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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 157, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/157\hfil Existence of solutions]
{Existence of solutions in the $\alpha$-norm for partial
differential equations of neutral type with finite delay}

\author[K. Ezzinbi, H. Megdiche, A. Rebey\hfil EJDE-2010/157\hfilneg]
{Khalil Ezzinbi, Hatem Megdiche, Amor Rebey}  % in alphabetical order

\address{Khalil Ezzinbi \newline
Universit\'e Cadi Ayyad, Facult\'e des Sciences Semlalia,
D\'epartement de Math\'ematiques, BP 2390, Marrakech, Maroc}
\email{ezzinbi@gmail.com}

\address{Hatem Megdiche \newline
D\'epartement de Math\'ematiques, 
Facult\'e des Sciences de Gab\`es, Cit\'e Erriadh 6072, Zrig, Gab\`es, Tunisie}
\email{megdichehatem@yahoo.fr}

\address{Amor Rebey \newline
Institut Sup\'erieur des Math\'ematiques Appliqu\'ees et de
l'Informatique de Kairouan, Avenue Assad Iben Fourat - 3100
Kairouan, Tunisie}
\email{rebey\_amor@yahoo.fr}

\thanks{Submitted June 28, 2010. Published October 29, 2010.}
\subjclass[2000]{34K30, 47D06}
\keywords{Neutral equation; analytic semigroup; fractional power;
\hfill\break\indent phase space; mild solution; Sadovskii's fixed point theorem}

\begin{abstract}
 In this work, we prove results on the local existence of mild
 solution and global continuation in the $\alpha$-norm for some
 class of partial neutral differential equations. We suppose that
 the linear part generates a compact analytic semigroup. The
 nonlinear part is just assumed to be continuous. We use the
 compactness method, to show the main result of this work.
\end{abstract}

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

\section{Introduction}

In this work, we study the existence and global continuation of
solutions in the $\alpha$-norm for partial differential equations
of neutral type with finite delay. The following model provides an
example of such a situation
\begin{equation}\label{explee}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}[v(t,x)-av(t-r,x)]\\
&=\frac{\partial^2}{\partial x^2}[v(t,x)- av(t-r,x)]
 +f(\frac{\partial}{\partial x}v(t-r,x))\quad \text{for }
 t\geq0,\;x\in[0,\pi]
\end{aligned}\\
v(t,0)=av(t-r,0),\quad v(t,\pi)=av(t-r,\pi)\quad \text{for } t\geq0,\\
v(t,x)=v_0(t,x) \quad \text{for } -r\leq t\leq0,\; x\in[0,\pi],
\end{gathered}
 \end{equation}
where $a$ and $r$ are positive constants,
$f:\mathbb{R}\to\mathbb{R}$ is a continuous function, and $v_0$ is
a given initial function from $[-r,0]\times [0,\pi]$ to
$\mathbb{R}$. Equation \eqref{explee} can be written in the
following abstract form for partial differential equations
\begin{equation}\label{edp}
\begin{gathered}
 \frac{d}{dt}Du_t = -ADu_t+F(t,\ u_t)\quad\text{for } t\geq0, \\
 u_0 = \varphi ,\quad \varphi\in C_\alpha,
\end{gathered}
\end{equation}
where $-A$ is the infinitesimal generator of an analytic
semigroup on a Banach space $X$,
$C_\alpha:=C([-r,0];D(A^{\alpha}))$, $0<\alpha<1$, denotes the
space of continuous functions from $[-r,0]$ into $D(A^\alpha)$,
and the operator $A^\alpha$ is the fractional $\alpha$-power of
$A$. This operator $(A^{\alpha},D(A^{\alpha}))$ will be described
 later. For $x\in C([-r,b];D(A^\alpha)),b>0$, and $t\in[0,b]$,
$x_t$ denotes, as usual, the element of $C_\alpha$ defined by
$x_t(\theta)=x(t+\theta)$ for $\theta\in [-r,0]$. $F$ is a
continuous function from $\mathbb{R}_+ \times C_\alpha$ with
values in $X$ and $D$ is a bounded linear operator from
$C_X:=C([-r,0];X)$ into $X$ defined by $D\varphi=\varphi(0)-
D_0\varphi$, for $\varphi\in C_X$, where $D_0$ is a bounded linear
operator given by:
$$
D_0\varphi= \int^0_{-r} d\eta(\theta)\varphi(\theta)\quad\text{for
}\varphi\in C_X,
$$
where $\eta:[-r,0]\to \mathcal{L}(X) $ is of bounded variation and
non-atomic at zero. That is, there is a continuous nondecreasing
function $\delta:[0,r]\to[0,+\infty[$ such that $\delta(0)=0$ and
\begin{equation}\label{non atomic}
 \big\| \int^0_{-s} d\eta(\theta)\varphi(\theta) \big\|
\leq \delta(s) \| \varphi\|\quad \text{for }
 \varphi\in C_X \,; s\in[0,r].
\end{equation}

There is an extensive literature of differential equations of
neutral type motivated by physical applications. Xia and Wu
(1996), Hale (1994), and Wu (1996) studied the neutral partial
functional differential equation
 \begin{equation}\label{edpk}
 \frac{\partial}{\partial t}Du_t
= K\frac{\partial^2}{\partial x^2}Du_t+f( u_t)\quad\text{for }
x\in S^1,
 \end{equation}
where $K$ is a positive constant and $X$ be the space
$C(S^1,\mathbb{R})$. Let $A=K\frac{\partial^2}{\partial x^2}$ with
domain $C^2(S^1,\mathbb{R})$, then $A$ is the infinitesimal
generator of an analytic semigroup $(T(t))_{t\geq0}$ on $X$ and
the associated integrated form of \eqref{edpk} subject to the
initial condition $u_0=\varphi\in C([-r,0],X)$ is
\begin{equation}\label{edpkk}
 D(u_t)=T(t)D(\varphi)+\int^t_0 T(t-s)f(u_s) ds\quad\text{for }
t\geq 0.
\end{equation}

Wu \cite{Wu} established the existence of mild solution of
\eqref{edpkk}. Travis and webb \cite{Tr-We2} considered
partial differential equations of the form
 \begin{equation}\label{edps}
\begin{gathered}
 \frac{d}{dt}u(t) = -Au(t)+F(t, u_t)\quad t\geq0, \\
 u_0 = \varphi ,\quad \varphi \in C_\alpha,
\end{gathered}
 \end{equation}
where $-A$ the infinitesimal generator of a compact analytic
semigroup and $F$ is only continuous with respect to a fractional
power of $A$ in the second variable.

This work is motivated by the paper of Travis and Webb \cite{Tr-We2},
where the authors studied the existence and continuability in
 the $\alpha$-norm for equation \eqref{edp} but in the case
where $D_0=0$, they assumed that $F:C_\alpha\to X$ is continuous.
In \cite{M-K-M} the authors obtained the local and the global
existence of solution of Eq. \eqref{edp} for $\alpha=0$ in the
case when the linear part is non densely defined Hille-Yosida.
Recently, in \cite{M-K} Adimy and Ezzinbi have developed a basic
theory of partial neutral functional differential equations in
fractional power spaces, they proved the existence and regularity
of the solution of Eq. \eqref{edp} where the nonlinear part
satisfies Lipschitz conditions.

 The present paper is organized as
follows. In the first section, we introduce some notations and
necessary preliminaries. In Section 2, we study the local
existence and global continuation of mild solutions of
\eqref{edp}. Finally, to illustrate our results, we give in
Section 3 an application.

\section{Existence of local mild solutions}

In this section we study the existence of mild solutions for the
abstract Cauchy problem \eqref{edp}. Before that, we state the
following assumption.

\begin{itemize}
\item[(H1)] $-A$ is the infinitesimal generator of an analytic
semigroup $(T(t))_{t\geq0}$ on a Banach space $X$ and
$0 \in \rho(A)$, where $\rho(A)$ is the resolvent set of $A$.
\end{itemize}

 Note that if $0\in \rho(A)$ is not satisfied, one can substitute
the operator $A$ by the operator ($A-\sigma I)$ with $\sigma$
large enough such that $0\in \rho(A-\sigma)$. This allows us
to define the fractional power $A^\alpha$ for $0<\alpha<1$,
as a closed linear invertible operator with domain $D(A^\alpha)$
dense in $X$. The closedness of $A^\alpha$ implies that
$D(A^\alpha)$, endowed with the graph norm of $A^\alpha$; i.e.,
the norm $|x|=\| x\|+\| A^\alpha x\|$, is a Banach space.
Since $A^\alpha$ is invertible, its graph norm $|\cdot|$ is
equivalent to the norm $\| x\|_\alpha=\| A^\alpha x\|$. Thus,
$D(A^\alpha)$ equipped with the norm $\|\cdot\|_\alpha$,
is a Banach space, which we denote by $X_\alpha$.
For $0<\beta \leq \alpha<1$, the imbedding
$X_\alpha\hookrightarrow X_\beta$ is compact if the resolvent
operator of $A$ is compact. Also, the following properties are
well known.

 \begin{theorem}[\cite{P}]\label{zzz}
Let $0<\alpha<1$ and assume that \textrm{(H1)} holds. Then
\begin{itemize}
 \item[(i)] $T(t):X \longrightarrow D(A^\alpha)$ for every $t>0$,
 \item[(ii)] $T(t)A^\alpha x =A^\alpha T(t)x$ for every
$x \in D(A^\alpha)$ and $t\geq0$,
 \item[(iii)] for every $t>0$ the operator $A^\alpha T(t)$
is bounded on $X$ and there exists $M_\alpha>0$ such that
 \begin{equation}\label{cc}
 \| A^\alpha T(t) \| \leq M_\alpha e^{\omega t} t^{-\alpha},
 \end{equation}
 \item[(iv)] There exists $N_\alpha>0$ such that
 \begin{equation}\label{cccc}
 \|(T(t)-I)A^{-\alpha} \| \leq N_\alpha t^\alpha \quad \text{for } t> 0.
 \end{equation}
 \end{itemize}
 \end{theorem}

 In the sequel, we denote by $C_\alpha:=C([-r,0]; X_\alpha)$
the Banach space of all continuous function from $[-r,0]$
to $X_\alpha$ endowed with the norm
 $$
\|\varphi\|_{C_\alpha}:=\sup_{\theta\in[-r,0]}
\|\varphi(\theta)\|_\alpha\quad \text{for } \varphi\in C_\alpha.
$$

 \begin{definition} \label{def1} \rm
 Let $\varphi\in C_\alpha$. A continuous function
$u:[-r,+\infty[\to X_\alpha$ is called a mild solution of
 \eqref{edp} if
 \begin{itemize}
 \item[(i)] $D(u_t)=T(t)D(\varphi)+\int^t_0 T(t-s)F(s,u_s) ds$
 for $t\geq0$,
 \item[(ii)] $u_0=\varphi$.
 \end{itemize}
 \end{definition}

 Besides (H1), we consider the  hypothesis:
 \begin{itemize}
 \item[(H2)] The semigroup $(T(t))_{t\geq0}$ is compact on $X$.
 \item[(H3)] If $x\in X_\alpha$ and $\theta\in [-r,0]$ then
$\eta(\theta)x \in X_\alpha$ and
$A^\alpha\eta(\theta)x=\eta(\theta)A^\alpha x$.
 \end{itemize}

 \begin{remark}\label{rem} \rm
 Assumption \textbf{(H3)} implies that if $\varphi\in C_\alpha$ then
 \begin{equation}\label{aaa}
 D_0(\varphi)\in X_\alpha, \quad
 A^\alpha D_0(\varphi)=D_0(A^\alpha\varphi),
 \end{equation}
 where
 $$
(A^\alpha\varphi)(\theta) =A^\alpha(\varphi(\theta)) \quad\text{for }
 \theta\in [-r,0], \; \varphi\in C_\alpha.
$$
 \end{remark}

 The main result of this section is the following theorem.

 \begin{theorem}\label{thm-exist}
 Assume that the hypothesis {\rm (H1)--(H3)} hold true.
 Let $U$ be an open subset of the Banach space $C_\alpha$.
If $F:[0,a] \times U\to X$ is continuous, then for each
$\varphi\in U$ there exist $t_1:=t_1(\varphi)$ with $0<t_1\leq a$ and
 a mild solution $u\in C([-r,t_1];X_\alpha)$ of  \eqref{edp}.
 \end{theorem}

 \begin{proof}
 The proof of this result is based on the Sadovskii's fixed-point
theorem. Let $\varphi\in U$ and $0<t_1\leq a$.
We choose $0<\rho\leq a$ to be small enough such that
$\{ \psi \in C_\alpha : \| \psi-\varphi
\|_{C_\alpha}\leq\rho \}\subset U$. Now we consider
the  set
 $$
\Omega :=\{u\in C([-r,t_1];X_\alpha): u_0=\varphi \text{ and }
\| u_t-\varphi \|_{C_\alpha}\leq\rho
 \text{ for } t\in[0,t_1] \},
$$
 where $C([-r,t_1];X_\alpha)$ is endowed with the uniform convergence
topology. It is easy to check that $\Omega$ is nonempty and bounded.
By using the triangular inequality, it is clear that
$\lambda u_1+(1-\lambda)u_2 \in \Omega$, for any $u_1, u_2 \in \Omega$
and $\lambda\in]0,1[$. Then $\Omega$ is convex.
Also, $\Omega$ is closed in $C([-r,t_1];X_\alpha)$. To prove that,
consider a convergent sequence $(u^n)_{n\geq0}$ of $\Omega$
with $\lim_{n\to+\infty}u^n=u$ in $\Omega$.
Then, for any $n$ in $ \mathbb{N}$, we have
 $$
\| u_0-\varphi \|_{C_\alpha} \leq \| u_0-u^n_0 \|_{C_\alpha}
+\| u^n_0-\varphi \|_{C_\alpha},
$$
 letting $n$ to $+\infty$, yields $ \| u_0-\varphi \|_{C_\alpha}=0$,
then $u_0=\varphi$.
In addition for any $t\in[0,t_1]$, $n \in \mathbb{N}$
 \begin{align*}
 \| u_t-\varphi \|_{C_\alpha}
&\leq  \| u_t-u^n_t \|_{C_\alpha}+ \| u^n_t-\varphi \|_{C_\alpha} \\
&\leq \| u_t-u^n_t \|_{C_\alpha}+\rho.
 \end{align*}
Letting $n$ to $+\infty$, we deduce that
$\| u_t-\varphi \|_{C_\alpha} \leq \rho$.
Consequently, $u\in\Omega$.
We have $\Omega$ is a nonempty, bounded, convex and closed subset
of $C([-r,t_1];X_\alpha)$ when $t_1$ is given by \eqref{s}.

Let the mapping $H:\Omega\to C([-r,t_1];X_\alpha)$ be defined by
 $$
 H(u)(t)= \begin{cases}
D_0(u_t)+T(t)D(\varphi)+\int^t_0 T(t-s) F(s,u_s) ds
&\text{if } t\in [0,t_1],\\
\varphi(t)&\text{if } t\in [-r,0].
\end{cases}
$$
 We will prove now the continuity of $H$. Let $(u^n)_{n\geq 1}$
be a convergent sequence in $\Omega$ with $\lim_{n\to \infty}u^n =u$.
Using \eqref{aaa}, and that $D_0$ is a bounded linear operator,
there exists a positive constant $M$ such that
\begin{equation}\label{inegalit}
 \| D_0(u^n_t) -D_0(u_t)\|_\alpha \leq M \| u^n_t - u_t \|_{C_\alpha}.
\end{equation}
On the other hand the set $\Lambda =\{(s,u^n_s),(s,u_s) :
s\in [0,t_1], n\geq 1 \}$ is compact in $[0,t_1]\times C_\alpha$.
By He\"{\i}ne's theorem implies that $F$ is uniformly continuous
in $\Lambda$. Accordingly, since $(u^n)_{n\geq 1}$ converge
to $u$, we have
\begin{equation}\label{integ}
 \|H(u^n)-H(u)\|_\infty \leq M_\alpha \int^{t_1}_0
\frac{e^{\omega s}}{s^\alpha} ds \sup_{s\in[0,t_1]}
\|F(s,u^n_s)-F(s,u_s)\|\to 0 \text{ as } n\to +\infty.
 \end{equation}
Using \eqref{integ} and the estimate \eqref{inegalit},
we obtain that $(Hu^n)_{n\geq 1}$ converge to $Hu$.
This yields the continuity of $H$.

 We will show that there exists $t_1:=t_1(\varphi) \in]0,a]$
such that $H(\Omega) \subseteq \Omega$. Let $u\in\Omega$.
We have the following translation property
\[
 (Hu)_t(\theta) = \begin{cases}
 D_0u_{t+\theta}+T(t+\theta)D\varphi\\
 + \int^{t+\theta}_0 T(t+\theta-s)F(s,u_s) ds &
 \text{if } t+\theta\in [0,t_1],\\[4pt]
 \varphi(t+\theta)& \text{if } t+\theta\in [-r,0].
\end{cases}
\]

Choose $\gamma >0$ such that
$$
\|\varphi(t+\theta)-\varphi(\theta)\|_\alpha
\leq \frac{\rho}{5} \min\{1,\frac{1}{{\rm var}_{[-r,0]}(\eta)}\},
$$
 for $t\in [0,\gamma]$ and $\theta\in [-r,0]$ such that
$t+\theta\in [-r,0]$. This implies in particular that
$\|(H(u))_t(\theta)-\varphi(\theta)\|_\alpha\leq \rho$,
for $t\in [0,\gamma]$ and $\theta\in [-r,0]$ such
that $t+\theta\in [-r,0]$.

Choose $s\in ]0,r]$ such that $\delta(s) \leq 1/5$
and $\| T(t)D\varphi-D\varphi \|_\alpha\leq \rho/5$,
for $t\in [0,s]$.

 If $0\leq t+\theta \leq s$, then
\begin{align*}
 (H(u))_t(\theta)-\varphi(\theta)
&=  \int^{-s}_{-r} d\eta(\tau)\Big(\varphi(t+\theta+\tau)
 -\varphi(\tau)\Big)\\
&\quad +\int^{0}_{-s} d\eta(\tau)\Big(u_{t+\theta}(\tau)
 -\varphi(\tau)\Big) +T(t+\theta)D(\varphi)- D(\varphi) \\
&\quad +\varphi(0)-\varphi(\theta) +\int^{t+\theta}_0
 T(t+\theta-s) F(s,u_s) ds.
\end{align*}
 As $F$ is continuous, we can choose $\rho>0$ small enough such that
there exists $N>0$ so that $\| F(t,\psi)\| \leq N$, for
$t\in [0,\rho]$ and $\| \psi-\varphi\|_{C_\alpha} \leq\rho$.
Then, if $t_1\leq\rho$ we obtain
$$
\|\int^{t+\theta}_0 T(t+\theta-s) F(s,u_s) ds\|_\alpha
\leq M_\alpha N \int^t_0 e^{\omega s}s^{-\alpha} ds.
$$
 We can take $\gamma$ such that
$\int^\gamma_0 e^{\omega s} s^{-\alpha} ds
\leq \frac{\rho}{5 M_\alpha N}$. We deduce that
\begin{align*}
 \|(H(u))_t(\theta)-\varphi(\theta)\|_\alpha
&\leq \operatorname{var}_{[-r,0]}(\eta)\sup_{\tau\in [-r,-s]}
 \| \varphi(t+\theta+\tau)-\varphi(\tau)\|_\alpha\\
&\quad +\delta(s)\| u_{t+\theta}-\varphi\|_{C_\alpha}
 +\|\varphi(0)-\varphi(\theta)\|_\alpha \\
&\quad + \| T(t+\theta)D(\varphi)- D(\varphi)\|_\alpha
 +M_\alpha N \int^t_0 e^{\omega s} s^{-\alpha} ds.
\end{align*}
 Finally, we choose
\begin{equation}\label{s}
 t_1=\min\{\gamma,s,\rho \}.
\end{equation}
 Then, for $0\leq t+ \theta \leq t_1$, we obtain
$\|(H(u))_t(\theta)-\varphi(\theta)\|_\alpha \leq \rho$.
So, we have proved that there exists $t_1:=t_1(\varphi) \in]0,a]$
such that $H(\Omega) \subseteq \Omega$.
 Consider now the mapping
 $H_1:\Omega\to C([-r,t_1];X_\alpha)$ defined by
 $$
H_1(u)(t)= \begin{cases}
D_0(u_t) & \text{if } t\in [0,t_1], \\
\varphi(t)-D\varphi &\text{if } t\in [-r,0].
\end{cases}
$$
Also define $H_2:\Omega\to C([-r,t_1];X_\alpha)$  by
$$
H_2(u)(t)= \begin{cases}
T(t)D(\varphi)+\int^t_0 T(t-s) F(s,u_s) ds & \text{if } t\in [0,t_1],\\
 D\varphi &\text{if } t\in [-r,0].
\end{cases}
$$
It is clear that $H=H_1 + H_2$. If we prove that $H_1$ is a strict
contraction and $H_2$ is compact. Apply the Sadovskii's fixed
theorem to obtain the existence of a fixed point of $H$ on $\Omega$.

(1) Let $u,v \in \Omega$. Then for each $t\in [0,t_1]$, we have
\begin{align*}
 H_1u(t)- H_1v(t) &=  D_0(u_t-v_t)\\
 &=  \int^0_{-r}d\eta(\theta)(u(t+\theta)-v(t+\theta))\\
 &=  \int^0_{-s}d\eta(\theta)(u(t+\theta)-v(t+\theta)).
\end{align*}
 According to \eqref{aaa}, we have
 $$
A^\alpha D_0(u_t-v_t)=\int^0_{-s}d\eta(\theta)
A^\alpha(u(t+\theta)-v(t+\theta)),
$$
 which implies
 $$
\| D_0(u_t-v_t)\|_\alpha\leq \delta(s)
\sup_{-r\leq t\leq t_1}\| u(t)-v(t)\|_\alpha
$$
Consequently,
 $$ \sup_{-r\leq t\leq t_1}\| H_1u(t)-H_1v(t)\|_\alpha
\leq \delta(s)\sup_{-r\leq t\leq t_1}\| u(t)-v(t)\|_\alpha.
$$
Since $\delta(s)\leq 1/5$, $H_1$ is therefore a strict contraction
in $\Omega$.

(2) We will show that the
$\operatorname{Im}(H_2):=\{H_2(u), u\in \Omega \}$,
is relatively compact. By the Arzela-Ascoli theorem it suffices
to prove that the set $\{H_2(u)(t): u\in \Omega \}$ is a relatively
compact in $X_\alpha$ for each $t\in [0,t_1]$, and
$H_2(\Omega)$ is an equicontinuous family of functions on $[0,t_1]$.

 (i) To prove the first assertion, it is sufficient to show that
the set $\{H_2u(t) : u\in \Omega \}$ is relatively compact for
each $t \in ]0,t_1]$.
 Let $t\in ]0,t_1]$ fixed, and $\beta>0$ such that $\alpha<\beta<1$,
we have
\begin{align*}
 \| (A^\beta H_2 u)(t) \|
&\leq \| A^{\beta-\alpha} T(t)A^\alpha D(\varphi) \|
 + \| \int^t_0 A^\beta T(t-s) F(s,u_s) ds \|\\
&\leq M_{\beta-\alpha} e^{\omega t}t^{\alpha-\beta}
 \| D(\varphi) \|_\alpha
 + M_\beta N \int^t_0 e^{\omega s}s^{-\beta} ds<+\infty.
\end{align*}
 Then for fixed $t\in ]0,t_1]$, $\{(A^\beta H_2 u)(t)\}$ is bounded
in $X$, and appealing to the compactness of $A^{-\beta}:X\to X_\alpha$,
we deduce that $\{ H_2(u)(t): u\in \Omega\}$ is relatively compact
set in $X_\alpha$.

 (ii) On the other hand, for every $0\leq t_0<t\leq t_1$, one has
\begin{align*}
 H_2u(t)- H_2u(t_0)
&=  (T(t)-T(t_0))D\varphi + \int^{t}_{t_0} T(t-s) F(s,u_s) ds \\
 &\quad + \int^{t_0}_0 (T(t-s)-T(t_0-s)) F(s,u_s) ds\\
 &=  (T(t)-T(t_0))D\varphi + \int^t_{t_0} T(t-s) F(s,u_s) ds\\
 &\quad + (T(t-t_0)-I) \int^{t_0}_0 T(t_0-s) F(s,u_s) ds.
\end{align*}
 We obtain that
 \begin{align*}
 \|H_2u(t)-H_2u(t_0)\|_\alpha
&\leq \|(T(t)-T(t_0)) D\varphi \|_\alpha+M_\alpha N\int^t_{t_0}
 \frac{e^{\omega s}}{s^\alpha}ds\\
&\quad + \|(T(t-t_0)-I)\int^{t_0}_0 A^\alpha T(t_0-s) F(s,u_s) ds\|.
\end{align*}
 It is clear that the first part tend to zero as $|t-t_0|\to 0$,
since for $t_0>0$ the set
 $$
\Big\{\int^{t_0}_0 A^\alpha T(t_0-s) F(s,u_s) ds,\; u\in \Omega \Big\}
$$
 is relatively compact in $X$, there is a compact set
$\widetilde{K}$ in $X$ such that
 $$
\int^{t_0}_0 A^\alpha T(t_0-s) F(s,u_s) ds \in \widetilde{K} \text{for }
u\in \Omega.
$$
 By Banach-Steinhaus's theorem, we have
$$
\big\|(T(t-t_0)-I)\int^{t_0}_0 A^\alpha T(t_0-s)F(s,u_s)ds\big\|\to 0
\quad\text{as } t\to t_0,
$$
uniformly in $u\in \Omega$. This implies
$$
\lim_{t\to t^+_0} \sup_{u\in \Omega} \| H_2(u)(t)
- H_2(u)(t_0) \|_\alpha =0.
$$
 Using similar argument for $0\leq t < t_0\leq b$, we can conclude
that $\{H_2u(t), u\in \Omega \}$ is equicontinuous.

 Finally, the Sadovskii's fixed-point theorem implies that $H$
has a fixed point $u$ in $\Omega$. The fact that $u$ is a mild
solutions of Equation \eqref{edp}. This completes the proof.
\end{proof}

To define the mild solution in its maximal interval of existence,
we add the following condition
 \begin{itemize}
 \item[(H4)] $F:[0,+\infty[ \times C_\alpha \to X$ is continuous
and takes bounded sets of $[0,+\infty[ \times C_\alpha$ into
bounded sets in $X$.
 \end{itemize}

 \begin{theorem}\label{prolon}
Assume that the hypotheses of Theorem \ref{thm-exist} hold
and $F$ satisfies \textbf{(H4)}. If $u$ is a mild solution of
\eqref{edp}  on $[-r,t_{\rm max}[$, then either
$t_{\rm max}=+\infty$ or $\limsup_{t\to t_{\rm max}} \|
u_t\|_{C_\alpha}=+\infty$.
\end{theorem}

 To prove this result, we need the following lemma.

\begin{lemma}[\cite{M-K}] \label{ineq}
Assume that {\rm (H1), (H3)} hold, and that there exist positive constants
$a,b,c$ such that, if $w\in C([-r,+\infty[;X_\alpha)$ is a solution
of
 \begin{equation}
\begin{gathered}
 Dw_t =  f(t)\quad\text{for } t\geq 0,\\
 w_0  =  \varphi , \quad \varphi\in C_\alpha,
 \end{gathered}
 \end{equation}
where $f$ is a continuous function from $[0,+\infty[$ to $X_\alpha$.
Then
\begin{equation}\label{lemma}
 \| w_t\|_{C_\alpha} \leq(a\|\varphi\|_{C_\alpha}
+\sup_{0\leq s\leq t}\| f(s)\|_\alpha)e^{ct}\quad\text{for } t\geq 0.
\end{equation}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{prolon}]
Assume that $t_{\rm max}<+\infty$ and $\limsup_{t\to t_{\rm max}} \|
u_t\|_{C_\alpha}<+\infty$.
 Let $R=\sup_{s\in[0,t_{\rm max}[ } \| F(s,u_s)\|$ and
$u:[t_0,t_{\rm max}[\to X_\alpha,\, t_0\in ]0,t_{\rm max}[$,
be the restriction of $u$ to $[t_0,t_{\rm max}[$.
Consider $t\in[t_0,t_{\rm max}[$ and $\beta$ such that
$\alpha<\beta<1$. Then
\begin{align*}
 \| D(u_t) \|_{\beta}
&\leq  \| A^{\beta-\alpha} T(t)A^\alpha D(\varphi) \|
  + \| \int^t_0 A^\beta T(t-s) F(s,u_s) ds \|\\
&\leq  M_{\beta-\alpha} e^{\omega t}
 t^{\alpha-\beta}\| D(\varphi) \|_\alpha+ M_\beta R
 \int^t_0 e^{\omega s}s^{-\beta} ds.
\end{align*}
 Thus, $\| D(u_t)\|_\beta $ is bounded on $[t_0,t_{\rm max}[$.
Now, for $t_0\leq t<t+h<t_{\rm max}$, we have
\begin{align*}
D(u_{t+h})-D(u_t)
&=  T(t+h)D\varphi- T(t)D\varphi+ \int^{t+h}_0 T(t+h-s)F(s,u_s) ds \\
&\quad - \int^t_0 T(t-s)F(s,u_s) ds\\
&=  T(t) [(T(h)-I)D\varphi] +(T(h)-I)\int^t_0 T(t-s)F(s,u_s) ds \\
&\quad + \int^{t+h}_t T(t+h-s)F(s,u_s) ds\\
&=  (T(h)-I)D(u_t)+ \int^{t+h}_t T(t+h-s)F(s,u_s) ds.
\end{align*}
We put, for $t\in[t_0,t_{\rm max}[$,
\[
 f(t)=(T(h)-I)D(u_t)+ \int^{t+h}_t T(t+h-s)F(s,u_s) ds.
\]
Using the estimate \eqref{lemma}, we obtain that
\[
 \| u_{t+h}-u_t\|_{C_\alpha} \leq (a\| u_h-u_0\|_{C_\alpha}
+b \sup_{0\leq s\leq t}\| f(s)\|_\alpha)e^{ct}.
\]
On the other hand and using \eqref{cccc}, for $t\in[t_0,t_{\rm max}[$,
we have
\begin{align*}
\|f(t)\|_\alpha
&\leq  \|(T(h)-I)A^{-(\beta-\alpha)} A^\beta D(u_t)\|
 + \|\int^{t+h}_t A^\alpha T(t+h-s)F(s,u_s) ds \|\\
&\leq N_{\beta-\alpha}h^{\beta-\alpha} \|D(u_t)\|_\beta
 +R M_\alpha\int^{t+h}_t e^{\omega(t+h-s)}(t+h-s)^{-\alpha} ds\\
 &\leq  N_{\beta-\alpha}h^{\beta-\alpha} \| D(u_t)\|_\beta
 + R M_\alpha \int^h_0 e^{\omega s} s^{-\alpha} ds\\
 &\leq  N_{\beta-\alpha}h^{\beta-\alpha} \| D(u_t)\|_\beta
 +R M_\alpha \max\{1,e^{\omega t_{\rm max}}\}\frac{ h^{1-\alpha
 }}{1-\alpha}\to 0 \quad \text{as } h\to 0.
\end{align*}
Since
$ \| u_h-u_0\|_{C_\alpha}\to 0$ as $h\to 0$,
\[
 \lim_{h\to 0} \| u_{t+h}-u_t\|_{C_\alpha}=0
\]
uniformly with respect to $t\in [t_0,t_{\rm max}[$.
Which implies that
$u$ is uniformly continuous and $\lim_{t\to t_{\rm max}}u(t)$ exists
in $X_\alpha$; the solution can be continued to the right to
$t_{\rm max}$, which contradicts the maximality of $[-r,t_{\rm max}[$.
This completes the proof of the theorem.
\end{proof}

The following result provides sufficient conditions for the existence
of global solutions to \eqref{edp}.

\begin{corollary} \label{coro1}
Under the same assumptions as in Theorem \ref{thm-exist}, if
there exists locally integrable functions $k_1$ and $k_2$ such
that $\| F(t,\varphi)\| \leq k_1(t) \|\varphi\|_{C_\alpha} +
k_2(t)$ for $\varphi \in C_\alpha$ and $t\geq 0$, then
\eqref{edp} admits global solutions.
\end{corollary}


\section{Example}

 Consider the  partial functional differential equation of neutral
type,
\begin{equation}\label{exple}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}[v(t,x)-av(t-r,x)]\\
&=\frac{\partial^2}{\partial x^2}[v(t,x)- av(t-r,x)]
+f(\frac{\partial}{\partial x}v(t-r,x))\quad \text{for } t\geq0 ,\;
 x\in[0,\pi],
\end{aligned}\\
v(t,0)=av(t-r,0),\quad  v(t,\pi)=av(t-r,\pi) \quad\text{for } t\geq0,\\
v(t,x)=v_0(t,x) \quad\text{for } -r\leq t\leq0,\; x\in[0,\pi].
\end{gathered}
\end{equation}
Where $a,r$ are positive constants, $f:\mathbb{R}\to\mathbb{R}$
and $v_0:[-r,0]\times[0,\pi]\to\mathbb{R}$ are continuous. Let
$X=\mathrm{L}^2([0,\pi];\mathbb{R})$ and $A:D(A)\subset X\to X$ be
defined by $Ay = -y''$ with domain $D(A)=H^2[0,\pi]\cap H^1_0
[0,\pi]$. Then
 $Ay=\sum^\infty _{n=1} n^2(y,e_n)e_n\quad \text{for }y\in D(A)$,
 where $\{e_n(s)=\sqrt{2/\pi} \sin ns, n\geq 1\}$,
is the orthonormal set of eigenvectors of $A$.
 For each $y\in D(A^{1/2}):=\{y\in X :
 \sum^\infty _{ n=1} n(y,e_n)e_n\,\in X \}$ the operator
 $A^{1/2}$ is given by $A^{1/2}y =\sum^\infty _{ n=1} n(y,e_n)e_n$.

 \begin{lemma}\cite{Tr-We1}
If $y\in D(A^{1/2})$, then $y$ is absolutely continuous,
$y'\in X$ and $\|y'\|_X= \|A^{1/2}y \|_X$.
\end{lemma}

 It is well known that $-A$ is the infinitesimal generator of
an analytic semigroup $(T(t))_{t\geq0}$ on $X$
given by $T(t)y = \sum^\infty _{n=1} e^{-n^2t}(y,e_n)e_n$, $y\in X$.
 It follows from this last expression that $(T(t))_{t\geq0}$ is a
compact semigroup on $X$
 (for any $t>0$, $T(t)$ is a Hilbert Schmidt operator).

Let $u(t)=v(t,.)$ for $t\geq 0$, $\varphi(\theta)=v_0(\theta,.)$
for $\theta \in [-r,0]$, $D:C_{1/2} \to X_{1/2}$
be defined by
 $$
D\varphi=\varphi(0)-a\varphi(-r)
=\varphi(0)-\int^0_{-r} d\eta(\theta)\varphi(\theta)\quad
 \text{for }\varphi \in C_{1/2},
$$
 with $\eta(\theta)=0$ for $-r<\theta\leq0$ and $\eta(-r)=aI$.
 Let  $F: C_{1/2} \longrightarrow X $ be given by
$$
(F(\varphi))(x)=f(\varphi(-r)'(x))\quad \text{for }
\varphi \in C_{1/2},\;  x\in [0,\pi].
$$
Then \eqref{exple} takes the  abstract form
 \begin{equation}\label{edpp}
\begin{gathered}
 \frac{d}{dt}Du_t =  -ADu_t+F(t, u_t)\quad\text{for } t\geq0, \\
 u_0 =  \varphi,\quad \varphi\in C_\alpha.
 \end{gathered}
\end{equation}

\begin{lemma} \label{lem1}
Operator $F$ is continuous from $ C_{1/2}$ to $X$.
\end{lemma}

\begin{proof}
Let $\varphi \in C_{1/2}$. We consider a sequence $(\varphi_n)_n$
convergent to $\varphi$ in $ C_{1/2}$. Then
\begin{align*}
\| A^{1/2}\varphi_n(-r)-A^{1/2}\varphi(-r)\|_X
&\leq \sup_{\theta\in[-r,0]}\| A^{1/2}\varphi_n(\theta)-A^{1/2}
 \varphi(\theta)\|_X\\
&= \|\varphi_n-\varphi\|_{C_\frac{1}{2}}\to 0\quad \text{as } n \to
+\infty.
\end{align*}
Then
\[
 \int^\pi_0|\frac{\partial}{\partial x}\varphi_n(-r)(x)
-\frac{\partial}{\partial x}\varphi(-r)(x)|^2 dx\to 0\quad
\text{as } n \to +\infty.
\]
This implies
\[
 \frac{\partial}{\partial x}\varphi_n(-r)\to
\frac{\partial}{\partial x}\varphi(-r)\quad \text{as } n\to\infty
\]
in $L^2[0,\pi]$. Consequently, there exists $(\varphi_{n_k})_k$,
$g\in L^2[0,\pi]$ such that
\[
 \frac{\partial}{\partial x}\varphi_{n_k}(-r)(x)\to
\frac{\partial}{\partial x}\varphi(-r)(x)\quad \text{a. e., as }
k\to\infty
\]
and
\[
 |\frac{\partial}{\partial x}\varphi_n(-r)(x)|
\leq |g(x)| \quad\text{a.e.}
\]
By the continuity of $f$,
\[
 f\Big(\frac{\partial}{\partial x}\varphi_{n_k}(-r)(x)\Big)\to
f\Big(\frac{\partial}{\partial x}\varphi(-r)(x)\Big) \quad
\text{as } k\to\infty.
\]
Assuming that $|f(t)|\leq b|t|+c$, by the Lebesgue's dominated
convergence theorem, we have
\[
 f\Big(\frac{\partial}{\partial x}\varphi_{n_k}(-r)\Big)\to
 f\Big(\frac{\partial}{\partial x}\varphi(-r)\Big) \quad
\text{as } k\to\infty
\]
in $L^2[0,\pi]$. Since the limit does not depend on the
subsequence $(\varphi_{n_k})_k$, then we obtain
\[
 F(\varphi_n)\to F(\varphi)
\]
in $L^2[0,\pi]$ as $n\to\infty$. We deduce that $F$ is continuous.
\end{proof}

Consequently, Theorem \ref{thm-exist} ensures the existence
of a maximal interval of existence $[-r,t_{\rm max}[$ and a mild
solution of  \eqref{exple}.

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\end{document}