\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 52, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/52\hfil Existence and uniqueness  of  strong solutions]
{Existence and uniqueness  of  strong solutions
 to nonlinear nonlocal functional differential equations}

\author[S. Agarwal \& D.  Bahuguna\hfil EJDE-2004/52\hfilneg]
{Shruti Agarwal \& Dhirendra  Bahuguna}  % in alphabetical order

\address{Shruti Agarwal \hfill\break
Department of Mathematics \\
Indian Institute of Technology, Kanpur - 208 016, India} 
\email{shrutiag@iitk.ac.in}

\address{Dhirendra  Bahuguna \hfill\break
Department of Mathematics \\
Indian Institute of Technology, Kanpur - 208 016, India}
\email{dhiren@iitk.ac.in}

\date{}
\thanks{Submitted October 6, 2003. Published April 8, 2004.}
\subjclass[2000]{34K30, 34G20, 47H06}
\keywords{Nonlocal problem, accretive operator, strong solution, method of lines}


\begin{abstract}
  In the present work we consider  a nonlinear nonlocal functional
  differential equations in a real reflexive Banach space.
  We apply the method of lines  to establish the existence and
  uniqueness of a strong solution.  We consider also some
  applications of the abstract results.
\end{abstract}

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



\section{Introduction}

 Consider the  following nonlocal
nonlinear functional differential  equation in a real reflexive
Banach space $X$,
\begin{equation}
\begin{gathered}
u' (t) +Au(t) = f(t,u(t),u(b_1(t)),u(b_2(t)),\dots,u(b_m(t))), \quad t \in (0,T],\\
h(u)=\phi_0, \quad \mbox{on } [-\tau,0],
\end{gathered} \label{cp}
\end{equation}
where  $0<T<\infty$, $\phi_0 \in \mathcal{C}_0:=C([-\tau,0];X)$, the
nonlinear operator $A$ is single-valued and $m$-accretive defined
from the domain $D(A) \subset X$ into $X$, the nonlinear map $f$
is defined from $[0,T] \times X^{m+1}$ into $X$ and the map $h$ is
defined from  $\mathcal{C}_T:=C([-\tau,T];X)$ into $\mathcal{C}_T$. Here
$\mathcal{C}_t:=C([-\tau,t];X)$ for $t \in [0,T]$ is the Banach space
of all continuous functions from $[-\tau,t]$ into $X$ endowed with
the supremum norm
$$
\|\phi\|_t :=\sup_{-\tau \le \eta \le t}\|\phi(\eta)\|, \quad \phi
\in \mathcal{C}_t,
$$
where $\|.\|$ is the norm in $X$. The existence and uniqueness
results for (\ref{cp}) may also be applied to the particular case,
namely, the retarded functional differential equation,
\begin{equation} \begin{gathered}
u' (t) +Au(t) =
f(t,u(t),u(t-\tau_1),u(t-\tau_2),\dots,u(t-\tau_m)), \quad t \in (0,T], \\
u=\phi_0, \quad \mbox{on } [-\tau,0],
\end{gathered} \label{cptau}
\end{equation}
where $\tau_i \ge 0$, and $\tau=\max{\{\tau_1,\tau_2,\dots,\tau_m\}}$.

The study of the nonlocal functional differential equation of the
type $(\ref{cp})$ is motivated by the paper of Byszewski and Akca
\cite{ba}. In \cite{ba} the authors have considered the nonlocal
Cauchy problem,
\begin{equation} \begin{gathered}
u' (t) +Au(t) =
f(t,u(t),u(a_1(t)),u(a_2(t)),\dots,u(a_m(t))),\quad t \in (0,T], \\
u(0)+g(u)=u_0, \end{gathered}\label{cpu0}
\end{equation}
where $-A$ is the generator of a compact semigroup in $X$,
$g:C([0,T];X)$ into $X$, $u_0 \in X$ and $a_i:[0,T] \to
[0,T]$. Although, in this case we may take $h(u)(t)\equiv u(0)+
g(u)$ on $[-\tau,T]$, $\phi_0(t)\equiv u_0$ on $[-\tau,0]$ and
$b_i(t)= a_i(t)$, for $ t \in [0,T]$ to write it as (\ref{cp}),
but the analysis presented here will not be applicable to
(\ref{cpu0}). We consider here a Volterra type operator $h$ which
is assumed to satisfy $h(\phi_1)=h(\phi_2)$ on $[-\tau,0]$ for any
$\phi_1$ and $\phi_2$ in $\mathcal{C}_T$ with $\phi_1=\phi_2$ on
$[-\tau,0]$ (cf. (A3) stated below). This condition will not hold
in general for the operator $h(u)(t) \equiv u(0)+g(u)$. We shall
treat this case differently in our subsequent work.

For the earlier works on existence, uniqueness and stability of
various types of solutions of differential and functional
differential equations with nonlocal conditions, we refer to
Byszewski and Lakshmikantham \cite{bl}, Byszewski \cite{b1},
Balachandran and Chandrasekaran \cite{bc}, Lin and Liu \cite{ll}
and references cited in these papers.

Our aim is to extend the application of the method of lines to
(\ref{cp}). For the applications of the method of lines to
nonlinear evolution and nonlinear functional evolution equations,
we refer to Kartsatos and Parrott \cite{kp}, Kartsatos \cite{k1}
Bahuguna and Raghavendra \cite{br} and references cited in these
papers.


Let $\tilde{T}$ be any number such that $0< \tilde{T} \le T$. Any
function in ${\mathcal C}_T$ is also considered belonging to the
space ${\mathcal C}_{\tilde{T}}$ as its restriction  on the
subinterval $[-\tau,\tilde{T}],\; 0<\tilde{T}\leq T$. For any
$\phi \in \mathcal{C}_{\tilde{T}}$, we consider the problem,
\begin{equation} \begin{gathered}
u' (t) +Au(t) =
f(t,u(t),u(b_1(t)),u(b_2(t)),\dots,u(b_m(t))), \quad t \in (0,\tilde{T}], \\
u=\phi, \quad \mbox{on } [-\tau,0].
\end{gathered}\label{cpphi}
\end{equation}
Suppose that there is $\psi_0 \in \mathcal{C}_T$ such that
$h(\psi_0)=\phi_0$ on $[-\tau,0]$ and $\psi_0(0) \in D(A)$. Let
$\mathcal{W}(\psi_0,{\tilde{T}}):=\{\psi \in
\mathcal{C}_{\tilde{T}}: \; \psi=\psi_0, \; \mbox{on }
[-\tau,0]\}.$  For any $\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$
we prove the existence and uniqueness of a {\em strong solution}
$u$ of (\ref{cpphi}) under the same assumptions of Theorem
\ref{mr}, stated in the next section, in the sense that there
exists a unique function $u \in \mathcal{C}_{\tilde{T}}$ such that
 $u(t) \in D(A)$ for  a.e. $t \in [0,{\tilde{T}}]$, $u$ is
differentiable a.e.  on $[0,{\tilde{T}}]$ and
\begin{equation} \begin{gathered}
u' (t) +Au(t) =
f(t,u(t),u(b_1(t)),\dots,u(b_m(t))), \quad \mbox{a.e. }t \in [0,{\tilde{T}}], \\
u=\phi, \quad \mbox{on } [-\tau,0].
\end{gathered}\label{cpphiae}
\end{equation}
Let $u_\phi \in \mathcal{C}_{\tilde{T}}$ be the  strong solution of
(\ref{cpphi}) corresponding to $\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$.
It can be shown that
$u_\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$. We define a map $S$ from
$\mathcal{W}(\psi_0,{\tilde{T}})$  into $\mathcal{W}(\psi_0,{\tilde{T}})$ given
by
$$S\phi=u_\phi, \quad \phi \in \mathcal{W}(\psi_0,{\tilde{T}}).$$
We then prove that $S$ is constant on
$\mathcal{W}(\psi_0,{\tilde{T}})$ and hence there exists a unique
$\chi_0 \in \mathcal{W}(\psi_0,{\tilde{T}})$ such that
$\chi_0=S\chi_0=u_{\chi_0}$. We then show that $u_{\chi_0}$ is a
strong solution of (\ref{cp}). Also, we establish that a strong
solution $u \in \mathcal{W}(\psi_0,\tilde{T})$ of (\ref{cp}) can
be continued uniquely to either the whole interval $[-\tau,T]$ or
there is the maximal interval $[-\tau,t_{\rm max})$, $0<t_{\rm max} \le
T$, such that for every $0<\tilde{T} <t_{\rm max}$, $u \in
\mathcal{W}(\psi_0,\tilde{T})$ is a strong solution of (\ref{cp})
on $[-\tau,\tilde{T}]$ and in the later case either
$$\lim_{t \to t_{\rm max}-}\|u(t)\| =\infty,
$$
or $u(t)$ goes to the boundary of $D(A)$ as $t \to t_{\rm max}-$.
Finally, we show that $u$ is unique if and only if $\psi_0\in
\mathcal{C}_T$ satisfying $h(\psi_0)=\phi_0$ is unique up to
$[-\tau,0]$. We also consider some applications of the abstract
results.

\section{Preliminaries and Main Result}

 Let $X$ be a real Banach space
such that its dual $X^*$ is uniformly convex.
 One of the consequences of the fact that $X^*$ is
uniformly convex is that the duality map $F:X \to
2^{X^*}$, given by
$$F(x)=\{x^* \in
X^*:\langle x,x^*\rangle =\|x\|^2=\|x^*\|_*^2\},
$$
is single-valued and is continuous on bounded subsets of $X$. Here
$2^{X^*}$ denotes the power set of $X^*$, $\|.\|$ and $\|.\|_*$
are the norms of $X$ and $X^*$, respectively, $\langle x,x^*
\rangle $ is the value of $x^* \in X^*$ at $x \in X$. Further, we
assume the following conditions:
\begin{itemize}
\item[(A1)] The operator $A:D(A)\subset X \to X$ is $m$-accretive, i.e.,
$\langle Ax-Ay,F(x-y)\rangle \ge 0$, for all $x,y \in D(A)$ and $R(I+A)=X$,
where $R(.)$ is the range of an operator.

\item[(A2)] The nonlinear map $f:[0,T] \times X^{m+1} \to X$
satisfies a local Lipschitz-like condition
\begin{align*}
 &\|f(t,u_1,u_2,\dots u_{m+1})-f(s,v_1,v_2,\dots,v_{m+1})\| \\
 &\le  L_f (r)[|t-s|+ \sum_{i=1}^{m+1}\|u_i-v_i\|],
\end{align*}
for all $(u_1,u_2,\dots,u_{m+1})$, $(v_1,v_2,\dots,v_{m+1})$ in
$B_r(X^{m+1},(x_0,x_0,\dots,x_0))$ and $t,s \in [0,T]$ where
$L_f:\mathbb{R}_+ \to \mathbb{R}_+$ is a nondecreasing function
and for $x_0 \in X$ and $r>0$
$$
B_r(X^{m+1},(x_0,x_0,\dots,x_0))=\{(u_1,\dots,u_{m+1}) \in
X^{m+1}:\sum_{i=1}^{m+1}\|u_i-x_0\| \le r\}.
$$

\item[(A3)] The nonlinear map $h:\mathcal{C}_T \to \mathcal{C}_T$ is
continuous and for any $\phi_1$ and $\phi_2$ in $\mathcal{C}_T$ with
$\phi_1=\phi_2$ on $[-\tau,0]$, $h(\phi_1)=h(\phi_2)$ on
$[-\tau,0]$.

\item[(A4)] For $i=1,2,\dots,m$, the maps $b_i:[0,T] \to
[-\tau,T]$ are continuous and $b_i(t) \le t$ for $t \in [0,T]$.
\end{itemize}

\begin{theorem} \label{mr}
Suppose that the conditions (A1)-(A4) are satisfied
and  there exists $\psi_0 \in \mathcal{C}_T$ such that $h(\psi_0)
=\phi_0$ on $[-\tau,0]$ and $\psi_0(0) \in D(A)$. Then (\ref{cp})
has a strong solution $u$ on $[-\tau,{\tilde{T}}]$, for some $0<
{\tilde{T}} \le T$, in the sense that there exists
 a function $u \in \mathcal{C}_{\tilde{T}}$ such that
$u(t) \in D(A)$ for a.e. $t \in [0,{\tilde{T}}]$, $u$ is
differentiable a.e. on $[0,{\tilde{T}}]$ and
\begin{equation} \begin{gathered}
u' (t) +Au(t) =
f(t,u(t),u(b_1(t)),\dots,u(b_m(t))), \quad \mbox{a.e. } t \in [0,{\tilde{T}}], \\
h(u)=\phi_0, \quad \mbox{on  } [-\tau,0].
\end{gathered}\label{cplim}
\end{equation}
Also, $u$ is unique in $\mathcal{W}(\psi_0,\tilde{T})$ and $u$ is
Lipschitz continuous on $[0,{\tilde{T}}]$. Furthermore, $u$ can be
continued uniquely either on the whole interval $[-\tau,T]$ or
there exists a maximal interval $[0,t_{\rm max})$, $0< t_{\rm max} \le T$,
such that $u$ is a strong solution of (\ref{cp}) on every
subinterval $[-\tau, {\tilde{T}}]$, $0< {\tilde{T}}<t_{\rm max}$. A
strong solution $u$ of (\ref{cp}) is unique on the interval of
existence if and only if $\psi_0 \in \mathcal{C}_T$ satisfying
$h(\psi_0)=\phi_0$ on $[-\tau,0]$ is unique up to $[-\tau,0]$.
 \end{theorem}

\section{Discretization Scheme and A Priori Estimates}

In this section we establish the existence and uniqueness of a
strong solution to (\ref{cpphi}) for a given
$\phi \in \mathcal{W}(\psi_0,{{T}})$. Let $\phi \in \mathcal{W}(\psi_0,{{T}})$.
Then $x_0:=\phi(0)=\psi_0(0) \in D(A)$.
 For the application of
the method of lines to (\ref{cpphi}), we proceed as follows. We
fix $R>0$ and let $R_0:=R+\sup_{t \in [-\tau,T]}\|\phi(t)-x_0\|$.
We choose $t_{0}$ such that
\begin{gather*}
0<t_{0} \le T, \\
t_{0}[\|Ax_0\| +3L_f(R_0)(T+(m+1)R_0)+\|f(0,x_0,x_0,\dots,x_0)\|] \le R.
\end{gather*}
 For $n \in \mathbb{N}$, let $h_{n}=t_{0}/n$. We set $u^{n}_0=x_0$ for all
$n \in \mathbb{N}$ and define each of $\{u^{n}_j \}_{j=1}^n$ as the
unique solution of the equation
\begin{equation}
{u-u^{n}_{j-1} \over
h_n}+Au=f(t^{n}_j,u^{n}_{j-1},\tilde{u}^{n}_{j-1}(b_1(t^{n}_j)),
\dots,\tilde{u}^{n}_{j-1}(b_m(t^{n}_j))), \label{scheme}
\end{equation}
where $\tilde{u}^n_0(t)=\phi(t)$ for $t \in [-\tau,0]$,
$\tilde{u}^n_0(t)=x_0$ for $t \in [0,t_0]$ and for $2 \le j \le n$,
\begin{equation}
\tilde{u}^{n}_{j-1}(t)= \begin{cases} \phi(t), & t \in [-\tau,0], \\
u^{n}_{i-1} +{1 \over h_n} (t-t^{n}_{i-1})(u^{n}_{i}-u^{n}_{i-1}),
& t \in [t^{n}_{i-1},t^n_i],\\
& i=1,2,\dots,j-1, \\
u^{n}_{j-1}, & t \in [t^n_{j-1},t_0]. \end{cases}
\label{fun}
\end{equation}
 The existence of a unique $u^n_j \in D(A)$ satisfying (\ref{scheme})
 is a consequence of the $m$-accretivity of $A$.
Using (A2) we first prove that the points $\{u^n_j\}_{j=0}^n$ lie
in a ball with its radius independent of the discretization
parameters $j$, $h_n$ and $n$. We then prove {\em a priori}
estimates on the difference quotients
$\{(u^{n}_j-u^{n}_{j-1})/h_n\}$ using (A2). We  define the
sequence $\{U^{n}\} \subset \mathcal{C}_{t_0}$ of polygonal functions
\begin{equation}
U^{n}(t) = \begin{cases} \phi(t), & t \in [-\tau,0], \\
u^{n}_{j-1} + {1 \over h_n}(t-t^n_{j-1})(u^{n}_j-u^{n}_{j-1}), & t
\in (t^n_{j-1},t^n_{j}], \end{cases}
\end{equation}
and prove the convergence of $\{U^{n}\}$  to a unique strong
solution $u$ of (\ref{cpphi}) in $\mathcal{C}_{t_0}$ as $n
\to \infty$.

Now, we  show that $\{u^{n}_j\}_{j=0}^n$ lie in a ball in $X$
of radius independent of $j$, $h_n$ and $n$.

\begin{lemma} \label{lem0} For $n \in \mathbb{N}$, $j=1,2,\dots,n,$
$$\|u^{n}_j-x_{0}\| \le R.$$
\end{lemma}

\begin{proof} From (\ref{scheme}) for $j=1$ and the
accretivity of $A$, we have
\[
 \|u^{n}_1 -x_{0}\|  \le   h_n[\|Ax_0\|+3L_f(R_0)(T+(m+1)R_0)
 +\|f(0,x_0,x_0,\dots,x_0)\|] \le R.
\]
Assume that $\|u^n_i-x_0\| \le R$ for $i=1,2,\dots,j-1$.
 Now, for $2 \le j \le n$,
\begin{align*}
 \|u^{n}_j -x_{0}\|  & \le   \|u^n_{j-1}-x_0\| + h_n[\|Ax_0\|+3L_f(R_0)(T+(m+1)R_0) \\
&\quad +\|f(0,x_0,x_0,\dots,x_0)\|].
\end{align*}
Repeating the above inequality, we obtain
\begin{align*}
\|u^{n}_j -x_{0}\| & \le  jh_n[\|Ax_0\|+3L_f(R_0)(T+(m+1)R_0) \\
&\quad + \|f(0,x_0,x_0,\dots,x_0)\|] \le R,
\end{align*}
 as $jh_n \le t_0$ for $0 \le j \le n$. This completes the proof
of the lemma.
\end{proof}

 Now, we  establish  {\em a priori} estimates for the
difference quotients  $\{{u^{n}_j-u^{n}_{j-1} \over h_n}\}$.

\begin{lemma} \label{lem1}
There exists a positive constant $K$ independent
of the discretization parameters $n$, $j$ and $h_n$ such that
$$
\big\|{u^{n}_j-u^{n}_{j-1} \over h_n}\big\| \le K,
\quad j=1,2,\dots,n, \; n=1,2,\dots.
$$ \end{lemma}

\begin{proof} In this proof and subsequently, $K$ will
represent a generic constant independent of $j$, $h_n$ and $n$.
Subtracting $Au^{n}_0=Ax_0$ from both the sides in (\ref{scheme})
and applying $F(u^{n}_1-u^{n}_0)$, using accretivity of $A$, we
get
$$
\big\|{u^{n}_1-u^{n}_0 \over h_n} \big\|
\le \|Ax_0\|+ \|f(0,x_0,x_0,\dots,x_0)\|+3L_f(R_0)(T+(m+1)R_0) \le K.
$$
Now, for $2 \le j \le n$ applying $F(u^{n}_{j}-u^{n}_{j-1})$ to
(\ref{scheme}) and using accretivity of $A$, we get
\begin{align*}
 \big\|{u^{n}_j -u^{n}_{j-1} \over h_n} \big\|
&\le \big\|{u^{n}_{j-1} -u^{n}_{j-2} \over h_n} \big\|
+\|f(t^n_j,u^{n}_{j-1},\tilde{u}^{n}_{j-1}(b_1(t^n_j)),\dots,\tilde{u}^{n}_{j-1}(b_m(t^n_j)))
\\
&\quad - f(t^n_{j-1},u^{n}_{j-2},\tilde{u}^{n}_{j-2}(b_1(t^n_{j-1})),\dots,
\tilde{u}^{n}_{j-2}(b_m(t^n_{j-1})))\|.
\end{align*}
 From the above inequality we get
$$
\big\|{u^{n}_j -u^{n}_{j-1} \over h_n} \big\|
\le (1+Ch_n) \big\|{u^{n}_{j-1} -u^{n}_{j-2} \over h_n}\big\| + Ch_n,
$$
where $C$ is a positive constant independent of
$j$, $h_n$ and $n$. Repeating the above inequality, we get
$$
\big\|{u^{n}_j -u^{n}_{j-1} \over h_n} \big\| \le (1+Ch_n)^j. C_1
\le C_1 e^{TC} \le K.$$ This completes the proof of the lemma.
\end{proof}

We introduce another sequence $\{X^{n}\}$ of step functions from
$[-h_n,t_0]$ into $X$ by
$$
X^{n}(t)=\begin{cases} x_0, & t \in [-h_n,0], \\
 u^{n}_{j}, & t \in (t^n_{j-1},t^n_j].
  \end{cases}$$

 \begin{remark} \label{rem1} \rm
 From Lemma \ref{lem1} it follows that the functions
 $U^{n}$ and $\tilde{u}^{n}_r$, $0 \le r \le n-1$, are Lipschitz continuous
 on $[0,t_0]$ with a uniform Lipschitz constant $K$. The
 sequence $U^{n}(t)-X^{n}(t) \to 0$ in $X$ as $n
 \to \infty$ uniformly on $[0,t_0]$. Furthermore, $X^{n}(t) \in D(A)$ for
 $t \in [0,t_0]$ and  the
 sequences $\{U^{n}(t)\}$ and $\{X^{n}(t)\}$
 are  bounded in $X$, uniformly in $n \in \mathbb{N}$ and $t \in
 [0,t_0]$. The sequence $\{AX^n(t)\}$ is  bounded uniformly in $n \in
 \mathbb{N}$
 and $t \in [0,t_0]$.
\end{remark}

For notational convenience, let
$$
f^{n}(t)=f(t^n_j,u^{n}_{j-1},\tilde{u}^{n}_{j-1}(b_1(t^n_j)),\dots,
\tilde{u}^{n}_{j-1}(b_m(t^n_j))), $$
$t \in (t^n_{j-1},t^n_j]$, $1 \le j \le n$.
Then (\ref{scheme}) may be rewritten as
\begin{equation}
{d^- \over dt} U^{n}(t)+AX^{n}(t)=f^{n}(t), \quad t \in (0,t_0],
\label{apde}
\end{equation}
where ${d^- \over dt}$ denotes the left derivative in $(0,t_0]$.
Also, for $t \in (0,t_0]$,  we have
\begin{equation}\int_0^tAX^{n}(s)\,ds=x_0-U^{n}(t)+\int_0^tf^{n}(s)\,ds.
\label{int}
\end{equation}

\begin{lemma} \label{lem2} There exists $u \in \mathcal{C}_{t_0}$ such that
$U^{n} \to u$ in $\mathcal{C}_{t_0}$ as $n \to
\infty$. Moreover, $u$ is Lipschitz continuous on $[0,t_0]$.
\end{lemma}

\begin{proof} From (\ref{apde}) for $t \in (0,t_0]$,  we have
\[
\big\langle {d^- \over dt}(U^{n}(t)-U^{k}(t)),F(X^{n}(t)-X^{k}(t))\big\rangle
\le \big\langle f^{n}(t)-f^{k}(t),F(X^{n}(t)-X^{k}(t)\big\rangle .
\]
 From the above inequality, we obtain
\begin{align*}
&{1\over 2} {d^- \over dt}\|U^{n}(t)-U^{k}(t)\|^2 \\
&\le \big\langle {d^- \over dt}(U^{n}(t)-U^{k}(t))-f^{n}(t)+f^{k}(t),
 F(U^{n}(t)-U^{k}(t))-F(X^{n}(t)-X^{k}(t))\big\rangle  \\
&\quad + \big\langle f^{n}(t)-f^{k}(t),F(U^{n}(t)-U^{k}(t))\big\rangle .
\end{align*}
Now,
$$\|f^{n}(t)-f^{k}(t)\| \le \epsilon_{nk}(t)+K \|U^{n}-U^{k}\|_t,
$$
where
\begin{align*}
 \epsilon_{nk}(t)&= K[|t^n_j-t^k_l|+(h_n+h_k) +
\|X^{n}(t-h_n)-U^{n}(t)\| +\|X^{k}(t-h_k)-U^{n}(t)\| \\
&\quad +\sum_{i=1}^{m}(|b_i(t_j^n)-b_i(t)|+|b_i(t_l^k)-b_i(t)|),
\end{align*}
for $t \in (t^n_{j-1},t^n_j]$ and $t \in (t^k_{l-1},t^k_l]$, $1
\le j \le n$, $1 \le l \le k$. Therefore, $\epsilon_{nk}(t)
\to 0$ as $n,k \to \infty$ uniformly on $[0,t_0]$.
This implies that for a.e. $t \in [0,t_0],$
$$
{d^- \over dt}\|U^n(t)-U^k(t)\|^2 \le
K[\epsilon_{nk}^1+\|U^n -U^k\|^2_t],$$
where $\epsilon_{nk}^1$ is
a sequence of numbers such that $\epsilon_{nk}^1 \to 0$ as
$n,k \to \infty$. Integrating the above inequality over
$(0,s)$, $0 <s \le t \le t_0$, taking the supremum over $(0,t)$
and using the fact that $U^n=\phi$ on $[-\tau,0]$ for all $n$, we
get
$$
\|U^n-U^k\|^2_t \le K[T \epsilon_{nk}^1 + \int_0^t \|U^n-U^k\|^2_s \; ds ].
$$
Applying Gronwall's inequality  we conclude that there exists
$u \in \mathcal{C}_{t_0}$ such that $U^n \to u$ in $\mathcal{C}_{t_0}$.
Clearly, $u=\phi$ on $[-\tau,0]$ and from Remark \ref{rem1} it
follows that $u$ is Lipschitz continuous on $[0,t_0]$. This
completes the proof of the lemma.
\end{proof}

\begin{proof}[Proof of Theorem \ref{mr}]
 First, we prove the existence on $[-\tau,t_0]$ and then prove the unique
 continuation
of the solution on $[-\tau,T]$. Proceeding similarly as in
\cite{br1}, we may show that $u(t) \in D(A)$ for $t \in [0,t_0]$,
$AX^n(t) \rightharpoonup Au(t)$ on $[0,t_0]$ and $Au(t)$ is weakly
continuous on $[0,t_0]$. Here $\rightharpoonup$ denotes the weak
convergence in $X$. For every $x^* \in X^*$ and $t \in (0,t_0]$,
we have
$$\int_0^t\langle AX^n(s),x^*\rangle
\,ds=\langle x_0,x^*\rangle -\langle U^n(t),x^*\rangle +\int_0^t<f^n(s),x^*\rangle \,ds.
$$
Using Lemma \ref{lem2} and the bounded convergence theorem, we obtain as
$n \to \infty$,
\begin{equation} \begin{aligned}
\int_0^t \langle Au(s),x^*\rangle \,ds
&=\langle x_0,x^*\rangle -\langle u(t),x^*\rangle \\
& \quad + \int_0^t\langle f(s,u(s),u(b_1(s)),\dots,u(b_m(s))),x^*\rangle \; ds.
\end{aligned}\label{inteq}
\end{equation}
Since $Au(t)$ is Bochner integrable (cf. \cite{br1}) on $[0,t_0]$,
from (\ref{inteq}) we get
\begin{equation}
{d \over dt}u(t)+Au(t)=f(t,u(t),u(b_1(t)),\dots,u(b_m(t))), \quad
\mbox{a.e.  } t \in [0,t_0]. \label{strong}
\end{equation}
 Clearly, $u$ is Lipschitz continuous on $[0,t_0]$ and $u(t) \in D(A)$ for $t \in [0,t_0]$.
 Now we prove the uniqueness
of a  function $u \in \mathcal{C}_{t_0}$ which is differentiable a.e.
on $[0,t_0]$ with $u(t) \in D(A)$ a.e. on $[0,t_0]$ and $u=\phi$
on $[-\tau,0]$ satisfying (\ref{strong}). Let $u_1,u_2 \in \mathcal{C}_{t_0}$ be two such functions. Let
$R=\max{\{\|u_1\|_{t_0},\|u_2\|_{t_0}\}}$. Then for
$u=u_1-u_2$, we have
$${d \over dt}\|u(t)\|^2 \le C_1(R) \|u\|^2_t, \quad \mbox{a.e. } t \in [0,t_0],
$$
where $C_1:\mathbb{R}_+\to \mathbb{R}_+$ is a nondecreasing function.
Integrating over $(0,s)$ for $0<s \le t \le t_0$, taking supremum over $(0,t)$
and using the fact that $u \equiv 0$ on $[-\tau,0]$, we get
$$
\|u\|^2_t  \le C_1(R) \int_0^t \|u\|^2_s \; ds.$$
Application of Gronwall's inequality implies that $u \equiv 0$ on $[-\tau,t_0]$.

Now, we prove the unique continuation of the solution $u$ on
$[-\tau,T]$. Suppose $t_0 < T$ and consider the  problem
\begin{equation} \begin{gathered}
w' (t) +Aw(t) =
\tilde{f}(t,w(t),w(\tilde{b}_1(t)),w(\tilde{b}_2(t)),\dots,w(\tilde{b}_m(t))),\quad
 0< t \le T-t_0, \\
w= \tilde{\phi}_0, \quad \mbox{on } [-\tau-t_0,0], \label{cont}
\end{gathered}
\end{equation}
where
$\tilde{f}(t,u_1,u_2,\dots,u_{m+1})=f(t+t_0,u_1,u_2,\dots,u_{m+1})$,
$0\le t \le T-t_0$,
\[
\tilde{\phi_0}(t)= \begin{cases} \phi(t+t_0), &  t \in [-\tau-t_0,-t_0], \\
u(t+t_0), & t \in [-t_0,0],  \end{cases}
\]
$\tilde{b}_i(t)=b_i(t+t_0)-t_0$, $t \in [0,T-t_0]$ $i=1,2,\dots,m$.

 Since $\tilde{\phi}_0(0)=u(t_0) \in D(A)$ and $\tilde{f}$
satisfies (A2) and $\tilde{b}_i$, $i=1,2,\dots,m$ satisfy (A4) on
$[0,T-t_0]$,  we may proceed as before and prove the existence of
a unique $w \in C([-\tau-t_0,t_1];X)$, $0<t_1\le T-t_0$, such that
$w$ is Lipschitz continuous on $[0,t_1]$, $w(t) \in D(A)$ for $t
\in [0,t_1]$ and $w$ satisfies
\begin{equation} \begin{gathered}
w' (t) +Aw(t) =
\tilde{f}(t,w(t),w(\tilde{b}_1(t)),w(\tilde{b}_2(t)),\dots,w(\tilde{b}_m(t))),\quad
\mbox{a.e. }  t \in [0,t_1], \\
w= \tilde{\phi}_0, \quad \mbox{on } [-\tau-t_0,0]. \label{cont+}
\end{gathered}
\end{equation}
 Then the function
$$\bar{u}(t) = \begin{cases} u(t), & t \in [-\tau,t_0], \\
w(t-t_0), & t \in [t_0,t_0+t_1],
\end{cases}$$
 is Lipschitz  continuous on $[0,t_0+t_1]$, $\bar{u}(t) \in D(A)$ for
$t \in [0,t_0+t_1]$ and satisfies (\ref{cpphiae}) a.e. on
$[0,t_0+t_1]$. Continuing this way we may prove the existence on
the whole interval $[-\tau,T]$ or there is the maximal interval
$[-\tau,t_{\rm max}),\;0<t_{\rm max}\leq T$, such that $u$ is a strong
solution of (\ref{cp}) on every subinterval $[-\tau,{\tilde{T}}]$,
$0<{\tilde{T}}<t_{\rm max}$. In the later case, if $\lim_{t\to
t_{max^-}}\|u(t)\|< \infty$ and $\lim_{t \to t_{\rm max}-}u(t)
\in D(A)$, then we may continue the solution beyond $t_{\rm max}$ but
this will contradict the definition of maximal interval of
existence. Therefore, either $\lim_{t\to
t_{max^-}}\|u(t)\|=\infty$ or $u(t)$ goes to the boundary of
$D(A)$ as $t\to t_{max^-}$.

Thus, for each $\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$, we have
proved the existence and uniqueness of a strong solution of
(\ref{cpphi}).

Now, let  $u_\phi$ be the  strong solution of (\ref{cpphi})
corresponding to $\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$. Since
$u_\phi= \phi$ on $[-\tau,0]$, it follows that
$u_\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$. We define a map
$S:\mathcal{W}(\psi_0,{\tilde{T}}) \to \mathcal{W}(\psi_0,{\tilde{T}})$
given by $S\phi = u_\phi$ for $\phi \in \mathcal{W}(\psi_0,{\tilde{T}})$.
Using similar arguments as used above in
the proof of uniqueness and the fact that $u_\phi=u_\psi=\psi_0$
on $[-\tau,0]$, we obtain
$$\|S\phi-S\psi\|_t^2 =\|u_\phi-u_\psi\|_t^2 \le C_2(R^{\phi \psi})
\int_0^t \|u_\phi-u_\psi\|^2_s \; ds,$$ where $R^{\phi
\psi}=\max{\{\|u_\phi\|_{\tilde{T}},\|u_\psi\|_{\tilde{T}} \}}$
and $C_2:\mathbb{R}_+ \to \mathbb{R}_+$ is a nondecreasing
function. Applying Gronwall's inequality we obtain that $S$ is
constant on $\mathcal{W}(\psi_0,{\tilde{T}})$ and therefore there
exists a unique $\chi_0 \in \mathcal{W}(\psi_0,{\tilde{T}})$ such
that $S\chi_0=\chi_0=u_{\chi_0}.$ It is easy to verify that
$u_{\chi_0}$ $(=\chi_0)$ is a strong solution to (\ref{cp}).
Clearly, if $\psi_0 \in \mathcal{C}_{{T}}$ satisfying
$h(\psi_0)=\phi_0$ on $[-\tau,0]$ is unique up to $[-\tau,0]$ then
$u$ is unique. If there are two $\psi_0$ and $\tilde{\psi}_0$ in
$\mathcal{C}_{{T}}$ satisfying
$h(\psi_0)=h(\tilde{\psi}_0)=\phi_0$ on $[-\tau,0]$, with $\psi_0
\neq \tilde{\psi}_0$ on $[-\tau,0]$, then
$\mathcal{W}(\psi_0,{\tilde{T}}) \cap
\mathcal{W}(\tilde{\psi}_0,{\tilde{T}}) =\emptyset$ and hence the
solutions $u$ and $\tilde{u}$ of (\ref{cp}) belonging to
$\mathcal{W}(\psi_0,{\tilde{T}})$ and
$\mathcal{W}(\tilde{\psi}_0,{\tilde{T}})$, respectively, are
different. This completes the proof of Theorem \ref{mr}.
\end{proof}

\section{Applications}

Theorem \ref{mr} may be applied to get the existence and
uniqueness results for (\ref{cp}) in the case when the operator
$A$, with the domain $D(A)=H^{2m}(\Omega)\cap H^{m}_0(\Omega)$
into $X:=L^2(\Omega)$, is associated with the nonlinear partial
differential operator
$$Au=\sum_{|\alpha|\le m} (-1)^{|\alpha|}
D^\alpha A_\alpha(x,u(x),Du,\dots,D^\alpha u),
$$
in a bounded domain
$\Omega$ in $\mathbb{R}^n$ with sufficiently smooth boundary
$\partial \Omega$, where $A_\alpha(x,\xi)$ are real functions
defined on $\Omega \times \mathbb{R}^N$ for some $N \in
\mathbb{N}$ and satisfying Caratheodory condition of measurability
and certain growth conditions (cf. Barbu \cite{vb} page 48).

In (\ref{cp}), we may take $f$ as the function $f: [0,T] \times
(L^2(\Omega))^{m+1} \to L^2(\Omega)$, given by
$$f(t,u_1,u_2,\dots,u_{m+1})=f_0(t)+a(t)
\sum_{i=1}^{m+1}\|u_i\|_{L^2(\Omega)} u_i,$$
where $f_0:[0,T] \to L^2(\Omega)$, and $a:[0,T]
\to \mathbb{R}$ are Lipschitz continuous functions on
$[0,T]$ and $\|.\|_{L^2(\Omega)}$ denotes the norm in
$L^2(\Omega)$. For the functions $b_i$, $i=1,2,\dots,n$ and $h$
we may have any of the following.

\noindent (b1) Let $\tau_i \ge 0$. For $i=1,2,\dots,m$, let
$b_i(t)=t-\tau_i$, $t \in [0,T]$.

\noindent (b2) Let $\tau_i$, $i=1,2,\dots,m$ be such that
$0<\tau_i <T$. For $t \in [0,T]$, let
$$
b_i(t) =\begin{cases} 0, & t \le \tau_i, \\
t-\tau_i, & t > \tau_i.
\end{cases}
$$

\noindent (b3) For $i=1,2,\dots,m$, let $b_i(t)=k_i t$, $t \in
[0,T]$, $0 <k_i
\le 1$.

\noindent (b4) Let $N \in \mathbb{N}$. Let $0 <k_i \le 1/(NT^N)$,
$i=1,2,\dots,m$. For $i=1,2,\dots,m$, let
$$b_i(t)=k_i  t^N, \quad t \in [0,T].$$

 Let $-\tau \le a_1<a_2 < \dots < a_r\le 0$, $c_i$
with $C:=\sum_{i=1}^rc_i \neq 0$ and $\epsilon_i>0$, for
$i=1,\dots,r$. Let $x \in D(A)$. Consider the conditions:

\begin{itemize}
\item[(h1)] $g_1(\chi):=\int_{-\tau}^{0} k(\theta) \chi (\theta) d \theta =x$
for  $\chi \in C([-\tau,0];X)$, where $k$ is in $L^1(-\tau,0)$
with $\kappa:=\int_{-\tau}^{0}k(s)ds \neq 0$

\item[(h2)] $g_2(\chi):=\sum_{i=1}^rc_i\chi(a_i)=x$ for
$\chi \in C([-\tau,0];X)$;

\item[(h3)] $g_3(\chi):=\sum_{i=1}^r {c_i \over \epsilon_i}
\int_{a_i-\epsilon_i}^{a_i} \chi(s) ds=x$ for  $\chi \in C([-\tau,0];X)$.
\end{itemize}
Clearly, $g_i:C([-\tau,0];X) \to X$, $i=1,2,3$. For
$i=1,2,3$, define $h_i(\psi)(t)\equiv g_i(\psi|_{[-\tau,0]})$ on
$[-\tau,T]$ for $\psi \in C([-\tau,T];X)$ where
$\psi|_{[-\tau,0]}$ is the restriction of $\psi$ on $[-\tau,0]$.
Let $\phi_0(t)\equiv x$ on $[-\tau,0].$ Then conditions (h1), (h2)
and (h3) are equivalent to $h_i(\psi)=\phi_0$ on $[-\tau,0]$,
$i=1,2,3$, respectively. For (h1), we may take $\psi_0(t)\equiv
x/\kappa$ and for (h2) as well as for (h3), we may take
$\psi_0(t)\equiv x/C$ on $[-\tau,T]$.

 \subsection*{Acknowledgements} The authors would like to
thank the National Board for Higher Mathematics for providing the
financial support to carry out this work under its research
project No. NBHM/2001/R\&D-II.

\begin{thebibliography}{99}
\bibitem{br} D. Bahuguna and V. Raghavendra, Application of
Rothe's method to nonlinear evolution equations in Hilbert spaces,
{\em Nonlinear Anal.}, 23 (1994), 75-81.

\bibitem{br1} D. Bahuguna and V. Raghavendra, Application of
Rothe's method to nonlinear Schrodinger type equations, {\em Appl.
Anal.}, 31 (1988), 149-160.

\bibitem{bc} K. Balachandran and M. Chandrasekaran,
Existence of solutions of a delay differential equation with
nonlocal condition, {\em Indian J. Pure Appl. Math.}, 27 (1996),
443-449.

\bibitem{vb} V. Barbu, {\em Nonlinear semigroups and differential
equations in Banach spaces}, Noordhoff International Publishing,
1976.

 \bibitem{b1} L. Byszewski, Theorems about the existence and
 uniqueness of solutions of a semilinear evolution nonlocal Cauchy
 problem, {\em J. Math. Anal. Appl.}, 162 (1991), 494-505.

 \bibitem{ba} L. Byszewski and H. Akca, Existence of solutions
 of a semilinear functional-differential evolution nonlocal
 problem, {\em Nonlinear Anal.}, 34 (1998), 65-72.

\bibitem{bl} L. Byszewski and V. Lakshmikantham, Theorem about
the existence and uniqueness of a solution of a nonlocal abstract
Cauchy problem in a Banach space, {\em Appl. Anal.}, 40 (1990),
11-19.

\bibitem{k1} A.G. Kartsatos, On the construction of methods
 of lines for functional evolution in general Banach spaces,
 {\em Nonlinear Anal.}, 25 (1995), 1321-1331.


 \bibitem{kp} A.G. Kartsatos and M.E. Parrott, A method of lines
 for a nonlinear abstract functional evolution equation, {\em Trans.
 Amer. Math. Soc.}, 286 (1984), 73-89.

 \item \label{kz} A.G. Kartsatos and W.R. Zigler, Rothe's method
 and weak solutions of perturbed evolution equations in reflexive
 Banach spaces, {\em Math. Annln.}, 219 (1976), 159-166.


 \bibitem{ll} Y. Lin and J.H. Liu, Semilinear
 integrodifferential equations with nonlocal Cauchy problem,
 {\em Nonlinear Anal.}, 26 (1996), 1023-1033.

\end{thebibliography}

\end{document}