f_{v,v'}\|_{C ([0,\infty);X)})  \\
&\quad + \mathcal{K} \| f_{u,u'} -f_{v,v'}\|_{C ([0,\infty);X)} ,
 \end{align*}
we obtain \eqref{des6}.
\smallskip


\noindent\textbf{Step 3.}
 The function $\mathcal{P}_{\Gamma u}$ belongs to
$ C^{\alpha }( [0,\infty);\mathcal{B}_{\beta}) $ and
 \begin{gather}
 \| P_{\Gamma u} - P_{\Gamma v} \|_{C ([0,b];\mathcal{B}_{\beta}) }
\leq \mathcal{K} \| (-A)^{\beta-1}\| ( \Lambda_1
+ \Lambda_{2}) \| f_{u,u'} -f_{v,v'}\|_{C^{\alpha} ([0,\infty);X)}) \label{des7},
\\
  [ P_{\Gamma u} - P_{\Gamma v} ]_{C^{\alpha}([0,b];\mathcal{B}_{\beta}) }  
\leq \mathcal{K} (\mathcal{K}+1) \| (-A)^{\beta-1}
\| \Lambda [ f_{u,u'} -f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)} . \label{des8}
 \end{gather}
From Proposition \ref{prop1} it follows that
 \begin{align*}
 \| P_{\Gamma u} - P_{\Gamma v} \|_{C ([0,b];\mathcal{B}_{\beta})}  
 &\leq \mathcal{K} \| \Gamma u- \Gamma v \|_{C([0,b];X_{\beta})} \\
 &\leq \mathcal{K} \| (-A)^{\beta-1}\| \| \Gamma u - \Gamma v \|_{C([0,b];X_1)}
 \\
 &\leq \mathcal{K} \| (-A)^{\beta-1}\| ( \Lambda_1
 + \Lambda_{2}) \| f_{u,u'} -f_{v,v'}\|_{C^{\alpha} ([0,\infty);X)},
 \end{align*}
which establish \eqref{des7}. Concerning \eqref{des8} we note that
 \begin{align*}
& \| P_{\Gamma u}(t+h) - P_{\Gamma v}(t+h) - ( P_{\Gamma u}(t)
- P_{\Gamma v}(t)) \|_{\mathcal{B}_{\beta} }  \\
 &\leq \mathcal{ K} \| P_{\Gamma u}(h) - P_{\Gamma v}(h)\|_{\mathcal{B}_{\beta}}
+  \mathcal{ K} \sup_{s\in [0,h]}\| \Gamma u(s+h) - \Gamma u(s+h)
-( \Gamma u(s) - \Gamma u(s)) \|_{\beta} \\
&\leq \mathcal{ K}^{2}
 \sup_{s\in [0,h]}\| \Gamma u(s) - \Gamma u(h) -( \Gamma u(0)
 - \Gamma u(0)) \|_{\beta} \\
&\quad +  \mathcal{ K} \sup_{s\in [0,t]}\| \Gamma u(s+h)
 - \Gamma u(s+h) -( \Gamma u(s) - \Gamma u(s)) \|_{\beta} \\
&\leq \mathcal{ K}^{2}\| (-A)^{\beta-1}\| \sup_{s\in [0,t]}\| \Gamma u(s+h)
- \Gamma u(s+h) -( \Gamma u(s) - \Gamma u(s)) \|_{X_1} \\
&\quad +  \mathcal{ K} \| (-A)^{\beta-1}\| \sup_{s\in [0,t]}\| \Gamma u(s+h)
- \Gamma u(s+h) -( \Gamma u(s) - \Gamma u(s)) \|_{X_1}
 \\
 &\leq \mathcal{ K}(\mathcal{ K}+1 )\| (-A)^{\beta-1} \|
 [ \Gamma u -\Gamma u ]_{C^{ \alpha}([0,\infty); X_1)},
 \end{align*}
which, via Proposition \ref{prop1}, implies
 \[
 { [ P_{\Gamma u} - P_{\Gamma v} ]_{C^{\alpha}([0,b];\mathcal{B}_{\beta}) } }
 \leq \mathcal{K} (\mathcal{K}+1) \| (-A)^{\beta-1} \| \Lambda
 [ f_{u,u'} -f_{v,v'} ]_{C^{\alpha} ([0,\infty);X)} .
 \]

From  Steps 2 and 3 we obtain that
$ \Phi( \Gamma u, \Gamma v) \leq \Xi \Phi( u, v) $ which proves that
 $\Gamma$ is a contraction on $\mathfrak{S}$. Finally, from the contraction
 mapping principle and Proposition \ref{prop1} we infer that there exists
a unique $\mathcal{S}$-asymptotically $\omega$-periodic strict solution
 $u\in C ^{1+\alpha}(J\cup \mathbb{R}^{+};X) \cap C(J\cup \mathbb{R}^{+} ;
 X_{\beta}) $ of the problem \eqref{1}--\eqref{2}.
\end{proof}

In the next theorem we prove the existence of a strict solution via the conditions
(H2) and (H3). In this result, $Y$ is the space in condition (H2) and $i_{Y,X}$
denotes the inclusion map from $Y$ into $X$.

 \begin{theorem}\label{thm2}
Assume the conditions {\rm (H2)} and {\rm (H3)} are satisfied,
$C(J;X)\hookrightarrow \mathcal{B}$, $M(t)\to 0 $ as $t\to\infty$,
$\varphi\in C^{1}(J;X) \cap \mathcal{B}_{\beta} $, $\varphi(0)\in X_1$,
 $\frac{d^{-}\varphi}{dt} (0)= A\varphi(0) $ and $ f(0,\varphi,\varphi')=0 $.
Suppose, in addition, there are $\psi_1 \in \mathcal{B}_{\mathcal{B}}$ and
$ \psi_{2}\in \mathcal{B}$ such that
$f(\cdot,\psi_1,\psi_{2})\in C([0,\infty);Y)$ and
 \begin{equation}\label{des1}
\begin{aligned}
 \Theta &= \mathcal{K} ( \| (-A)^{\beta-1}\| +1)
 \| H \ast L_{f}\| _{C([0,\infty);\mathbb{R}^{+} )} \\
&\quad  + \mathcal{K} \| L_{f}\| _{C([0,\infty);\mathbb{R}^{+} )}
 \| i_{Y,X}\|_{\mathcal{L}(Y,X)}( \frac{ C_0}{\gamma}+1) <1,
\end{aligned}
 \end{equation}
where $ H \ast L_{f} (t)=\int^{t}_0 H(t-s)L_{f}(s)ds$.
 Then there exists a unique $\mathcal{S}$-asymptotically $\omega$-periodic
strict solution $ u\in C ^{1}(J\cup \mathbb{R}^{+};X) \cap C(J\cup \mathbb{R}^{+} ;
X_{\beta}) $ of \eqref{1}--\eqref{2}.
 \end{theorem}

\begin{proof}
 Let $\mathcal{S}= C^{1} (J\cup [0,\infty); X) \cap C (J\cup [0,\infty); X_{\beta})$
and
\[
 \mathfrak{F}=\{u\in \mathcal{S}: u\big|_{[0,\infty)}, u'\big|_{[0,\infty)}
\in SAP_{\omega}(X) , \mathcal{P}_{u}\in C([0,\infty);\mathcal{B}_{\beta})
\cap C^{1} ([0,\infty);\mathcal{B} ) \},
\]
endowed with the metric
$\Phi(u,v)= \| \mathcal{P}_{u} - \mathcal{P}_{v}\|_{C^{1}([0,\infty);\mathcal{B}) }
+ \| \mathcal{P}_{u} - \mathcal{P}_{v}\|_{C([0,\infty);\mathcal{B}_{\beta}) } $.
 Let $\Gamma : \mathfrak{F} \to\mathfrak{F} $ the map defined in the proof of
Theorem \ref{thm1}.
 Next, we show that $\Gamma$ is a contraction on $ \mathfrak{F}$.
To begin, we show that $\Gamma$ is a well defined function from $ \mathfrak{F}$
into $\mathfrak{F}$. In the remainder of this proof we assume that
$u,v\in \mathfrak{F} $.

 By noting that $ f_{u,u'}\in C([0,\infty);Y)$, from condition (H2)
it is easy to see that
 \begin{align*}
& \| (-A)^{\beta} \Gamma u (t)\|\\
&\leq  C_0 e^{-\gamma t} \| (-A)^{\beta} \varphi(0) \|
+ \| (-A)^{\beta-1}\|_{\mathcal{L}(X)} \int_0^{t}
 \| AT(t-s) \|_{\mathcal{L}(Y,X)} \| f_{u,u'}(s) \|_{Y} ds \\
&\leq  C_0 e^{-\gamma t} \| (-A)^{\beta} \varphi(0) \|
+  \| (-A)^{\beta-1}\|_{\mathcal{L}(X)}\| f_{u,u'} \|_{C([0,\infty); Y)}
\| H\|_{L^{1}([0,\infty),\mathbb{R}^{+})} ,
 \end{align*}
which implies that $ \Gamma u \in C(J\cup [0,\infty);X_{\beta})$ and
$ \mathcal{P}_{\Gamma u} \in C([0,\infty);\mathcal{B}_{\beta})$ since
$\varphi \in \mathcal{B}_{\beta} $. Moreover, from Lemma \ref{lemma3}
we have that $ \Gamma u $ is continuously differentiable and
$$
\| ( \Gamma u)'(t) \| \leq e^{-\gamma t}\| A\varphi(0)\|
 + \| f_{u,u'} \|_{C([0,\infty); Y)} ( \| H\|_{L^{1}([0,\infty),\mathbb{R}^{+})} +1) ,
$$
which shows that $ ( \Gamma u)' \in C( [0,\infty);X)$. In addition to the above,
from Lemma \ref{lemma2}, the compatibility condition
$ \frac{d^{-}\varphi}{dt} (0) = A \varphi(0)+ f(0,\varphi, \varphi' )= A \varphi(0) $
and the fact that
\[
 ( \Gamma u)'(t) = AT(t)\varphi(0)+ \int_0^{t}AT(t-s)f_{u,u'}(s) ds
+ f(t,u_{t},u_{t}') , \quad \forall t\geq 0,
\]
we infer that $ \mathcal{P}_{\Gamma u} \in C^{1}([0,\infty);\mathcal{B}) $
and $ (\mathcal{P}_{\Gamma u})' =\mathcal{P}_{ ( \Gamma u)'} $.

 To complete the proof that $\Gamma$ has values in $\mathfrak{F}$, it remain to
prove that the functions $ \Gamma u\big|_{[0,\infty)} $ and
$( \Gamma u)'\big|_{[0,\infty)} $ belong to $SAP_{\omega}(X) $.
 From Lemma \ref{lemma8} we have that
$\mathcal{P}_{u} \in SAP_{\omega}(\mathcal{B}_{\beta})$,
$ \mathcal{P}_{u'}\in SAP_{\omega}(\mathcal{B} )$ and from the conditions
(H2) and (H3) is easy to show that
$ f(\cdot,\mathcal{P}_{u }(\cdot), \mathcal{P}_{u'}(\cdot)) \in SAP_{\omega}(Y) $.
In addition, by using Lemma \ref{lemma6} with $Z=X$ and $ \mathcal{Q}(t)=T(t)$
we obtain that $ \Gamma u \in SAP_{\omega}(X) $. Moreover, arguing as above,
but using Lemma \ref{lemma6} with $Z=Y$ and
 $ \mathcal{Q}(t)=AT(t)$ it follows that
$ (\Gamma u)'\big|_{[0,\infty)} \in SAP_{\omega}(X) $.
This completes the proof that $\Gamma$ is a $ \mathfrak{F}$-valued function.

On the other hand, for $u,v\in \mathfrak{F} $ and $t\geq 0 $ we obtain
\[
 \| \Gamma u(t)- \Gamma v(t)\|_{\beta}
\leq  \| (-A)^{\beta-1}\| \int_0^{t} \| AT(t-s)\|_{\mathcal{L}(Y,X)} L_{f}(s)
\Phi(u,v) ds,
\]
and hence,
\begin{equation}
 \| \Gamma u- \Gamma v\|_{C([0,\infty);X_{\beta})}
\leq \| (-A)^{\beta-1}\| \| H \ast L_{f}\| _{C([0,\infty);\mathbb{R}^{+} )}
\Phi(u,v). \label{31}
\end{equation}
Similarly, from the inequality
\[ %\label{33}
 \| \Gamma u(t)- \Gamma v(t)\| \leq
 C_0 \int_0^{t} e^{-\gamma(t-s)} \| i_{Y,X}\|_{\mathcal{L}(Y,X)}
\| f_{u,u'}(s) -f_{v,v'}(s) \|_{Y} ds
\]
we obtain
\begin{equation}
 \| \Gamma u- \Gamma v\|_{C([0,\infty);X)}
\leq \frac{ C_0 }{\gamma} \| i_{Y,X}\|_{\mathcal{L}(Y,X)}
\| L_{f}\| _{C([0,\infty);\mathbb{R}^{+})} \Phi(u,v) \label{43}.
\end{equation}
In addition, from Lemma \ref{lemma3}, it follows that
\begin{align*}
 \| ( \Gamma u)'(t)- ( \Gamma v)'(t)\|
&\leq  \int_0^{t} H(t-s) L_{f}(s) (\| u_{s}-v_{s}\|_{\mathcal{B}_{\beta}}
 + \| u_{s}' - v_{s}' \|_{\mathcal{B}}) ds \\
&\quad + L_{f}(t) \| i_{Y,X}\|_{\mathcal{L}(Y,X)}
(\| u_{t}- v_{t} \|_{\mathcal{B}_{\beta}} + \| u_{t}'- v_{t}'\|_{\mathcal{B}} ) ,
\end{align*}
from which we infer that
\begin{equation}
\begin{aligned}
&\| ( \Gamma u)' - ( \Gamma v)' \|_{C([0,\infty);X)}  \\
& \leq \big( \| H \ast L_{f}\| _{C([0,\infty);\mathbb{R}^{+} )}
+ \| L_{f}\| _{C([0,\infty);\mathbb{R}^{+})}
\| i_{Y,X}\|_{\mathcal{L}(Y,X)} \big) \Phi( u, v) .
\end{aligned} \label{32}
\end{equation}
From inequalities \eqref{31}-\eqref{32} we obtain that $\Phi( \Gamma u, \Gamma v)
 \leq \Theta \Phi(u,v) $ which proves that $\Gamma$ is a contraction on
$ \mathfrak{F} $ and there exits a unique mild solution $u \in \mathfrak{F} $
of \eqref{1}--\eqref{2}. Finally, from Lemma \ref{lemma3} it follows that
 $u(\cdot)$ is a strict solution of the problem \eqref{1}--\eqref{2}.
This completes the proof.
\end{proof}

\section{Applications}\label{application}

Motivated by the examples presented in the introduction, in this section we
discuss the existence of a $\mathcal{S}$-asymptotically $\omega$-periodic
strict solution for the partial neutral differential problem
 \begin{gather} \label{eq7}
 u'(t,\xi)  = \Delta u(t,\xi) - \int_{-\infty}^{t} a(t)k(t-s)u' (s,\xi)ds + g(t) ,
 \\
u(t,0) = u(t,\pi )=0, \label{eq8} \\
u(s,\xi ) = \varphi(s,\xi) , \quad  s \leq 0, \label{eq9}
\end{gather}
 for $ (t,\xi)\in [0,\infty)\times [0,\pi]$ where
$g\in SAP_{\omega}(\mathbb{R})$ and $\varphi(\cdot)$ is a function defined from
$(-\infty,0]\times [0,\pi]$ into $\mathbb{R}$.

To treat the problem \eqref{eq7}-\eqref{eq9} under the abstract framework
in Section \ref{introduction}, we take $X = L^{2}([0, \pi])$ and
 $A:D(A) \subset X\to X$ be the operator $Ax= x''$ on $D(A)
 = \{x\in X: x'' \in X, \ x(0) = x(\pi)=0\} $. It is well known that
$A$ is the generator of an analytic
semigroup $(T(t))_{t\geq 0}$ on $X$, $A$ has discrete spectrum with
 eigenvalues $- n^{2},\;n \in \mathbb{N},$ and associated normalized
eigenvectors $z_{n} (\xi) = (\frac{2}{\pi})^{1/2} \sin (n \xi)$.
We note that $ \| T(t) \|\leq e^{-\frac{t}{2}}$,
$\| AT(t) \| \leq e^{-\frac{t}{2}}t^{-1} $ and
$\| A^{2}T(t) \| \leq 4e^{-\frac{t}{2} } t^{-2} $ for $t>0$.


As a phase space we consider the space
 $ \mathcal{B}= C_{r} \times L^{p}(\rho, X )$. Let $r \geq 0, 1 \leq p < \infty$
and $ \rho:(-\infty,-r] \to \mathbb{R} $ be a nonnegative
measurable function which satisfies the conditions (g-5)-(g-7) in
the terminology of \cite{HMN}. The space $ C_{r} \times
L^{p}(\rho,X)$ is formed by all classes of functions
$ \psi : (-\infty , 0] \to X$ such that $ \psi_{\mid_{[-r,0]} } \in C([-r,0],X) $,
$\psi(\cdot)$ is Lebesgue-measurable and
$ \rho^{\frac{1}{p}}\psi \in L^{p}((-\infty,-r],X)$. The norm in
$ C_{r}\times L^{p}(\rho,\mathcal{D})$ is given by
$
\| \psi \|_{\mathcal{B}} = \| \psi \|_{C([-r,0]; X)}
+ \| \rho^{\frac{1}{p}}\psi\|_{L^{p}((-\infty,-r],X)}.
$
From \cite{HMN}, we know that $\mathcal{B}$ satisfy the conditions in
Section \ref{section1} and $\mathcal{B}$ is a uniform fading memory space,
which implies that $M(t)\to 0$ as $t\to \infty$ and $K(\cdot)$ is bounded,
see \cite[pp.190]{HMN} for details.

Next we assume $k \in C([0,\infty);\mathbb{R})$,
$a\in C^{\alpha}([0,\infty),\mathbb{R})\cap SAP_{\omega}(\mathbb{R})$
for some $\alpha\in (0,1)$ and $\omega>0$, and
\[
\Theta = 2\| a\| _{C([0,\infty);\mathbb{R})}
\Big(\int_{-\infty}^{0} \frac{ k^{2}(-\tau)}{\rho(\tau)}d\tau \Big)^{1/2} <\infty.
\]
 Under these conditions, the function $f:[0,\infty)\times \mathcal{B} \to X $
given by
$ f(t,\psi )(\xi) = \int_{-\infty}^{0} a(t)k(-\tau) \psi (\tau,\xi) d\tau$
 is well defined,
$f\in C^{\alpha}([0,\infty);\mathcal{L}(\mathcal{B} ,X))
\cap SAP_{\omega}(\mathcal{L}(\mathcal{B}))$ and
$\| f\|_{C^{\alpha}([0,\infty); \mathcal{L }( \mathcal{B} ,X))}\leq \Theta $.

Next, we say that a function
$u\in C ^{1}(J\cup \mathbb{R}^{+};X) \cap C(J\cup \mathbb{R}^{+} ; X_1) $
is a strict solution of \eqref{eq7}-\eqref{eq9} if $u(\cdot)$ is a strict
solution of the associated problem \eqref{1}--\eqref{2}.
In the next result, which follows directly from Theorem \ref{thm1},
$\Lambda$ is the number in Proposition \ref{prop1} and $D_{A}(\alpha,\infty)$,
 $\mathcal{K}$ are as in Section \ref{introduction}. We also note that
$\Lambda$ appears explicitly in the end of the proof of Proposition \ref{prop1}
and that in the current case
$\Lambda =( \frac{2 }{\alpha }+ 4 + \frac{ 4}{ \alpha(1-\alpha)}) $,
$\Lambda_1 = (2+\frac{1}{\alpha} )$ and $\Lambda_{2} =2$.

\begin{proposition}\label{prop2}
Assume that $\varphi(0,\cdot)\in X_1$,
$\frac{d^{-}\varphi}{dt}(0,\cdot)= A\varphi(0,\cdot) \in D_{A}(\alpha,\infty)$,
$ f(0,\varphi)+f_{2}(0,\varphi')=0 $,
$\varphi\in C^{1+\alpha }(J;X) \cap C ^{ \alpha }(J;X_{\beta}) $ and
$\mathcal{P}_{\tilde{\varphi}} \in C^{1+\alpha}([0,\infty);\mathcal{B})$. If
 \begin{equation}
 \mathcal{K}\big[ (\mathcal{K}+1)( \mathcal{A } \Lambda +1)
+ \mathcal{A }(4+\frac{1}{\alpha} )+1 \big]\Theta <1,
 \end{equation}
where $\mathcal{A}= \| (-A)^{\beta-1} \| + \| A^{ -1}\| +1$,
then there exists a unique $\mathcal{S}$-asymptotically $\omega$-periodic
strict solution of the problem \eqref{eq7}--\eqref{eq9}.
\end{proposition}

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\end{document}