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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 110, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/110\hfil Existence of solutions]
{Existence of solutions for non-autonomous  functional
 evolution equations with nonlocal conditions}

\author[X. Fu \hfil EJDE-2012/110\hfilneg]
{Xianlong Fu} 

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

\thanks{Submitted August 22, 2011. Published July 2, 2012.}
\thanks{Supported by grants 11171110 from the NSF of China,
and B407 from Shanghai \hfill\break\indent  Leading Academic Discipline}
\subjclass[2000]{34K30, 34K05, 47D06, 47N20}
\keywords{Functional evolution equation; nonlocal condition; 
\hfill\break\indent linear evolution system; fractional power operator}

\begin{abstract}
 In this work, we study the existence of mild solutions and strict
 solutions of semilinear functional evolution equations with nonlocal
 conditions, where the linear part is non-autonomous and generates a
 linear evolution system. The fraction power theory and $\alpha$-norm
 are used to discuss the problems so that the obtained results can be
 applied to the equations in which the nonlinear terms involve
 spatial derivatives. In particular, the compactness condition or
 Lipschitz condition for the function $g$ in the nonlocal conditions
 appearing in various literatures is not required here. An example is
 presented to show the applications of the obtained results.
\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 of solutions for
semilinear neutral functional evolution equations with nonlocal
conditions. More precisely, we consider the  nonlocal Cauchy problem
\begin{equation}\label{pr}
\begin{gathered}
\frac{d}{dt}[x(t)+F(t, x(t))]  +A(t)x(t) = G(t, x(r (t))) ,\quad 0\leq t\leq T, \\
x(0)+g(x)=x_0,
\end{gathered}
\end{equation}
   where the family $\{A(t):{0}\leq{t}\leq{T}\}$ of linear operators
generates a linear evolution system, and $F,G $ are given functions
to be specified later. The nonlocal Cauchy problem was considered by
Byszewski \cite{Bys} and the importance of nonlocal conditions in
different fields has been discussed in \cite{Bys} and the references
therein. In the past several years theorems about existence,
uniqueness and stability of differential and functional differential
abstract evolution Cauchy problems with nonlocal conditions have
been studied extensively, see, for example, papers
\cite{AL}-\cite{FaL} and the references therein.

When $F(\cdot,\cdot)=0$ and $A$ generating a $C_0-$ semigroup in Eq.
\eqref{pr}, Byszewski and Akca have investigated the existence of
mild solutions and classical solutions in paper \cite{BA} by using
Schauder's fixed point theorem. To take away an unsatisfactory
condition on solutions and extend the results in \cite{BA} to
neutral equations, in \cite{FE} the authors have studied the
existence of mild solutions and strong solutions for the equations
with the form
\begin{gather*}
\frac{d}{dt}[x(t)+F(t,x(h_1(t)))]+Ax(t)=G(t, x(h_2 (t))) ,\quad 0\leq t\leq T, \\
x(0)+g(x)=x_0\in X,
\end{gather*}
where the operator $-A$ generates a compact analytic semigroup.
The main tools and techniques in \cite{FE} are the properties of
fractional power and Sadovskii fixed point theorem. Papers
\cite{BGHN}, \cite{EL} and \cite{Fu} have established the
corresponding results for the situation in which the linear operator
$A$ is non-densely defined. Paper \cite{FaL} and \cite{FC} have
investigated the existence topics on impulsive nonlocal problems,
and in Papers \cite{AL} and \cite{Xue} the authors have studied the
nonlocal Cauchy problems for the case that $A$ generates a nonlinear
semigroup. Existence of solutions for quasilinear delay
integrodifferential equations with nonlocal conditions has been
established in Paper \cite{BS}. In order to establish the existence
results for the non-autonomous equations, Paper \cite{FL} has
considered existence of solutions for  \eqref{pr} which
is a more general situation as $A(t)$ is non-autonomous. However,
although the system
\begin{equation}\label{pr2}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}[u(t,x)+f(t,
u(b(t),x), \frac{\partial}{\partial
x}u(b_1(t),x))]+c(t)\frac{\partial^2}{\partial x^2 }u(t,x)\\
& =h(t,u(a(t),x),\frac{\partial}{\partial x}u(a_1(t),x)),
\end{aligned}\\
u(0)+g(x)=u_0.
\end{gathered}
\end{equation}
can also be rewritten as an abstract equation of form \eqref{pr},
those results founded in \cite{FL} become invalid for it, since the
functions $f,h$ in \eqref{pr2} involve spatial derivatives.

The purpose of the present note is
to extend and develop the work in \cite{FL} and \cite{FC}. We shall
discuss this problem by using fractional power operators theory and
$\alpha-$norm; i.e., we will restrict this equation in a Banach
space $X_\alpha(t_0)(\subset X)$ and investigate the existence and
regularity of mild solutions for \eqref{pr}. In Particular,
borrowing the idea from \cite{LLX2}  we do not require the function
$g$ in the nonlocal condition satisfy the compactness condition or
Lipschitz condition, instead, it is continuous and is completely
determined on $[\tau, T]$ for some small $\tau>0$. The compactness
condition or Lipschitz condition for $g$ appear, respectively, in
almost all the papers on the topics of nonlocal problem, for example
in \cite{BGHN,BA,EFH,FC,LC, Xue}. Although paper \cite{Xue} has also discussed the case
that the function $g$ is continuous, it assumed additionally some
pre-compact condition relative to $g$. In addition, the obtained
results extend also the ones in \cite{Ran} and \cite{TW2}.

This article is organized as follows: we firstly introduce some
preliminaries about the linear evolution operator and fractional
power operator theory in Section 2. The main results are arranged in
Section 3. In Subsection 3.1  we discuss the existence of mild
solutions by Sadovskii fixed point theorem and limit arguments, and
in Subsection 3.2 we show the regularity of mild solutions. Finally,
an examples is presented in Section 4 to show the applications of
our obtained results.

\section{Preliminaries}

Throughout this paper $X$ will be a Banach space with norm
$\|\cdot\|$. For the family $\{A(t):{0}\leq{t}\leq T\}$ of linear
operators, we impose the following restrictions:
\begin{itemize}
\item[(B1)] The domain $D(A)$ of $\{A(t):{0}\leq{t}\leq{T}\}$ is dense
in $X$ and independent of $t$, $A(t)$ is closed linear operator;

\item[(B2)] For each $t\in [0,T]$, the resolvent $R(\lambda,A(t))$
exists for all $\lambda$ with $Re\lambda\leq 0$ and there exists
$K>0$ so that $\|R(\lambda,A(t))\|\leq K/(|\lambda|+1)$;

\item[(B3)] There exists $0<\delta \le 1$ and $K>0$ such that
$\|(A(t)-A(s))A^{-1}(\tau )\|\leq K|t-s|^{\delta}$ for all
$t,s,\tau\in [0,T]$;

\item[(B4)]For each $t\in [0,T]$ and some $\lambda\in\rho
(A(t))$, the resolvent set of $A(t)$, the resolvent
$R(\lambda,A(t))$, is a compact operator.

\end{itemize}
Under these assumptions, the family $\{A(t):{0}\leq{t}\le T\}$
generates a unique linear evolution system, or called linear
evolution operator, $U(t,s),$ ${0}\leq{s}\leq{t}\le T$, and there
exists a family of bounded linear operators
$\{R (t,\tau)|0\leq\tau\leq t\leq T\}$ with
$\|R(t,\tau )\|\leq K|t-\tau |^{\delta -1}$ such that
$U(t,s)$ has the representation
\begin{equation}\label{U}
U(t,s)=e^{-(t-s)A(t)}+\int_s^te^{-(t-\tau )A(\tau )}R(\tau
,s)d\tau,
\end{equation}
 where $\exp(-\tau A(t))$ denotes the
analytic semigroup having infinitesimal generator $-A(t)$ (note that
Assumption (B2) guarantees that $-A(t)$ generates an analytic
semigroup on $X$). The family of the linear
operator$\{U(t,s): {0}\leq{s}\leq{t}\leq{T}\}$ satisfies the
following properties:
\begin{itemize}

\item[(a)] $U(t,s)\in L(X)$, the space of bounded linear transformations on
$X$, whenever $0\leq s\leq t\leq T$ and for each $x\in X$, the
mapping $(t,s)\to U(t,s)x$ is continuous;

\item[(b)] $U(t,s)U(s,\tau)=U(t,\tau)$ for $0\leq\tau\leq s\leq t\leq T$;

\item[(c)] $U(t,t)=I$;

\item[(d)] $U(t,s)$ is a compact operator whenever $t-s>0;$

\item[(e)] $\frac{\partial}{\partial t} U(t,s)=-A(t)U(t,s),$ for $s<t$.
\end{itemize}
Condition (B4) ensures the generated evolution operator satisfies
(d) (see \cite[Proposition 2.1]{Fit}). We have also the following
inequalities:
\begin{gather*}
\|e^{-tA(s)}\|\le K,\quad t\ge 0,\; s\in [0,T],\\
\|A(s)e^{-tA(s)}\|\le \frac{K}{t},\quad t,\; s\in [0,T],\\
\|A(t)U(t,s)\|\le \frac{K}{|t-s|},\quad 0\le s\le t\le T.
\end{gather*}
Furthermore,  Assumptions (B1)--(B3) imply that for each
$t\in[0,T]$, the integral
$$
A^{-\alpha}(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{+\infty}s^{\alpha-1}
e^{-sA(t)}ds
$$
exists for each $\alpha\in (0,1]$. The operator defined by this
formula is a bounded linear operator and yields
$A^{-\alpha}(t)A^{-\beta}(t)=A^{-(\alpha+\beta)}(t)$. Thus, we can
define the fractional power as
$$
A^{\alpha}(t)=[A^{-\alpha}(t)]^{-1},
$$
which is a closed linear operator with $D(A^{\alpha}(t))$ dense in
$X$ and $D(A^{\alpha}(t))\subset D(A^{\beta}(t))$ for
$\alpha\ge \beta$. $D(A^{\alpha}(t))$ becomes a Banach space endowed with the
norm $\|x\|_{\alpha,t}=\|A^{\alpha}(t)x\|$, and is denoted by
$X_{\alpha}(t)$.

The following estimates and lemmas are from (\cite[Part II]{Fri}):
\begin{equation}\label{AA}
\|A^{\alpha}(t)A^{-\beta}(s)\|\le C_{(\alpha, \beta)},
\end{equation}
 where $C_{(\alpha, \beta)}$ is a constant related
to $T$ and $\delta$, $t,~s\in [0,T]$ and $0\le \alpha<\beta$.
\begin{gather}\label{Ae}
\|A^{\beta}(t)e^{-sA(t)}\|\le \frac{C_\beta}{s^{\beta}}e^{-w
s},\quad t>0,\; \beta\le 0,\; w>0, \\
\label{AU}
\|A^{\beta}(t)U(t,s)\|\le
\frac{C_\beta}{|t-s|^{\beta}},\quad 0<\beta<\delta+1, \\
\label{AUA}
\|A^{\beta}(t)U(t,s)A^{-\beta}(s)\|\le
C'_{\beta},\quad 0<\beta<\delta+1,
\end{gather}
 for some $t>0$,
where $C_{(\alpha, \beta)}$ $C_{\beta}$ and $C'_{\beta}$ indicate
their dependence on the constants $\alpha$, $\beta$.

\begin{lemma}\label{Le01}
  Assume that {\rm (B1)--(B3)} hold. If $0\le
\gamma\le 1$, $0\le \beta\le \alpha<1+\delta$, $0<\alpha-\gamma\le 1$,
then for any $0\le\tau<t+\Delta t\le t_0$, $0\le\zeta\le T$,
\begin{equation}\label{leAU}
\|A^{\gamma}(\zeta)(U(t+\Delta
t,\tau)-U(t,\tau))A^{-\beta}(\tau)\|
\le C(\beta,\gamma,\alpha)(\Delta t)^{\alpha-\gamma}|t-\tau|^{\beta-\alpha}.
\end{equation}
\end{lemma}

\begin{lemma}\label{Le02}
 Assume that {\rm(B1)--(B3)} hold and
let $0\le\gamma<1$. Then for any $0\le\tau<t+\Delta t\le t_0$ and
for any continuous function $f(s)$,
\begin{equation}\label{leAU2}
\begin{split}
&\|A^{\gamma}(\zeta)[\int_{t}^{t+\Delta t}U(t+\Delta t,s)f(s)ds-
\int_{\tau}^tU(t,s)f(s)ds]\|\\
&\le C_\gamma(\Delta t)^{1-\gamma}(|\log(\Delta
t)|+1)\max_{\tau\le s\le t+\Delta t}\|f(s)\|.
\end{split}
\end{equation}
\end{lemma}

For more details about the theory of linear evolution system,
operator semigroups and fraction powers of operators, we  refer
the reader to \cite{Fri,Paz,So}.

The considerations of this paper are based on the following result.

\begin{theorem}[\cite{Sad}] \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}

\section{Main results}

The main results of this note are presented in this section. We
shall study the existence and regularity of mild solutions for
\eqref{pr}, and we consider this problem on the Banach
subspace $X_\alpha (t_0)$ defined in Section 2 for some $0<\alpha<1$
and $t_0\in [0,T]$.

\subsection{Existence of mild solutions}

Firstly we consider the existence of mild solutions for \eqref{pr}.
The mild solutions are defined as follows.

\begin{definition} \label{def3.1} \rm
A continuous function $x(\cdot):[0,T]\to X_\alpha (t_0)$
is said to be a mild solution of the nonlocal Cauchy problem
\eqref{pr}, if the function
$$
U(t,s)A(s)F(s, x(s)), \quad s\in [0,t)
$$
is integrable on $[0,t)$ and the following integral equation is
verified:
\begin{equation}\label{so}
\begin{split}
x(t) &=U(t,0)[x_0+F(0, x(0))-g(x)]-F(t, x(t))\\
 &\quad +\int_{0}^t U(t,s) A(s)F(s, x(s))ds\\
 &\quad +\int_{0}^tU(t,s) G(s, x(r (s))) ds,  \quad0\leq t\leq T.
\end{split}
\end{equation}
\end{definition}

  Now we present the basic assumptions on \eqref{pr}.

\begin{itemize}
\item[(H1)] $F:[0,T]\times X_\alpha(t_0) \to X$ is a continuous
function, $F([0,T]\times X_\alpha(t_0))\subset D(A)$, and there
exist constants $L,L_1>0$ such that the function $A(t)F$ satisfies
the Lipschitz condition
\begin{equation}\label{AF1}
\|A(t)F(s_1,x_1)-A (t) F(s_2, x_2)\| \leq L(|s_1
-s_2|+\|x_{1}-{x}_{2}\|_\alpha)
\end{equation}
for every $0\leq s_1$, $s_2\leq T$, $x_1,x_2 \in X_\alpha(t_0)$; and
\begin{equation}\label{AF2}
\|A (t)F(t, x)\|\leq L_{1}(\|x\|_\alpha+1)\,.
\end{equation}

\item[(H2)] The function $G:[0,T]\times X_\alpha(t_0)\to
X$ satisfies the following conditions:
\begin{itemize}
\item[(i)] For each $ t \in[0,T]$, the function $G(t,\cdot):X_\alpha(t_0)\to X$
   is continuous, and for each $x\in X_\alpha(t_0)$, the function
    $G(\cdot,x):[0,T]\to X$ is strongly measurable;
\item[(ii)] There is a positive function
 $w(\cdot)\in C([0,T])$ such that
 $$
 \sup_{ \|x\|_\alpha\leq k}\|G(t, x)\|\leq w(k),\quad
 \liminf_{k \to +\infty}\frac{w(k)}{k}=\gamma <\infty.
 $$
\end{itemize}

\item[(H3)] $r \in C([0,T];[0,T])$. $g: E\to D(A)$ is a function
satisfying that $A(t)g$ is continuous on $E$ and there exists a
constant $L_2
>0$ such that $\|A(t)g(u)\|\leq L_2 \|u\|_E$ for each $x \in
E$, where $E=C([0,T];X_\alpha(t_0))$. In addition, there is a
$\tau(k)>0$ such that $g(u)=g(v)$ for any $u,~v\in B_k$ with
$u(s)=v(s)$, $s\in[\tau,~a]$, where $B_k=\{u\in E:\|u(\cdot)\|_E\leq
k\},$.
\end{itemize}

 \begin{theorem} \label{th1}
If  {\rm (B1)--(B4), (H1)--(H3)} are satisfied,
$x_0\in X_\beta(t_0)$ for some $\beta$,
$0<\alpha<\beta\le 1$. Then the nonlocal Cauchy problem \eqref{pr}
has a mild solution provided that $L$, $L_1$ and $\gamma$ are small
enough; more precisely,
\begin{equation}\label{L0}
L_0:=\Big[C'_\beta C_{(\alpha,\beta)}C_{(\beta,1)} +C_{(\alpha,1)}
+C_{(\alpha,\beta)} \frac{C_\beta T^{1-\beta}}{1-\beta}\Big]L <1
\end{equation}
and
\begin{equation}\label{L1}
\begin{aligned}
&C'_\beta C'_1 C_{(\alpha,\beta)}C_{(\beta,1)}L_2+ \Big[C'_\beta
C_{(\alpha,\beta)}C_{(\beta,1)} +C_{(\alpha,1) } +
C_{(\alpha,\beta)}  \frac{C_\beta T^{1-\beta}}{1-\beta}\Big]L_1\\
&+C_{(\alpha,\beta)} \frac{C_\beta T^{1-\beta}}{1-\beta}\gamma<1.
\end{aligned}
\end{equation}
\end{theorem}

We remark that inequalities \eqref{L0} and \eqref{L1} are
verified explicitly by the example given in Section 4, which shows
that they are applicable.

\begin{proof}[Proof of Theorem \ref{th1}]
The  proof  is divided into two steps.

\textbf{Step 1.}  We first consider that, for any $\epsilon>0$
very small, the existence of mild solutions for the
equation
\begin{equation}\label{epsr}
\begin{gathered}
\frac{d}{dt}[x(t)+F(t, x(t))]  +A(t)x(t) = G(t, x(r (t))) ,\quad 0\leq t\leq T, \\
x(0)+U(\epsilon,0)g(x)=x_0.
\end{gathered}
\end{equation}
Define the operator $P$ on $E$ by the formula
\begin{align*}
(Px)(t)
&= U(t,0)[x_0+F(0,x(0))-U(\epsilon,0)g(x)]-F(t,x(t))\\
&\quad +\int_{0}^t U(t,s) A(s)F(s,x(s))ds
 +\int_{0}^tU(t,s)G(s,x(r(s))) ds,  \quad0\leq t\leq T.
\end{align*}

For each positive number $k$, $B_k$ is clearly a nonempty bounded
closed convex set in $E$. We claim that there exists a positive
number $k$ such that $P(B_k) \sqsubseteq B_k$. If it is not true,
then for each positive number $k$, there is a function
$x _k (\cdot)\in B_k$, but $Px_k \not \in B_k$, that is
$\|Px_k (t)\|_\alpha> k$ for some $t(k)\in[0,T]$. On the other hand,
however, we have by conditions (H1)--(H3) and \eqref{AA}
\eqref{AU} \eqref{AUA} that
\begin{align*}
k&<\|(Px_k )(t)\|_\alpha\\
&=\Big\|A^{\alpha}(t_0)U(t,0)[x_0 -U(\epsilon,0)g(x_k )+F(0,x_k (0))]-F(t, x_{k}(t)) \\
&\quad +\int_{0}^t A^{\alpha}(t_0)U(t,s) A(s)F(s. x_{k}(s))ds
 +\int_{0}^tA^{\alpha}(t_0)U(t,s)G(s,x_k (r(s)))ds \Big\|\\
&\le \|A^{\alpha}(t_0)A^{-\beta}(t)\|\,\|A^{\beta}(t)U(t,0)
 A^{-\beta}(0)\|\\
&\quad\times \big[\|A^{\beta}(0)x_0\|+\|A^{\beta}(0)A^{-1}(t)\|\,
\|A(t)F(0,x_k (0))\|\big]\\
&\quad +\|A^{\alpha}(t_0)A^{-\beta}(t)\|\, \|A^{\beta}(t)U(t,0) A^{-\beta}(0)\|\, 
\|A^{\beta}(0)A^{-1}(\epsilon)\|\\
&\quad\times \| A(\epsilon)U(\epsilon,0) A^{-1}(0)\|\, \|A(0)g(x_k)\|
\\
&\quad +\|A^{\alpha}(t_0)A^{-1}(t)\|\, \|A(t)F(t, x_{k}(t))\| \\
&\quad +\int_{0}^t
\|A^{\alpha}(t_0)A^{-\beta}(t)\|\,\|A^{\beta}(t)U(t,s)\|\, \|A(s)F(s.
x_{k}(s))\|ds\\
&\quad +\int_{0}^t\|A^{\alpha}(t_0)A^{-\beta}(t)\|\|A^{\beta}(t)
U(t,s)\|\,\|G(s,x_k (r(s)))\|ds \\
&\leq C'_\beta C_{(\alpha,\beta)}C_{(\beta,1)}\|A^{\beta}(0)x_0\|
+C'_\beta C'_1 C_{(\alpha,\beta)}C_{(\beta,1)}L_2k\\
&\quad +[C'_\beta C_{(\alpha,\beta)}C_{(\beta,1)} +C_{(\alpha,1)}
+ C_{(\alpha,\beta)}  \frac{C_\beta T^{1-\beta}}{1-\beta}]L_1 (k+1)
+C_{(\alpha,\beta)}  \frac{C_\beta T^{1-\beta}}{1-\beta}w(k)
\end{align*}
Dividing on both sides by $k$ and taking the lower limit as
$k \to +\infty$, we get that
\begin{align*}
&C'_\beta C'_1 C_{(\alpha,\beta)}C_{(\beta,1)}L_2+
\big[C'_\beta C_{(\alpha,\beta)}C_{(\beta,1)} +C_{(\alpha,1) } +
C_{(\alpha,\beta)} \frac{C_\beta T^{1-\beta}}{1-\beta}\big]L_1\\
&+C_{(\alpha,\beta)} \frac{C_\beta T^{1-\beta}}{1-\beta}\gamma\geq
1.
\end{align*}
This contradicts \eqref{L1}. Hence for some positive number
$k$, $PB_k \sqsubseteq B_k$.

Next we will show that the operator $P$ has a fixed point on $B_k$,
which implies that  \eqref{epsr} has a mild solution. To
this end, we decompose $P$ into $P=P_1 +P_2$, where the operators
$P_1,P_2$ are defined on $B_k$ respectively by
\begin{gather*}
(P_1 x)(t)=U(t,0)F(0,x(0))-F(t, x(t))+\int_{0}^t U(t,s) A(s)F(s,x(s))ds,\\
(P_2 x)(t)=U(t,0)[x_0 -U(\epsilon ,0)g(x)]+\int_{0}^tU(t,s)
G(s, x(r(s)))ds,
\end{gather*}
for $0\leq t \leq T$ and we will verify that $P_1$ is a contraction
while $P_2$ is a compact operator.

To prove that $P_1$ is a contraction, we take $x_1,x_2\in B_k$, then
for each $t \in[0,T]$ and by condition (H1), \eqref{AA},
\eqref{AU}, \eqref{AUA} and \eqref{L0}, we have
\begin{align*}
&\|(P_1 x_1)(t)-(P_1 x_2)(t)\|_\alpha\\
& \leq \|A^{\alpha}(t_0)A^{-\beta}(t)\|\|A^{\beta}(t)U(t,0)
A^{-\beta}(0)\|\|A^{\beta}(0)A^{-1}(t)\|\\
&\quad\times \|A(t)[F(0,x_1 (0))-F(0,x_2 (0))]\|
\\
& \quad +\|A^{\alpha}(t_0)A^{-1}(t)\|
\|A(t)[F(t, x_{1}(t))-F(t, x_{2}(t))]\|
\\
& \quad +\|\int_{0}^t A^{\alpha}(t_0)A^{-\beta}(t)\|
\|A^{\beta}(t)U(t,s) \|\|A(s) [F(s. x_{1}(s))-F(s. x_{2}(s))]\|ds\\
& \leq  [C'_\beta C_{(\alpha,\beta)}C_{(\beta,1)} +C_{(\alpha,1)}]
 L \sup_{0\leq s \leq T}\|x_{1}(s)-x_{2}(s)\|_\alpha
 \\
&\quad +C_{(\alpha,\beta)} \frac{C_\beta T^{1-\beta}}{1-\beta}L
 \sup_{0\leq s\leq T}\|x_{1}(s)-x_{2}(s)\|_\alpha\\
&\leq L[C'_\beta C_{(\alpha,\beta)}C_{(\beta,1)} +C_{(\alpha,1)}
+C_{(\alpha,\beta)} \frac{C_\beta T^{1-\beta}}{1-\beta}] \sup
_{0\leq
 s\leq T}\|x_1 (s)-x_2 (s)\|_\alpha\\
&=L_0 \sup_{0\leq s\leq T}\|x_1 (s)-x_2 (s)\|_\alpha.
\end{align*}
Thus
$$
\|P_1x_1 -P_1x_2\|_\alpha\leq L_0\|x_1 -x_2\|_\alpha,
$$
which shows $P_1$ is a contraction.

To prove that $P_2$ is compact, firstly we prove that $P_2$ is
continuous on $B_k$. Let $\{x_n\}\sqsubseteq B_k$ with $x_n \to x$ in $B_k$,
then by (H2)(i), we have
$$
G(s,x_{n}(r(s)))\to G(s,x(r(s))),\quad n \to \infty.
$$
Since
$$
\|G(s,x_{n}(r(s)))-G(s,x(r(s)))\|\leq 2w(k ),
$$
by the dominated convergence theorem we have
\begin{align*}
&\|P_2 x_n -P_2 x\|_\alpha \\
&= \sup_{0\leq t\leq T}
\|A^{\alpha}(t_0)U(t,0)U(\epsilon,0)[g(x_n )-g(x)]
\\
&\quad +\int_{0}^tA^{\alpha}(t_0)U(t,s)[G(s,x_{n}(r(s)))-G(s,x(r(s)))]ds\|
\\
&\le \sup_{0\leq t\leq T} \|A^{\alpha}(t_0)U(t,0)U(\epsilon,0)[g(x_n )-g(x)]
 \\
&\quad +\int_{0}^t\|A^{\alpha}(t_0)A^{-\beta}(t)\|
 \|A^{\beta}(t)U(t,s)\|\|[G(s,x_{n}(r(s)))-G(s,x(r(s)))]\|ds\}\\
&\to 0,\quad \text{as } n \to +\infty;
\end{align*}
i.e., $P_2$ is continuous.

Next we prove that the family $\{P_{2}x:x \in B_k\}$ is a family of
equi-continuous functions. To do this, let $0< t< T$, $h\neq 0$ with
$t+h\in [0,T]$, then
\begin{align*}
&\|(P_{2}x)(t+h)-(P_{2}x)(t)\|_{\alpha}\\
&=\|A^{\alpha}(t_0)[U(t+h ,0)-U(t
,0)](x_0-U(\epsilon ,0)g(x))\\
&\quad +\int_{0}^{t+h}A^{\alpha}(t_0)U(t+h ,s)G(s,x(r(s)))ds-
\int_{0}^tA^{\alpha}(t_0)U(t ,s)G(s,x(r(s)))ds\|\\
&\leq\|A^{\alpha}(t_0)[U(t_2 ,0)-U(t_1
,0)](x_0-U(\epsilon ,0)g(x))\|\\
&\quad +\int_{0}^{t -\varepsilon}\|A^{\alpha}(t_0)(U(t+h ,s)-U(t,s))\|
\|G(s,x(r(s)))\|ds\\
&\quad +\int_{t -\varepsilon}^t\|A^{\alpha}(t_0)(U(t+h ,s)
-U(t,s))\|\|G(s,x(r(s)))\|ds\\
&\quad +\int_{t}^{t+h}\|A^{\alpha}(t_0)U(t+h ,s)\|\|G(s,x(r(s)))\|ds.
\end{align*}
Formula \eqref{U} gives that
\begin{align*}
&\|(P_{2}x)(t+h)-(P_{2}x)(t)\|_{\alpha}\\
&\le\|A^{\alpha}(t_0)[U(t_2 ,0)-U(t_1,0)](x_0-U(\epsilon ,0)g(x))\|\\
&\quad + w(k))\int_{0}^{t-\varepsilon}\|A^{\alpha}(t_0)[e^{-(t+h-s)A(t+h)}-e^{-(t-s)A(t)}]
\|ds\\
&\quad  +w(k)\int_{0}^{t-\varepsilon}\|A^{\alpha}(t_0)\int^t_s[e^{-(t+h-\tau)
A(\tau)}-e^{-(t-\tau)A(\tau)}]R(\tau,s)d\tau \|ds\\
&\quad  +w(k)\int_{0}^{t-\varepsilon}\|A^{\alpha}(t_0)\int^{t+h}_se^{-(t+h-\tau)
A(\tau)}R(\tau,s)d\tau \|ds\|\\
&\quad +w(k)\int_{t -\varepsilon}^t\|A^{\alpha}(t_0)(U(t+h
,s)-U(t ,s))\|ds\\
&\quad +w(k)\int_{t}^{t+h}\|A^{\alpha}(t_0)U(t+h ,s)\|ds
: =\sum_{i=1}^6I_i.
\end{align*}
By Lemma \ref{Le01} we deduce easily that $I_1\to 0$ as $h\to 0$.
Since $A(t)e^{-\tau A(s)}$ is uniformly continuous in $(t,\tau,s)$
for $0\le t\le T$, $m\le \tau\le T$ and $0\le s\le T$, where $m$ is
any positive number (cf. \cite{Fri} and \cite{Paz} ), we see that
$I_2$ also tends to $0$ as $h\to 0$. And
\begin{align*}
I_3&=w(k)\int_{0}^{t-\varepsilon}\|\int^{t-\varepsilon}_sA^{\alpha}(t_0)[e^{-(t+h-\tau)
A(\tau)}-e^{-(t-\tau)A(\tau)}]R(\tau,s)d\tau \|ds\\
&\quad  +w(k)\int_{0}^{t-\varepsilon}\|\int^t_{t-\varepsilon}A^{\alpha}(t_0)[e^{-(t+h-\tau)
A(\tau)}-e^{-(t-\tau)A(\tau)}]R(\tau,s)d\tau \|ds\\
&:= I_{3i}+I_{32}.
\end{align*}
Again from the uniform continuity of $A(t)e^{-\tau A(s)}$ and the
estimate of $R(\tau,s)$ it is easy to infer that $I_{31}\to 0$ as
$h\to 0$. For $I_{32}$, there yields by \eqref{Ae} that
\begin{align*}
I_{32}
&\le w(k)\int_{0}^{t-\varepsilon}\int^t_{t-\varepsilon}
C_{(\alpha,\beta)}
[\frac{C_\beta}{t+h-\tau}+\frac{C_\beta}{t-\tau}]\frac{K}{|\tau-s
|^{1-\delta }}d\tau ds\\
&\le w(k)K C_\beta
C_{(\alpha,\beta)}\int_{0}^{t-\varepsilon}\frac{1}{|t-\varepsilon-s
|^{1-\delta }}ds\int^t_{t-\varepsilon}
[\frac{1}{t+h-\tau}+\frac{1}{t-\tau}]d\tau \\
&=w(k)K C_\beta
C_{(\alpha,\beta)}\frac{1}{\delta}\frac{1}{1-\beta}(t-\varepsilon
)^{\delta
}[(h+\varepsilon)^{1-\beta}-h^{1-\beta}+\varepsilon^{1-\beta}]
\to 0.
\end{align*}
Similarly, one can verify by the estimate of $R(\tau,s)$ and
\eqref{AA}-\eqref{AU} that $I_4$, $I_5$ and $I_6$ all tend to $0$ as
$h\to 0$. Therefore, $\|(P_{2}x)(t+h)-(P_{2}x)(t)\|_\alpha$ tends to
zero independently of $x \in B_k$ as $h\to0$ with $\varepsilon$
sufficiently small. Observe that $U(\epsilon,0)g$ is compact in $X$,
by the similar method as above we also get that
$\|(P_{2}x)(t)-(P_{2}x)(0)\|_\alpha\to 0$ as $t\to 0^+$. Hence,
$P_2$ maps $B_k$ into a family of equi-continuous functions..

 It remains to prove that $V (t)=\{(P_2 x)(t):x \in B_k\}$ is
relatively compact in $X_\alpha(t_0)$. It is easy to verify that
$V(0)$ is relatively compact in $X_\alpha(t_0)$.
 Now, for any $\beta$, $0\le \alpha<\beta<1$, and
$t\in (0, T]$,
\begin{align*}
\|(A^{\beta}(t_0)P_{x}x)(t)\|
&\leq\int_{0}^t \|A^{\beta}(t_0)U(t,s) G(s,x(r(s)))\|ds\\
&\leq w(k)\int_{0}^t\|A^{\beta}(t_0)U(t,s) \| ds\\
&\le w(k)C_{(\beta,\beta')} \frac{C_{\beta'}}{1-\beta'}a^{\beta'},
\end{align*}
where $\beta<\beta'<1$. This shows that $A^{\beta}(t_0)V(t)$ is
bounded in $X$. On the other hand, $A^{-\beta}(t_0)$ is compact
since $A^{-1}(t_0)$ is compact by Assumption (B4), thus
$A^{-\beta}(t_0): X\to X_\alpha(t_0)$ is compact for each
$\beta>\alpha$ (note that the imbedding $X_\beta(t_0)\hookrightarrow
X_\alpha(t_0)$ is compact). Therefore, we infer that $V(t)$ is
relatively compact in $X_\alpha(t_0)$. Thus, by Arzela-Ascoli
theorem $P_2$ is a compact operator.  These arguments above enable
us to conclude that $P=P_1+P_2$ is a condense mapping on $B_k$, and
by Theorem \ref{Th0} there exists a fixed point $x_\epsilon(\cdot)$
for $P$ on $B_k$, which is a mild solution for the problem
\eqref{epsr}.

\textbf{Step 2.} We prove that there is a subsequence
$x_\epsilon(\cdot)$ converging to a mild solution of \eqref{pr}.
We denote by $\Sigma$ the set of all the fixed points
$x_\epsilon(\cdot)$ of operator $P$ on $B_k$ obtained above for
$\epsilon>0$, that is,
$$
\Sigma=\left\{x_\epsilon(\cdot)\in E:x_\epsilon(\cdot)=(Px_\epsilon)(\cdot)\right\}.
$$
We shall prove that $\Sigma$ is relatively compact in $E$.

For $\epsilon>0$, each $x_\epsilon(\cdot)\in \Sigma$ satisfies
\begin{align*}
{x}_\epsilon(t)
&=U(t,0)[x_0-U(\epsilon,0)g(x_\epsilon)-F(0,x_\epsilon(0))]
+F(t,x_\epsilon(t))\\
&\quad +\int_0^t
 U(t,s)A(s)F(s,x_\epsilon(s))ds
+\int_0^t U(t,s)G(s, x_\epsilon(r (s)))ds.
\end{align*}
Let $0< t<T$, $h>0$ very small, then
\begin{align*}
&\|{x}_\epsilon(t+h)-{x}_\epsilon(t)\|_\alpha\\
&=\|A^\alpha(t_0)\left(U(t+h,0)-U(t,0)\right)[x_0-U(\epsilon,0)
 g(x_\epsilon)-F(0,x_\epsilon(0))]\|\\
&\quad +\|A^\alpha(t_0)A^{-1}(t)A(t)[F(t+h,x_\epsilon(t+h))
 -F(t,x_\epsilon(t))]\|\\
&\quad +\|A^\alpha(t_0)[\int_0^{t+h}
 U(t+h,s)A(s)F(s,x_\epsilon(s))ds-\int_0^t
 U(t,s)A(s)F(s,x_\epsilon(s))ds]\|\\
&\quad +\|A^\alpha(t_0)[\int_0^{t+h} U(t+h,s)G(s, x_\epsilon(r
(s)))ds-\int_0^t U(t,s)G(s, x_\epsilon(r (s)))ds]\|.
\end{align*}
From \eqref{AF1} and \eqref{L0} it follows that
\begin{align*}
&\|(1-C_{(\alpha,1)}L){x}_\epsilon(t+h)-{x}_\epsilon(t)\|_\alpha\\
&\le\|A^\alpha(t_0)U(t+h,0)-U(t,0)[x_0-U(\epsilon,0)g(x_\epsilon)
-F(0,x_\epsilon(0))]\| +C_{(\alpha,1)}Lh\\
&\quad +\|A^\alpha(t_0)[\int_0^{t+h}
 U(t+h,s)A(s)F(s,x_\epsilon(s))ds-\int_0^t
 U(t,s)A(s)F(s,x_\epsilon(s))ds]\|\\
&\quad +\|A^\alpha(t_0)[\int_0^{t+h} U(t+h,s)G(s, x_\epsilon(r
(s)))ds-\int_0^t U(t,s)G(s, x_\epsilon(r (s)))ds]\|.
\end{align*}
 Thus, using the similar
arguments as proving the equi-continuity for the family
$\{P_{2}x:x \in B_k\}$ in Step 1, one can easily prove that $\Sigma$ is an
equi-continuous family on $C([\tau,T],X_\alpha(t_0))$ for
$\tau(k)>0$.

Next we show that, for each fixed $t\in [\tau,T]$, $\Sigma(t)$ is
relatively compact in $X_\alpha(t_0)$. From
\[
 \|A^\beta(t_0)F(t, x_\epsilon(t))\|=\|A^\beta(t_0)A^{-1}A(t)F(t,
 x_\epsilon(t))\|
 \le C_{(\beta,1)}L_1(\|x_\epsilon\|+1)
\]
 and the compactness of $A^{-\beta}(t_0): X\to X_\beta(t_0)$
 ($\subset X_\alpha(t_0)$) it follows that, for each $t\in[\tau,T]$,
 $\{F(t,x_{\epsilon}(t)):x_{\epsilon}\in \Sigma\}$ is relatively compact in
 $X_\alpha(t_0)$. Hence, we can also prove that $\Sigma(t)$ is relatively
compact in $X_\alpha(t_0)$ by the same techniques as in Step 1.

Hence, again by Arzela-Ascoli theorem we deduce that
$\Sigma|_{[\tau,T]}$ is relatively compact in the space
$C([\tau,T],X_\alpha(t_0))$.
Set
$$
\tilde{x}_{\epsilon}(t)=\begin{cases}
x_{\epsilon}(t), &t\in [\tau,a],\\
x_{\epsilon}(\tau), &t\in [0,\tau],
\end{cases}
$$
then $g({\tilde{x}}_{\epsilon})=g({x}_{\epsilon})$ by $(H'_3)$ and
we may assume without loss of generality that
${\tilde{x}}_{\epsilon}(\cdot)\to x(\cdot)$ on interval $[\tau, T]$.

Next we need to prove that $\Sigma(0)=\left\{x_0-U({\epsilon},0)
g(x_{\epsilon})\right\}$ is relatively compact in $X_\alpha(t_0)$.
In fact, by \eqref{AA}, \eqref{AUA}, \eqref{leAU} and condition
(H3), we obtain
\begin{align*}
&\left\|A^{\alpha}(t_0)U({\epsilon},0)
g(x_{\epsilon})-A^{\alpha}(t_0) g(x)\right\|\\
&\le \left\|A^{\alpha}(t_0)U({\epsilon},0)g(
x_{\epsilon})-A^{\alpha}(t_0)U({\epsilon},0)
g(x)\right\|\\
&\quad +\left\|A^{\alpha}(t_0)U({\epsilon},0)
A^{\alpha}(t_0)A^{\alpha}(t_0)g(x)-A^{\alpha}(t_0) g(x)\right\|\\
&~~\le
\left\|A^{\alpha}(t_0)A^{-1}(\epsilon)\right\|\left\|A(\epsilon)U({\epsilon},0)
A^{-1}(0)\right\|\left\|A(0)g( {\tilde{x}}_{\epsilon})-A(0)
g(x)\right\|\\
&\quad +\left\|A^{\alpha}(t_0)\left[U({\epsilon},0)-I\right]A^{-1}(0)\right\|\left\|A(0)
g(x)-A(0) g(x)\right\|
\to 0,\quad\text{as }\epsilon\to 0^+.
\end{align*}
To complete the proof for the relative compactness of $\Sigma$ in
$E$ it remains to verify that $\Sigma$ is equi-continuous at $t=0$,
while this can be reached readily by the relative compactness of
$\{U(\epsilon,0) g(x_\epsilon): \epsilon>0\}$.

Therefore, $\Sigma$ is relatively compact in $E$ and we may assume
that $x_\epsilon(\cdot)\to x(\cdot)$ in $E$ for some $x(\cdot)$.
Then, by taking the limit as $\epsilon\to 0^+$ in
$x_\epsilon(\cdot)=Px_\epsilon(\cdot)$ and using the Lebesgue
dominated convergence theorem, we deduce without difficulty that
$x(\cdot)$ is a mild solution to System \eqref{pr}. The proof is
complete.
\end{proof}

\subsection{Existence of strict solutions}

In this subsection, we provide conditions which allow the differentiability
 of the mild solutions obtained in Section 3.1.

\begin{definition}\label{def3.2} \rm
 A function $x(\cdot):[0,T]\to X_\alpha(t_0)$ is
said to be a strict solution of the nonlocal Cauchy problem
\eqref{pr}, if
\begin{itemize}
\item[(1)] $x$ belongs to $C([0,T];X_\alpha(t_0))\cap C^1((0,T];X)$;
\item[(2)] $x$ satisfies
$$
\frac{d}{dt}[x(t)+F(t, x(t))]  +A(t)x(t) = G(t, x(r (t)))
$$ on $(0,T]$, and
$x(0)+g(x)=x_0$.
\end{itemize}
\end{definition}

For the next theorem, we define the following assumptions:
\begin{itemize}
\item[(H1')] For any function $y \in E$, the mapping
$t \to F(t, y(t))$ is H\"older continuous on $[0,T]$;

\item[(H4)] $G(\cdot,\cdot)$ is H\"older continuous; i.e. for each
$(t^0,x^0) \in [0,T]\times X_\alpha(t_0)$, there exist a
neighborhood $W$ of $(t^0,x^0)$ in $[0,T]\times X_\alpha(t_0)$ and
constants $L_3 >0$, $0<\theta \leq1$ such that
$$
\|G(s, x)- G(\bar{s},\bar{x})\|
\leq L_3 [|s-\bar{s}|^{\theta}+ \|x- \bar{x}\|^{\theta}_\alpha]
$$
for $(s, x),\; (\bar s,\bar{x})\in W$;

\item[(H5)] There is a constant $l >0$, such that for all
$s,\bar{s}\in [0,T]$,
$$
\|r (s)-r(\bar{s})\|\leq l|s-\bar{s}|\,;
$$

\item[(H6)] $x_0 \in D(A)$.
\end{itemize}

We remark that (H1') is also verified by the
example presented in Section 4.

\begin{theorem}\label{th2}
Suppose that {\rm (B1)--(B4), (H1), (H2)(ii), (H3), (H1'), (H4)--(H6)}
are satisfied.
Then the nonlocal Cauchy problem \eqref{pr} has a strict solution on
$[0,T]$ provided that \eqref{L0} and \eqref{L1} hold.
\end{theorem}

\begin{proof}
By Theorem \ref{th1}, we see that\eqref{pr} has a mild solution
$x(\cdot)$ on $[0,T]$. We now consider the differentiability of
$x(t)$. Let
\begin{gather*}
f(t)=F(t,x(t)),\\
o(t)=U(t,0)[x_0+F(0, x(0))-g(x)]
= U(t,0)[x(0)+F(0, x(0))],\\
p(t)= \int_{0}^t U(t,s) A(s)F(s, x(s))ds,\\
q(t)= \int_{0}^tU(t,s) G(s, x(r (s))) ds.
\end{gather*}
It follows from Lemma \ref{Le01}, Lemma \ref{Le02}, \eqref{leAU} and
\eqref{leAU2} that
$$
\|o(t+h)-o(t)\|_{\alpha}\le
C(\alpha,1)\|A((0)[x(0)+F(0,x(0))]\|h^{1-\alpha},
$$
and
\begin{align*}
\|p(t+h)-p(t)\|_{\alpha}
&\le C(\alpha)(|\log h|+1))\max_{0\le s\le t+h}\|A(s)F(s, x(s))\|h^{1-\alpha}\\
&\le C(\alpha)h^{\beta}(|\log h|+1))\max_{0\le s\le
t+h}\|A(s)F(s, x(s))\|h^{1-\alpha-\beta},
\end{align*}
where we have chosen $0<\beta<1$ such that $1-\alpha-\beta>0$.
Observing $h^{\beta}(|\log h|+1))$ is bounded we see that $o(t)$ and
$p(t)$ are both H\"{o}lder continuous on $[0,T ]$ with exponent
$1-\alpha-\beta$, and similarly, this holds for $q(t)$.
 So by (H1') it is easy to deduce that $ x(\cdot)$ is
H\"{o}lder continuous on $[0, T]$. On the other hand, it has been
shown in \cite{FL} that the Lipschitz continuity of
$A(t)F(\cdot,\cdot,\cdot)$ (condition (H1)) implies
$A(\cdot)F(\cdot,\cdot,\cdot)$ is locally H\"{o}lder continuous.
Hence conditions (H4), (H5) assure that
$$
s \mapsto A(s)F(s,x(s))\quad\text{and}\quad
s \mapsto G(s,x(r(s)))
$$
are both H\"older continuous in $X$ on $[0, T]$. Thus, from the
proof of \cite[Theorem 5.7.1]{Paz} it is not difficult to see that
$p(t) \in D(A)$, $q(t) \in D(A)$, and
\begin{gather*}
p'(t)= A(t)F(t, x(t))-A(t)\int_{0}^t U(t,s) A(s)F(s, x(s))ds\\
q'(t)= G(t, x(r (t)))- A(t)\int_{0}^tU(t,s) G(s,  x(r (s))) ds,
\end{gather*}
Moreover, $p(t),q(t)\in C^{1}([\epsilon,T]:X)$. On the other hand,
$f(t) \in C^{1}([0,T]$. Consequently, $x$ is differentiable on
$(0,T]$ and satisfies
\begin{align*}
&\frac{d}{dt}[x(t)+ F(t, x(t))] \\
&=\frac{d}{dt}U(t,0)[x_0+F(0, x(0))-g(x)]+p'(t) +q'(t) \\
&=A(t) U(t,0)[x_0+F(0, x(0))-g(x)]\\
&\quad +A(t) F(t, x(t))-A(t)p(t)+G(t,x(r (t)))-A(t)q(t)\\
&=-A(t)x(t)+ G(t,x(r (t)))
\end{align*}
This shows that $x(\cdot)$ is a strict solution of the nonlocal
Cauchy problem \eqref{pr}. Thus the proof is complete.
\end{proof}

\section{ Example}

 To illustrate the applications of Theorems \ref{th1} and
 \ref{th2}, we consider the following example:
\begin{equation}\label{ex}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}[z(t,x)+ \int_{0}^{\pi}\int_{0}^t
b(s,y,x)(z(s,y)+\frac{\partial} {\partial y}z(s,y))dsdy] \\
&= \frac{\partial^2}{\partial x^2}z(t,x)+a(t)z(t,x)
 +h(t,z(t \sin t,x),\frac{\partial}{\partial y}z(t \sin t,x)),
\end{aligned}\\
0\leq t \leq T,\; 0\leq x \leq \pi ,\\
z(t,0)=z(t,\pi)=0,\\
z(0,x)+\sum_{i=1}^{p}g_1(z(t_i,x))=z_0 (x), \quad 0\leq x \leq \pi,
\end{gathered}
\end{equation}
where $a(t)<0$ is a continuous function and is H\"older continuous in
$t$ with parameter $0<\delta<1$. $T \leq \pi$, $p$ a positive
integer, $0<t_0 <t_1 <\cdots<t_p <T$.  $ z_0 (x)\in
X:=L^{2}([0,\pi])$.

Let $A(t)$ be defined by
$$
A(t)f=-f''-a(t)f
$$
with  domain
$$
D(A)=\{f(\cdot)\in X: f, f' \mbox {absolutely
continuous,} f''\in X,f(0)=f(\pi)=0\}.
$$
Then it is not difficult to
verify that $A(t)$ generates an evolution operator $U(t,s)$
satisfying assumptions $(B_1 )-(B_4 )$ and
$$
U(t,s)=T(t-s)\exp\Big(\int_s^ta(\tau)d\tau\Big),
$$
where $T(t)$ is the compact analytic semigroup generated by the operator
$-A$ with
$-Af=-f''$ for $f\in D(A)$. It is easy to compute that, $A$ has a
discrete spectrum, the eigenvalues are $n^2 ,n\in \mathbb{N}$, with
the corresponding normalized eigenvectors
$z_n (x)=\sqrt{\frac{2}{\pi}}\sin(nx)$. Thus for $f\in D(A)$, there holds
$$
-A(t)f=\sum_{n=1}^{\infty}(-n^2+a(t))\langle f,z_n\rangle z_n,
$$
and clearly the common domain coincides with that of the operator
$A$. Furthermore, we may define $A^\alpha(t_0)$ ($t_0\in [0,T]$) for
self-adjoint operator $A(t_0)$ by the classical spectral theorem and
it is easy to deduce that
$$
A^{\alpha}(t_0)f=\sum_{n=1}^{\infty}(n^2-a(t_0))^{\alpha}\langle f,z_n\rangle z_n
$$
on the domain $D[A^{\alpha}]=\{f(\cdot)\in
X,\sum_{n=1}^{\infty}(n^2-a(t_0))^{\alpha}\langle
f,z_n\rangle z_n\in X\}$. Particularly,
$$
A^{1/2}(t_0)f=\sum_{n=1}^{\infty}\sqrt{n^2-a(t_0)}
\langle f,z_n\rangle z_n .
$$
Therefore, for each $f\in X$,
\begin{gather*}
U(t,s) f=\sum_{n=1}^{\infty}e^{-n^2(t-s)+\int_s^ta(\tau)d\tau}\langle
f,z_n\rangle z_n,
\\
A^{\alpha}(t_0)A^{-\beta}(t_0)f=\sum_{n=1}^{\infty}
(n^2-a(t_0))^{\alpha-\beta}\langle f,z_n\rangle z_n ,
\\
A^{\alpha}(t_0)U(t,s)
f=\sum_{n=1}^{\infty}(n^2-a(t_0))^{\alpha}e^{-n^2(t-s)+\int_s^ta(\tau)d\tau}\langle
f,z_n\rangle z_n.
\end{gather*}
Then
\begin{equation}\label{aa1}
\|A^{\alpha}(t)A^{-\beta}(s)\|\le (1+\|a(\cdot)\|)^{\alpha},\quad
\|A^{\beta}(t)U(t,s)A^{-\beta}(s)\|\le (1+\|a(\cdot)\|)^{\beta},
\end{equation}
 for  $t,s\in[0,T]$, $0<\alpha<\beta$. Also
\begin{align*}
&\|A^{\beta}(t)U(t,s) f\|^2\\
&=\sum_{n=1}^{\infty}(n^2-a(t))^{2\beta} e^{-2n^2(t-s)
+2\int_s^ta(\tau)d\tau} |\langle f,z_n\rangle |^2
\\
&=(t-s)^{-2\beta} \sum_{n=1}^{\infty}[(n^2-a(t))(t-s)]
 ^{2\beta}e^{-2(n^2-a(t))(t-s)-2a(t)(t-s)+2\int_s^ta(\tau)d\tau}
 |\langle f,z_n\rangle |^2
\\
&=(t-s)^{-2\beta}\sum_{n=1}^{\infty}e^{{2\beta}\log[(n^2-a(t))(t-s)]
 -2(n^2-a(t))(t-s)-2a(t)(t-s)+2\int_s^ta(\tau)d\tau}
 |\langle f,z_n\rangle |^2
 \\
&\le (t-s)^{-2\beta}\sum_{n=1}^{\infty}\beta^{2\beta}
 e^{-2a(t)(t-s)+2\int_s^ta(\tau)d\tau} |\langle f,z_n\rangle |^2;
\end{align*}
(note that $c\log x-x\le c\log c-c$),
which shows that
\begin{equation}\label{au1}
\|A^{\beta}(t)U(t,s) \|\le \frac{C_\beta}{(t-s)^{\beta}}
\end{equation}
for ${C_\beta}=\beta^{\beta}\max
\left\{e^{-2a(t)(t-s)+2\int_s^ta(\tau)d\tau}:t,s\in[0,T]\right\}>0$.

Now define the abstract functions $F, G: X_{1/2}(t_0)\to X$
 by
\begin{gather*}
F(t, Z(t,\cdot))(x) =\int_{0}^{\pi}\int_{0}^t
b(s,y,x)[Z(s,y)+\frac{\partial} {\partial y}Z(s,y)]dsdy, \\
G(t,Z(t,x))(x)=h(t,Z(t,x),\frac{\partial} {\partial x}Z(t,x)),
\end{gather*}
and $g: C([0,T];X_{1/2}(t_0)\to X$ by
$$
g(Z(t,x))(x)=\sum_{i=1}^{p}g_1(z(t_i,x)).
$$
Then system \eqref{ex} is rewritten in the form \eqref{pr}.


For System \eqref{ex} we assume that
the following conditions hold:
\begin{itemize}
\item[(C1)] The function $b(\cdot,\cdot,\cdot)$ is a $C^2$ function,
and $b(y,0)=b(y,\pi)=0$;

\item[(C2)]
For the function
$h: [0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ the
following three conditions are satisfied:
\begin{itemize}
 \item[(1)]  For each $t\in [0,T]$, $h(t,\cdot,\cdot)$ is continuous, and
 $h(\cdot,\cdot,\cdot)$ is measurable in $t$,
 \item[(2)] There are positive functions $h_1,h_2\in C([0,T])$ such
 that for all $(t,z)\in [0,T]\times X$,
 $$
 |h(t,z)|\leq h_1(t)|z|+h_2(t)
 $$
 \end{itemize}

\item[(C3)] $g_1$ takes values in $D(A)$ and $A(t_0)g_1$ is a
continuous map and there is a positive constants $L$ such that
$\|g_1(x)\|_{1/2}\le L$.
\end{itemize}

Condition (C1) implies that $R(F)\subset D(A)$. Clearly,
$A(t)F(\cdot)$ the Lipschitz continuous on $X$. Observe that, for
any $z_1,~z_2\in X_{1/2}(t_0)$,
\begin{align*}
\|z_2(x)-z_1(x)\|^2
&=\sum_{n=1}^{\infty} {\langle z_2-z_1,z_n\rangle} ^2\\
& \leq  \sum_{n=1}^{\infty} (n^2+a(t_0)) {\langle z_2-z_1,z_n\rangle}^2\\
& \leq {\|z_2(x)-z_1(x)\|_{1/2}}^2,
\end{align*}
 it follows that the above conditions ensure that  $F$, $G$ and $g$
verify Assumptions (H1)--(H3) respectively.
Consequently, for any $z_0\in X_\beta(t_0)$
($\frac{1}{2}<\beta\le 1$), by Theorem \ref{th1},
system \eqref{ex} has a mild solution on $[0,T]$ under these
assumptions,  provided that \eqref{L0} and \eqref{L1}
hold (note that the constants
$C_{\beta},C'_{\beta},C_{(\alpha,\beta)}$ are given by \eqref{aa1}
and \eqref{au1} explicitly).

Furthermore, if we suppose that
\begin{itemize}
\item[(C4)] The function $h(t,z)$ is Lipschitz continuous.
\end{itemize}
Then it is not difficult to verify that the conditions (including
condition (H1')) of Theorem \ref{th2} are  satisfied and so
the mild solution is also a strict solution of  \eqref{ex} for
given $z_0\in D(A)$.

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