\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2003(2003), No. 31, pp. 1--16.
\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/31\hfil Approximations of solutions]
{Approximations of solutions to nonlinear Sobolev type evolution equations}

\author[Dhirendra Bahuguna \& Reeta Shukla \hfil EJDE--2003/31\hfilneg]
{Dhirendra Bahuguna \& Reeta Shukla}


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

\address{Reeta  Shukla \hfill\break
Department of Mathematics,
Indian Institute of Technology Kanpur,
Kanpur - 208 016, India }
\email{reetas@lycos.com}

\date{}
\thanks{Submitted October 10, 2001. Published march 17, 2003.}
\subjclass[2000]{34A12, 34A45, 34G20, 47D06, 47J35}
\keywords{Faedo-Galerkin approximation, analytic semigroup, \hfill\break\indent
 mild solution, contraction mapping theorem, fixed points}

\begin{abstract}
 In the present work we study the approximations of solutions to a
 class of nonlinear Sobolev type evolution equations in a Hilbert space.
 These equations arise in the analysis of the partial neutral functional
 differential equations with unbounded delay. We consider an associated
 integral equation and a sequence of approximate integral equations.
 We establish the existence and uniqueness of the solutions to every
 approximate integral equation using the fixed point arguments.
 We then prove the convergence of the solutions of the approximate
 integral equations to the solution of the associated integral equation.
 Next we consider the Faedo-Galerkin approximations of the solutions and
 prove some convergence results. Finally we demonstrate some of the
 applications of the results established.
\end{abstract}

\maketitle

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

\section{Introduction}

In the present work we are concerned with the approximation of
solutions to the nonlinear Sobolev type evolution equation
\begin{equation}
\begin{gathered}
{d \over dt}(u(t)+g(t,u(t)))+Au(t)=f(t,u(t)), \quad t > 0,\\
u(0)=\phi,
\end{gathered} \label{bs1}
\end{equation}
in a separable Hilbert space $(H,\|.\|,(.,.))$, where the linear
operator $A$ satisfies the assumption (H1) stated later in this
section so that $-A$ generates an analytic semigroup. The functions
$f$ and $g$ are the appropriate continuous functions of their
arguments in $H$.

The case of (\ref{bs1}) in which $g \equiv 0$ has been extensively studied
in literature, see for instance, the books of
Krein \cite{ksg}, Pazy \cite{ap}, Goldstein \cite{gj} and the references
cited in these books.

The study of (\ref{bs1}) with linear $g$ was initiated by
Showalter \cite{sre0,sre1,sre2,sre3,sre4} with the applications to
the degenerate parabolic equations. Brill \cite{brill} has
reformulated a class of pseudoparabolic partial differential
equations as (\ref{bs1}) with linear $g$ and has considered the
applications to a variety of physical problems, for example, in
the thermodynamics \cite{chen&gurtin}, in the flow of fluid
through fissured rocks \cite{barenblatt}, in the shear in
second-order fluids \cite{ting} and in the soil mechanics
\cite{taylor}.


The nonlinear Sobolev type equations of the form (\ref{bs1}) arise
in the study of partial neutral functional differential equations
with an unbounded delay which can be modelled in the form (cf.
\cite{hernandez0+,hernandez1})
\begin{equation}
{d \over dt}(u(t)+G(t,u_t))=Au(t) +F(t,u_t), \quad t >0,
\end{equation}
in a Banach space $X$ where $A$ is the infinitesimal generator of
an analytic semigroup in $X$, $F$ and $G$ are appropriate
nonlinear functions from $[0,T] \times W$ into $X$ and for any
function $u \in C((-\infty,\infty),X)$ the history function
$u_t\in C((-\infty,0],X)$ of $u$ is given by
$u_t(\theta)=u(t+\theta)$.

In the present work we are interested in the Faedo-Galerkin
approximations of solutions to (\ref{bs1}). The Faedo-Galerkin
approximations of solutions to the particular case of (\ref{bs1})
where $g \equiv 0$ and $f(t,u)=M(u)$ has been considered by
Miletta \cite{mpd}. The more general case has been dealt with by
Bahuguna, Srivastava and Singh \cite{bss}. The existence and
uniqueness of solutions to (\ref{bs1}) has been studied by
Hern$\acute{a}$ndez \cite{hernandez0} under the assumptions that
$-A$ is the infinitesimal generator of an analytic semigroup of
bounded linear operators defined on a Banach space $X$ and $f$ and
$g$ are appropriate continuous functions on $[0,T] \times W$ into
$X$ where $W$ is an open subset of $X$.

Now, we consider some assumptions on $A$, $f$ and $g$.  We assume
that the operator $A$ satisfies the following.
\begin{itemize}
\item[(H1)] $A$ is a closed, positive definite, self-adjoint, linear
operator from the domain $D(A) \subset H$ of $A$ into $H$ such
that $D(A)$ is dense in $H$, $A$ has the pure point spectrum
$$
0 <\lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots
$$ and a
corresponding complete orthonormal system of eigenfunctions
$\{u_i\}$, i.e.,
$Au_i=\lambda_iu_i$ and $(u_i,u_j)=\delta_{ij}$,
where $\delta_{ij} = 1$ if $i=j$ and zero
otherwise.
\end{itemize}
These assumptions on $A$ guarantee that $-A$ generates an analytic
semigroup, denoted by $e^{-tA}$, $t \ge 0$.

We mention some notions and preliminaries essential for our
purpose. It is well known that there exist constants
$\tilde{M}\geq 1$ and $\omega \ge 0$ such that
$$
\|e^{-tA}\| \leq\tilde{M}e^{\omega t}, \quad t \geq 0.
$$
Since $-A$ generates the analytic semigroup $e^{-tA}$, $t\geq 0$,
we may add $cI$ to $-A$
for some constant $c$, if necessary, and in what follows we may
assume without loss of generality that $\|e^{-tA}\|$ is uniformly
bounded by $M$, i.e., $\|e^{-tA}\|\leq M$ and $0 \in \rho(A)$. In
this case it is possible to define the fractional power $A^{\eta}$
for $0 \le \eta \le 1$ as closed linear operator with domain
$D(A^{\eta})\subseteq H$ (cf. Pazy \cite{ap}, pp. 69-75 and p.
195). Furthermore, $D(A^{\eta})$ is dense in $H$ and the
expression
$$ \|x\|_{\eta}=\|A^{\eta}x\|, $$
defines a norm on
$D(A^{\eta})$. Henceforth we represent by $X_{\eta}$ the space
$D(A^{\eta})$ endowed with the norm $\|.\|_{\eta}$. In the view of the
facts mentioned above we have the following result for an analytic
semigroup $e^{-tA}$, $t \ge 0$ (cf. Pazy \cite{ap} pp. 195-196).

\begin{lemma}
\label{bslem1} Suppose that $-A$ is the infinitesimal generator of
an analytic semigroup $e^{-tA}$, $t \ge 0$  with $\|e^{-tA}\| \le
M$ for $t \ge 0$ and $0 \in \rho(-A)$. Then we have the following
properties.
\begin{itemize}
\item[(i)] $X_{\eta}$ is a Banach space for $0 \le \eta \le 1$.

\item[(ii)] For $0 < \delta \leq \eta <1$, the embedding $X_{\eta}
\hookrightarrow X_{\delta}$ is continuous.

\item[(iii)] $A^{\eta}$ commutes with $e^{-tA}$ and there exists a
constant $C_{\eta} > 0$ depending on $0 \le \eta \le 1$ such that
$$\|A^{\eta}e^{-tA}\|\leq C_{\eta}t^{-\eta}, \quad t > 0.$$
\end{itemize}
\end{lemma}

We assume the following assumptions on the nonlinear maps $f$ and
$g$.
\begin{itemize}
\item[(H2)] There exist positive constants $0 < \alpha < \beta < 1$ and
$R$ such that the functions $f$ and $A^{\beta}g$ are continuous
for $(t,u) \in [0,\infty)\times B_R(X_{\alpha},\phi)$, where
$B_R(Z,z_0)=\{z \in Z\;|\;\|z-z_0\|_Z\leq R\}$ for any Banach
space $Z$ with its norm $\|.\|_Z$ and there exist constants $L$,
$0< \gamma \le 1$ and a nondecreasing function $F_R$ from
$[0,\infty)$ into $[0,\infty)$ depending on $R > 0$ such that for
every $(t,u)$, $(t,u_1)$ and $(t,u_2)$ in $[0,\infty) \times
B_R(X_{\alpha},\phi)$,
\begin{gather*}
\|A^{\beta}g(t,u_1)-A^{\beta}g(s,u_2)\|
\leq L \{|t-s|^{\gamma}+\|u_1-u_2\|_{\alpha}\},\\
\|f(t,u)\|\leq F_R(t),\\
\|f(t,u_1)-f(t,u_2)\| \leq F_R(t) \|u_1-u_2\|_{\alpha},\\
L\|A^{\alpha-\beta}\|<1.
\end{gather*}
\end{itemize}

The plan of this paper is as follows. In the second section, we
consider an integral equation associated with (\ref{bs1}).  We
then consider a sequence of approximate integral equations and
establish the existence and uniqueness of solutions to each of the
approximate integral equations. In the third section we prove the
convergence of the solutions of the approximate integral equations
and show that the limiting function satisfies the associated
integral equation. In the fourth section we consider the
Faedo-Galerkin approximations of solutions and prove some
convergence results for such approximations.
Finally in the last section we demonstrate some of the applications
of the results established in earlier sections.

\section{Approximate Integral Equations}

We continue to use the notions and notations of the earlier section.
The existence of solutions to (\ref{bs1}) is closely associated with the existence of solutions to the integral equation
\begin{equation}
\begin{aligned}
u(t)=&e^{-tA}(\phi + g(0,\phi))-g(t,u(t))+\int_0^tAe^{-(t-s)A}g(s,u(s))ds \nonumber \\
& + \int_0^te^{-(t-s)A}f(s,u(s))ds, \quad t \ge 0.
\end{aligned}\label{bs2}
\end{equation}
In this section we will consider an approximate integral equation
associated with (\ref{bs2}) and establish the existence and
uniqueness of the solutions to the approximate integral equations.
By a solution $u$ to (\ref{bs2}) on $[0,T]$, $0 < T < \infty$, we
mean a function $u \in X_{\alpha}(T)$ satisfying (\ref{bs2}) on
$[0,T]$ where $X_{\alpha}(T)$ is the Banach space
$C([0,T],X_{\alpha})$ of all continuous functions from $[0,T]$
into $X_{\alpha}$ endowed with the supremum norm
$$
\|u\|_{X_{\alpha}(T)}= \sup_{0 \leq t \leq T}\|u(t)\|_{\alpha}.
$$
By a solution $u$ to (\ref{bs2}) on
$[0,\tilde{T})$, $0 < \tilde{T} \leq \infty$, we mean a function $u$
such that $u \in X_{\alpha}(T)$  satisfying (\ref{bs2}) on $[0,T]$
for every $0 < T < \tilde{T}$.

Let $H_n$ denote the finite dimensional subspace of the Hilbert
space $H$ spanned by $\{u_0,u_1,\dots, u_n\}$ and let
$P^n:H\to H_n$ for $n=1,2,\cdots,$ be the corresponding
projection operators.

Let $0 < T_0 < \infty$ be arbitrarily fixed and let
$$
B=\max_{0 \leq t \leq T_0}\|A^{\beta}g(t,\phi)\|.
$$
We choose $0 < T \leq T_0$ such that
\begin{gather*}
\|(e^{-tA}-I)A^{\alpha}(\phi+g(0,P^n\phi))\| \leq (1-\mu){R\over 3},\\
\|A^{\alpha - \beta}\|LT^{\gamma}
+C_{1+\alpha-\beta}(L\tilde{R}+B){T^{\beta-\alpha}\over \beta-\alpha}
+C_{\alpha}F_{\tilde{R}}(T_0){T^{1-\alpha}\over 1-\alpha}< (1-\mu){R\over 6}, \\
C_{1+\alpha-\beta}L{T^{\beta-\alpha}\over \beta-\alpha}
+C_{\alpha}F_{\tilde{R}}(T_0){T^{1-\alpha}\over 1-\alpha}<1-\mu,
\end{gather*}
where
$\mu=\|A^{\alpha - \beta}\|L$, $\tilde{R}=\sqrt{R^2+\|\phi\|^2_{\alpha}}$
and $C_{\alpha}$ and $C_{1+\alpha-\beta}$ are the constants in
Lemma \ref{bslem1}.

For each $n$, we define
\begin{gather*}
f_n:[0,T]\times X_{\alpha}(T)\to H \quad \mbox{by}
\quad f_n(t,u)=f(t,P^nu(t)), \\
g_n:[0,T]\times X_{\alpha}(T)\to X_{\beta}(T) \quad
\mbox{by} \quad g_n(t,u)=g(t,P^nu(t)).
\end{gather*}
We set $\tilde{\phi}(t)=\phi$ for $t \in [0,T]$ and define a map $S_n$ on
$B_R(X_{\alpha}(T),\tilde{\phi})$ by
\begin{equation}
\begin{aligned}
(S_nu)(t)=&e^{-tA}(\phi+g_n(0,\tilde{\phi}))-g_n(t,u)
+\int_0^tAe^{-(t-s)A}g_n(s,u)ds \\
&+ \int_0^te^{-(t-s)A}f_n(s,u)ds.
\end{aligned} \label{bs3}
\end{equation}

\begin{proposition} \label{bspr1}
Let (H1) and (H2) hold. Then there exists a unique
function $u_n \in B_R(X_{\alpha}(T),\tilde{\phi})$ such that
$S_nu_n=u_n$ for each $n=0,1,2,\dots $; i.e., $u_n$ satisfies the
approximate integral equation
\begin{equation}
\begin{aligned}
u_n(t)=&e^{-tA}(\phi+g_n(0,\tilde{\phi}))-g_n(t,u_n)
+\int_0^tAe^{-(t-s)A}g_n(s,u_n)ds  \\
& + \int_0^te^{-(t-s)A}f_n(s,u_n)ds.
\end{aligned}\label{bs4}
\end{equation}
\end{proposition}

\begin{proof} First we show that the  map $t\mapsto (S_nu)(t)$ is
continuous from $[0,T]$ into $X_{\alpha}$ with respect to norm
$\|.\|_{\alpha}$. For $t \in [0,T]$ and sufficiently small $h>0$,
we have
\begin{align}
&\|(S_nu)(t+h)-(S_nu)(t)\|_{\alpha} \nonumber\\
&\leq \|(e^{-hA}-I)A^{\alpha}e^{-tA}\|(\|\phi\|+\|g(0,P^n\phi)\|)\nonumber\\
&\quad+ \|A^{\alpha-\beta}\|\,\|A^{\beta}g_n(t+h,u)-A^{\beta}g_n(t,u)\|
\nonumber\\
&\quad + \int_0^t\|(e^{-hA}-I)A^{1+\alpha-\beta}e^{-(t-s)A}\|\,
\|A^{\beta}g_n(s,u)\|ds \nonumber \\
&\quad + \int_t^{t+h}\|e^{-(t+h-s)A}A^{1+\alpha-\beta}\|\,
\|A^{\beta}g_n(s,u)\|ds  \label{bs5} \\
&\quad + \int_0^t\|(e^{-hA}-I)A^{\alpha}e^{-(t-s)A}\|\,\|f_n(s,u)\|ds\nonumber\\
&\quad + \int_t^{t+h}\|e^{-(t+h-s)A}A^{\alpha}\|\,\|f_n(s,u)\|ds. \nonumber
\end{align}
Using (H2) we obtain
\begin{equation}
\begin{aligned}
\|A^{\beta}g_n(t+h,u)-A^{\beta}g_n(t,u)\|
&\leq L(h^{\gamma}+\|P^nu(t+h)-P^nu(t)\|_{\alpha})\\
&\leq L(h^{\gamma}+\|u(t+h)-u(t)\|_{\alpha})
\end{aligned}\label{bs55}
\end{equation}
and
\begin{equation}
\int_t^{t+h}\|e^{-(t+h-s)A}A^{1+\alpha-\beta}\|\,\|A^{\beta}g_n(s,u)\|ds
\leq {(L\tilde{R}+B)C_{1+\alpha-\beta}h^{\beta-\alpha} \over \beta-\alpha},
\label{bs56}
\end{equation}
since
\begin{equation}
\begin{aligned}
\|A^{\beta}g_n(s,u)\|&\leq \|A^{\beta}g_n(s,u)-A^{\beta}g(s,\phi)\|
  +\|A^{\beta}g(s,\phi)\| \\
&\leq L\|P^nu(s)-\phi\|_{\alpha}+B \leq L\tilde{R}+B
\end{aligned}\label{bs57}
\end{equation}
and
\begin{equation}
\int_t^{t+h}\|e^{-(t+h-s)A}A^{\alpha}\|\,\|f_n(s,u)\|ds
\leq {C_{\alpha}F_{\tilde{R}}(T_0)h^{1-\alpha}\over 1-\alpha}.
\label{bs58}
\end{equation}
Part (d) of Theorem 2.6.13 in Pazy \cite{ap} implies that for
$0 < \vartheta \leq 1$ and $x \in D(A^{\vartheta})$,
\begin{equation}
\|(e^{-tA}-I)x\| \leq C'_{\vartheta}t^{\vartheta}\|x\|_{\vartheta}.
\label{bs6}
\end{equation}
Let $\vartheta$ be a real number with
$0 < \vartheta < \min\{1-\alpha,\beta-\alpha\}$,
then $A^{\alpha}y \in D(A^{\vartheta})$ for any
$y \in D(A^{\alpha+\vartheta})$. For all $t,s \in [0,T]$, $t \geq s$
and $0 < h < 1$, we get the following inequalities:
\begin{gather}
\|(e^{-hA}-I)A^{\alpha}e^{-tA}\|
\leq C'_{\vartheta}h^{\vartheta}\|A^{\alpha+\vartheta}e^{-tA}\|
\leq {\tilde{C}h^{\vartheta}\over t^{\alpha+\vartheta}},
\label{bs6+} \\
\|(e^{-hA}-I)A^{\alpha}e^{-(t-s)A}\|
\leq {\tilde{C}h^{\vartheta}\over (t-s)^{\alpha+\vartheta}},
\label{bs7} \\
\|(e^{-hA}-I)A^{1+\alpha-\beta}e^{-(t-s)A}\|
\leq {\tilde{C}h^{\vartheta}\over t^{1+\alpha+\vartheta-\beta}},
\label{bs8}
\end{gather}
where $\tilde{C}=C'_{\vartheta}
 \max{\{C_{\alpha+\vartheta},C_{1+\alpha+\vartheta-\beta}\}}$.
Using the estimates (\ref{bs57}), (\ref{bs7}) and (\ref{bs8}), we get
\begin{equation}
\int_0^t\|(e^{-hA}-I)A^{1+\alpha-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u)\|ds
\leq \tilde{C}h^{\vartheta}(L\tilde{R}+B){T_0^{\beta-(\alpha+\vartheta)}\over
\beta-(\alpha+\vartheta)}
\label{bs9}
\end{equation}
and
\begin{equation}
\int_0^t\|(e^{-hA}-I)A^{\alpha}e^{-(t-s)A}\|\,\|f_n(s,u)\|ds
\leq \tilde{C}h^{\vartheta}F_{\tilde{R}}(T_0){T_0^{1-(\alpha+\vartheta)}\over 1-(\alpha+\vartheta)}.
\label{bs10}
\end{equation}
 From the inequalities (\ref{bs5}), (\ref{bs55}), (\ref{bs56}),
(\ref{bs58}), (\ref{bs6+}), (\ref{bs9}) and (\ref{bs10}), it
follows that $(S_nu)(t)$ is continuous from $[0,T]$ into
$X_{\alpha}$ with respect to the norm $\|.\|_{\alpha}$. Now, we
show $S_nu \in B_R(X_{\alpha}(T),\tilde{\phi})$. Consider
\begin{align*}
&\|(S_nu)(t)-\phi\|_{\alpha}\\
&\leq \|(e^{-tA}-I)A^{\alpha}(\phi+g_n(0,\tilde{\phi}))\|
 +  \|A^{\alpha-\beta}\|\,\|A^{\beta}g_n(0,\tilde{\phi})-A^{\beta}g_n(t,u)\| \\
&\quad + \int_0^t\|A^{1+\alpha-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u)\|ds
+ \int_0^t\|e^{-(t-s)A}A^{\alpha}\|\,\|f_n(s,u)\|ds \\
& \leq (1-\mu){R \over 3}+\|A^{\alpha-\beta}\|L\{T^{\gamma}+\|u(t)-\phi\|_{\alpha}\} \\
& \quad +  C_{1+\alpha-\beta}(L\tilde{R}+B){T^{\beta-\alpha}\over \beta-\alpha}+C_{\alpha}F_{\tilde{R}}(T_0){T^{1-\alpha}\over 1-\alpha}\\
& \leq (1-\mu){R \over 3}+(1-\mu){R \over 6}+\mu R \leq  R.
\end{align*}
Taking the supremum over $[0,T]$, we obtain
$$ \|S_nu-\tilde{\phi}\|_{X_{\alpha}(T)} \leq R.
$$
 Hence $S_n$ maps
$B_R(X_{\alpha}(T),\tilde{\phi})$ into
$B_R(X_{\alpha}(T),\tilde{\phi})$. Now we show that $S_n$ is  a
strict contraction on $B_R(X_{\alpha}(T),\tilde{\phi})$. For $u,v
\in B_R(X_{\alpha}(T),\tilde{\phi})$, we have
\begin{equation}
\begin{aligned}
&\|(S_nu)(t)-(S_nv)(t)\|_{\alpha}\\
&\leq \|A^{\alpha-\beta}\|\,\|A^{\beta}g_n(t,u)-A^{\beta}g_n(t,v)\|_{\alpha}\\
&\quad + \int_0^t\|A^{1+\alpha-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u)
-A^{\beta}g_n(s,v)\|ds \\
&\quad +\int_0^t\|e^{-(t-s)A}A^{\alpha}\|\,\|f_n(s,u)-f_n(s,v)\|ds.
\end{aligned}\label{bs11}
\end{equation}
Now,
\begin{equation}
\|A^{\beta}g_n(t,u)-A^{\beta}g_n(t,v)\|\leq L\|u(t)-v(t)\|_{\alpha}
\leq L\|u-v\|_{X_{\alpha}(T)}.
\label{bs12}
\end{equation}
Also, we have
\begin{equation}
 \|f_n(s,u)-f_n(s,v)\| \leq F_{\tilde{R}}(T_0)\|u(s)-v(s)\|_{\alpha}
\leq F_{\tilde{R}}(T_0)\|u-v\|_{X_{\alpha}(T)}.
\label{bs14}
\end{equation}
Using (\ref{bs12}) and (\ref{bs14}) in (\ref{bs11}) and taking supremum
over $[0,T]$, we get
\[
\|S_nu-S_nv\|_{X_{\alpha}(T)}
\leq  \big(\|A^{\alpha-\beta}\|L
+C_{1+\alpha-\beta}L{T^{\beta-\alpha}\over \beta-\alpha}
+C_{\alpha}F_{\tilde{R}}(T_0){T^{1-\alpha}\over 1 - \alpha}\big)
\|u-v\|_{X_{\alpha}(T)}.
\]
The above estimate and the definition of $T$ imply that $S_n$ is a
strict contraction on $B_R(X_{\alpha}(T),\tilde{\phi})$.
Hence there exists a unique $u_n \in B_R(X_{\alpha}(T),\tilde{\phi})$
such that $S_nu_n=u_n$. Clearly $u_n$ satisfies (\ref{bs4}).
This completes the proof of the proposition.
\end{proof}

\begin{proposition} \label{bspr2}
Let (H1) and (H2) hold. If $\phi \in D(A^{\alpha})$
then $u_n(t) \in D(A^{\vartheta})$ for all $t \in (0,T]$ where $0
\leq \vartheta \leq \beta < 1$. Furthermore, if $\phi \in D(A)$
then $u_n(t) \in D(A^{\vartheta})$ for all $t \in [0,T]$ where $0
\leq \vartheta \leq \beta < 1$.
\end{proposition}

\begin{proof} From Proposition \ref{bspr1}, we have the existence of
a unique $u_n \in B_R(X_{\alpha}(T),\tilde{\phi})$ satisfying
(\ref{bs4}). Part (a) of Theorem 2.6.13 in Pazy \cite{ap} implies
that for $t>0$ and $0 \le \vartheta<1$, $e^{-tA}:H \to
D(A^{\vartheta})$ and for $0 \leq \vartheta \leq \beta <1$,
$D(A^{\beta}) \subseteq D(A^{\vartheta})$. (H2) implies that the
map $t \mapsto A^\beta g(t,u_n(t))$ is H\"older
continuous on $[0,T]$ with the exponent
$\rho=\min\{\gamma,\vartheta\}$ since the H\"older
continuity of $u_n$ can be easily established using the similar
arguments from (\ref{bs5}) to (\ref{bs10}). It follows that (cf.
Theorem 4.3.2 in \cite{ap})
$$
\int_0^te^{-(t-s)A}A^\beta g_n(s,u_n)ds \in D(A).
$$
Also from Theorem 1.2.4 in Pazy \cite{ap}, we have $e^{-tA}x \in D(A)$ if
$x \in D(A)$. The required result follows from these facts and the fact
that $D(A) \subseteq D(A^{\vartheta})$ for $0 \leq \vartheta \leq 1$.
\end{proof}

\begin{proposition} \label{bspr3}
Let (H1) and (H2) hold. If $\phi \in D(A^{\alpha})$
and $t_0 \in (0,T]$ then $$\|u_n(t)\|_{\vartheta} \leq U_{t_0},
\quad \alpha < \vartheta < \beta, \quad t \in [t_0,T], \quad n=1,2, \cdots,$$ for some
constant $U_{t_0}$, dependent of $t_0$ and
$$\|u_n(t)\|_{\vartheta} \leq U_0, \quad 0 < \vartheta \le \alpha,
\quad t \in [0,T], \quad n=1,2, \cdots,$$ for some constant $U_0$. Moreover, if $\phi \in D(A^{\beta})$, then there exists a
constant $U_0$, such that
$$\|u_n(t)\|_{\vartheta} \leq U_0, \quad 0 < \vartheta < \beta, \quad t \in [0,T], \quad n=1,2, \cdots.$$
\end{proposition}

\begin{proof} First, we assume that $\phi \in D(A^{\alpha})$.
Applying $A^{\vartheta}$ on both the sides of (\ref{bs4}) and
using (iii) of Lemma \ref{bslem1}, for $t\in [t_0,T]$ and $\alpha
< \vartheta < \beta$, we have
\begin{align*}
\|u_n(t)\|_{\vartheta} \leq& \|A^{\vartheta}e^{-tA}(\phi+g_n(0,\tilde{\phi}))\|+\|A^{\vartheta-\beta}\|\,\|A^{\beta}g_n(t,u_n)\| \\
& + \int_0^t\|A^{1+\vartheta-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u_n)\|ds \\
&+\int_0^t\|e^{-(t-s)A}A^{\vartheta}\|\,\|f_n(s,u_n)\|ds \\
\leq& C_{\vartheta}t_0^{-\vartheta}(\|\phi\|+\|g_n(0,\tilde{\phi})\|) + \|A^{\vartheta-\beta}\|(L\tilde{R}+B) \\
&  + \; C_{1+\vartheta-\beta}(L\tilde{R}+B){T^{\beta-\vartheta} \over \beta-\vartheta} + C_{\vartheta}F_{\tilde{R}}(T_0){T^{1-\vartheta}\over 1 - \vartheta}\leq U_{t_0}.
\end{align*}
Again, for $t \in [0,T]$ and $0 < \vartheta \le \alpha$, $\phi \in
D(A^{\vartheta})$ and
\begin{align*}
\|u_n(t)\|_{\vartheta}
\leq& M(\|A^{\vartheta}\phi\|+\|g_n(0,\tilde{\phi}\|_{\vartheta})
+ \|A^{\vartheta-\beta}\|(L\tilde{R}+B) \\
&+  C_{1+\vartheta-\beta}(L\tilde{R}+B)
{T^{\beta-\vartheta} \over \beta-\vartheta}
+ C_{\vartheta}F_{\tilde{R}}(T_0){T^{1-\vartheta}\over 1 - \vartheta}\leq U_0.
\end{align*}
Furthermore, If $\phi \in D(A^{\beta})$ then $\phi \in
D(A^{\vartheta})$ for $0 < \vartheta \leq \beta$ and we can easily
get the required estimate. This completes the proof of the
proposition. \end{proof}

\section{Convergence of Solutions}

In this section we establish the convergence of the solution $u_n
\in X_{\alpha}(T)$ of the approximate integral equation
(\ref{bs4}). to a unique solution $u$ of (\ref{bs2}).

\begin{proposition}\label{bspr4}
Let (H1) and (H2) hold. If $\phi \in D(A^{\alpha})$,
then for any $t_0\in (0,T]$,
$$
\lim_{m \to \infty} \sup_{\{n\geq m, ~ t_0 \leq t \leq
T\}}\|u_n(t)-u_m(t)\|_{\alpha}=0.
$$
\end{proposition}

\begin{proof} Let $0 < \alpha < \vartheta < \beta$. For $n \geq m$, we have
\begin{align*}
\|f_n(t,u_n)-f_m(t,u_m)\|
&\leq \|f_n(t,u_n)-f_n(t,u_m)\|+\|f_n(t,u_m)-f_m(t,u_m)\|\\
&\leq F_{\tilde{R}}(T_0)[\|u_n(t)-u_m(t)\|_{\alpha}
+\|(P^n-P^m)u_m(t)\|_{\alpha}].
\end{align*}
Also,
\[
\|(P^n-P^m)u_m(t)\|_{\alpha}\leq
\|A^{\alpha-\vartheta}(P^n-P^m)A^{\vartheta}u_m(t)\|
\leq {1 \over \lambda_m^{\vartheta-\alpha}}\|A^{\vartheta}u_m(t)\|.
\]
Thus, we have
\[
 \|f_n(t,u_n)-f_m(t,u_m)\|\leq
 F_{\tilde{R}}(T_0)[\|u_n(t)-u_m(t)\|_{\alpha}
 +{1 \over \lambda_m^{\vartheta-\alpha}}\|A^{\vartheta}u_m(t)\|].
\]
Similarly
\begin{align*}
&\|A^{\beta}g_n(t,u_n)-A^{\beta}g_m(t,u_m)\|\\
&\leq\|A^{\beta}g_n(t,u_n)-A^{\beta}g_n(t,u_m)\|
+\|A^{\beta}g_n(t,u_m)-A^{\beta}g_m(t,u_m)\|\\
&\leq L[\|u_n(t)-u_m(t)\|_{\alpha}
+{1 \over \lambda_m^{\vartheta-\alpha}}\|A^{\vartheta}u_m(t)\|].
\end{align*}
Now, for $0 < t'_0 < t_0$, we may write
\begin{align*}
&\|u_n(t)-u_m(t)\|_{\alpha} \\
& \leq  \|e^{-tA}A^{\alpha}(g_n(0,\tilde{\phi})-g_m(0,\tilde{\phi}))\|
+ \|A^{\alpha-\beta}\|\,\|A^{\beta}g_n(t,u_n)-A^{\beta}g_m(t,u_m)\| \\
& \quad + \Big(\int_0^{t'_0}+\int^t_{t'_0}\Big)
\|A^{1+\alpha-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u_n)-A^{\beta}g_m(s,u_m)\|ds  \\
& \quad + \Big(\int_0^{t'_0}+\int^t_{t'_0}\Big)
\|A^{\alpha}e^{-(t-s)A}\|\,\|f_n(s,u_n)-f_m(s,u_m)\|ds.
\label{bs18}
\end{align*}
We estimate the first term as
\begin{align*}
\|e^{-tA}A^{\alpha}(g_n(0,\tilde{\phi})-g_m(0,\tilde{\phi}))\|
&\leq M\|A^{\alpha-\beta}\|\,\|A^{\beta}g(0,P^n\phi)-A^{\beta}g(0,P^m\phi)\|\\
&\leq M\|A^{\alpha-\beta}\|L\|(P^n-P^m)A^{\alpha}\phi\|.
\end{align*}
The first and the third integrals are estimated as
\begin{gather*}
\int_0^{t'_0}\|A^{1+\alpha-\beta}e^{-(t-s)A}\|\,
\|A^{\beta}g_n(s,u_n)-A^{\beta}g_m(s,u_m)\|ds\\
\leq 2C_{1+\alpha-\beta}(L\tilde{R}+B)(t_0-t'_0)^{-(1+\alpha-\beta)}t'_0,\\
\int_0^{t'_0}\|A^{\alpha}e^{-(t-s)A}\|\,\|f_n(s,u_n)-f_m(s,u_m)\|ds
\leq 2C_{\alpha}F_{\tilde{R}}(T_0)(t_0-t'_0)^{-\alpha}t'_0.
\end{gather*}
For the second and the fourth integrals, we have
\begin{align*}
&\int^t_{t'_0}\|A^{1+\alpha-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u_n)-A^{\beta}g_m(s,u_m)\|ds \\
&  \leq C_{1+\alpha-\beta}L\int^t_{t'_0}(t-s)^{-(1+\alpha-\beta)}[\|u_n(s)-u_m(s)\|_{\alpha}+{1 \over \lambda_m^{\vartheta-\alpha}}\|A^{\vartheta}u_m(s)\|]ds \\
&  \leq
C_{1+\alpha-\beta}L\Big({U_{t'_0}T^{\beta-\alpha}\over
\lambda_m^{\vartheta-\alpha}(\beta-\alpha)}+\int^t_{t'_0}(t-s)^{-
(1+\alpha-\beta)}\|u_n(s)-u_m(s)\|_{\alpha}ds\Big),
\end{align*}
\begin{align*}
&\int^t_{t'_0}\|A^{\alpha}e^{-(t-s)A}\|\,\|f_n(s,u_n)-f_m(s,u_m)\|ds \\
& \leq C_{\alpha}F_{\tilde{R}}(T_0)\int^t_{t'_0}(t-s)^{-\alpha}[\|u_n(s)-u_m(s)\|_{\alpha}+{1 \over \lambda_m^{\vartheta-\alpha}}\|A^{\vartheta}u_m(s)\|]ds \\
& \leq
C_{\alpha}F_{\tilde{R}}(T_0)\Big({U_{t'_0}T^{1-\alpha}\over
\lambda_m^{\vartheta-\alpha}(1-\alpha)}+\int^t_{t'_0}(t-s)^{-\alpha}
\|u_n(s)-u_m(s)\|_{\alpha}ds\Big).
\end{align*}
Therefore,
\begin{align*}
\|u_n(t)-u_m(t)\|_{\alpha}
\leq &M\|A^{\alpha-\beta}\|L\|(P^n-P^m)A^{\alpha}\phi\| \\
&+\|A^{\alpha-\beta}\|L\Big(\|u_n(t)-u_m(t)\|_{\alpha}
+{U_{t'_0} \over \lambda_m^{\vartheta-\alpha}}\Big) \\
&+ 2\Big({C_{1+\alpha-\beta}(L\tilde{R}+B)\over (t_0-t'_0)^{1
+\alpha-\beta}}+{C_{\alpha}F_{\tilde{R}}(T_0)\over (t_0-t'_0)^{\alpha}}\Big)t'_0 + C_{\alpha,\beta}{U_{t'_0}\over \lambda_m^{\vartheta-\alpha}}\\
&+ \int^t_{t'_0}\Big({C_{\alpha}F_{\tilde{R}}(T_0)\over (t-s)^{\alpha}}
+{C_{1+\alpha-\beta}L\over (t-s)^{1+\alpha-\beta}}\Big)\|u_n(s)
-u_m(s)\|_{\alpha}ds,
\end{align*}
where $$C_{\alpha,\beta}=C_{\alpha}F_{\tilde{R}}(T_0){T^{1-\alpha}\over 1-\alpha}+C_{1+\alpha-\beta}L{T^{\beta-\alpha}\over \beta-\alpha}.$$
Since $\|A^{\alpha-\beta}\|L < 1$, we have
\begin{align*}
\|u_n(t)-u_m(t)\|_{\alpha}
 \leq & {1\over (1-\|A^{\alpha-\beta}\|L)}\Big\{M\|(P^n-P^m)A^{\alpha}\phi\|
 + \|A^{\alpha-\beta}\|L{U_{t'_0} \over \lambda_m^{\vartheta-\alpha}} \\
& + 2\Big({C_{1+\alpha-\beta}(L\tilde{R}+B)\over (t_0-t'_0)^{1+\alpha-\beta}}
+{C_{\alpha}F_{\tilde{R}}(T_0)\over (t_0-t'_0)^{\alpha}}\Big)t'_0
+C_{\alpha,\beta}{U_{t'_0}\over \lambda_m^{\vartheta-\alpha}}  \\
& +\int^t_{t'_0}\Big({C_{\alpha}F_{\tilde{R}}(T_0)\over (t-s)^{\alpha}}+{C_{1+\alpha-\beta}L\over (t-s)^{1+\alpha-\beta}}
\Big) \|u_n(s)-u_m(s)\|_{\alpha}ds \Big\}.
\end{align*}
Lemma 5.6.7 in \cite{ap} implies that there exists a constant C
such that
\begin{align*}
&\|u_n(t)-u_m(t)\|_{\alpha}\\
& \leq {1\over (1-\|A^{\alpha-\beta}\|L)}\Big\{M\|(P^n-P^m)A^{\alpha}\phi\|
+ (\|A^{\alpha-\beta}\|L+C_{\alpha,\beta}){U_{t'_0} \over
\lambda_m^{\vartheta-\alpha}} \\
& + 2\Big({C_{1+\alpha-\beta}(L\tilde{R}+B)\over
(t_0-t'_0)^{1+\alpha-\beta}}+{C_{\alpha}F_{\tilde{R}}(T_0)\over
(t_0-t'_0)^{\alpha}}\Big)t'_0 \Big\}C.
\end{align*}
Taking supremum over $[t_0,T]$ and letting $m \to \infty$, we obtain
\begin{align*}
&\lim_{m \to \infty} \sup_{\{n \ge m,t \in [t_0,T]\}}\|u_n(t)-u_m(t)\|_{\alpha}\\
& \leq {2\over
(1-\|A^{\alpha-\beta}\|L)}\Big({C_{1+\alpha-\beta}(L\tilde{R}+B)\over
(t_0-t'_0)^{1+\alpha-\beta}}+{C_{\alpha}F_{\tilde{R}}(T_0)\over
(t_0-t'_0)^{\alpha}}\Big)C.
\end{align*}
As $t'_0$ is arbitrary, the right hand side may be made as small as desired by taking $t'_0$ sufficiently small. This completes the proof of the proposition.
\end{proof}

\begin{corollary} \label{bscor1}
If $\phi \in D(A^{\beta})$ then
$$
\lim_{m\to \infty} \sup_{\{n\geq m, ~ 0 \leq t \leq
T\}}\|u_n(t)-u_m(t)\|_{\alpha}=0.
$$
\end{corollary}

\begin{proof} Propositions \ref{bspr2} and \ref{bspr3} imply that
in the proof of  Proposition \ref{bspr4} we may take $t_0=0$.
\end{proof}

For the convergence of the solution $u_n(t)$ of the approximate
integral equation (\ref{bs4}) we have the following result.

\begin{theorem} \label{bsthm1}
Let (H1) and (H2) hold and let $\phi \in
D(A^{\alpha})$. Then there exists a unique function $u \in
X_{\alpha}(T)$ such that $u_n \to u$ as $n \to
\infty$ in $X_{\alpha}(T)$ and $u$ satisfies (\ref{bs2}) on
$[0,T]$. Furthermore $u$ can be extended to the maximal interval
of existence $[0,t_{\rm max})$, $0 < t_{\rm max} \leq \infty$ satisfying
(\ref{bs2}) on $[0,t_{\rm max})$ and $u$ is a unique solution to
(\ref{bs2}) on $[0,t_{\rm max})$.
\end{theorem}

\begin{proof}
 Let us assume that $\phi \in D(A^{\alpha})$. Since,
for $0 < t \leq T$, $A^{\alpha}u_n(t)$ converges to
$A^{\alpha}u(t)$ as $n \to \infty$ and $u_n(0)=u(0)=\phi$
for all $n$,  we have, for $0 \leq t \leq T$, $A^{\alpha}u_n(t)$
converges to $A^{\alpha}u(t)$ in $H$ as $n \to \infty$.
Since $u_n \in B_R(X_{\alpha}(T),\tilde{\phi})$, it follows that
$u \in B_R(X_{\alpha}(T),\tilde{\phi})$ and for any $0 < t_0 \leq T$,
$$
\lim_{n \to \infty}\sup_{\{t_0 \leq t \leq T\}}
\|u_n(t)-u(t)\|_{\alpha}=0.
$$
Also,
\[
\sup_{t_0 \leq t \leq T}\|f_n(t,u_n)-f(t,u(t))\|
\leq  F_{\tilde{R}}(T_0)(\|u_n-u\|_{X_{\alpha}(T)}
+\|(P^n-I)u\|_{X_{\alpha}(T)})
\to 0
\]
as $n \to \infty$ and
\[
\sup_{t_0 \leq t \leq T}\|A^{\beta}g_n(t,u_n)-A^{\beta}g(t,u(t))\|
\leq L(\|u_n-u\|_{X_{\alpha}(T)}+\|(P^n-I)u\|_{X_{\alpha}(T)}) \to 0
\]
as $n \to \infty$.
Now, for $0 < t_0 < t$, we may rewrite (\ref{bs4}) as
\begin{align*}
u_n(t) &=  e^{-tA}(\phi+g_n(0,\tilde{\phi}))-g_n(t,u_n)+\left(\int_0^{t_0}+\int_{t_0}^t\right)Ae^{-(t-s)A}g_n(s,u_n)ds  \\
&\quad+\Big(\int_0^{t_0}+\int_{t_0}^t\Big)e^{-(t-s)A}f_n(s,u_n)ds
\end{align*}
The first and third integrals are estimated as
\begin{align*}
\|\int_0^{t_0}Ae^{-(t-s)A}g_n(s,u)ds\|
&\leq\int_0^{t_0}\|A^{1-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g_n(s,u_n)\|ds\\
&\leq C_{1-\beta}(L\tilde{R}+B)T^{1-\beta}t_0,
\end{align*}
\[
\|\int_0^{t_0}e^{-(t-s)A}f_n(s,u_n)ds\|\leq MF_{\tilde{R}}(T_0)t_0.
\]
Thus, we have
\begin{align*}
&\Big\|u_n(t)-e^{-tA}(\phi+g_n(0,\tilde{\phi}))+g_n(t,u_n)\\
&-\int^t_{t_0}Ae^{-(t-s)A}g_n(s,u_n)ds
- \int^t_{t_0}e^{-(t-s)A}f_n(s,u_n)ds \Big \| \\
& \leq  (C_{1-\beta}(L\tilde{R}+B)T^{1-\beta}+MF_{\tilde{R}}(T_0))t_0.
\end{align*}
Letting $n \to \infty$ in the above inequality, we get
\begin{align*}
&\Big\|u(t)-e^{-tA}(\phi+g(0,\phi))+g(t,u(t))\\
&-\int^t_{t_0}Ae^{-(t-s)A}g(s,u(s))ds
- \int^t_{t_0}e^{-(t-s)A}f(s,u(s))ds\Big\| \\
&\leq  (C_{1-\beta}(L\tilde{R}+B)T^{1-\beta}+MF_{\tilde{R}}(T_0))t_0.
\end{align*}
Since $0 < t_0 \leq T$ is arbitrary, we obtain that $u$ satisfies the
integral equation (\ref{bs2}).

If $u$ satisfies (\ref{bs2}) on $[0,T_1]$ for some $0 < T_1 \le T_0$,
then we show that, $u$ can be extended further. Since $0 < T_0 < \infty$,
was arbitrary, we assume that $0 < T_1 < T_0$. We consider the equation
\begin{gather*}
{d\over dt}(w(t)+G(t,w(t)))+Aw(t)= F(t,w(t)), \quad
0 \leq t \leq T_0 < \infty, \\
w(0)=u(T_1),
\end{gather*}
where, $F,G:[0,T_0-T_1]\times D(A^{\alpha})\to H$ are defined by
$$
F(t,x)=f(t+T_1,x),\quad G(t,x)=g(t+T_1,x),
$$
for $(t,x) \in [0,T_0-T_1]\times D(A^{\alpha})$. We note that $F$
and $G$ satisfy (H2), where $T_0$ is
replaced by $T_0-T_1$. Hence, there exists a unique $w \in
C([0,T_2],D(A^{\alpha}))$ for some $0 < T_2 < T_0-T_1$ satisfying
the integral equation
\begin{align*}
w(t)&=e^{-tA}(u(T_1)+G(0,u(T_1))-G(t,w(t)) \\
&\quad +\int_0^tAe^{-(t-s)A}G(s,w(s))ds
 +  \int_0^te^{-(t-s)A}F(s,w(s))ds, \quad 0 \leq t \leq T_2.
\end{align*}
We define
\[
\tilde{u}(t)=\begin{cases} u(t), & 0 \leq t \leq T_1, \\
w(t-T_1), & T_1 \leq t \leq T_1+T_2. \end{cases}
\]
Then $\tilde{u}$ satisfies the integral equation
\begin{equation}
\begin{aligned}
\tilde{u}(t)&=e^{-tA}(\phi+g(0,\phi))-g(t,\tilde{u}(t))
+\int_0^tAe^{-(t-s)A}g(s,\tilde{u}(s))ds  \\
&\quad+  \int_0^te^{-(t-s)A}f(s,\tilde{u}(s))ds, \quad 0 \leq t \leq T_1+T_2.
\end{aligned}
\label{bs20}
\end{equation}
To see this, we need to verify (\ref{bs20}) only on $[T_1,T_1+T_2]$. For $t \in [T_1,T_1+T_2]$,
\begin{align*}
\tilde{u}(t)&=w(t-T_1) \\
&=e^{-(t-T_1)A}(u(T_1)+G(0,u(T_1)))-G(t-T_1,w(t-T_1))\\
&\quad +  \int_0^{t-T_1}Ae^{-(t-T_1-s)A}G(s,w(s))ds
+ \int_0^{t-T_1}e^{-(t-T_1-s)A}F(s,w(s))ds.
\end{align*}
Putting $T_1+s=\eta$, we get
\begin{align*}
\tilde{u}(t)&=e^{-(t-T_1)A}(\{e^{-T_1A}(\phi + g(0,\phi))-g(T_1,u(T_1))  \\
&\quad+ \int_0^{T_1}Ae^{-(T_1-s)A}g(s,u(s))ds  + \int_0^{T_1}e^{-(T_1-s)A}f(s,u(s))ds\}\\
& \quad+ G(0,u(T_1)))-G(t-T_1,w(t-T_1))\\
& \quad+ \int_{T_1}^tAe^{-(t-\eta)A}G(\eta-T_1,w(\eta-T_1))d\eta \\
& \quad+ \int_{T_1}^te^{-(t-\eta)A}F(\eta-T_1,w(\eta-T_1))ds\\
&=e^{-tA}(\phi+g(0,\phi))-g(t,w(t-T_1))+\int_0^{T_1}Ae^{-(t-s)A}g(s,u(s))ds \\
&\quad+ \int_{T_1}^tAe^{-(t-s)A}g(s,w(s-T_1))ds+\int_0^{T_1}e^{-(t-s)A}f(s,u(s))ds \\
&\quad+ \int_{T_1}^te^{-(t-s)A}f(s,w(s-T_1))ds,
\end{align*}
as $G(0,u(T_1))=g(T_1,u(T_1))$, $G(t-T_1,w(t-T_1))=g(t,w(t-T_1))$ and $F(t-T_1,w(t-T_1))=f(t,w(t-T_1))$.
Hence, we have
\begin{align*}
\tilde{u}(t)&= e^{-tA}(\phi+g(0,\phi))-g(t,\tilde{u}(t)) +\int_0^tAe^{-(t-s)A}g(s,\tilde{u}(s))ds \\
&\quad+  \int_0^te^{-(t-s)A}f(s,\tilde{u}(s))ds,
\end{align*}
for $t \in [0,T_1+T_2]$. Thus, we see $\tilde{u}(t)$ satisfy
(\ref{bs20}) on $[0,T_1+T_2]$. hence, we may extend $u(t)$ to
maximal interval $[0,t_{\rm max})$ satisfying (\ref{bs20}) on
$[0,t_{\rm max})$ with $0 < t_{\rm max} \le \infty$.

Now, we show the uniqueness of solutions to (\ref{bs2}). Let $u_1$
and $u_2$ be two solutions to (\ref{bs2}) on some interval
$[0,T_3]$, where $T_3$ be any number such that $0 < T_3 < t_{\rm max}$.
Then, for $0 \leq t \leq T_3$, we have
\begin{align*}
\|u_1(t)-u_2(t)\|_{\alpha}
&\leq \|A^{\alpha-\beta}\|\,\|A^{\beta}g(t,u_1(t))-A^{\beta}g(t,u_2(t))\| \\
&\quad+  \int_0^{t}\|A^{1+\alpha-\beta}e^{-(t-s)A}\|\,\|A^{\beta}g(s,u_1(s))-A^{\beta}g(s,u_2(s))\|ds\\
&\quad + \int_0^{t}\|e^{-(t-s)A}A^{\alpha}\|\,\|f(s,u_1(s))-f(s,u_2(s))\|ds \\
& \leq \|A^{\alpha-\beta}\|L\|u_1(t)-u_2(t)\|_{\alpha}\\
&\quad +C_{1+\alpha-\beta}L\int_0^{t}(t-s)^{-(1+\alpha-\beta)}\|u_1(s)-u_2(s)\|_{\alpha}ds\\
&\quad+C_{\alpha}F_{\tilde{R}}(T_3)\int_0^{t}(t-s)^{-\alpha}\|u_1(s)-u_2(s)\|_{\alpha}ds.
\end{align*}
Since, $\|A^{\alpha-\beta}\|L<1$, we have
\begin{align*}
&\|u_1(t)-u_2(t)\|_{\alpha}\\
&\leq {1 \over
(1-\|A^{\alpha-\beta}\|L)}\int_0^{t}\Big({C_{1+\alpha-\beta}L\over
(t-s)^{1+\alpha-\beta}}+{C_{\alpha}F_{\tilde{R}}(T_3)\over
(t-s)^{\alpha}}\Big)\|u_1(s)-u_2(s)\|_{\alpha}ds.
\end{align*}
Using Lemma 5.6.7 in Pazy \cite{ap}, we get
\begin{align*}
\|u_1(t)-u_2(t)\|_{\alpha}=0
\end{align*}
for all $0 \le t \le T_3$. From the fact that
$$\|u_1(t)-u_2(t)\| \leq {1 \over
\lambda_0^{\alpha}}\|u_1(t)-u_2(t)\|_{\alpha},$$ it follows that
$u_1=u_2$ on $[0,T_3]$. Since $0 < T_3 < t_{\rm max}$ was arbitrary, we
have $u_1=u_2$ on $[0,t_{\rm max})$. This completes the proof of the
theorem. \end{proof}

\section{Faedo-Galerkin Approximations}

For any $0 < T < t_{\rm max}$, we have a unique $u \in X_{\alpha}(T)$ satisfying the integral equation
\begin{equation}
\begin{aligned}
u(t)&=e^{-tA}(\phi + g(0,\phi))-g(t,u(t))+\int_0^tAe^{-(t-s)A}g(s,u(s))ds \nonumber \\
& \quad+ \int_0^te^{-(t-s)A}f(s,u(s))ds.
\end{aligned}\label{bs20+}
\end{equation}
Also, we have a unique solution $u_n \in X_{\alpha}(T)$ of the
approximate integral equation
\begin{equation}
\begin{aligned}
u_n(t)&=e^{-tA}(\phi+g_n(0,\tilde{\phi}))-g_n(t,u_n)+\int_0^tAe^{-(t-s)A}g_n(s,u_n)ds \nonumber \\
&\quad+ \int_0^te^{-(t-s)A}f_n(s,u_n)ds.
\end{aligned} \label{bs21}
\end{equation}
If we project (\ref{bs21}) onto $H_n$, we get the Faedo-Galerkin approximation
$\hat{u}_n(t)=P^nu_n(t)$ satisfying
\begin{equation}
\begin{aligned}
\hat{u}_n(t)&= e^{-tA}(P^n\phi+P^ng(0,P^n\phi))-P^ng(t,\hat{u}_n(t)) \\
&\quad + \int_0^tAe^{-(t-s)A}P^ng(s,\hat{u}_n(s))ds
+ \int_0^te^{-(t-s)A}P^nf(s,\hat{u}_n(s))ds
\end{aligned}\label{bs22}
\end{equation}
The solution $u$ of (\ref{bs20+}) and $\hat{u}_n$ of (\ref{bs22}),
have the representation
\begin{gather}
u(t)=\sum_{i=0}^{\infty}\alpha_i(t)u_i, \quad \alpha_i(t)=(u(t),u_i),
\quad i=0,1,\ldots;
\label{bs23} \\
\hat{u}_n(t)=\sum_{i=0}^n\alpha^n_i(t)u_i, \quad
\alpha^n_i(t)=(\hat{u}_n(t),u_i), \quad i=0,1,\ldots;
\label{bs24}
\end{gather}
Using (\ref{bs24}) in (\ref{bs22}), we get the following system of
first order ordinary differential equations
\begin{equation}
\begin{gathered}
{d\over dt}\left(\alpha^n_i(t)+G^n_i(t,\alpha_0^n(t),\dots,
\alpha_n^n(t))\right) +\lambda_i\alpha_i^n(t)
=F^n_i(t,\alpha_0^n(t),\dots,\alpha_n^n(t)),\\
\alpha_i^n(0)=\phi_i,
\end{gathered}\label{bs27}
\end{equation}
where
\begin{gather*}
G^n_i(t,\alpha_0^n(t),\dots,\alpha_n^n(t))
=\big(g(t,\sum_{i=0}^n\alpha^n_i(t)u_i),u_i\big),\\
F^n_i(t,\alpha_0^n(t),\dots,\alpha_n^n(t))
=\big(f(t,\sum_{i=0}^n\alpha^n_i(t)u_i),u_i\big),
\end{gather*}
and $\phi_i=(\phi,u_i)$ for $i = 1,2,\dots n$.

The system (\ref{bs27}) determines the $\alpha^n_i(t)$'s. Now, we
shall show the convergence of $\alpha^n_i(t) \to
\alpha_i(t)$. It can easily be checked that
\[
A^{\alpha}[u(t)-\hat{u}(t)]
=A^{\alpha}\Big[\sum_{i=0}^{\infty}(\alpha_i(t)-\alpha^n_i(t))u_i\Big ]
=\sum_{i=0}^{\infty}\lambda_i^{\alpha}(\alpha_i(t)-\alpha_i^n(t))u_i.
\]
Thus, we have
$$\|A^{\alpha}[u(t)-\hat{u}(t)]\|^2\geq\sum_{i=0}^n\lambda_i^{2\alpha}(\alpha_i(t)-\alpha_i^n(t))^2.$$
We have the following convergence theorem.

\begin{theorem} \label{bsthm2}
Let (H1) and (H2) hold. Then we have the following.
\begin{itemize}
\item[(a)] If $\phi \in D(A^{\alpha})$, then for any $0 < t_0 \leq T$,
$$
\lim_{n \to \infty}\sup_{t_0 \leq t \leq T}
\Big[\sum_{i=0}^n\lambda_i^{2\alpha}(\alpha_i(t)-\alpha_i^n(t))^2\Big]=0.
$$
\item[(b)] If $\phi \in D(A^{\beta})$, then
$$
\lim_{n \to \infty}\sup_{0 \leq t \leq T}
\Big[\sum_{i=0}^n\lambda_i^{2\alpha}(\alpha_i(t)-\alpha_i^n(t))^2\Big]=0.
$$
\end{itemize}
\end{theorem}

The assertion of this theorem follows from the facts mentioned
above and the following result.

\begin{proposition} \label{bspr5}
Let (H1) and (H2) hold and let $T$ be any number
such that $0 < T < t_{\rm max}$, then we have the following.
\begin{itemize}
\item[(a)] If $\phi \in D(A^{\alpha})$, then for any $0 < t_0 \leq T$,
$$
\lim_{n \to \infty}\sup_{\{n \geq m, t_0 \leq t \leq T\}}
\|A^{\alpha}[\hat{u}_n(t)-\hat{u}_m(t)]\|=0.
$$
\item[(b)] If $\phi \in D(A^{\beta})$, then
$$
\lim_{n \to \infty}\sup_{\{n \geq m, 0 \leq t \leq T\}}
\|A^{\alpha}[\hat{u}_n(t)-\hat{u}_m(t)]\|=0.
$$
\end{itemize}
\end{proposition}

\begin{proof} For $n \geq m$, we have
\begin{align*}
\|A^{\alpha}[\hat{u}_n(t)-\hat{u}_m(t)]\|
&=\|A^{\alpha}[P^nu_n(t)-P^mu_m(t)]\| \\
&\le \|P^n[u_n(t)-u_m(t)]\|_{\alpha}+\|(P^n-P^m)u_m\|_{\alpha}\\
&\le \|u_n(t)-u_m(t)\|_{\alpha}+\frac{1}{\lambda_m^{\vartheta-\alpha}}
\|A^{\vartheta}u_m\|.
\end{align*}
If $\phi \in D(A^{\alpha})$ then the result in (a) follows from
Proposition \ref{bspr4}. If $\phi \in D(A^{\beta})$, (b) follows from
Corollary \ref{bscor1}.
\end{proof}

\section{Applications}

In this section we give some applications of the results
established in the earlier sections.
Consider the initial boundary value problem
\begin{equation}
\begin{gathered}
\frac{\partial}{\partial t}(w(x,t)-\Delta
w(x,t))+\Delta^2w(x,t)=h(x,t,w(x,t)),  \\
w(x,0)=w_0(x), \quad x \in \Omega,
\end{gathered}\label{appl1}
\end{equation}
with the homogeneous boundary conditions
where $\Omega$ is a  bounded domain in the $\mathbb{R}^N$ with the
sufficiently smooth boundary $\partial \Omega$ and  $\Delta$ is
$N$-dimensional Laplacian. The nonlinear function $h$ is
sufficiently smooth in all its arguments.

Let $X=L^2(\Omega)$ and define the operator $A$ by
$$
D(A)=H^1_0(\Omega)\cap H^2(\Omega), \quad  Au=-\Delta u, \quad u \in D(A),
$$
then we can reformulate (\ref{appl1}) in the abstract form
\begin{equation}
\begin{gathered}
\frac{d}{dt}(u(t)+Au(t))+A^2u(t)=h(t,u(t)), \\
 u(0)=w_0.
\end{gathered}\label{appl2}
\end{equation}
The operator $A$ is not invertible but for $c>0$ large enough $(A+cI)$
is invertible and $\|(A+cI)^{-1}\|\le C$. Therefore, we can
write (\ref{appl2}) as a Sobolev
type evolution equation of the form (\ref{bs1}) where
$$ g(t,u)=(1-c)(A+cI)^{-1}u $$
and
$$ f(t,u)=cA(A+cI)^{-1}u+h(t,(A+cI)^{-1}u). $$
We see that the operator $A$ satisfies (H1). Also we can easily check
that $g$ and $f$ satisfy (H2). Thus, we may apply the results of the earlier
sections to guarantee the existence of Faedo-Galerkin
approximations and their convergence to the unique solution of
(\ref{appl1}).

A particular example of (\ref{appl1}) is the meta-parabolic (cf.
Carroll and Showalter \cite{cs}, Showalter \cite{sre4} and Brown \cite{brown})
problem
\begin{equation}
\begin{gathered}
\frac{\partial}{\partial t}(u(x,t)-\frac{\partial^2u(x,t)}{\partial x^2}
)+\frac{\partial^4u(x,t)}{\partial x^4} =
f(x,t,u(x,t)),\quad 0 < x < 1,  \\
u(0,t)=u(1,t)=\frac{\partial u}{\partial x}(0,t)=\frac{\partial
u}{\partial x}(1,t)=0, \quad t>0,  \\
 u(x,0)=u_0(x), \quad 0 < x < 1.
\end{gathered}\label{appl3}
\end{equation}

\subsection*{Acknowledgements}
The authors would like to thank the referee for his
valuable comments and suggestions. The authors also wish to express their
sincere gratitude to Prof R.E. Showalter for bringing some of the references
to their notice. The first author would like to acknowledge the financial
support from National Board of Higher Mathematics (NBHM), INDIA under its
Research Project No. 48/3/2001/R\&D-II. The second author would like to
thank the Council of Scientific \& Industrial Research (CSIR), INDIA for
the financial support under its Research Project No.9/92(198)/2000-EMR-I.

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\end{document}