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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 55, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/55\hfil Existence and stability of solutions]
{Existence and stability of solutions to neutral
equations with infinite delay}

\author[Xianlong Fu\hfil EJDE-2013/55\hfilneg]
{Xianlong Fu}  % in alphabetical order

\address{Xianlong Fu \newline
 Department of Mathematics, East China Normal University\\
Shanghai, 200241,  China}
\email{xlfu@math.ecnu.edu.cn}

\thanks{Submitted September 16, 2012. Published February 21, 2013.}
\subjclass[2000]{34K25, 34K30, 34G20}
\keywords{Neutral functional differential equation; analytic semigroup;
\hfill\break\indent fractional power operator; linearized stability;
 infinite delay}

\begin{abstract}
 In this article, by using a fixed point theorem, we study the existence
 and regularity of mild solutions for a class of abstract neutral
 functional differential equations with infinite delay. The fraction
 power theory and $\alpha$-norm is used to discuss the problem so
 that the obtained results can be applied to equations with terms
 involving spatial derivatives. A stability result for the autonomous case
 is also established. We conclude with an example that illustrates
 the applications of the results obtained.
\end{abstract}

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

\section{Introduction}

In this article, we study the existence, regularity and stability of
mild solutions for the following abstract neutral functional
evolution equation with infinite delay:
\begin{equation}\label{pr}
\begin{gathered}
\frac{d}{dt}[x(t)+F(t, x_t)]+Ax(t) = G(t,x_t) , \quad 0{\leq}t{\leq}a,\\
x_0=\phi\in {\mathscr{B}_\alpha}.
\end{gathered} 
\end{equation}
where $x(\cdot)$ takes values in a subspace of
Banach space $X$, the operator $-A:D(A)\to X$ generates an analytic
semigroup $(S(t))_{t\ge 0}$, and $F,G:[0,a]\times
{\mathscr{B}}_\alpha\to X$ are appropriate functions,
${\mathscr{B}}_\alpha\subset {\mathscr{B}}$, and ${\mathscr{B}}$ is
the phase space to be specified later.

Since many practical functional differential models can be
studied by rewritten to abstract equation \eqref{pr}, in these years
there has been an increasing interest in the study of semilinear
evolution equations of form \eqref{pr}, such as existence and
asymptotic behavior of solutions (mild solutions, strong solutions
and classical solutions), and existence of (almost) periodic
solutions, etc. Here we mention the work of Travis and
Webb \cite{TW}, Rankin III \cite{Ra}, B\'atkai1 and
Piazzera \cite{BP} for the case of finite delay, and
Henr\'iquez \cite{He}, Adimy et al \cite{ABE}-\cite{AE},
 Liu \cite{Li}, Diagana and
Hern\'andez\cite{DH}, Hern\'andez et al \cite{HRH, HO, HO2} and Ye \cite{Ye} 
for the case of infinite delay. In \cite{HH} and
\cite{HH2} Hern\'andez and Henr\'iquez
have extended the problem studied in \cite{He} to neutral equations
and established the corresponding existence results of solutions and
periodic solutions. In their work, the operator $A$ generates an
analytic semigroup so that the theory of fractional power has been
used effectively there to obtain the existence of mild solutions,
strong solutions and periodic solutions for  \eqref{pr}. In the
subsequent years, various similar results have been established by
many mathematicians. In paper \cite{ABE}-\cite{AE} Adimy et al
have discussed this topic for the equations where the linear parts
are non-densely defined and have also achieved some similar results.
Particularly, in paper \cite{ABE2} the authors have discussed the
following functional differential system with infinite delay:
\begin{equation}\label{pr2}
\begin{gathered}
\frac{d}{dt}[x(t)+F(t, x_t)]=A[x(t)+F(t,x_t)]+  G(t,x_t), \quad
0{\leq}t{\leq}a,\\
x_0=\phi\in {\mathscr B},
\end{gathered}
\end{equation}
where $A$ is non-densely defined Hille-Yosida operator and generates
an integrated semigroup. The authors have proved there the
existence, uniqueness and the regularity of integral solutions, and
have investigated the stability near an equilibrium associated to
the autonomous case of \eqref{pr2}.

 The purpose of this article is
to extend the work in \cite{HH} and \cite{ABE2} so that the
corresponding results can be applied to the system
\begin{equation}\label{pr3}
\begin{gathered}
\frac{\partial}{\partial t}[u(t,x)+f(t, u(\cdot,x),\frac{\partial}
{\partial x}u(\cdot,x))]+\frac{\partial^2}{\partial x^2}u(t,x) =g(t,
u(\cdot,x),\frac{\partial}{\partial
x}u(\cdot,x)),\\
z(t)=z(t,\pi)=0,\\
z(\theta,x)=\phi(\theta,x),\theta\leq 0,\quad 0\leq x\leq\pi.
\end{gathered}
\end{equation}
Evidently, this system can be treated as the abstract equation
\eqref{pr}, however, the results established in \cite{HH} become
invalid for this situation, since the functions $f,~g$ in
\eqref{pr3} involve spatial derivatives. As one will see in Section
5, if take $X=L^2([0,\pi])$, then the third variables of $f$ and $g$
 are defined on $X_{\frac{1}{2}}$ and so the solutions can not be
 discussed on $X$ like in \cite{HH}. In this paper, inspired by the work in
\cite{TW1},\cite{TW2} and \cite{EFH}, we shall discuss this problem
by using fractional power operators theory and $\alpha-$norm, that
is, we shall restrict this equation in a Banach space
$X_\alpha(\subset X)$ and investigate the existence and regularity
of mild solutions for \eqref{pr}, as well as the stability for
the autonomous equation via $\|\cdot\|_\alpha$. We mention here
that, for the regularity of mild solutions, other than paper
\cite{HH}, we obtain the existence of strict solutions (not strong
solutions) for Eq. \eqref{pr} under H\"{o}lder continuous
conditions, see Section 3.2.

This article is organized as follows: we firstly introduce some
preliminaries about analytic semigroup and phase space for infinite
delay in Section 2, particularly, to make them to be still valid in
our situation, we have restated the axioms of phase space on the
space $X_\alpha$. The existence and uniqueness results of mild
solutions are discussed in Section 3 by applying fixed point
theorem. In this section we also provide some sufficient conditions
to guarantee the regularity of mild solutions, that is, we obtain
the existence of strict solutions. In section 4, we are concerned
with the stability of mild solutions. As in \cite{ABE2}, we state in
this part some properties of the solution operator associated to the
autonomous case of \eqref{pr}. Also, we investigate here the
stability near an equilibrium for this situation by using
linearized technique.  Finally, an example is presented
in Section 5 to show the applications of the results obtained.

\section{Preliminaries}

 Throughout this paper $X$ is a Banach space with norm
$\|\cdot\|$. And, $-A: D(-A)\to X$ is the infinitesimal generator of
a compact analytic semigroup $(S(t))_{t\ge 0}$ of uniformly bounded
linear operators. Let $0 \in \rho(A)$. Then it is possible to define
the fractional power $A^{\alpha}$ , for $0 <\alpha \leq 1$ , as a
closed linear operator on its domain $D(A^{\alpha})$ . Furthermore,
the subspace $D(A^{\alpha})$ is dense in $X$ and the expression
\[
\|x\|_{\alpha} =\|A^{\alpha}x\|,\quad x \in D(A^{\alpha}),
\]
defines a norm on $D(A^{\alpha})$. Hereafter we denote by
$X_{\alpha}$ the Banach space $D(A^{\alpha})$ normed with
$\|x\|_{\alpha}$. Then for each $\alpha >0,\;X_{\alpha}$ is a Banach
space, and $X_{\alpha} \hookrightarrow X_{\beta} $ for $0 <\beta
<\alpha $ and the imbedding is compact whenever the resolvent
operator of $A$ is compact.

 For the semigroup $(S(t))_{t\geq 0}$, the following properties 
will be used:
\begin{itemize}
\item[(a)] There exist $M \geq 1$ and $\omega\in \mathbb{R}$ such that
\begin{equation}\label{c0}\|S(t)\|
     \leq Me^{\omega t}, \quad \text{for all }t \ge 0 ;
\end{equation}
\item[(b)] For any $\alpha>0$, there exists a constant
$C_{\alpha}>0$ such that
\begin{equation}\label{c00}
    \|A^{\alpha}S(t)\| \leq
    \frac{C_{\alpha}}{t^{\alpha}}e^{\omega t}\,,\quad t>0.
\end{equation}
\item[(c)] For every $\alpha>0$, there exists a constant $C'_\alpha>0$
 such that
\begin{equation}\label{c01}
    \|(S(t)-I)A^{-\alpha}\| \leq
    {C'_{\alpha}}{t^{\alpha}}\,,\quad 0<t\le a.
\end{equation}
\end{itemize}
 In the sequel, we will use directly the estimates 
$\|S(t)\| \leq M$ and $\|A^{\alpha}S(t)\| \leq
    \frac{C_{\alpha}}{t^{\alpha}}$ 
on finite intervals. For
more details about the theory of operator semigroups and fraction
powers of operators, we refer to \cite{EN} and \cite{Pa}.

To study \eqref{pr}, we assume that the histories
$x_{t}:(-\infty,0]\to X$, $x_{t}(\theta)=x(t+\theta)$, belong to
some abstract phase space ${\mathscr{B}}$, which is defined
axiomatically. In this article, we employ an axiomatic definition of
the phase space ${\mathscr{B}}$ introduced by Hale and Kato
\cite{HK} and follow the terminology used in \cite{HMN}. Thus,
${\mathscr{B}}$ will be a linear space of functions mapping
$(-\infty,0]$ into $X$ endowed with a seminorm
$\|\cdot\|_{\mathscr{B}}$. We assume that ${\mathscr{B}}$ satisfies the
following axioms:
\begin{enumerate}
\item[(A1)]
 If $x:(-\infty,\sigma +a)\to X$, $a>0$, is continuous on
$[\sigma,\sigma +a)$ and $x_\sigma\in {\mathscr{B}}$, then for every
$t\in[\sigma,\sigma +a)$ the following statements hold:
\begin{itemize}
 \item[(i)] $x_t$ is in ${\mathscr{B}}$;

 \item[(ii)] $\|x(t)\|\leq H\|x_t\|_{\mathscr B}$;

\item[(iii)] $\|x_t\|_{\mathscr{B}}\leq K(t-\sigma)\sup\{\|x(s)\|:
\sigma\leq s\leq t\}+M(t-\sigma)\|x_\sigma\|_{\mathscr{B}_\alpha}$.
Here $H\geq 0$ is a constant, $K,M :[0,+\infty)\to [0,+\infty)$,
$K(\cdot)$ is continuous and $M(\cdot)$ is locally bounded, and $H$,
$K(\cdot)$, $M(\cdot)$ are independent of $x(t)$.
\end{itemize}

 \item[(A2)] For the function $x(\cdot)$ in (A1), $x_t$ is a 
$\mathscr{B}$-valued continuous
function on $[\sigma,\sigma +a]$.

\item[(B1)] The space $\mathscr{B}$ is complete.
\end{enumerate}
We denote by $\mathscr{B}_\alpha$ the set of all the elements in
$\mathscr{B}$ that take values in space $X_\alpha$; that is,
$$
\mathscr{B}_\alpha:=\{\phi\in \mathscr{B}:\phi(\theta)\in
X_\alpha\text{ for all }\theta\le 0 \}.
$$ 
Then$\mathscr{B}_\alpha$
becomes a subspace of $\mathscr{B}$ endowed with the seminorm
$\|\cdot\|_{\mathscr{B}_\alpha}$ which is induced by
$\|\cdot\|_{\mathscr{B}}$ through $\|\cdot\|_{\alpha}$. More
precisely, for any $\phi\in \mathscr{B}_\alpha$, the seminorm
$\|\cdot\|_{\mathscr{B}_\alpha}$ is defined by
$\|A^\alpha\phi(\theta)\|$, instead of $\|\phi(\theta)\|$. For
example, let the phase space $\mathscr{B}=C_r\times L^p(g:X)$, 
$r\ge 0$, $1\le p<\infty$ (cf. \cite{HMN}), which consists of all classes of
functions $\phi:(\infty,0]\to X$ such that $\phi$ is continuous on
$[-r,0]$, Lebesgue-measurable, and $g \|\phi(\cdot)\|^p$ is Lebesgue
integrable on $(-\infty,-r)$, where $g:(-\infty,-r)\to \mathbb{R}$
is a positive Lebesgue integrable function. The seminorm in
$\mathscr{B}$ is defined by
$$
\|\phi\|_{\mathscr{B}}=\sup \{\phi(\theta):-r\le\theta \le0\}+
\Big(\int_{-\infty}^{-r}g(\theta)
\|\phi(\theta)\|^pd\theta\Big)^{1/p}.
$$ 
Then the seminorm in $\mathscr{B}_\alpha$ is defined by
$$
\|\phi\|_{\mathscr{B}_\alpha}
=\sup\{\|A^\alpha\phi(\theta)\|:-r\le\theta \le0\}+
\Big(\int_{-\infty}^{-r}g(\theta)
\|A^\alpha\phi(\theta)\|^pd\theta\Big)^{1/p}.
$$ 
See also the space $\mathscr{C}_{g,\frac{1}{2}}$ presented in Section 5.
Hence, since $X_\alpha$ is still a Banach space, we will assume that
the subspace ${\mathscr{B}_\alpha}$ also satisfies the following
conditions:
\begin{itemize}
\item[(A1')]
 If $x:(-\infty,\sigma +a)\to X_\alpha$, $a>0$, is continuous on
$[\sigma,\sigma +a)$ (in $\alpha-$norm) and $x_\sigma\in
{\mathscr{B}}_\alpha$, then for every $t\in[\sigma,\sigma +a)$ the
followings hold:
\begin{itemize}
 \item[(i)] $x_t$ is in ${\mathscr{B}_\alpha}$;

 \item[(ii)] $\|x(t)\|_\alpha\leq H\|x_t\|_{\mathscr B_\alpha}$;\par

\item[(iii)] $\|x_t\|_{\mathscr{B}_\alpha}\leq K(t-\sigma)
 \sup\{\|x(s)\|_\alpha:\sigma\leq s\leq t\}
 +M(t-\sigma)\|x_\sigma\|_{\mathscr{B}_\alpha}$.
\end{itemize}
Here $H$, $K(\cdot)$ and $M(\cdot)$ are as in $(A)(iii)$ above.

 \item[(A2')] For the function $x(\cdot)$ in (A), $x_t$ is a 
${\mathscr{B}_\alpha}$-valued continuous
function on $[\sigma,\sigma +a]$.

\item[(B1')] The space $\mathscr{B}_\alpha$ is complete.
\end{itemize}

Finally we conclude this section by stating the following two theorems,
which play an essential role for our proofs in the next section.

\begin{theorem}[\cite{Sa}]\label{Th0}
Let $P$ be a condensing operator on a Banach space
$X$; i.e., $P$ is continuous and takes bounded sets into bounded
sets, and $\alpha(P(B))\leq \alpha(B)$ for every bounded set $B$ of
$X$ with $\alpha(B)>0$. If $P(H)\subset H$ for a convex, closed and
bounded set $H$ of $X$, then $P$ has a fixed point in $H$ (where
$\alpha(\cdot)$ denotes Kuratowski's measure of non-compactness).
\end{theorem}

\begin{theorem}[\cite{DS}] \label{th01}
 Let $(V(t))_{t\ge 0}$ be a nonlinear strongly
continuous semigroup on subset $\Omega $ of a Banach space $X$.
Assume that $x_0\in \Omega$ is an equilibrium of $(V(t))_{t\ge 0}$
and $V(t)$ is Fr\'echet-differentiable at $x_0$ for $
t\ge 0 $, with  $W(t)$ the Fr\'echet derivative at
$x_0$ of $V(t)$. Then $(W(t))_{t\ge 0}$ is a strongly continuous
semigroup of bounded linear operators on $X$. Moreover,  if the zero
equilibrium of $(W(t))_{t\ge 0}$ is exponentially stable, then $x_0$
is a locally exponentially stable equilibrium of $(U(t))_{t\ge 0}$.
\end{theorem}

\section{Existence results}

We devote this section to study the existence and regularity
of mild solutions for \eqref{pr}.

\subsection{Existence of mild solutions}

A mild solution of \eqref{pr} is defined as follows.

\begin{definition}\rm
 A function $x(\cdot):(-\infty,b]\to
D(A^{\alpha})$, $b>0$, is a mild solution of  \eqref{pr},
if $x_{0}=\phi$, the restriction of $x(\cdot)$ to the interval
$[0,b]$ is continuous and for each $0\leq t\leq b$, the function
$AS(t-s)F(s,x_s)$, $s\in [0,t)$ is integrable and the following
integral equality is satisfied:
\begin{equation}\label{sol}
\begin{aligned}
x(t) &=S(t)[\phi (0)+F(0,\phi)]-F(t,x_t)+\int_{0}^{t} AS(t-s)
F(s,x_s)ds\\
  &\quad +\int_{0}^{t}S(t-s) G(s,x_{s})ds,  \quad 0\leq t\leq b.
\end{aligned}
\end{equation}
 The last two terms are integrals in sense of Bocher
(see \cite{Ma}).
\end{definition}

We now give the basic assumptions for \eqref{pr} in our
discussion. Let $\Omega\subset {\mathscr{B}_\alpha}$ be an open set.
\begin{itemize}
\item[(H1)] $F:[0,a]\times \Omega\to D(A^{\alpha+\beta})$ is a continuous
function for some $\beta\in (0,1)$ with $\alpha+\beta\le 1$, and
there exists $l>0$ such
 that the function $A^\beta F$ satisfies:
\begin{equation}\label{c1}
\|A^\beta F(s_1,\phi_{1})-A^\beta F(s_2,\phi_{2})\|_\alpha\leq
l(|s_1 -s_2 |+\|\phi_{1}-\phi_{2}\|_{\mathscr{B}_\alpha } )
\end{equation}
 for any $0\leq s_1 ,s_2\leq a$, $\phi_{1},\phi_{2}\in \Omega$, 
and the inequality
\begin{equation}\label{c2}
M_1l K(0)<1
\end{equation}
holds, where $M_1:=\|A^{-\beta}\|$.

\item[(H2)]  The function $G:[0,a]\times \Omega \to X$ is continuous.
\end{itemize}

\begin{theorem}\label{th1} 
Let $\phi\in\Omega$. If  assumptions {\rm (H1), (H2)}
are satisfied, then  \eqref{pr} admits at least one mild
solution on $(-\infty,b_{\phi}]$ for some $b_\phi <a$.
\end{theorem}

\begin{proof}
 Let $y(\cdot):(-\infty,a]\to X_\alpha$ be the function defined by
\[
  y(t):= \begin{cases}
         S(t)\phi(0), &t\geq 0,\\
         \phi(t), &-\infty<t<0,
        \end{cases}
\]
then $y_0=\phi$, $y_t\in \mathscr{B}_\alpha$ for any $t\in [0,a]$,
and it is easy to prove that the map $t\to y(t)$ is continuous in
$\alpha-$ norm on $[0,a]$, hence $t\to y_t$ is continuous in
seminorm $\|\cdot\|_{\mathscr{B}_\alpha }$. We denote
$N_1:=\sup\{\|y_t\|_{\mathscr{B}_\alpha } :0\leq t\leq a\}$. Since
$A^\beta F(\cdot,\cdot)$ satisfies Lipschitz condition, $G$ is
continuous and $\Omega$ is open, there exists $r>0$ such that 
$B_r(\phi)\subset\Omega$ and $\|A^\beta F(t,\psi )\|\leq N_2$ and
$\|G(t,\psi )\|\leq N_3$ for constants $N_2,~N_3\geq 0$ and all
$(t,\psi )\in [0,a ]\times B_r (\phi)$. In the sequel, we always
denote
$$
K_{t}:=\sup_{s\in [0,t]} K(s), \quad 
M_t := \sup_{s\in [0,t]} M(s).
$$ 
As $y_0 =\phi$,  we may choose $0<b_1 <a$ such that 
$\|y_t -\phi\|_{\mathscr{B}_\alpha }\leq r/2$ for all $0\leq t\leq b_1$.

Let $\rho =\frac{r}{2K_{b_1}}$, and define the set
$$
S(\rho):= \{z\in C([0, b_{\phi}];X_{\alpha}): z(0)=0,\;
\|z(t)\|_{\alpha}\leq\rho,~0\leq t\leq b_{\phi}\},
$$ 
where
$b_\phi$ $(<b_1)$ will be determined below. Then $S(\rho)$ is clearly
a non-empty bounded, closed and convex subset of 
$C([0,b_\phi ];X_{\alpha})$. For each $z\in S(\rho)$, we denote by 
$\bar z$ the function defined by
\[
  \bar{z}(t):= \begin{cases}
        z(t),&0\leq t\leq b_{\phi},\\
        0,&-\infty<t<0.
        \end{cases}
 \]
Obviously, if $x(\cdot)$ satisfies \eqref{sol}, we can decompose it
as $x(t)=z(t)+y(t)$, $0\leq t\leq b_{\phi}$, which implies
$x_t=\bar{z}_t+y_t$ for every $0\leq t\leq b_{\phi}$ and the
function $z(\cdot)$ satisfies
\begin{align*}
z(t)&=S(t)F(0,\phi)-F(t, \bar{z}_t+y_t)+\int_{0}^{t} AS(t-s) 
 F(s, \bar{z}_s +y_s)ds\\
&\quad +\int_{0}^{t}S(t-s)G(s, \bar{z}_s +y_s)ds,\quad 0\leq t\leq
b_{\phi}.
\end{align*}
Let $P$, $P_1$, $P_2$ be the operators on $ S(\rho)$ defined,
respectively, by
\begin{gather*}
\begin{aligned}
(Pz)(t)&:=S(t)F(0,\phi)-F(t, \bar{z}_t +y_t)+\int_{0}^{t} AS(t-s) F(s, \bar{z}_s +y_s)ds\\
&\quad +\int_{0}^{t}S(t-s)G(s, \bar{z}_s +y_s)ds,
\end{aligned}
\\
 (P_1 z)(t):=S(t)F(0,\phi)-F(t, \bar{z}_t+y_t)+\int_{0}^{t} AS(t-s)
 F(s, \bar{z}_s +y_s)ds
\end{gather*}
and
$$
(P_2 z)(t):= \int_{0}^{t}S(t-s)G(s, \bar{z}_s +y_s)ds.
$$
Then, the assertion that  \eqref{pr} admits a mild solution is
equivalent to $P=P_1 +P_2$ has a fixed point. Next we prove that $P$
has a fixed point by using Theorem \ref{Th0}. For this purpose, we
will show that $P$ maps $S(\rho)$ into itself and $P_1$ verifies a
contraction condition while $P_2$ is a completely continuous
operator.

Initially, we see that if $z(t)\in S(\rho)$, then 
$\bar{z}_t +y_t \in B_r (\phi)$ for all $0\leq t\leq b_\phi$. 
In fact, Axiom (A1') of
the phase space $\mathscr{B}_\alpha $ yields that 
\begin{align*}
\| \bar{z}_t +y_t -\phi\|_{\mathscr{B}_\alpha } 
& \leq \|\bar{z}_t \|_{\mathscr{B}_\alpha } 
 +\| y_t -\phi \|_{\mathscr{B}_\alpha }\\
&\le K(t)\sup_{0\le s\le t}\|z(s)\|_{\alpha}
 +\| y_t -\phi \|_{\mathscr{B}_\alpha }\\
&\leq K_{ b_{\phi}} \rho+\frac{r}{2} \leq r.
\end{align*}
To show that $P$ maps $S(\rho)$ into $S(\rho)$, let $z\in S(\rho)$.
Then
\begin{align*}
(P_1 z)(t)&= S(t) A^{-\beta} [ A^{\beta}  F(0,\phi)- A^{\beta}  F(t,y_t )] \\
&\quad +(S(t)-I)F(t,y_t )\\
&\quad +A^{-\beta}[A^{\beta} F(t,y_t )- A^{\beta} F(t,\bar{z}_t +y_t )]\\
&\quad +\int_{0}^{t} AS(t-s) F(s,\bar{z}_s +y_s )ds,
\end{align*}
then from assumption (H1), \eqref{c0} and \eqref{c01} it follows
that
\begin{align*}
\|(P_1 z)(t)\|_{\alpha} 
&\leq \|S(t)\|
\|A^{-\beta}\|\|A^{\alpha}( A^{\beta}  F(0,\phi)- A^{\beta}  F(t,y_t ))\|\\
&\quad+\|A^{\alpha}(S(t)-I)A^{-\beta}A^{\beta}F(t,y_t )\|\\
&\quad +\|A^{-\beta}\|\|A^{\alpha}(A^{\beta} F(t,y_t )- A^\beta
F(t,\bar{z}_t +y_t))\|\\
&\quad+\int_{0}^{t}\|A^{1-\beta} S(t-s) \|
\| A^{\alpha}A^{\beta}F(s,\bar{z}_s +y_s )\|ds\\
&\leq MM_1l(t+\|y_t -\phi\|_{\mathscr{B}_\alpha })+M_1
 C'_{\beta}  N_2t^{\beta}\\
&\quad+M_1lK_{ b_{\phi}}\rho+
C_{1-\beta}N_2\int_{0}^{t}\frac{1}{(t-s)^{\beta}}ds.
\end{align*}
for $0\leq t\leq b_{\phi}$. And we have also that
$$
\|(P_2z)(t)\|_{\alpha} 
 =\|\int_{0}^{t}A^{\alpha}S(t-s)G(s, \bar{z}_{s} +y_s)ds\|\\
\leq C_\alpha N_3\int_{0}^{t}\frac{1}{(t-s)^{\alpha}}ds.
$$
Therefore, by \eqref{c2} we may choose $b_{\phi}$, $0< b_{\phi}<b_1$
such that 
\begin{equation}\label{L1}
\begin{aligned}
&MM_1l(t+\|y_t -\phi\|_{\mathscr{B}_\alpha })+M_1
 C'_{\beta}  N_2t^{\beta}+
\frac{C_{1-\beta}N_2}{\beta}{t^{\beta}}+\frac{C_\alpha
N_3}{1-\alpha}{t^{1-\alpha}}\\
&\le (1-M_1lK_{ b_{\phi}})\rho
\end{aligned}
\end{equation}
 for all
$0<t \le b_{\phi}$,   and
\begin{equation}\label{L2}
l^* := lK_{ b_{\phi}}(M_1+C_{1-\beta}
\frac{b_{\phi}^{1-\beta}}{1-\beta})<1.
\end{equation} 
 Hence from \eqref{L1} we obtain that
\begin{align*}
\|(Pz)(t)\|_{\alpha} 
& \le\|(P_1 z)(t)\|_{\alpha}+\|(P_2 z)(t)\|_{\alpha}\\
&\leq (1-M_1l K_{ b_{\phi}})\rho+M_1l K_{ b_{\phi}}\rho
=\rho,
\end{align*}
which shows $P$ maps $S(\rho)$ into itself.

Now we prove that $P_1$ is a contraction map. 
Take $z_1,z_2\in S(\rho)$, then for each $t\in[0,b_\phi ]$ and 
by Axiom (A1)(ii) and
\eqref{c1}, we have
\begin{align*}
&\|(P_1 z_1)(t)-(P_1 z_2)(t)\|_{\alpha}\\
&\quad \leq\| F(t, \bar{z}_{1,t}+y_t)-F(t, \bar{z}_{2,t}+y_t)\|_{\alpha}\\
&\quad +\|\int_{0}^{t} AS(t-s)
[F(s, \bar{z}_{1,s}+y_s)-F(s, \bar{z}_{2,s}+y_s)]ds\|_{\alpha}\\
&\quad \leq M_1l\|\bar{z}_{1,t}-\bar{z}_{2,t}\|_{\mathscr{B}_\alpha }
+\int_{0}^{t}\frac{C_{1-\beta}}{(t-s)^{\beta}}l\|\bar{z}_{1,s}
-\bar{z}_{2,s}\|_{\mathscr{B}_\alpha } ds\\
&\quad \leq lK_{ b_{\phi}}(M_1+C_{1-\beta}
\frac{b_{\phi}^{1-\beta}}{1-\beta})\sup_{0\leq s\leq b_{\phi}}\|z_1
(s)-z_2 (s)\|_\alpha\\
&\quad =l^* \sup_{0\leq s\leq b_{\phi}}\|z_1 (s)-z_2 (s)\|_\alpha,
\end{align*}
where $l^*<1$ by \eqref{L2}. Thus
 $$
\|P_1 z_1 -P_1 z_2\|_\alpha<l^* \|z_1 -z_2\|_\alpha,
$$ 
and so $P_1$ is a contraction.

To prove that $P_2$ is a completely continuous operator, first we
note that $P_2$ is obviously continuous on $ S(\rho) $. Then we
prove that the family $\{P_{2}z:z\in S(\rho)\}$ is a family of
equi-continuous functions. To do this, let $0< t\leq b_{\phi}$,
$h>0$ be sufficient small, then
\begin{align*}
&\|(P_{2}z)(t+h)-(P_{2}z)(t)\|_{\alpha}\\
&=\|\int_{0}^{t+h}A^{\alpha}S(t+h-s)G(s,\bar{z}_s+y_s)ds-
\int_{0}^{t}A^{\alpha}S(t-s)G(s,\bar{z}_s+y_s)ds\|\\
&\leq\int_{0}^{t -\epsilon}\|A^{\alpha}(S(t+h-s)-S(t-s))\|
\|G(s,\bar{z}_s+y_s)\|ds\\
&\quad +\int_{t -\epsilon}^{t}\|A^{\alpha}(S(t+h-s)
-S(t-s))\|\|G(s,\bar{z}_s+y_s)\|ds\\
&\quad +\int_{t}^{t+h}\|A^{\alpha}S(t+h -s)\|\|G(s,\bar{z}_s+y_s)\|ds.\\
&\le N_3\|S(h +\epsilon) -S(\epsilon)\|\int_0^{t-\epsilon}
\|A^{\alpha}S(t
-s-\epsilon)\|ds\\
&\quad +N_3\int_{t-\epsilon}^{t}
       \|A^{\alpha}[S(t+h -s) -S(t -s)]\|ds\\
&\quad  +N_3\|\int_{t}^{t+h} A^{\alpha}T(t+h
-s)ds\|\\
&\le \frac{C_{\alpha}}{1-\alpha}
N_3(t-\epsilon)^{1-\alpha}\|S(h+\epsilon)
-S(\epsilon)\|\\
&\quad +\frac{C_{\alpha}}{1-\alpha}
N_3[h^{1-\alpha}-(h-\epsilon)^{1-\alpha}
+\epsilon^{1-\alpha}]
 +\frac{C_{\alpha}}{1-\alpha} N_3h^{1-\alpha}.
\end{align*}
The right-hand side tends to zero as $h \to 0$ with $\epsilon$
sufficiently small, since $S(t)$ is strongly continuous, and the
compactness of $S(t)$, $t >0$, implies the continuity in the uniform
operator topology. Hence, $P_2$ maps $S(\rho)$ into a family of
equi-continuous functions.

It remains to prove that $V(t)=\{(P_2 z)(t):z\in S(\rho)\}$ is
relatively compact in $X_\alpha$. Obviously it is true in the case
$t=0$. Observe that for $0<\alpha<\alpha_1<1$, $t>0$,
\begin{align*}
\|A^{\alpha_1}(P_{2}z)(t) \|
&=\big\|\int_{0}^{t}A^{\alpha_1}S(t-s)G(s, \bar{z}_{s} +y_s)ds\big\|\\
&\leq C_{\alpha_1} N_3\int_{0}^{t}\frac{1}{(t-s)^{\alpha_1}}ds,
\end{align*}
which implies that $A^{\alpha_1}(P_{2}z)(t)$ is bounded in $X$,
Hence, by the compactness of operator $A^{-\alpha_1} : X\to
X_{\alpha}$ (note the imbedding $X_{-\alpha_1}\hookrightarrow
X_{\alpha}$ is compact), we infer that the set $V(t)$ is relatively
compact in $X_\alpha$. Thus, by Arzela-Ascoli theorem $P_2$ is a
completely continuous operator. These arguments enable us to
conclude that $P=P_1+P_2$ is a condensing map on $S(\rho)$, and by
Theorem \ref{Th0} there exists a fixed point $z(\cdot)$ for $P$ on
$S(\rho)$, which implies equation \eqref{pr} admits a mild solution
on $(-\infty,b_\phi ]$. Then the proof is complete.
\end{proof}

 We can easily prove the following result on uniqueness of solutions.

\begin{theorem}\label{th2}
Assume the condition {\rm (H1)} of the preceding theorem holds. If
there exists $l' >0$ such that
$$
\|G(t ,\phi_1)-G(t,\phi_2)\|\leq l' \|\phi_1 -\phi_2\|_{\mathscr{B}_\alpha }
 $$
for all $0\leq t\leq a$, and $\phi_1,\phi_2\in \Omega$. Then, for
any $\phi\in\Omega$, the problem \eqref{pr} has a unique mild
solution on $(-\infty,b_\phi ]$ for some $b_\phi \in (0,a)$.
\end{theorem}

The extension of solutions to  \eqref{pr} can also be obtained by
standard arguments. Here we only state the result as a theorem, the
proof is very similar to that in Paper \cite{He} and \cite{TW1}.

\begin{theorem} 
Assume that the conditions of Theorem \ref{th1} or
Theorem \ref{th2} are satisfied. Then, for any $\phi\in\Omega$, the
equation \eqref{pr} has a solution $x(t)$ on a maximal interval of
existence $(-\infty,b_{\rm max} )$. And, if $b_{\rm max} <\infty$, then
$\overline{\lim}_{t\to b^-_{\rm max}}\|x(t)\|_{\alpha}=\infty$.
\end{theorem}

\subsection{Existence of strict solutions}

In this subsection, we discuss the regularity of mild solutions for
\eqref{pr}; that is, we will provide conditions to allow the
differentiability of mild solutions of \eqref{pr}. 
For this purpose we need some additional properties of the phase subspace
${\mathscr{B}_\alpha}$. Let $\mathcal{BC}_\alpha$ be the set of bounded
and continuous functions mapping $(-\infty,0]$ into $X_\alpha$, and
$C_{00}$ its subset consisting of functions with compact support. If
$\mathscr{B}_\alpha$ also satisfies the additional axiom:
\begin{itemize}
\item[(C1)] If a uniformly bounded sequence $\{\phi^n(\theta)\}$ in $C_{00}$
converges to a function $\phi(\theta)$ uniformly on every compact
set on
$(-\infty,0]$, then $\phi\in \mathscr{B}_\alpha$ and
 $$ 
\lim_{n\to +\infty}\|\phi^n-\phi\|_{\mathscr{B}_\alpha}=0.
$$
\end{itemize}
Then $\mathcal{BC}_\alpha$ is continuously imbedded into
$\mathscr{B}_\alpha$. Put
$$
\|\phi \|_{\infty}=\sup \{ \|\phi (\theta) \|_\alpha: \theta\leq 0\},
$$
for $\phi\in \mathcal{BC}_\alpha$, then one has the following result.

\begin{lemma}[\cite{HK}] \label{Le04} 
If the phase space $\mathscr{B}_\alpha$
satisfies the axiom {\rm (C1)}, then $\mathcal{BC}_\alpha \subset
\mathscr{B}_\alpha $, and there exists a constant $J>0$ such that
$\|\phi \|_{\mathscr{B}_\alpha}\leq J\|\phi \|_{\infty}$ for all
$\phi\in \mathcal{BC}_\alpha$
\end{lemma}
 
\begin{definition} \rm
A function $x(\cdot):(-\infty,b]\to X_{\alpha}$, $b>0$, is said to be a
strict solution of problem \eqref{pr}, if
\begin{itemize}
\item[(1)] $x(t)+F(t, x_t )~\in C([0,b];X_\alpha)\cap C^1((0,b];X)$;

\item[(2)] $x(\cdot)\in D(A)$ satisfies
$$
\frac{d}{dt}[x(t)+F(t, x_t )]  +Ax(t) = G(t, x_t ) ,
 $$ 
on $[0,b]$ and
$$
x_0=\phi\in\mathscr{B}_\alpha.
$$
\end{itemize}
\end{definition}

\begin{theorem} \label{th3}
Let the phase space $\mathscr{B}_\alpha$ satisfies the axiom {\rm (C1)}
 additionally. Suppose
that condition {\rm (H1)} and {\rm (H2)} are satisfied. Also the following
conditions hold:
\begin{itemize}
\item[(H1')] Let $x:(-\infty,a]\to X_{\alpha}$ be a function such
that $x_t \in \Omega$ for $t \in [0,a]$ and $x(\cdot)$ is continuous
on $[0,a]$, then the map $t \to A^\beta F(t,x_t)$ is H\"{o}lder
continuous in $\alpha-$norm with exponent $0<\theta_1<1$ satisfying
$\theta_1>1-\alpha-\beta$.

\item[(H2')] Function $G(\cdot,\cdot)$ is locally H\"{o}lder
continuous; i.e., for each $(t^0,\phi^0 )\in [0,a]\times \Omega$,
there exists a neighborhood $W$ of $(t^0,\phi^0 )$, and constants
$l_2>0,~ 0<\theta_2 <1$, such that
 $$
\|G(s_2 ,\phi_2 )-G(s_1 ,\phi_1 )\|\leq l_2 [|s_2 - s_1
|^{\theta_2}+\|\phi_{2}-\phi_{1}\|^{\theta_2}_{\mathscr{B}_\alpha
}]
$$ 
for $(s_i ,\phi_i ) \in W\subset ([0,a]\times \Omega )$,
$i=1,2$;
\item[(H3)] The initial function $\phi \in\Omega$ is H\"{o}lder
continuous, and  $\phi(0)+F(0,\phi)\in D(A)$.
\end{itemize}
Then the equation \eqref{pr} has a strict solution on
$(-\infty,b_{\phi}] $ for some $b_\phi >0$.
\end{theorem}

\begin{proof}
By Theorem \ref{th1}, we see that  \eqref{pr} has a mild solution
$x(\cdot)$ on $(-\infty,b_{\phi}]$. For this $x(\cdot)$, let
\begin{gather*}
m(t)=S(t)[\phi (0)+F(0, \phi )],\\
p(t)= \int_{0}^{t}  AS(t-s)F(s, x_s ) ds,\\
q(t)= \int_{0}^{t}S(t-s) G(s, x_s ) ds.
\end{gather*}
It follows from \eqref{c0} and \eqref{c01} that
\begin{align*}
\|m(t+h)-m(t)\|_{\alpha}
&=\|S(t)(S(h)-I)A^{-(\alpha'-\alpha)}A^{\alpha'}
[\phi(0)+F(0,\phi)]\|\\
&\le MC'_\alpha h^{\alpha'-\alpha} \|\phi(0)+F(0,\phi)\|_{\alpha'},
\end{align*}
where $\alpha'>0 $ is a constant chosen to satisfy
$\alpha<\alpha'<\alpha+\beta$,
\begin{align*}
&\|p(t+h)-p(t)\|_{\alpha}\\
&\le \|\int_{0}^{t} AS(t-s)[S(h)-I]F(s, x_s )
ds\|_\alpha+\|\int_{t}^{t+h} AS(t+h-s)F(s,
x_s ) ds\|_\alpha\\
&\le  \|\int_{0}^{t}
A^{1-(\alpha'-\alpha)}S(t-s)[S(h)-I]A^{-(\alpha+\beta-\alpha')}A^{\alpha}A^{\beta}F(s,
x_s ) ds\|\\
&\quad +\|\int_{t}^{t+h}
A^{1-\beta}S(t+h-s)A^{\alpha}A^{\beta}F(s, x_s ) ds\|\\
 &\le [\int_{0}^{t}
C_{1-(\alpha'-\alpha)}C'_{\alpha+\beta-\alpha'}(t-s)^{(\alpha'-\alpha)-1}
h^{\alpha+\beta-\alpha'}ds\\
&\quad + \int_{t}^{t+h}
C_{1-\beta}(t+h-s)^{\beta-1} ds]\max_{0\le
s\le b_\phi }\|A^\beta F(s, x_s)\|_\alpha\\
&\le [\frac{C_{1-(\alpha'-\alpha)}
 C'_{\alpha+\beta-\alpha'}}{\alpha'-\alpha}t^{\alpha'-\alpha}h^{
\alpha+\beta-\alpha'}+
\frac{C_{1-\beta}}{\beta}h^{\beta}]\max_{0\le s\le
b_\phi}\|A^\beta F(s, x_s)\|_\alpha,
\end{align*}
and
\begin{align*}
&\|q(t+h)-q(t)\|_{\alpha}\\
&\le \|\int_{0}^{t} S(t-s)[S(h)-I]G(s, x_s )
ds\|_\alpha+\|\int_{t}^{t+h} S(t+h-s)G(s,
x_s ) ds\|_\alpha\\
&\le \|\int_{0}^{t} A^{\alpha'}S(t-s)[S(h)-I]A^{\alpha-\alpha'}G(s,
x_s ) ds\|\\
&\quad +\|\int_{t}^{t+h}
A^{\alpha}S(t+h-s)G(s, x_s ) ds\|\\
&\le \Big[\int_{0}^{t}
C_{\alpha'}C'_{\alpha'-\alpha}(t-s)^{-\alpha'}h^{\alpha'-\alpha}ds+
\int_{t}^{t+h} C_{\alpha}(t+h-s)^{-\alpha} ds\Big]\max_{0\le
s\le b_\phi }\|G(s, x_s)\|\\
&\le [\frac{C_{\alpha'}C/-{\alpha'-\alpha}}{1-\alpha'}t^{1-\alpha'}h^{
\alpha'-\alpha}+ \frac{C_{\alpha}}{1-\alpha}h^{1-\alpha}
]\max_{0\le s\le b_\phi}\|G(s, x_s)\|,
\end{align*}
from which we see that $m(t)$, $p(t)$ and $q(t)$ are all H\"{o}lder
continuous on $[0 ,b_\phi ]$.
 So combined condition $(H_1 ') $ it is easy to deduce that $ x(\cdot)$ is
H\"{o}lder continuous on $[0, b_\phi]$. Since $\phi$ is H\"{o}lder
continuous on $(-\infty, 0]$ we infer that $ x(\cdot)$ is H\"{o}lder
continuous on $(-\infty, b_\phi]$. Thus, by Lemma \ref{Le04} the map
$t\to x_t (\cdot,\phi )$ is also H\"{o}lder continuous on $[0
,b_\phi ]$. Hence the map
$$
s\to G(s, x_s ) 
$$ 
is H\"{o}lder continuous on $[0,b_\phi ]$. Therefore, from the proof 
of \cite[Corollary 4.3.3]{Pa} 
it is not difficult to see that $q(t)\in D(A)$, and
$$
q'(t)= G(t, x_t ) - A\int_{0}^{t}S(t-s) G(s, x_s ) ds.
$$
On the other hand, we can also show $p(t)$ has the similar property
as $q(t)$. Indeed, let $t\in [0,b_\phi)$ and $h>0$, then
\begin{equation}\label{p1}
\begin{aligned}
&\frac{S(h)p(t)-p(t)}{h}\\
&=\frac{1}{h}[S(h)\int^t_0AS (t-s)F(s,x_s)ds-\int^t_0AS (t-s)F(s,x_s )ds]\\
&=\frac{1}{h}(p(t+h)-p(t))-\frac{1}{h}\int^{t+h}_{t}AS
(t+h-s)F(s,x_s )ds, 
\end{aligned}
\end{equation}
and

\begin{equation} \label{p2}
\begin{aligned}
&\|\frac{1}{h}\int^{t+h}_{t}AS (t+h-s)F(s,x_s )ds-AF(t,x_t)\| \\
&\le \|\frac{1}{h}\int^{t+h}_{t}AS (t+h-s)[F(s,x_s
)-F(t,x_t)]ds\|\\
&\quad  +\|\frac{1}{h}\int^{t+h}_{t}AS (t+h-s)F(t,x_t
)ds-AF(t,x_t)\|
\to 0, \quad \text{as }h\to0^+.
\end{aligned}
\end{equation}
Let
\begin{align*}
p(t)&=p_1(t)+p_2(t) \\
&:=\int_{0}^{t}  AS(t-s)[F(s, x_s ) -F(t, x_t )]ds +\int_{0}^{t}
AS(t-s)F(t, x_t )ds.
\end{align*}
Then $S(t)X\subset {\cap}_{n=1}^{+\infty}D(A^n)$ and $A^\alpha$ is
closed for any $\alpha>0$ imply that $p_2(t)\in D(A)$. Since
\begin{equation}\label{p3}
Ap_{2}(t)=\int_{\delta}^{t} A^2S(t-s)F(t, x_t )ds+\int_{0}^{\delta}
A^2 S(t-s)F(t, x_t )ds,
\end{equation}
the first term on the right side of \eqref{p3} is clearly continuous
and the second term is $O(\delta)$, this means $Ap_2(t)$ is
continuous. Set
$$
p_{1,\epsilon}(t):=\int_{0}^{t-\epsilon} AS(t-s)[F(s, x_s ) -F(t, x_t )]ds,
$$
then  condition (H1')  yields 
\begin{align*}
Ap_{1,\epsilon}(t)
&=\int_{0}^{t-\epsilon} A^2S(t-s)[F(s, x_s ) -F(t, x_t )]ds \\
&=\int_{0}^{t-\epsilon} A^{2-\alpha-\beta}S(t-s)
A^{\alpha+\beta}[F(s, x_s ) -F(t, x_t )]ds \\
&\to \int_{0}^{t} A^2S(t-s)[F(s, x_s ) -F(t, x_t )]ds,\quad
\text{as }\epsilon\to 0.
\end{align*}
Hence from the closure of $A$ it follows that $p_1(t)\in D(A)$ and
$$
Ap_{1}(t)=\int_{0}^{t} A^2S(t-s)[F(s, x_s ) -F(t, x_t )]ds.
$$
The continuity of $Ap(t)$ can be shown as that of $Ap_2(t)$. Hence
we deduce that $p(t)\in D(A)$ and $Ap(t)$ is continuous. Thus,
\eqref{p1} and \eqref{p2} indicate that $p(t) $ is differentiable
and
$$
p'(t)=Ap(t)+AF(t,x_t).
$$
Therefore, $ x(t)+ F(t, x_t )$ is differentiable in $t$ on
$[0,b_\phi ]$ and satisfies that
\begin{align*}
\frac{d}{dt}[x(t)+ F(t, x_t )] 
&=\frac{d}{dt}S(t)[\phi (0)+F(0, \phi )]+p'(t) +q'(t) \\
&=-A(t) S(t)[ \phi (0)+F(0, \phi )]\\
&\quad +A(t) F(t, x_t )-A(t)p(t) +G(t, x_t )-A(t)q(t)\\
&=-A(t)x(t)+ G(t,x_t ).
\end{align*}
This shows that $x(\cdot)$ is a strict solution of the Cauchy
problem \eqref{pr}. Thus the proof is complete.
\end{proof}

\section{Stability of mild solutions}

In this section, we study the  stability of mild solutions
for  \eqref{pr} with $F$ and $G$ autonomous; namely, we 
discuss the stability of the equilibrium of the  autonomous
equation
 \begin{equation}\label{apr}
\begin{gathered}
\frac{d}{dt}[x(t)+F( x_t)]+Ax(t) = G(x_t) , \quad t{\geq}0,\\
x_0=\phi\in \mathscr{B}_\alpha.
\end{gathered}
\end{equation}
In this equation $F$ and $G$ satisfy  the following
conditions:
\begin{itemize}
\item[(H4)] $F,G$ satisfy the Lipschitz condition; i.e,
\begin{gather*}
\|A^\beta F(\phi_1)-A^\beta F(\phi_2)\|_\alpha\leq l_3 \|\phi_1
-\phi_2\|_{\mathscr{B}_\alpha}, \\
 \|G(\phi_1)-G(\phi_2)\|\leq l_4 \|\phi_1
-\phi_2\|_{\mathscr{B}_\alpha} ,
\end{gather*}
 for $\phi_1,\phi_2\in {\mathscr{B}_\alpha}$.
\item[(H5)] There holds
\begin{equation}\label{p4}
K(0)(l_3\|A^{-\beta}\|+l_3C_{1-\beta}\Gamma(\beta)
+l_4C_{\alpha}\Gamma(1-\alpha) )<1,
\end{equation}
where $\Gamma(\cdot)$ is the gamma function satisfying the formula
$\int_0^{+\infty}e^{-\beta t}t^{-\alpha}dt=\Gamma(1-\alpha)\beta^{\alpha-1}$,
 for $0<\alpha<1$ and $\beta>0$.
\end{itemize}

 First we consider the solution semigroup for  \eqref{apr}.
For each $t\ge 0$, define the nonlinear operator semigroup 
$(U(t))_{t\ge 0}$ as
$$
U(t)(\phi)=x_t(\cdot,\phi),
$$
where $x_t(\cdot,\phi)$ denotes the unique mild solution of 
\eqref{apr} through $(0,\phi)$. Then $(U(t))_{t\ge 0}$ is a
nonlinear strongly continuous semigroup on $\mathscr{B}_\alpha$;
that is,
\begin{itemize}
\item[(i)] $U(0)=I$;
\item[(ii)] $U(t+s)=U(t)U(s)$ for all $t,~s\ge 0$;
\item[(iii)] For all $\phi\in \mathscr{B}_\alpha$, the map $t\to U(t)(\phi)$
is continuous in $\mathscr{B}_\alpha$.
\end{itemize}
And it satisfies the  translation property
\[
  (U(t)(\phi))(\theta)= \begin{cases}
         (U(t+\theta)(\phi))(0),&t+\theta\geq 0,\\
         \phi(t+\theta),&-\infty<t+\theta<0,
        \end{cases}
\]
for $t\ge 0$ and $\theta\in (-\infty,0]$.
Moreover, we have the following result.

\begin{theorem}\label{th5}
For the nonlinear semigroup $(U(t))_{t\ge 0}$, there exist a 
$\mu\in \mathbb{R}^+$ and a function
 $P(\cdot,\mu) \in L^\infty( (0,+\infty);\mathbb{R}^+)$ such that, 
for $\phi_1,\phi_2\in \mathscr{B}_\alpha$,
$$
\|U(t)\phi_1 -U(t)\phi_2\|_{\mathscr{B}_\alpha}\le P(t,\mu)e^{\mu t}\|\phi_1
-\phi_2\|_{\mathscr{B}_\alpha}.
$$
\end{theorem}

\begin{proof}
Let $t_0>0$, $K_{t_0}=\max_{0\le s\le t_0}K(s)$, 
$M_{t_0}=\sup_{0\le s\le t_0}M(s)$, and $x^1(\cdot)=x(\cdot,\phi_1)$,
$x^2(\cdot)=x(\cdot,\phi_2)$. For $t\in[0,t_0]$, there holds
\begin{align*}
&\|U(t)\phi_1 -U(t)\phi_2\|_{\mathscr{B}_\alpha}\\
&=\|x^1_t-x^2_t \|_{\mathscr{B}_\alpha} \\
&\le K(t)\sup_{0\le s\le t}\|x^1(s)-x^2(s)
\|_{\alpha}+M(t)\|\phi_1-\phi_2 \|_{\mathscr{B}_\alpha}\\
&\le K_{t_0}\sup_{0\le s\le t}\{\|S(s)[\phi_1(0)-\phi_2(0)
 +F(\phi_1)-F(\phi_2)]\|_\alpha
 +\|F(x^1_s)-F(x^2_s)\|_{\alpha}\\
&\quad +\|\int_0^sAS(s-\tau)[F(x^1_\tau)-F(x^2_\tau)]d\tau\|_{\alpha}\\
&\quad +\|\int_0^sS(s-\tau)[G(x^1_\tau)-G(x^2_\tau)]d\tau\|_{\alpha}\}
 +M_{t_0}\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}
\\
&\le K_{t_0}\sup_{0\le s\le t}(Me^{\omega s}H+l_3\|A^{-\beta}\|)\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}\\
&\quad  +K_{t_0}\|A^{-\beta}\|l_3\sup_{0\le s\le t}\|x^1_s-x^2_s\|_{\mathscr{B}_\alpha}\\
&\quad  +K_{t_0}l_3\sup_{0\le s\le
t}[\int_0^sC_{1-\beta}(s-\tau)^{\beta-1}
e^{\omega(s-\tau)}\|x^1_\tau-x^2_\tau\|_{\mathscr{B}_\alpha}d\tau\\
&\quad  +K_{t_0}l_4\sup_{0\le s\le
t}\int_0^sC_{\alpha}(s-\tau)^{-\alpha}
e^{\omega(s-\tau)}\|x^1_\tau-x^2_\tau\|_{\mathscr{B}_\alpha}d\tau]
 +M_{t_0}\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}.\\
\end{align*}
Choose $\mu\in \mathbb{R}^+$ such that $\omega-\mu<-1$, then the
above estimate implies that
\begin{align*}
e^{-\mu t}\|x^1_t-x^2_t \|_{\mathscr{B}_\alpha} 
&\le e^{-\mu t}K_{t_0}\sup_{0\le s\le t}(Me^{\omega s}H+l_3\|A^{-\beta}\|)
\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}\\
&\quad  +K_{t_0}l_3\|A^{-\beta}\|e^{-\mu t}\sup_{0\le s\le t}
\|x^1_s-x^2_s\|_{\mathscr{B}_\alpha}\\
&\quad +K_{t_0}l_3\sup_{0\le s\le
t}[\int_0^sC_{1-\beta}(s-\tau)^{\beta-1}
e^{\omega(s-\tau)}e^{-\mu t}\|x^1_\tau-x^2_\tau\|_{\mathscr{B}_\alpha}d\tau\\
&\quad  +K_{t_0}l_4\sup_{0\le s\le
t}\int_0^sC_{\alpha}(s-\tau)^{-\alpha}
e^{\omega(s-\tau)}e^{-\mu t}\|x^1_\tau-x^2_\tau\|_{\mathscr{B}_\alpha}d\tau]\\
&\quad +M_{t_0}e^{-\mu t}\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}.
\end{align*}
Put $W(s):=e^{-\mu s}\|x^1_s-x^2_s \|_{\mathscr{B}_\alpha}$, then
\begin{align*}
&\sup_{0\le s\le t}W(s) \\
&\le K_{t_0}\sup_{0\le s\le t}\left(Me^{\omega
s}H+l_3\|A^{-\beta}\|\right)
\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}
  +K_{t_0}l_3\|A^{-\beta}\|\sup_{0\le s\le t}W(s)\\
&\quad \quad +K_{t_0}l_3\sup_{0\le s\le
t}\Big[\int_0^sC_{1-\beta}(s-\tau)^{\beta-1}
e^{(\omega-\mu)(s-\tau)}W(\tau)d\tau\\
&\quad \quad +K_{t_0}l_4\sup_{0\le s\le
t}\int_0^sC_{\alpha}(s-\tau)^{-\alpha}
e^{(\omega-\mu)(s-\tau)}W(\tau)d\tau\Big]
 +M_{t_0}\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}
\\
&\le K_{t_0}\sup_{0\le s\le t}\left(Me^{\omega s}H+l_3\|A^{-\beta}\|\right)
\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}\\
&\quad  +K_{t_0}l_3\|A^{-\beta}\|\sup_{0\le s\le t}W(s)
 +K_{t_0}l_3C_{1-\beta}\Gamma(\beta)(\mu-\omega)^{-\beta}\sup_{0\le s\le t}W(s)\\
&\quad  +K_{t_0}l_4C_{\alpha}\Gamma(1-\alpha)(\mu-\omega)^{\alpha-1}
 \sup_{0\le s\le t}W(s)
 +M_{t_0}\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}.
\end{align*}
So, by \eqref{p4} we may take $t_0>0$ sufficiently small such that
$1-K_{t_0}(l_3\|A^{-\beta}\|+l_3C_{1-\beta}\Gamma(\beta)
(\mu-\omega)^{-\beta}+l_4C_{\alpha}\Gamma(1-\alpha)
(\mu-\omega)^{\alpha-1})>0$, then
\begin{align*}
&\sup_{0\le s\le t}W(s) \\
&\le \Big(K_{t_0}\sup_{0\le s\le
t}(Me^{\omega s}H+l_3\|A^{-\beta}\|)
+M_{t_0}\Big)\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha}
\\
&\div\Big(1-K_{t_0}(l_3\|A^{-\beta}\|+l_3C_{1-\beta}\Gamma(\beta)
(\mu-\omega)^{-\beta}+l_4C_{\alpha}\Gamma(1-\alpha)
(\mu-\omega)^{\alpha-1})\Big)
\\
&:=P(t,\mu)\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha},
\end{align*}
or
\begin{equation}\label{pu}
\|x^1_t-x^2_t \|_{\mathscr{B}_\alpha} \le P(t,\mu)e^{\mu
t}\|\phi_1-\phi_2\|_{\mathscr{B}_\alpha},
\end{equation}
for all $t\in[0,t_0]$. For any $t>t_0$, find an $n\in \mathbb{N}$
 such that $t\in (nt_0,(n+1)t_0]$, then we may repeat the above computation
 for $n$ times and obtain that \eqref{pu} holds for $t>t_0$. Thus we
 complete the proof of the assertion.
\end{proof}

 In what follows, we investigate the stability of an equilibrium of 
 \eqref{apr}. For each $u\in X_\alpha$, the corresponding constant
 function $\tilde{u}\in {\mathscr{B}_\alpha}$ is defined by 
$\tilde{u}(\theta) \equiv u$, $\theta\in (-\infty,0]$.
 Here by an equilibrium of  \eqref{apr} we mean a constant
 function $\tilde{u}\in \mathscr{B}_\alpha$ satisfying
 $$
u\in D(A) \quad\text{and}\quad
-A{u}+ G(\tilde{u})=0.
$$
If $\tilde{u}$ is an equilibrium of 
 \eqref{apr}, then it is trivial to verify that $0$ is the equilibrium 
solution of the equation
$$
\frac{d}{dt}[x(t)+F_1( x_t)]+Ax(t) = G_1(x_t),
$$
where $F_1(\phi)=F(\phi+\tilde{u})-F(\tilde{u})$, 
$G_1(\phi)=  G(\phi+\tilde{u})-G(\tilde{u})$.
 Accordingly, without loss of generality we may assume that $\tilde{u}=0$
 and $ G(0)=F(0)=0$. Moreover, we suppose that
\begin{itemize}
\item[(H6)] $A^\beta F$ is Fr\'echet-differentiable at
$0$ in space $X_\alpha$,  $G$ is
Fr\'echet-differentiable at $0$ in $X$.
\end{itemize}
Let $L_1=A^{-\beta}(A^\beta F)'(0),~L_2=G'(0)$. Then the linearized
equation of Equation \eqref{apr} around the equilibrium $0$ is
 \begin{equation}\label{lpr}
\begin{gathered}
\frac{d}{dt}[x(t)+L_1 x_t]+Ax(t) = L_2x_t , \quad t{\geq}0,\\
x_0=\phi\in \mathscr{B}_\alpha.
\end{gathered}
\end{equation}
Denote by $(T(t))_{t\ge 0}$ the linear solution semigroup associated
to  \eqref{lpr}. Then we have the following result.

\begin{theorem}\label{th6}
 Suppose that conditions {\rm (H4)--(H6)} are satisfied.
 Then, for $t\ge 0$, the
Fr\'echet derivative of $U(t)$ at zero is $T(t)$.
\end{theorem}

\begin{proof}
It suffices to prove that for any $\phi\in \mathscr{B}_\alpha$,
$t>0$ and $\epsilon>0$, there exists a $\delta>0$ such that
\begin{equation}\label{u}
\|U(t)\phi-T(t)\phi\|_{\mathscr{B}_\alpha}\le \epsilon\|\phi\|
_{\mathscr{B}_\alpha},\quad  \text{for }
\|\phi\|_{\mathscr{B}_\alpha}<\delta. 
\end{equation} 
In fact, we have
\begin{align*}
&\|U(t)\phi -T(t)\phi\|_{\mathscr{B}_\alpha} \\
&\le K(t)\sup_{0\le s\le t}\|(U(s)(\phi))(0)-(T(s)(\phi))(0)
\|_{\alpha}\\
&\le K_{t}\sup_{0\le s\le t}\Big\{\|S(s)[F(\phi)-L_1(\phi)]\|_\alpha
\\
&\quad  +\|F(U(s)\phi)-F(T(s)\phi)+F(T(s)\phi)-L_1(T(s)\phi)\|_{\alpha}
\\
&\quad +\big\|\int_0^sAS(s-\tau)[F(U(\tau)\phi)-F(T(\tau)\phi)
 +F(T(\tau)\phi)-L_1(T(\tau)\phi)]d\tau\big\|
_{\alpha}\\
&\quad +\big\|\int_0^sS(s-\tau)[G(U(\tau)\phi))-G(T(\tau)\phi))
+G(T(\tau)\phi))-L_2((\tau)\phi))]d\tau\big\|_{\alpha}\Big\}.
\end{align*}
Take $t_0>0$ such that
$1-K_{t_0}(l_3\|A^{-\beta}\|+l_3C_{1-\beta}\Gamma(\beta)
(\mu-\omega)^{-\beta}+l_4C_{\alpha}\Gamma(1-\alpha)
(\mu-\omega)^{\alpha-1})>0$, and by virtue of the continuous
differentiability of $A^\beta F$ and $G$ at $0$ and from Theorem
\ref{th5} we infer that, for any $\epsilon>0$, there is a
$\delta_0>0$ such that, for each $0<t<t_0$ and any
$\|\phi\|_{\mathscr{B}_\alpha}<\delta_0$,
\begin{gather*}
\|S(s)[F(\phi)-L_1(\phi)]\|_\alpha\le \epsilon \|\phi\|_{\mathscr{B}_\alpha},\\
\big\|F(T(s)\phi)-L_1(T(s)\phi)\big\|_{\alpha}\le \epsilon 
\|\phi\|_{\mathscr{B}_\alpha},\\
\big\|\int_0^sAS(s-\tau)[F(T(\tau)\phi)-L_1(T(\tau)\phi)]d\tau\big\|_\alpha
\le \epsilon \|\phi\|_{\mathscr{B}_\alpha},\\
\big\|\int_0^sS(s-\tau)[G(T(\tau)\phi))-L_2((\tau)\phi))]d\tau\big\|_\alpha\le
 \epsilon \|\phi\|_{\mathscr{B}_\alpha}.
\end{gather*}
Hence,
\begin{align*}
&e^{-\mu t}\|U(t)\phi -T(t)\phi\|_{\mathscr{B}_\alpha}\\
&\le 4\epsilon K_{t}\|\phi\|_{\mathscr{B}_\alpha}
+K_{t}\|A^{-\beta}\|l_3e^{-\mu t}\sup_{0\le s\le
t}\|U(s)\phi -T(s)\phi\|_
{\mathscr{B}_\alpha}\\
&\quad  +K_{t}l_3\sup_{0\le s\le
t}\Big[\int_0^sC_{1-\beta}(s-\tau)^{\beta-1}
e^{\omega(s-\tau)}e^{-\mu t}\|U(\tau)\phi -T(\tau)\phi\|_{\mathscr{B}_\alpha}d\tau\\
&\quad  +K_{t}l_4\sup_{0\le s\le
t}\int_0^sC_{\alpha}(s-\tau)^{-\alpha}
e^{\omega(s-\tau)}e^{-\mu t}\|U(\tau)\phi -T(\tau)\phi\tau\|_{\mathscr{B}_\alpha}d\tau\Big]\\
&\le 4
\epsilon K_{t}\|\phi\|_{\mathscr{B}_\alpha}
 +K_{t}l_3\|A^{-\beta}\|\sup_{0\le s\le t}e^{-\mu s}\|U(s)\phi -T(s)\phi\|
_{\mathscr{B}_\alpha}\\
&\quad +K_{t}l_3C_{1-\beta}\Gamma(\beta)(\mu-\omega)^{-\beta}
\sup_{0\le s\le t}e^{-\mu s}\|U(s)\phi -T(s)\phi\|_{\mathscr{B}_\alpha}\\
&\quad  +K_{t}l_4C_{\alpha}\Gamma(1-\alpha)
(\mu-\omega)^{\alpha-1}\sup_{0\le s\le t}e^{-\mu s}\|U(\tau)\phi
-T(\tau)\phi \|_{\mathscr{B}_\alpha}.
\end{align*}
So, using \eqref{p4} again we obtain that \eqref{u} is true for all
$0<t\le t_0$, and then as in the Proof of Theorem \ref{th5} we can
conclude that \eqref{u} holds for all $t>0$.
\end{proof}
 As a consequence of the above two results we
obtain the following theorem.

\begin{theorem} 
Under the assumptions of Theorems \ref{th5} and \ref{th6}, 
if the zero equilibrium of $(T(t))_{t\ge 0}$ is
exponentially stable, then the zero equilibrium of $(U(t))_{t\ge 0}$
is locally exponentially stable in the sense that there exist
 $\mu,\mu>0$ and $k\ge 1$ such that, for $t\ge 0$ and
any $\phi\in{\mathscr{B}_\alpha}$ with $\|\phi\|_{\mathscr{B}_\alpha}
<\mu$,
$$
\|U(t)\phi\|_{\mathscr{B}_\alpha}\le ke^{-\mu t}\|\phi\|
_{\mathscr{B}_\alpha}.
$$
\end{theorem}

\begin{proof} 
Based on Theorems \ref{th5},\ref{th6} and \ref{th01} 
this theorem can be proved  by using the similar
method as that in \cite{HV} and \cite{AE}, and we omit the proof
here.
\end{proof}


\section{An Example}
 To apply Theorems \ref{th1} and  \ref{th3},
we consider the  system
\begin{equation}\label{ex}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}
\Big[z(t,x)+\int_{-\infty}^{t}\int_{0}^{\pi}
b(s-t,x,y)\Big[z(s,y)+\frac{\partial}{\partial
y}z(s,y)\Big]\,dy\,ds\Big] \\
&=\frac{\partial^2}{\partial x^2}z(t,x)
+h\Big(z(\cdot,x), \frac{\partial}{\partial x}z(\cdot,x)\Big),
\quad 0\leq t\leq a,\; 0\leq x\leq \pi,
\end{aligned} \\
z(t,0)=z(t,\pi)=0,\\
z(\theta,x)=\phi(\theta,x),\quad \theta\leq 0,\quad 0\leq x\leq\pi,
\end{gathered}
\end{equation}
where the functions $b$ and $h$ will be described below.

Let $X=L^{2}([0,\pi])$ and operator $A$ be defined by
$$
Af=-f''
$$ 
with the domain
\[
D(A)= H_0^2([0,\pi])= \{f(\cdot) \in X:f',f''\in X,\;f(0)=f(\pi)=0\}.
\]
then $-A$ generates a strongly continuous semigroup
$(S(\cdot))_{t\ge 0}$ which  is analytic, compact and self-adjoint.
Furthermore, $-A$ has a discrete spectrum, the eigenvalues are
$-n^2$, $\in N$, with the corresponding normalized eigenvectors
$z_n(x)= \sqrt{\frac{2}{\pi}}\sin(nx)$. Then the following
properties hold:
\begin{itemize}
\item[(a)] If $f\in D(A)$, then 
\[
Af= \sum_{n=1}^{\infty} n^2\langle f,z_n \rangle z_n.
\]

\item[(b)] For every $f\in X$, 
\begin{gather*}
S(t)f=\sum_{n=1}^{\infty} e^{-n^2t}\langle f,z_n \rangle z_n,\\
A^{-1/2}f=\sum_{n=1}^{\infty} \frac{1}{n}\langle f,z_n
\rangle z_n.
\end{gather*}
In particular, $\|S(t)\|\le e^{-t}$, $\|A^{-1/2}\|=1$.

\item[(c)] The operator $A^{1/2}$ is given by
\[
A^{1/2}f= \sum_{n=1}^{\infty} n\langle f,z_n \rangle z_n.
\]
on the space $D(A^{1/2})= \{f(\cdot) \in X,\;\sum_{n=1}^{\infty}
n\langle f,z_n \rangle z_n \in X\}$.
\end{itemize}
 Here we take $\alpha =\beta=1/2$ and the phase space
 ${\mathscr{B}}={\mathscr C}_{g}$,
 where the space ${\mathscr C}_{g}$ is defined as: let $g$ be a continuous
function on $(-\infty , 0]$ with $g(0)=1$, 
$\lim_{\theta\to -\infty}g(\theta)=\infty$, and $g$ is decreasing on
 $(-\infty,0]$,
then
$$
{\mathscr C}_{g} =\{\phi\in C((-\infty,0];X):\sup_{s\leq 0}
\frac{\|\phi (s)\|}{g(s)}< \infty\},
$$
 and the norm is defined by, for $\phi\in {\mathscr C}_{g}$,
 $$
|\phi|_g =   \sup_{s\leq 0}\frac{\|\phi
 (s)\|}{g(s)}.
$$
 It is known that ${\mathscr C}_{g}$ satisfies the axioms
 $(A1), (A2)$, and (B1), see \cite{HMN}. 
Further, the subspace ${\mathscr C}_{g,\frac{1}{2}}$ is defined by
$$
{\mathscr C}_{g,\frac{1}{2}} =\{\phi\in C((-\infty,0];X_\frac{1}{2}):
\sup_{s\leq 0}\frac{\|A^{\frac{1}{2}}\phi (s)\|}{g(s)}< \infty\},
$$
endowed with the norm 
$|\phi|_{g,\frac{1}{2}}=\sup_{s\leq 0}
\frac{\|A^{\frac{1}{2}}\phi (s)\|}{g(s)}$. Clearly,
 ${\mathscr C}_{g\frac{1}{2}}$ satisfies correspondingly the axioms
 (A1'),(A2'), and (B1'), and we may choose a proper $g$ such 
that $H,K(\cdot),M(\cdot)\le 1$
  (also see \cite{FL}).

We assume that the following conditions hold:
\begin{itemize}
\item[(i)] The function $b(\cdot,\cdot,\cdot)\in C^2$ with $b(\cdot,\cdot,0)
=b(\cdot,\cdot,\pi)\equiv 0$, and
 $$
c:=\{\int_0^\pi[\int_{-\infty}^0g(\theta)(\int_0^\pi
(\frac{\partial^2}{\partial x^2}b(\theta,y,x))^2dy)^{1/2} d\theta]^2
dx\}^{1/2}<\infty.
$$

\item[(ii)] The function $h:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is
continuous in the two variables.

\item[(iii)] The function $\phi$ defined by
$\phi(\theta)(x)=\phi(\theta,x)$ belongs to ${\mathscr C}_{g,\frac{1}{2}}$.
\end{itemize}

Now define the abstract functions 
$F, G:  {\mathscr C}_{g,\frac{1}{2}}\to X_{\frac{1}{2}}$ by
\begin{gather*}
F(\phi)=\int_{-\infty}^{0}\int_{0}^{\pi} b(\theta,y,x)[\phi(\theta)(y)
 +\phi(\theta)'(y)]\,dy\,d\theta,\\
G(\phi)= h(\phi(\cdot)(x),\phi(\cdot)'(x)).
\end{gather*}
Then the system \eqref{ex} is rewritten as the abstract form
\eqref{pr}, and condition (i) implies that $R(F)\subset D(A)$, since
\begin{align*}
\langle F(\phi),z_n\rangle
&=-\frac{1}{n}\Big\langle\int_{-\infty}^{0}
\int_{0}^{\pi} \frac{\partial}{\partial x}b(\theta,y,x)
[\phi(\theta)(y)+\phi(\theta)'(y)]dyd\theta,\bar{z}_n(x)\Big\rangle\\
&=\frac{1}{n^2}\Big\langle\int_{-\infty}^{0} \int_{0}^{\pi}
\frac{\partial^2}{\partial x^2}b(\theta,y,x)
[\phi(\theta)(y)+\phi(\theta)'(y)]dyd\theta,{z}_n(x)\Big\rangle,
\end{align*}
where $\bar{z}_n(x)= \sqrt{\frac{2}{\pi}}\cos(nx)$, $n=1,2,\dots$.
Observe that, for any $\theta\in (-\infty,0]$,
\begin{align*}
\|\phi_2(\theta)(x)-\phi_1(\theta)(x)\|^2
& =\sum_{n=1}^{\infty} {\langle \phi_2-\phi_1,z_n\rangle} ^2\\
& \leq  \sum_{n=1}^{\infty} n^2 {\langle \phi_2-\phi_1,z_n\rangle}^2\\
&\leq {\|\phi_2(\theta)(x)-\phi_1(\theta)(x)\|_{\frac12}}^2,
\end{align*}
and
\begin{align*}
\|\phi_2(\theta)'(x)-\phi_1(\theta)'(x)\|^2 
& =\sum_{n=1}^{\infty} {\langle \phi'_2-\phi'_1,z_n\rangle} ^2\\
& =  \sum_{n=1}^{\infty} {\langle \phi_2-\phi_1,z'_n\rangle}^2\\
& =  \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} n^2 \langle
\phi_2-\phi_1,z_n\rangle
\langle \phi_2-\phi_1,z_m\rangle\langle -z''_n,z'_m\rangle\\
&\leq {\|\phi_2(\theta)(x)-\phi_1(\theta)(x)\|_{\frac12}}^2,
\end{align*}
we see that
\begin{gather*}
|\phi_2(\cdot)-\phi_1(\cdot)|_{g}\le
|\phi_2(\cdot)-\phi_1(\cdot)|_{g,\frac{1}{2}},\\
|\phi_2(\cdot)'-\phi_1(\cdot)'|_{g}\le
|\phi_2(\cdot)-\phi_1(\cdot)|_{g,\frac{1}{2}}.
\end{gather*}
Thus, conditions
(i) and (ii) ensure that $A^{\frac{1}{2}}F(\cdot)$ satisfies the
Lipschitz continuous on ${\mathscr C}_{g,\frac{1}{2}}$,
 $F$ and $G$ verify assumption $(H_1 )$ and
(H2 ) respectively. Consequently, by Theorem \ref{th1} the system
\eqref{ex} has a mild solution on $(-\infty,b_\phi]$ for some
$b_\phi>0$.

Furthermore, if take $\mathscr B={\mathscr C}_{g,\frac{1}{2}}^0$,
where
$$
{\mathscr C}^0_{g,\frac{1}{2}} =\{\phi\in {\mathscr C}_{g,\frac{1}{2}};
X_\frac{1}{2}):
\lim_{s\leq 0}\frac{\|A^{\frac{1}{2}}\phi (s)\|}{g(s)}=0\},
$$ so
that Axiom (C1) is satisfied (see \cite{HMN}) and assume that $h$ is
H\"{o}lder continuous in two variables, then condition (H1') and
(H2') are satisfied. Therefore, if $\phi(\cdot,x)$ is uniformly
H\"{o}lder continuous and $\phi (0,x)\in D(A)$, then the system
\eqref{ex} has a strict solution on $(-\infty,b_\phi]$.

\subsection*{Acknowledgments}
This research was supported by grants 11171110 from the NSF of China,
 and B407 from the Shanghai Leading Academic Discipline Project.

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\end{document}