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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 137, pp. 1--7.\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-2013/137\hfil Generalized Bohl-Perron principle]
{Generalized Bohl-Perron principle for differential equations
 with delay in a Banach spaces}

\author[M.  Gil' \hfil EJDE-2013/137\hfilneg]
{Michael  Gil'}  % in alphabetical order

\address{Michael  Gil' \newline
Department of Mathematics \\
Ben Gurion University of the Negev \\
P.0. Box 653, Beer-Sheva 84105, Israel}
\email{gilmi@bezeqint.net}


\thanks{Submitted February 27, 2013. Published June 20, 2013.}
\subjclass[2000]{34K30, 34K06, 34K20}
\keywords{Banach space; differential equation with delay;
linear  equation; \hfill\break\indent exponential stability}

\begin{abstract}
 We consider a linear homogeneous functional differential equation
 with delay in a Banach space.  It is proved that if the corresponding 
 non-homogeneous equation,  with an arbitrary free term bounded on 
 the positive half-line  and with the zero initial condition, 
 has a bounded solution,  then the considered homogeneous equation 
 is exponentially  stable. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction and statement of the main result}

Recall  that the Bohl-Perron principle states that the homogeneous ordinary 
differential equation (ODE)
$dy/dt=A(t)y$ $(t\ge 0)$ with a variable $n\times n$-matrix $A(t)$,
bounded on $[0,\infty)$  is exponentially stable, provided the
nonhomogeneous ODE $dx/dt=A(t)x+f(t)$ with the zero initial condition 
has a bounded solution for any bounded vector valued function $f$ \cite{Daleckii}.

In  \cite[Theorem 4.15]{halan},
the Bohl-Perron principle was generalized to a class of retarded systems 
with finite delays; also the asymptotic (not exponential) stability was proved. 
The result from \cite{halan} was a considerable development afterwards, cf. 
the book \cite{az} and  the very interesting papers \cite{ber1, ber2}, 
in which the generalized Bohl-Perron principle
was  effectively used for the stability analysis
of  the first and second order scalar equations.
In particular, in \cite{ber1} the scalar non-autonomous linear functional 
differential equation $\dot{x}(t)+a(t)x(h(t))=0$  is considered. 
The authors give sharp conditions for exponential stability, 
which are suitable in the case that the coefficient function $a(t)$ is periodic,
almost periodic or asymptotically almost periodic, as often encountered 
in applications.
In  \cite{ber2}, the authors provide sufficient conditions for the stability 
of rather general second-order delay differential equations.
In  \cite{gi11-del-perron,gi13} a result similar to
 the Bohl-Perron principle  has been  derived
 in  terms of the norm of the space
$L^p$, which is  called
the $L^p$-version of the generalized Bohl-Perron principle.

In this article, we  extend  the Bohl-Perron principle  to
a class of  functional differential
equations with delay in a Banach space. 
In Section 3 below, we show that our results
can be effectively used for the stability analysis.
As it is well-known, the basic method for the stability analysis 
of functional differential  equations is the direct Lyapunov
method. By  this method  very strong results are
obtained. But finding Lyapunov's type functionals for
nonautonomous vector equations with  delay is usually difficult.
In Section 3 we suggest explicit sharp stability conditions,
which supplement the well-known results
on stability of equations  with delay in a Banach space; see
\cite{Adimy, Arino, Ezzinbi,Matsui, Pao, Wu} and references given therein.

Let $X$ be a complex Banach  space with
a norm $\|\cdot\|_X$ and the unit operator  $I$.
Denote by $C(\omega)\equiv C (\omega, X)$
the space of  continuous  functions $u$
defined on a set $\omega\subseteq \mathbb{R}$  with values in $X$    
and the finite sup-norm $\|\cdot\|_{C(\omega)}$.
For a bounded linear operator $T$ acting from  $X$ into a normed space $Y$
we put
$\|T\|_{X\to Y}=\sup_{u\in X}\|Tu\|_{Y}/\|u\|_{X}$.

Let $A(t)$ be a linear generally unbounded operator in $X$ with a
constant dense domain $\operatorname{Dom}(A)$.
In $X$, for  a positive constant $\eta<\infty$ consider  the equation
\begin{equation}
\dot y(t)=A(t)y(t)+ \int_0^\eta  B(t, s)y(t-s) ds
 + \sum_1^m  B_k(t)y(t-h_k(t)),
\label{e1.1}
\end{equation}
where $\dot y(t)$ is a strong derivative of $y$;
$B_k(t)$ $(k=1, \dots, m)$ are  bounded  continuous operator
functions on $[0,\infty)$;
$B(t,s)$  is an operator function defined and bounded on
$[0,\infty)\times [0,\eta]$, which is
 continuous in $t$ and integrable in $s$;
 $0\le h_k(t)\le \eta$ are continuous functions.
Let   the initial condition be
\begin{equation}
y(t)=\phi (t)(  -\eta \leq t \le 0)
\label{e1.2}
\end{equation}
for a given $\phi\in C(-\eta, 0)\cap  \operatorname{Dom}(A)$.
For $w\in C(-\eta, \infty)$, put
$$
Ew=
\int_0^\eta  B(t, s)w(t-s) ds + \sum_1^m  B_k(t)w(t-h_k(t))\,.
$$
Then \eqref{e1.1} takes the form
\begin{equation}
\dot y(t)=A(t)y(t)+ Ey(t). \label{e1.3}
\end{equation}
It is assumed that $A(t)$ generates a strongly continuous evolution family
$\{U(t,s)\}$ $(t\ge s\ge 0)$ of bounded operators in $X$. That is, $U(t,s)$  is the evolution operator of the equation
\begin{equation}
\dot \zeta(t)=A(t)\zeta(t) \label{e1.4}
\end{equation}
cf. \cite{Chicone}. Following the Browder terminology \cite{Henry}, a continuous function
$y$ satisfying
\begin{equation}
y(t)=U(t,0)\phi(0)+\int_0^t  U(t,t_1)Ey(t_1)dt_1 \label{e1.5}
\end{equation}
and \eqref{e1.2} we will be called  a \emph{mild solution} to \eqref{e1.1}, 
\eqref{e1.2}.
Consider also the non-homogeneous equation
\begin{equation}
\dot x(t)=A(t)x(t)+Ex(t)+f(t),\quad t> 0 \label{e1.6}
\end{equation}
with a  given  function $f(t)\in C(0,\infty)$, and the zero initial condition
\begin{equation}
x(t)=0, \quad -\eta\le t \le 0. \label{e1.7}
\end{equation}
Then a continuous function
$x$ satisfying
\begin{equation}
x(t)=\int_0^t  U(t,t_1)(Ex(t_1)+f(t_1))dt_1
\label{e1.8}
\end{equation}
and  \eqref{e1.7}  will be called a mild solution to \eqref{e1.6}, \eqref{e1.7}.
Below we show that ,
problems \eqref{e1.1}, \eqref{e1.2} and \eqref{e1.6}, \eqref{e1.7}
have unique mild solutions.

We will say that \eqref{e1.1} is exponentially stable, if there are 
positive constants $M_1, \epsilon$, such that 
$\|y(t)\|\le M_1e^{-\epsilon t}\|\phi\|_{C(-\eta, 0)}$
$(t\ge  0)$ for any mild solution $y(t)$ of  \eqref{e1.1}, \eqref{e1.2}.


We assume that there are positive constants $\alpha_0$ and $M$,
such that
\begin{gather}
\|U(t,s)\|_X\le Me^{-\alpha_0 (t-s)}\quad \forall t\ge s\ge 0, \label{e1.9}\\
A(t)z\in C(0,\infty)\;\text{for any }z\in \operatorname{Dom}(A).
\label{e1.10}
\end{gather}

\begin{theorem}  \label{thm1.1}
If conditions \eqref{e1.9} and \eqref{e1.10} hold, and  
for any $f\in C(0,\infty)$, problem \eqref{e1.6}, \eqref{e1.7} 
has a  bounded  mild solution on $[0,\infty)$,
 then \eqref{e1.1} is exponentially  stable.
\end{theorem}

This theorem is proved in the next section.


Suppose $1\le p<\infty$, then  for an exponentially bounded and
strongly continuous evolution family $U(t, s)$ of bounded linear operators
acting in $X$, the following  condition is  equivalent to \eqref{e1.9}:
 there exists a constant $M_p > 0$, such that
\begin{equation}
\sup_{s\ge 0}
\int_s^\infty   \|U(t, s)z\|_X^pdt\le M_p\|z\|_X^p, \forall z\in X,
\label{e1.11}
\end{equation}
cf.  \cite[p. 75]{Chicone}. Other conditions equivalent to  \eqref{e1.9}
can be found in \cite[p. 77]{Chicone}.

\section{Proofs}


It is not difficult to check that for all $\tau>0$, 
\begin{equation}
\|Ew\|_{C(0, \tau)}\le v_0\|w\|_{C(-\eta,\tau)}\quad\text{for }w\in C(-\eta,\tau),
\label{e2.1}
\end{equation}
 where
$$
v_0=\sup_t \Big(\int_0^\eta  \|B(t, s)\|_X ds + \sum_1^m  \|B_k(t)\|_X\Big).
$$
For brevity, in this section, sometimes we  use
$\|\cdot\|_{C(0, \tau)}=|\cdot|_\tau$
for  $\tau>0$. Let us define the operator $V$ by
$$
Vw(t)=\int_0^t  U(t,t_1)(Ew)(t_1)dt_1
$$
for any integrable function $w(t)$ $(t\ge 0)$ with values in $X$.
According to \eqref{e1.9} and \eqref{e2.1} it is easy to check that
for any finite $T$ and
$u\in C(-\eta, T)$  with $u(t)=0$  for $t\le 0$,
$V$ satisfies
\begin{align*}
|V^k u|_{T}&\le Mv_0\int_0^T   |V^{k-1}u|_{t} dt\\
& \le(Mv_0)^2 \int_0^T \int_0^t  |V^{k-2}u|_{t_1}dt_1 dt\\
&\le \dots \le \frac{(TMv_0)^k}{k!}|u|_{T}.
\end{align*}
Hence, it follows that

\begin{corollary} \label{coro2.1}
 For any continuous $f$, problem \eqref{e1.6}, \eqref{e1.7}
has a unique mild solution $x(t)$, which can be represented
as
\begin{equation}
x=\sum_1^\infty  V^k f_1,\quad \text{where }
f_1(t)=\int_0^t  U(t,t_1)f(t_1)dt_1. \label{e2.2}
\end{equation}
\end{corollary}

\begin{lemma} \label{lem2.2}
Under condition \eqref{e1.10}, if for any $f\in C(0,\infty)$, 
problem \eqref{e1.6}, \eqref{e1.7} has a  bounded  
 mild solution on $[0,\infty)$, then for any 
 $\phi\in C(-\eta, 0)\cap  \operatorname{Dom}(A)$  problem  
\eqref{e1.1}, \eqref{e1.2}  has a unique mild solution  bounded 
on $(0,\infty)$.
\end{lemma}

\begin{proof}
 Put
$$
\hat \phi(t)=\begin{cases}
\phi(0)  &\text{if } t\ge 0, \\
\phi(t)  &\text{if }-\eta\le  t<  0\,.
\end{cases} 
$$
Then $d\hat \phi(t)/dt=0$ for $t\ge 0$. Consider the equation
$$
dx(t)/dt=A(t)(x(t) +\phi(0))+E(x(t)+\hat \phi(t ))\;(t>0),
$$
with condition \eqref{e1.7}. According to \eqref{e1.10} and 
\eqref{e2.1}, $A(t)\phi(0)+ E\hat \phi(t)\in  C(-\eta,\infty)$.
Due to the hypotheses of this lemma, the latter equation has
a solution $x\in C(0,\infty)$.
Then the function  $y(t)=x(t)+\hat \phi(t)\in C(-\eta,\infty)$ 
and satisfies problem \eqref{e1.1}, \eqref{e1.2}.
As claimed.
\end{proof}



\begin{proof}[Proof of Theorem \ref{thm1.1}]
Substituting
\begin{equation}
y(t)=y_\epsilon(t) e^{-\epsilon t} \label{e2.3}
\end{equation}
with an  $\epsilon>0$ in \eqref{e1.1},  we obtain
\begin{equation}
dy_\epsilon(t)/dt=(A(t)+\epsilon)y_\epsilon(t) +E_\epsilon y_\epsilon(t)\quad (t>0), \label{e2.4}
\end{equation}
where
$$
E_\epsilon w(t)=\int_0^\eta  B(t,s)e^{\epsilon s}w(t-s) ds +
\sum_1^m  e^{\epsilon h_k(t)}B_k(t)w(t-h_k(t))
$$
for a continuous $w$.
It is easy to check that $E_\epsilon\to E$ in the operator norm
of $C(0, \infty)$ as $\epsilon\to 0$.

Furthermore, due to \eqref{e2.2} we obtain
$x=\hat Gf$, where
$$
 \hat G:=(I-V)^{-1} W=\sum_1^\infty  V^kW, \quad
\text{with } Wf(t)= \int_0^t  U(t,t_1)f(t_1)dt_1.
$$
By the hypothesis of the theorem, we have
$$
x=\hat Gf\in C(0,\infty)\quad \text{for any }f\in C(0,\infty).
$$
So $\hat G$ is defined on the whole space $C(0,\infty)$.
It is closed, since problem \eqref{e1.6}, \eqref{e1.7} has a unique solution.
Therefore,  $\hat G$ is bounded  according to the Closed Graph theorem
\cite{Dunford}.

Consider now the equation
\begin{equation}
\dot x_\epsilon(t)=
(A(t)+\epsilon I)x_\epsilon(t) +E_\epsilon x_\epsilon(t)+f(t) \label{e2.5}
\end{equation}
with the zero initial condition.  Its mild solution is defined by
\begin{equation}
x_\epsilon(t)=\int_0^t  U(t,t_1)(\epsilon x_\epsilon(t_1)+ E_\epsilon x_\epsilon(t_1)dt_1)+f_1
\label{e2.6}
\end{equation}
where $f_1$ is defined as in  \eqref{e2.2}.
For solutions $x$ and $x_\epsilon$ of \eqref{e1.8} and \eqref{e2.6}, respectively, 
we obtain
\begin{align*}
x_\epsilon(t)-x(t)
&= \int_0^t  U(t,t_1)(\epsilon x_\epsilon(t_1)+ E_\epsilon x_\epsilon(t_1)-Ex(t_1))dt_1\\
&= V(x_\epsilon(t)-x(t))+f_\epsilon(t),
\end{align*}
where
$$
f_\epsilon(t)=\int_0^t  U(t,t_1)(\epsilon x_\epsilon(t_1)+ (E_\epsilon-E) x_\epsilon(t_1))dt_1.
$$
Consequently,
\begin{equation}
x-x_\epsilon=\hat G f_\epsilon. \label{e2.7}
\end{equation}
However  $|\hat G|_T\le \|\hat G\|_{C(0,\infty)}$, and
\begin{align*}
|(E_\epsilon-E) w|_T
&\le \sup_{t\ge 0} (\int_0^\eta  \|B(t,s)\|_X
|e^{\epsilon s}-1|\:\|w(t-s)\|_X ds \\
&\quad + \sum_1^m  |e^{\epsilon h_k(t)}-1|\|B_k(t)w(t-h_k(t))\|_X)\\
& \le v_0(e^{\epsilon \eta}-1)|w|_T.
\end{align*}
By \eqref{e1.9} and \eqref{e2.1}, $|V|_T\le Mv_0/\alpha_0$.
So $|f_\epsilon|_T\le  |x_\epsilon|_T (\epsilon + Mv_0\alpha_0^{-1}(e^{\epsilon \eta}-1))$
and
$$
|x_\epsilon|_T \le |x|_T+ \|\hat G\|_{C(0,\infty)}|x_\epsilon|_T
(\epsilon +Mv_0\alpha_0^{-1}(e^{\epsilon \eta}-1)).
$$
Thus, for a sufficiently small $\epsilon$,
$$
|x_\epsilon|_T \le \frac{|x|_T}{1-
\|\hat G\|(\epsilon +M\|\hat G\|_{C(0,\infty)}v_0\alpha_0^{-1}(e^{\epsilon \eta}-1)) }.
$$
Letting $T\to\infty$, we obtain $x_\epsilon\in C(0,\infty)$.
Hence, by  Lemma \ref{lem2.2}, a solution $y_\epsilon$ of \eqref{e2.4}
is bounded. Now \eqref{e2.3}  proves the exponential stability,
as claimed.
\end{proof}


\section{Equations in a Hilbert space}

In this section we illustrate Theorem \ref{thm1.1} in  a Hilbert space.
Let $X=H$ be a Hilbert space with a scalar product
$(.,.)$, and the norm $\|\cdot\|_H=\sqrt{(.,.)}$. Let $A(t)$ map 
$\operatorname{Dom}(A)$ into itself and
\begin{equation}
\sup_{z\in \operatorname{Dom}(A)} \frac{Re (A(t)z,z)}{(z,z)}
\le -\alpha(t)\le -\alpha_0\quad\forall t\ge 0, \label{e3.1}
\end{equation}
where $\alpha(t)$ is a positive continuous function and $\alpha_0$
is a positive constant.
From \eqref{e1.4} it follows that
$$
\frac d{dt}(\zeta (t), \zeta (t))=
(\dot \zeta (t), \zeta (t))+(\zeta (t), \dot \zeta (t))=
2 Re (\dot \zeta (t), \zeta (t))=2 Re (A(t)\zeta (t), \zeta (t)).
$$
Thus
$$
\frac d{dt}(\zeta (t), \zeta (t))=2 \|\zeta (t)\|_H \frac d{dt}\|\zeta (t)\|_H
\le -2\alpha(t)(\zeta (t), \zeta (t)),
$$
or
$$
\frac d{dt}\|\zeta (t)\|_H\le -\alpha(t)\|\zeta (t)\|_H.
$$
Solving this inequality with $\zeta (s)\in \operatorname{Dom}(A)$, we obtain
$$
\|U(t,s)\|_H\le e^{-\int_s^t \alpha (\tau) d\tau}\le  e^{-\alpha_0 (t-s)},\quad
\text{for }t\ge s\ge 0.
$$
Hence,
$$
\sup_t \int_0^t  \|U(t,t_1)\|_H dt_1\le J,
$$
where
$$
J:=\sup_t \int_0^t  e^{-\int_{t_1}^t \alpha (\tau) d\tau} dt_1.
$$
 From \eqref{e1.8} and \eqref{e2.1} it follows that
$$
\|x\|_{C(0,\infty)}\le v_0J\|x\|_{C(0,\infty)}+ \|f_1\|_{C(0,\infty)}.
$$
Consequently, if
\begin{equation}
v_0J<1, \label{e3.2}
\end{equation}
then
$$
\|x\|_{C(0,\infty)}\le \frac{\|f_1\|_{C(0,\infty)}}{1-v_0J}.
$$
Using Theorem \ref{thm1.1} we  arrive at the following result.

\begin{corollary} \label{coro3.1}
Suppose $E$ maps $\operatorname{Dom}(A)$
into itself, and conditions \eqref{e1.10},  \eqref{e3.1}, and  \eqref{e3.2} hold.
Then \eqref{e1.1} is exponentially  stable.
\end{corollary}

Some additional stability
criteria can be found, for instance, in \cite{Matsui,gi05-padun,gi00-padlihi}.
In particular, in  \cite{Matsui}, the authors prove important  results on 
the asymptotic behavior of solutions for semilinear autonomous functional 
differential equations with infinite delay.
In \cite{gi05-padun}, the authors considered equations with unbounded 
history response. Article \cite{gi00-padlihi}
is devoted to the stability of  linear time-variant functional differential 
equations in a Hilbert space.
The generalized Aizerman-Myshkis problem  for abstract
differential-delay equations is considered in  \cite{gi02-padaiz, gi03-patenaiz}.
A criterion for global stability of parabolic systems with delay
is suggested in \cite{gi98-pardesi}.



\begin{thebibliography}{99}

\bibitem{Adimy} M. Adimy, K. Ezzinbi; 
Local existence and linearized stability
for partial functional differential equations, \emph{Dynamic Systems and
Applications, 7}, (1998), 389-404.

\bibitem{Arino} O. Arino, M. L. Hbid, E. Ait Dads (eds);
\emph{Delay Differential Equations and Applications},
Springer, Dordrecht, 2006

\bibitem{az} N. V. Azbelev, P. M. Simonov;
 \emph{Stability of Differential Equations with Aftereffects},
Stability Control Theory Methods Appl. v. 20,  Taylor \& Francis, London, 2003.

\bibitem{ber1}  L. Berezansky, E. Braverman;
On exponential stability of a linear delay differential equation with 
an oscillating coefficient. \emph{Appl. Math. Lett. 22} (2009),
 no. 12, 1833-1837. 
 
\bibitem{ber2}  L. Berezansky,  E. Braverman, A. Domoshnitsky;
Stability of the second order delay differential equations with a damping term.
\emph{Differ. Equ. Dyn. Syst. 16} (2008), no. 3, 185-205.

\bibitem{Chicone}  C. Chicone, Latushkin Yu;
\emph{Evolution Semigrous in Dynamical Systems and Differential
Equations}, Amer. Math. Soc., Math. Surv. and Monographs, 70, 1999.

\bibitem{Daleckii} L. Yu, Daleckii, M. G. Krein;
\emph{Stability of Solutions of Differential Equations in Banach Space}, 
Amer. Math. Soc., Providence,  RI,  1971.

\bibitem{Dunford} N. Dunford, J.T. Schwartz;
 \emph{Linear Operators, part I},  Interscience Publishers, Inc., New York, 1966.

\bibitem{Ezzinbi} K. Ezzinbi, S. L. Rhali;
 Positivity and stability for some partial functional differential equations, 
 \emph{Nonlinear Differential Equations and Applications},  
 10, no. 1, (2003), 1-13.
 
\bibitem{gi98-pardesi} M. I.  Gil';
 On global stability of parabolic  systems with delay,
\emph{Applicable Analysis,  69},   (1998), 57-71.

\bibitem{gi00-padlihi} M. I.  Gil';
 Stability of  linear time-variant functional differential equations 
 in a Hilbert space, \emph{Funkcialaj Ekvacioj,   43}, (2000), 31-38.
 
\bibitem{gi02-padaiz} M. I.  Gil';
 On the generalized Aizerman-Myshkis problem  for retarded distributed 
 parameters systems,
 \emph{IMA Journal of Mat. Control and Information} {\bf 19} (2002),  1-8

\bibitem{gi03-patenaiz} M. I.  Gil';
 Generalized Aizerman - Myshkis problem  for abstract differential-delay 
 equations, \emph{Nonlinear Analysis, TMA,  55}, no. 6   (2003),  771-784.

\bibitem{gi05-padun} M. I.  Gil';
Stability of abstract nonlinear non-autonomous equations with unbounded 
history-response operators, \emph{Math. Analysis and Appl.},
308, no. 1 (2005), 140-158.

\bibitem{gi11-del-perron} M. I. Gil';
The $L^p$- version of the generalized Bohl - Perron principle for vector 
equations with delay, \emph{Int. J. Dynamical Systems  and
Differential Equations, 3}, no. 4 (2011) 448-458.

\bibitem{gi13} M. I. Gil';
\emph{Stability of Vector  Differential Delay Equations}, 
Birkh\"auser Verlag, Basel,  2013.

\bibitem{Henry} D. Henry;
\emph{Geometric Theory of Semilinear Parabolic Equations}.
Lecture Notes in Mathematics, No 840. Springer-Verlag, New York, 1981.

\bibitem{halan} A. Halanay;
 \emph{Differential Equations: Stability, Oscillation, Time Lags}, 
Academic Press, New York, 1966.

\bibitem{Matsui} Matsui Kazuyuki, Matsunaga Hideaki, Murakami Satoru;
Perron type theorems for functional differential equations with 
infinite delay in a Banach space.
\emph{Nonlinear Anal., Theory Methods Appl. 69,} no. 11A, (2008) 3821--3837.

\bibitem{Pao} C. V. Pao;
Parabolic systems in unbounded domains. II. Equations with
time delays, \emph{J. of Math. Anal. Appl.,  225}, (1998), 557-586

\bibitem{Wu}  J. Wu;
\emph{Theory and Applications of Partial Functional
Differential Equations}, Springer-Verlag, New York, 1996.

\end{thebibliography}

\end{document}