\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conference 19 (2010), pp. 99--121.\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{8mm}}
\begin{document} \setcounter{page}{99}
\title[\hfilneg EJDE-2010/Conf/19/\hfil Continuous dependence of solutions]
{Continuous dependence of solutions for ill-posed evolution problems}
\author[M. Fury, R. J. Hughes\hfil EJDE/Conf/19 \hfilneg]
{Matthew Fury, Rhonda J. Hughes} % in alphabetical order
\address{Matthew Fury \newline
Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{mfury@brynmawr.edu}
\address{Rhonda J. Hughes \newline
Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{rhughes@brynmawr.edu}
\thanks{Published September 25, 2010.}
\subjclass[2000]{47A52, 42C40}
\keywords{Continuous dependence on modelling; time-dependent problems;
\hfill\break\indent Ill-posed problems}
\begin{abstract}
We prove H\"older-continuous dependence results for the difference
between certain ill-posed and well-posed evolution problems in
a Hilbert space. Specifically, given a positive self-adjoint
operator $D$ in a Hilbert space, we consider the ill-posed evolution
problem
\begin{gather*}
\frac{du(t)}{dt} = A(t,D)u(t) \quad 0\leq t\omega
$$
and every finite sequence $0\leq t_1\leq t_2,\dots ,t_k\leq T$,
$k=1,2,\dots $.
\end{definition}
\noindent \textbf{Remark.} If for $t\in [0,T]$, $A(t)$ is the
infinitesimal generator of a $C_0$ semigroup $\{S_t(s)\}_{s\geq
0}$ satisfying $\|S_t(s)\|\leq e^{\omega s}$, then by the
Hille-Yosida theorem (cf. \cite{p1}), the family $\{A(t)\}_{t\in
[0,T]}$ is stable with constants $M=1$ and $\omega$.
We now use the theory of stable families of operators to gain
well-posedness of \eqref{star} in the following way. Let $X$ and
$Y$ be Banach spaces with norms $\|\cdot\|$ and $\|\cdot\|_Y$
respectively. Assume that $Y$ is densely and continuously
imbedded in $X$, that is $Y$ is a dense subspace of $X$ and there
is a constant $C$ such that
$$
\|y\|\leq C\|y\|_Y \quad \text{for } y\in Y.
$$
For each $t\in [0,T]$, let $A(t)$ be the
infinitesimal generator of a $C_0$ semigroup
$\{S_t(s)\}_{s\geq 0}$ on $X$. Assume the following
conditions (cf. \cite{k1,p1}):
\begin{itemize}
\item[(H1)] $\{A(t)\}_{t\in [0,T]}$ is a stable family with
stability constants $M$, $\omega$.
\item[(H2)] For each $t\in [0,T]$, $Y$ is an invariant subspace
of $S_t(s),s\geq 0$, the restriction $\tilde{S}_t(s)$ of $S_t(s)$
to $Y$ is a $C_0$ semigroup in $Y$, and the family
$\{\tilde{A}(t)\}_{t\in [0,T]}$ of parts $\tilde{A}(t)$ of $A(t)$
in $Y$, is a stable family in $Y$.
\item[(H3)] For $t\in [0,T]$,
$\operatorname{Dom}(A(t))\supseteq Y$, $A(t)$ is a bounded
operator from $Y$ into $X$, and $t\mapsto A(t)$ is continuous
in the $B(Y,X)$ norm $\|\cdot\|_{Y\to X}$.
\end{itemize}
\begin{theorem}[{\cite[Theorem 4.1]{k1}, \cite[Theorem 5.3.1]{p1}}]
\label{thm3}
For each $t\in [0,T]$, let $A(t)$ be the infinitesimal generator
of a $C_0$ semigroup $\{S_t(s)\}_{s\geq 0}$ on $X$.
If the family $\{A(t)\}_{t\in [0,T]}$ satisfies conditions
{\rm (H1)--(H3)}, then there exists a unique evolution system
$U(t,s)$, $0\leq s\leq t\leq T$, in $X$ satisfying
\begin{itemize}
\item[(E1)] $\|U(t,s)\|\leq Me^{\omega (t-s)}$ for
$0\leq s\leq t\leq T$.
\item[(E2)] $\frac{\partial^+}{\partial t}U(t,s)y\big|_{t=s}=A(s)y$
for $y\in Y$, $0\leq s\leq T$.
\item[(E3)] $\frac{\partial}{\partial s}U(t,s)y=-U(t,s)A(s)y$
for $y\in Y$, $0\leq s\leq t\leq T$,
\end{itemize}
where the derivative from the right in (E2) and the derivative
in (E3) are in the strong sense in $X$.
\end{theorem}
This theorem will help in obtaining a certain kind of classical
solution of \eqref{star} in the case where the family
$\{A(t)\}_{t\in [0,T]}$ of infinitesimal generators of $C_0$
semigroups on $X$ satisfies conditions (H1)--(H3).
\begin{definition}[{\cite[Definition 5.4.1]{p1}}] \label{def5} \rm
Let $X$ and $Y$ be Banach spaces such
that $Y$ is densely and continuously imbedded in $X$ and let
$\{A(t)\}_{t\in [0,T]}$ be a family of infinitesimal generators of
$C_0$ semigroups on $X$ satisfying the assumptions (H1)--(H3).
A function $u\in C([s,T]:Y)$ is a \emph{Y-valued solution} of
\eqref{star} if $u\in C^1((s,T):X)$ and $u$ satisfies \eqref{star}
in $X$.
\end{definition}
\noindent \textbf{Remark.} A $Y$-valued solution $u$ of
\eqref{star} is a classical solution of \eqref{star} such that
$u(t)\in Y\subseteq \operatorname{Dom}(A(t))$ for $t\in [s,T]$ and
$u(t)$ is continuous in the stronger $Y$-norm rather than merely
in the $X$-norm.
\begin{theorem}[{\cite[Thm. 5.4.3]{p1}}] \label{thm4}
Let $\{A(t)\}_{t\in [0,T]}$ satisfy the conditions of Theorem
\ref{thm3} and let $U(t,s)$, $0\leq s\leq t\leq T$ be the
evolution system given in Theorem \ref{thm3}. If
\begin{itemize}
\item[(E4)] $U(t,s)Y\subseteq Y$ for $0\leq s\leq t\leq T$ and
\item[(E5)] For $x\in Y$, $U(t,s)x$ is continuous in $Y$ for
$0\leq s\leq t\leq T$,
\end{itemize} then for every $x\in Y$, $U(t,s)x$ is the unique
$Y$-valued solution of \eqref{star}.
\end{theorem}
We now use the above theory of stable families of generators to
give criteria for well-posedness of the evolution problem
\eqref{e2}. Let \eqref{e2'} denote the initial value problem
\eqref{e2} with $0$ replaced by $s$ for $s\in [0,T)$; i.e.,
\begin{equation}
\begin{gathered} \label{e2'}
\frac{dv(t)}{dt}
= f(t,D)v(t) \quad 0\leq s\leq t< T \\
v(s) = \chi.
\end{gathered}
\end{equation}
We determine conditions on $f$ so that the family of operators
$\{f(t,D)\}_{t\in [0,T]}$ is stable and such that \eqref{e2'} is
well-posed.
\begin{proposition} \label{prop1}
Let $f:[0,T]\times [0,\infty)\to \mathbb{R}$ be continuous in $t$
and Borel in $\lambda$. Assume there exist $\omega\in \mathbb{R}$
such that $f(t,\lambda)\leq \omega$ for all $(t,\lambda)\in
[0,T]\times [0,\infty)$ and a Borel function $r:[0,\infty)\to
[0,\infty)$ such that $|f(t,\lambda)|\leq r(\lambda)$ and
$\operatorname{Dom}(f(t,D))=\operatorname{Dom}(r(D))$ for all
$t\in [0,T]$. Set $Y=\operatorname{Dom}(r(D))$ and let $\|\cdot
\|_Y$ denote the graph norm associated with the operator $r(D)$.
Further, assume $t\mapsto f(t,D)$ is continuous in the $B(Y,H)$
norm $\|\cdot \|_{Y\to H}$. Then \eqref{e2'} is well-posed and
for $\chi \in Y$, $V(t,s)\chi=e^{\int_s^tf(\tau,D)d\tau}\chi$ is a
unique $Y$-valued solution of \eqref{e2'}.
\end{proposition}
\begin{proof}
By \cite[Theorem XII.2.6]{d1}, $r(D)$ is a closed operator in $H$
with dense domain. Set $Y=\operatorname{Dom}(r(D))$ and endow
$Y$ with the graph norm $\|\cdot\|_Y$ given by
$$
\|y\|_Y=\|y\|+\|r(D)y\|
$$
for all $y\in Y$. Since $r(D)$ is a closed operator, it follows
that $(Y,\|\cdot\|_Y)$ is a Banach space. It is also clear
that $Y$ is densely and continuously imbedded in $H$.
Since $f(t,\lambda)\leq \omega$ for all $(t,\lambda)\in
[0,T]\times [0,\infty)$, we have that for each $t\in [0,T]$,
$f(t,D)$ is the infinitesimal generator of the $C_0$ semigroup
$\{S_t(s)\}_{s\geq 0}$ on $H$ given by $S_t(s)=e^{sf(t,D)}$. We
show that the family $\{f(t,D)\}_{t\in [0,T]}$ satisfies
conditions (H1)--(H3).
Let $t\in [0,T]$, $x\in H$. Then
\[
\|e^{sf(t,D)}x\|^2
= \int_0^{\infty}|e^{sf(t,\lambda)}|^2d(E(\lambda)x,x)
\leq (e^{s\omega})^2\int_0^{\infty}d(E(\lambda)x,x)
= (e^{s\omega})^2\|x\|^2,
\]
showing that $\|S_t(s)\|=\|e^{sf(t,D)}\|\leq e^{\omega s}$. Thus,
$\{f(t,D)\}_{t\in [0,T]}$ is a stable family with stability
constants $M=1$ and $\omega$, and so (H1) is satisfied.
Next, let $t\in [0,T]$, $y\in Y$. For any $s\geq 0$, since $y\in
Y=\operatorname{Dom}(r(D))$, we have
$$
\int_0^{\infty}|r(\lambda)e^{sf(t,\lambda)}|^2d(E(\lambda)y,y)\leq
(e^{s\omega})^2\int_0^{\infty}|r(\lambda)|^2d(E(\lambda)y,y)<\infty.
$$
Thus, $S_t(s)y\in \operatorname{Dom}(r(D))$ and so $Y$ is an
invariant subspace of $S_t(s)$. Let $\tilde{S}_t(s)$ be the
restriction of $S_t(s)$ to $Y$. For any positive constant $c$,
for $0\leq s\leq c$,
$$
|r(\lambda)(e^{sf(t,\lambda)}-1)|^2 \leq
|r(\lambda)|^2(e^{c\omega}+1)^2\in L^1(E(\cdot)y,y).
$$
Therefore, by Lebesgue's Dominated Convergence Theorem,
\begin{align*}
\lim_{s\to 0^+}\|r(D)(S_t(s)-I)y\|^2
&= \lim_{s\to 0^+}\int_0^{\infty}|r(\lambda)(e^{sf(t,\lambda)}-1)
|^2d(E(\lambda)y,y) \\
&= \int_0^{\infty}\lim_{s\to 0^+}|r(\lambda)(e^{sf(t,\lambda)}-1)
|^2d(E(\lambda)y,y)
= 0,
\end{align*}
and so
\begin{align*}
\|\tilde{S}_t(s)y-y\|_Y
&= \|\tilde{S}_t(s)y-y\|+\|r(D)(\tilde{S}_t(s)y-y)\| \\
&= \|S_t(s)y-y\|+\|r(D)(S_t(s)-I)y\| \\
&\to 0 \quad \text{as } s\to 0^+.
\end{align*}
Thus, $\tilde{S}_t(s)$ is a $C_0$ semigroup on $Y$.
Next, consider the family $\{\tilde{f}(t,D)\}_{t\in
[0,T]}$ of parts $\tilde{f}(t,D)$ of $f(t,D)$ in $Y$. For each
$t\in [0,T]$, $\tilde{f}(t,D)$ is defined by
$$
\operatorname{Dom}(\tilde{f}(t,D))=\{x\in
\operatorname{Dom}(f(t,D))\cap Y : f(t,D)x\in Y\}
$$
and
$$
\tilde{f}(t,D)x=f(t,D)x \quad \text{for } x\in
\operatorname{Dom}(\tilde{f}(t,D)).
$$
It is seen \cite[Theorem 4.5.5]{p1} that $\tilde{f}(t,D)$ is the
infinitesimal generator of the $C_0$ semigroup $\tilde{S}_t(s)$.
Moreover, for
$y\in Y$,
\begin{align*}
\|\tilde{S}_t(s)y\|_Y
&= \|\tilde{S}_t(s)y\| +\|r(D)\tilde{S}_t(s)y\| \\
&= \|S_t(s)y\| +\|r(D)S_t(s)y\| \\
&\leq e^{s\omega}\|y\| + e^{s\omega}\|r(D)y\| \\
&= e^{s\omega}\|y\|_Y.
\end{align*}
Thus, $\|\tilde{S}_t(s)\|_Y\leq e^{\omega s}$ for all $t\in [0,T]$
and so the family $\{\tilde{f}(t,D)\}_{t\in [0,T]}$ is stable with
stability constants $\tilde{M}=1$ and $\omega$. We have shown
that (H2) is satisfied.
Finally, let $t\in [0,T]$. Since $|f(t,\lambda)|\leq r(\lambda)$,
we have for $y\in Y$,
$$
\int_0^{\infty}|f(t,\lambda)|^2d(E(\lambda)y,y)
\leq \int_0^{\infty}|r(\lambda)|^2d(E(\lambda)y,y)<\infty.
$$
Thus $\operatorname{Dom}(f(t,D))\supseteq Y$. Also, for $y\in Y$,
\[
\|f(t,D)y\| \leq \|y\| + \|f(t,D)y\|
\leq \|y\|+\|r(D)y\| = \|y\|_Y,
\]
showing that $f(t,D)$ is a bounded operator from $Y$ into $H$.
By assumption, $t\mapsto f(t,D)$ is continuous in the $B(Y,H)$
norm $\|\cdot \|_{Y \to H}$ and so (H3) is satisfied.
By Theorem \ref{thm3}, there exists a unique evolution system $V(t,s)$,
$0\leq s\leq t\leq T$, in $H$ satisfying conditions (E1)-(E3)
with the operators $f(t,D)$, $t\in [0,T]$, and $M=1$ in the
condition (E1); that is we have
\begin{gather*}
\|V(t,s)\|\leq e^{\omega (t-s)} \quad \text{for }
0\leq s\leq t\leq T, \\
\frac{\partial^+}{\partial t}V(t,s)y\big|_{t=s}=f(s,D)y \quad
\text{for } y\in Y, \; 0\leq s\leq T,\\
\frac{\partial}{\partial s}V(t,s)y=-V(t,s)f(s,D)y \quad \text{for }
y\in Y, \; 0\leq s\leq t\leq T,
\end{gather*}
where the derivatives are in the strong sense in $H$.
It can be shown using the Spectral Theorem that
$e^{\int_s^tf(\tau,D)d\tau}$ is such an evolution system,
and so by uniqueness we must have $V(t,s)=e^{\int_s^tf(\tau,D)d\tau}$.
It is also readily seen that $V(t,s)=e^{\int_s^tf(\tau,D)d\tau}$
satisfies (E4) and (E5). Therefore, by Theorem \ref{thm4}, for every
$\chi\in Y$, $V(t,s)\chi=e^{\int_s^tf(\tau,D)d\tau}\chi$
is the unique $Y$-valued solution of \eqref{e2'}.
Finally, suppose $v_1$ is a classical solution of
\eqref{e2'}. Then $v_1(q)\in
\operatorname{Dom}(f(q,D))=\operatorname{Dom}(r(D))$ for
$q\in (s,T)$. As $V(t,s), \; 0\leq s\leq t\leq T$, satisfies
condition (E3) with the operators $f(t,D)$, $t\in [0,T]$, the function
$q\mapsto V(t,q)v_1(q)$ is then differentiable and
\begin{align*}
\frac{\partial}{\partial q} V(t,q)v_1(q)
&= -V(t,q)f(q,D)v_1(q)+V(t,q)\frac{d}{dq}v_1(q) \\
&= -V(t,q)f(q,D)v_1(q)+V(t,q)f(q,D)v_1(q) = 0.
\end{align*}
Thus $V(t,q)v_1(q)$ is constant for $q\in (s,t)$. Since $v_1$ is
a classical solution, the function $V(t,q)v_1(q)$ is also continuous for
$q\in [s,t]$. Thus we have
$$
v_1(t)=V(t,t)v_1(t)=V(t,s)v_1(s)=V(t,s)\chi.
$$
Thus condition
(ii) of Definition \ref{def2} is satisfied and we see that \eqref{e2'}
is well-posed with unique classical solution given by
$v(t)=V(t,s)\chi$.
\end{proof}
\section{The Approximation Theorem}
In order that solutions of \eqref{e2} approximate known solutions
of \eqref{e1}, we will require additional conditions on $f$. The
following definition is inspired by results obtained by Ames and
Hughes \cite[Definition 1]{a3} for continuous dependence on
modelling in the autonomous case, that is when $A(t)=A$ is
independent of $t$.
\begin{definition} \label{def6} \rm
Let $f:[0,T]\times [0,\infty)\to \mathbb{R}$ be a function
continuous in $t$ and Borel in $\lambda $ and assume the hypotheses
of Proposition \ref{prop1}. Then $f$ is said to satisfy the
\emph{approximation condition with polynomial p} or simply
\emph{Condition} $(\mathcal{A},p)$ if there exist a constant
$\beta$, with $0<\beta<1$, and a nonzero polynomial $p(\lambda)$
independent of $\beta$ such that for each $t\in [0,T]$,
$\operatorname{Dom}(p(D))\subseteq \operatorname{Dom}(A(t,D))
\cap \operatorname{Dom}(f(t,D))$, and
$$
\|(-A(t,D)+f(t,D))\psi\|\leq \beta\|p(D)\psi\|,
$$
for all $\psi\in \operatorname{Dom}(p(D))$.
\end{definition}
Now assume $f$ satisfies Condition $(\mathcal{A},p)$.
For each $t\in [0,T]$, set
$$
g(t,\lambda)=-A(t,\lambda)+f(t,\lambda),
$$
and for each $n\geq |\omega|$, set
$$
e_n=\{\lambda\in [0,\infty): \max_{t\in [0,T]}|g(t,\lambda)|\leq n\}.
$$
Then
\begin{gather*}
\lambda\in e_n
\Rightarrow \max_{t\in [0,T]}|g(t,\lambda)|\leq n \\
\Rightarrow |g(t,\lambda)|\leq n \quad \forall t\in [0,T] \\
\Rightarrow A(t,\lambda)\leq n+f(t,\lambda) \quad \forall t\in [0,T].
\end{gather*}
Since $A(t,\lambda)\geq 0$ and $f(t,\lambda)\leq \omega$ for all
$(t,\lambda)\in [0,T]\times [0,\infty)$, we have that on $e_n$,
$$
\max_{t\in [0,T]}|A(t,\lambda)|\leq n+\omega.
$$
Since $f(t,\lambda)=A(t,\lambda)+g(t,\lambda)$, it then follows
that on $e_n$,
$$
\max_{t\in [0,T]}|f(t,\lambda)|\leq 2n+\omega.
$$
Set $E_n=E(e_n)$ and let $\psi\in H$ be arbitrary. Consider
the following three evolution problems:
\begin{equation} \label{e3}
\begin{gathered}
\frac{du_n(t)}{dt} = A(t,D)E_nu_n(t) \quad 0\leq s \leq t < T \\
u_n(s) = \psi,
\end{gathered}
\end{equation}
\begin{equation} \label{e4}
\begin{gathered}
\frac{dv_n(t)}{dt} = f(t,D)E_nv_n(t) \quad 0\leq s \leq t < T \\
v_n(s) = \psi,
\end{gathered}
\end{equation}
\begin{equation} \label{e5}
\begin{gathered}
\frac{dw_n(t)}{dt} = g(t,D)E_nw_n(t) \quad 0\leq s \leq t < T \\
w_n(s) = \psi.
\end{gathered}
\end{equation}
Problems \eqref{e3}--\eqref{e5}, as we will see, are well-posed due to
the action of $E_n$ and their solutions will aid in approximating
known solutions of the ill-posed problem \eqref{e1}.
\begin{lemma} \label{lem1}
For each $t\in [0,T]$, $A(t,D)E_n$ is a bounded operator on
$H$ such that
$$
\|A(t,D)E_n\|\leq n+\omega,
$$
and \eqref{e3} has a unique classical solution
$u_n(t)=U_n(t,s)\psi$. The solution operator $U_n(t,s)$
is a bounded operator on $H$ with
$$
\|U_n(t,s)\|\leq e^{T(n+\omega)}
$$
for all $s,t$ such that $0\leq s\leq t\leq T$.
Furthermore, if $\psi$ is replaced by $\psi_n=E_n\psi$
in \eqref{e3}, then
$$
U_n(t,s)\psi_n=e^{\int_s^tA(\tau,D)d\tau}\psi_n.
$$
\end{lemma}
\begin{proof}
Fix $t\in [0,T]$. For all $x\in H$, by \cite[Theorem XII.2.6]{d1},
\begin{align*}
\|A(t,D)E_nx\|^2
&= \int_0^{\infty}|A(t,\lambda)|^2d(E(\lambda)E_nx,E_nx) \\
&= \int_{e_n}|A(t,\lambda)|^2d(E(\lambda)x,x) \\
&\leq (n+\omega)^2\int_{e_n}d(E(\lambda)x,x) \\
&\leq (n+\omega)^2\int_0^{\infty}d(E(\lambda)x,x) \\
&= (n+\omega)^2\|x\|^2,
\end{align*}
showing that $A(t,D)E_n$ is a bounded operator on $H$
with $\|A(t,D)E_n\|\leq n+\omega$.
Next, let $t_0\in [0,T]$. Since $e_n$ is a bounded subset
of $[0,\infty)$, we have that $D^jE_n\in B(H)$ for each
$1\leq j\leq k$. Then by continuity of $a_j$ for each
$1\leq j\leq k$, we have
\begin{align*}
\|A(t,D)E_n-A(t_0,D)E_n\|
&= \|\sum_{j=1}^k(a_j(t)-a_j(t_0))D^jE_n\| \\
&\leq \sum_{j=1}^k|a_j(t)-a_j(t_0)|\;\|D^jE_n\|
\to 0 \quad \text{as} \;\; t\to t_0,
\end{align*}
showing that $t\mapsto A(t,D)E_n$ is continuous in the uniform
operator topology. It follows from Theorem \ref{thm1} that \eqref{e3}
has a unique classical solution $u_n(t)=U_n(t,s)\psi$. That
$$
\|U_n(t,s)\|\leq e^{T(n+\omega)}
$$
follows directly from Theorem \ref{thm2} (i) and the fact that
$\|A(t,D)E_n\|\leq n+\omega$ for all $t\in [0,T]$.
Next, set $\psi_n=E_n\psi$ and let \eqref{e3'}
denote the evolution problem \eqref{e3} with $\psi$ replaced
by $\psi_n$; i.e.,
\begin{equation} \label{e3'}
\begin{gathered}
\frac{du_n(t)}{dt} = A(t,D)E_nu_n(t) \quad 0\leq s \leq t < T,\\
u_n(s) = \psi_n.
\end{gathered}
\end{equation}
Using the Spectral Theorem it can be shown that
$e^{\int_s^tA(\tau,D)d\tau}\psi_n$ is a classical solution
of \eqref{e3'}. In particular, using properties of the projection
operator $E_n$, we have
\begin{align*}
\frac{d}{dt}e^{\int_s^tA(\tau,D)d\tau}\psi_n
&= A(t,D)e^{\int_s^tA(\tau,D)d\tau}\psi_n \\
&= A(t,D)E_ne^{\int_s^tA(\tau,D)d\tau}\psi_n,
\end{align*}
and
$$
e^{\int_s^sA(\tau,D)d\tau}\psi_n=\psi_n.
$$
Therefore, by uniqueness guaranteed by Theorem \ref{thm1}, we have
\[
U_n(t,s)\psi_n=e^{\int_s^tA(\tau,D)d\tau}\psi_n.
\]
\end{proof}
\begin{lemma} \label{lem2}
For each $t\in [0,T]$, $f(t,D)E_n$ is a bounded operator on $H$
such that
$$
\|f(t,D)E_n\|\leq 2n+\omega,
$$
and
\eqref{e4} has a unique classical solution $v_n(t)=V_n(t,s)\psi$.
The solution operator $V_n(t,s)$ is a bounded operator on
$H$ with
$$
\|V_n(t,s)\|\leq e^{T(2n+\omega)}
$$
for all $s,t$ such that $0\leq s\leq t\leq T$.
Furthermore, if $\psi$ is replaced by $\psi_n=E_n\psi$
in \eqref{e4}, then
$$
V_n(t,s)\psi_n=e^{\int_s^tf(\tau,D)d\tau}\psi_n.
$$
\end{lemma}
\begin{proof}
Using the fact that on $e_n$,
$\max_{t\in [0,T]}|f(t,\lambda)|\leq 2n+\omega$, it is easily
shown that for each $t\in [0,T]$, $f(t,D)E_n$ is a bounded
operator on $H$ such that $\|f(t,D)E_n\|\leq 2n+\omega$.
Next, let $t_0\in [0,T]$. Since
$E_nH\subseteq \operatorname{Dom}(f(t,D))=\operatorname{Dom}(r(D))$
for all $t\in [0,T]$, we have $r(D)E_n \in B(H)$, and so
\begin{align*}
&\|f(t,D)E_n-f(t_0,D)E_n\|\\
&= \sup_{x\in H,\,\|x\|\leq 1} \|(f(t,D)-f(t_0,D))E_nx\| \\
&\leq \sup_{x\in H,\, \|x\|\leq 1} \|f(t,D)-f(t_0,D)
\|_{Y\to H}\|E_nx\|_Y \\
&= \sup_{x\in H,\, \|x\|\leq 1} \|f(t,D)-f(t_0,D)\|_{Y\to H}
(\|E_nx\|+\|r(D)E_nx\|) \\
&\leq \|f(t,D)-f(t_0,D)\|_{Y\to H}(\|E_n\|+\|r(D)E_n\|)
\to 0 \quad \text{as } t\to t_0
\end{align*}
by the assumption that $t\mapsto f(t,D)$ is continuous in the
$B(Y,H)$ norm $\|\cdot \|_{Y\to H}$. Therefore,
$t\mapsto f(t,D)E_n$ is continuous in the uniform operator topology.
It follows from Theorem \ref{thm1} that \eqref{e4}
has a unique classical solution $v_n(t)=V_n(t,s)\psi$. That
$$
\|V_n(t,s)\|\leq e^{T(2n+\omega)}
$$
follows directly from Theorem \ref{thm2} (i) and the fact that
$\|f(t,D)E_n\|\leq 2n+\omega$ for all $t\in [0,T]$.
The rest of the proof is similar to that of Lemma \ref{lem1}.
\end{proof}
\begin{lemma} \label{lem3}
For each $t\in [0,T]$, $g(t,D)E_n$ is a bounded operator
on $H$ such that $$\|g(t,D)E_n\|\leq n,$$ and \eqref{e5} has a
unique classical solution $w_n(t)=W_n(t,s)\psi$.
The solution operator $W_n(t,s)$ is a bounded operator on
$H$ with
$$
\|W_n(t,s)\|\leq e^{Tn}
$$
for all $s,t$ such that $0\leq s\leq t\leq T$.
Furthermore, if $\psi$ is replaced by $\psi_n=E_n\psi$ in \eqref{e5},
then
$$
W_n(t,s)\psi_n=e^{\int_s^tg(\tau,D)d\tau}\psi_n.
$$
\end{lemma}
\begin{proof}
Using the fact that on $e_n$,
$\max_{t\in [0,T]}|g(t,\lambda)|\leq n$, it is easily shown that
for each $t\in [0,T]$, $g(t,D)E_n$ is a bounded operator on $H$
such that $\|g(t,D)E_n\|\leq n$. Also, by the relation
$g(t,D)E_n=-A(t,D)E_n+f(t,D)E_n$, it follows that
$t\mapsto g(t,D)E_n$ is continuous in the uniform operator topology.
Therefore, by Theorem \ref{thm1}, \eqref{e5} has a unique classical solution
$w_n(t)=W_n(t,s)\psi$. That
$$
\|W_n(t,s)\|\leq e^{Tn}
$$
follows directly from Theorem \ref{thm2} (i) and the fact that
$\|g(t,D)E_n\|\leq n$ for all $t\in [0,T]$.
The rest of the proof is similar to that of Lemma \ref{lem1}.
\end{proof}
\begin{corollary} \label{coro1}
Let $\psi \in H$ and $\psi_n=E_n\psi$. Then
$$
U_n(t,s)W_n(t,s)\psi_n = V_n(t,s)\psi_n
= W_n(t,s)U_n(t,s)\psi_n
$$
for all $0\leq s\leq t\leq T$.
\end{corollary}
The corollary above follows immediately from Lemmas
\ref{lem1}, \ref{lem2}, and \ref{lem3}, and
from properties of the functional calculus for unbounded
self-adjoint operators \cite[Corollary XII.2.7]{d1}.
We now have all the necessary machinery to prove our
approximation theorem. Our strategy will be to extend the
solutions $u_n(t)$ of \eqref{e3} with $\psi=\chi_n$,
and $v_n(t)$ of \eqref{e4} with $\psi = \chi_n$, into the complex
strip $S=\{t+i\eta:t\in [0,T],\;\eta\in \mathbb{R}\}$,
and eventually employ Hadamard's Three Lines Theorem (cf. \cite{r1}).
To make use of such extensions we will need the following results.
Our approach is motivated by work of Agmon and Nirenberg \cite{a1}.
\begin{definition}[{\cite[Definition 11.1]{r1}}] \label{def7} \rm
Let $\phi(\alpha)$ be a complex function defined in a plane open
set $\Omega$. Assume all partial derivatives of $\phi$ exist
and are continuous. Define the \emph{Cauchy-Riemann operator}
$\bar{\partial}$ as
$$
\bar{\partial}=\frac{1}{2}
\Big(\frac{\partial}{\partial t}+i\frac{\partial}{\partial \eta}\Big),
$$
where $\alpha=t+i\eta$.
\end{definition}
\begin{theorem}[{\cite[Theorem 11.2]{r1}}] \label{thm5}
Suppose $\phi(\alpha)$ is a complex function in $\Omega$ such
that all partial derivatives of $\phi$ exist and are continuous.
Then $\phi$ is analytic in $\Omega$ if and only if the Cauchy-Riemann
equation
$$
\bar{\partial}\phi(\alpha)=0
$$
holds for every $\alpha\in \Omega$.
\end{theorem}
\begin{lemma}[\cite{a1}] \label{lem4}
Let $\phi(z)$ be a complex function with $z = x+iy$.
Assume $\phi(z)$ is continuous and bounded on
$S=\{z=x+iy : x\in [0,T],y\in \mathbb{R}\}$.
For $\alpha=t+i\eta \in S$, define
$$
\Phi(\alpha)=-\frac{1}{\pi}\int \int_S \phi(z)
\Big(\frac{1}{z-\alpha}+\frac{1}{\bar{z}+1+\alpha}\Big)\,dx\,dy.
$$
Then $\Phi(\alpha)$ is absolutely convergent,
$\bar{\partial}\Phi(\alpha)=\phi(\alpha)$, and there exists a
constant $K$ such that
$$
\int_{-\infty}^{\infty}\big|\frac{1}{z-\alpha}
+\frac{1}{\bar{z}+1+\alpha}\big|dy
\leq K\Big(1+{\rm{log}}\frac{1}{|x-t|}\Big)
$$
if $x\neq t$.
\end{lemma}
We now state and prove our approximation theorem.
\begin{theorem} \label{thm6}
Let $D$ be a positive self-adjoint operator acting on $H$ and let
$A(t,D)$ be defined as above for all $t\in [0,T]$. Let $f$
satisfy Condition $(\mathcal{A},p)$, and assume that there exists
a constant $\gamma$, independent of $\beta$, $\omega$, and $t$
such that $g(t,\lambda)\leq \gamma$, for all $(t,\lambda)\in
[0,T]\times [0,\infty)$. Then if $u(t)$ and $v(t)$ are classical
solutions of $\eqref{e1}$ and $\eqref{e2}$ respectively, and if
there exist constants $M',M'',M'''\geq 0$ such that $\|u(T)\|\leq
M'$, $\|p(D)\chi\|\leq M''$, and $\|p(D)A(t,D)u(T)\|\leq M'''$ for
all $t\in [0,T]$, then there exist constants $C$ and $M$
independent of $\beta$ such that for $0\leq t