\documentstyle[twoside]{article}
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\markboth{\hfil Quasireversibility Methods  \hfil EJDE--1994/08}% 
{EJDE--1994/08\hfil G.W. Clark \& S. F. Oppenheimer\hfil}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
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\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}\newline
Vol. {\bf 1994}(1994), No. 08, pp. 1-9. Published November 29, 1994.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
Quasireversibility Methods for Non-Well-Posed Problems
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35A35, 35R25.\newline\indent
{\em Key words and phrases:} Quasireversibility, Final Value Problems, 
Ill-Posed Problems.
\newline\indent
\copyright 1994 Southwest Texas State University  and University of 
North Texas.\newline\indent
Submitted: November 14, 1994.\newline\indent
Partially supported by Army contract DACA 39-94-K-0018 (S. F. O.)} }
\date{}
\author{Gordon W. Clark \\ and \\ Seth F. Oppenheimer}

\maketitle

\begin{abstract}
The final value problem, 
$$
\left\{ \begin{array}{ll}
u_t+Au=0\,, & 0<t<T \\ 
u(T)=f & 
\end{array}\right.  
$$
with positive self-adjoint unbounded $A$
is known to be ill-posed. One approach to dealing with this has been the
method of quasireversibility, where the operator is perturbed to obtain a
well-posed problem which approximates the original problem. In this work,
we will use a quasi-boundary-value method, where we perturb the final 
condition to form an approximate non-local problem depending on a small 
parameter $\alpha$. We show that the approximate problems are well posed 
and that their solutions $u_\alpha $ converge on $[0,T]$ if and only if 
the original problem has a classical solution. We obtain several other 
results, including some explicit convergence rates.
\end{abstract}

\section{Introduction}

Let $A$ be a self-adjoint operator on a Hilbert space $H$ such that $-A$
generates a compact contraction semi-group on $H$. We consider the problem
of finding a $u:[0,T]\longrightarrow H$ such that
$$
\left\{ 
\begin{array}{ll}
u^{\prime }(t)+Au(t)=0\,, & 0<t<T \\ 
u(T)=f & 
\end{array}
\right.  \eqno (FVP)
$$
for some prescribed final value $f$ in $H$. Such problems are not well
posed, that is, even if a unique solution exists on $[0,T]$ it need not
depend continuously on the final value $f$. One method for approaching such
problems is quasi reversibility, introduced by Lattes and Lions in the
1960's. The idea is to replace (FVP) with an approximate problem which is
well posed, then use the solutions of this new problem to construct
approximate solutions to (FVP). In the original method of quasi
reversibility [2] Lattes and Lions approximate (FVP) with 
$$
\left\{ 
\begin{array}{cc}
v_\alpha ^{\prime }(t)+Av_\alpha (t)-\alpha A^2v_\alpha (t)=0\,, & 0<t<T \\ 
v_\alpha (T)=f\,, & 
\end{array}
\right.  
$$
where the operator $A$ is replaced by a perturbation, in this case by ${%
A-\alpha A}^2.$ For each $\alpha >0$, they use the {\it initial value\/} $u_0=v_a(0)$
in 
$$
\left\{ 
\begin{array}{cc}
u_\alpha ^{\prime }(t)+Au_\alpha (t)=0\,, & 0<t<T \\ 
u_\alpha(0)=v_\alpha(0)\,.  & 
\end{array}
\right.
$$
Finally they show that the $u_\alpha (T)$ converge to $f$ as {}$\alpha $
tends to zero. The method does not consider $u(t)$ for $t<T$ and the
operator carrying $f$ into $v_\alpha (0)$ has large norm for small $\alpha $
(on the order of {}$e^{\frac c\alpha }$)[3].

In [6], Showalter approximates (FVP) with%
$$
\left\{ 
\begin{array}{cc}
v_\alpha ^{\prime }(t)+\alpha Av_\alpha ^{\prime }(t)+Av_\alpha (t)=0\,, & 
0<t<T \\ 
v_\alpha (T)=f & 
\end{array}
\right.  
$$
and as above for each $\alpha >0$, uses the initial value $u_0=v_\alpha (0)$
in%
$$
\left\{ 
\begin{array}{cc}
u_\alpha ^{\prime }(t)+Au_\alpha (t)=0\,, & 0<t<T \\ 
u_a(0)=v_\alpha (0)\,.  & 
\end{array}
\right. 
$$
The solutions $u_a$ are shown to approximate (FVP) in the sense that $%
u_\alpha (T)$ converges to $f$ as $\alpha $ tends to zero. Also the $%
u_\alpha (t)$ are shown to converge to the solution $u(t)$ of (FVP) if
and only if such
exists, but again the norm of the function carrying $f$ to $v_\alpha \left(
0\right) $ is quite large for small $\alpha $.

Miller [3] addresses this problem of large norm by finding optimal
perturbations of the operator A. He states that it should be possible to
make the norm on the order of $\frac c\alpha $ rather than $exp(\frac
c\alpha )$ and derives conditions on the perturbation $f(A)$ to achieve
{\it best possible\/} results. As in the methods above he approximates (FVP) with%
$$
\left\{ 
\begin{array}{cc}
v_{}^{\prime }(t)+f(A)v_{}(t)=0\,, & 0<t<T \\ 
v_{}(T)=f & 
\end{array}
\right.  
$$
and again solves the problem forward using $v(0)$ as an initial condition.
Miller calls this {\it stabilized quasi reversibility\/}.

Finally Showalter [7] addresses a more general problem in a different way. He
approximates the problem%
$$
\left\{ 
\begin{array}{cc}
u^{\prime }(t)+Au(t)-Bu(t)=0\,, & 0<t<T \\ 
u(0)=f\,. & 
\end{array}
\right.  
$$
with%
$$
\left\{ 
\begin{array}{cc}
u^{\prime }(t)+Au(t)-Bu(t)=0\,, & 0<t<T \\ 
u(0)+\alpha u(T)=f\,. & 
\end{array}
\right.  
$$
He calls this the {\it quasi-boundary-value method\/}, and he suggests that this
method gives a better approximation than many other quasireversibility type
methods. In this work we study this method to approximate (FVP) and prove
results analogous to the ones stated in [7].  We note that (FVP) is a special
case of the problem studied in [7].  However, ore results are proved directly
and this allows us to obtain explicit estimates for the convergence rate of 
the approximations.

\section{Perturbing the final conditions} 
We approximate (FVP) with the {\it quasi-boundary value problem\/} 
$$
\left\{ \begin{array}{cc}
u^{\prime }(t)+Au(t)=0\,, & 0<t<T \\ 
\alpha u(0)+u(T)=f\,. & 
\end{array}
\right. \eqno (QBVP)
$$
One superficial advantage of this method is that there is no need to 
{\it solve forward\/} here. More importantly, the error introduced by small changes in
the final value $f$ is not exponential, but of the order $\frac 1\alpha $.
We will show that this problem is well posed for each $\alpha >0,$ and that
the approximations $u_\alpha $ are stable. We show that $u_\alpha (T)$
converges to $f$ as $\alpha $ goes to zero and that the values $u_\alpha (t)$
converge on $[0,T]$ if and only if (FVP) has a solution.

In the following, assume that $H$ is a separable Hilbert space and $A$ is as
above and that $0$ is in the resolvent set of $A$. Let $S(t)$ be the compact
contraction semi-group generated by $-A$. Since $A^{-1}$ is compact, there
is an orthonormal eigenbasis ${\phi }_n$ for $H$ and eigenvalues $\frac
1{\lambda _n}$ of $A^{-1}$ such that $A^{-1}\phi _n=\frac 1{\lambda _n}\phi
_n.$ Then the eigenvalues of $-A$ are $-\lambda _n$ and those for $S(t)$ are 
${e}^{-t\lambda _n}${\ } (and possibly zero) [5]. In particular, for each
positive $\alpha $, $\alpha I+S(T)$ is invertible. Also, if $%
u=\sum_{i=1}^\infty a_i\phi _i,$ then $S(T)u=\sum_{i=1}^\infty {e}%
^{-T\lambda _i}a_i\phi _i$ and 
$$
\left( S(T)u,u\right) =\sum_{i=1}^\infty {e}^{-T\lambda i}a_i^2\geq 0\,. 
$$
From this accretive type condition we obtain%
$$
\| \left( \alpha I+S(T)\right) ^{-1}\| \leq \frac 1\alpha\,. 
$$

It is useful to know exactly when (FVP) has a solution. The following lemma
answers this question.

\begin{lemma}
If $f=\sum_{i=1}^\infty b_i\phi _i$, then (FVP) has a solution if and only
 if \newline $\sum_{i=1}^\infty b_i^2{e}^{2T\lambda _i}$ converges.
\end{lemma}

\paragraph{Proof.} If $\sum_{i=1}^\infty b_i^2{e}^{2T\lambda _i}$
converges, we merely define $u(t)=\sum_{i=1}^\infty {e}^{\left( T-t\right)
\lambda _i}b_i\phi _i$. Let $u$ be a solution to (FVP). Then $u(0)$ has an
eigenfunction expansion $u=\sum_{i=1}^\infty a_i\phi _i,$ and 
$$
S(T)u=\sum_{i=1}^\infty e^{-T\lambda _i}a_i\phi _i=f=\sum_{i=1}^\infty
b_i\phi _i\,.
$$
This implies that ${e}^{-T\lambda _i}a_i=b_i$ and thus $a_i=b_i{%
e}^{T\lambda _i}.$ Since $u(0)$ is in $H$, we have $\left| \left| u\right|
\right| ^2=\sum_{i=1}^\infty a_i^2<\infty $ and we are done. $\Box$

We wish to show that our approximate problem is well-posed and the following
gives us what we need.

\paragraph{Definition.}
Define $u_\alpha (t)=S(t)(\alpha I+S(T))^{-1}f$, for $f$ in $H$, $\alpha >0$
and $t$ in $[0,T]$.


\begin{theorem}
The function $u_\alpha (t)$ is the unique solution of (QBVP) and it depends
continuously on f.
\end{theorem}

\paragraph{Proof.} Since $(\alpha I+S(T))^{-1}f$ is in the domain of $%
A$, it is clear that $u_\alpha $ is a classical solution of the differential
equation. Furthermore, 
\begin{eqnarray*}
\alpha u_\alpha (0)+u_\alpha (T)&=&
\alpha (\alpha I+S(T))^{-1}f+S(T)(\alpha I+S(T))^{-1}f \\
&=&(\alpha I+S(T))(\alpha I+S(T))^{-1}f=f. 
\end{eqnarray*}
To see the continuous dependence of $u_{\alpha}$ on $f$, compute 
 \begin{eqnarray*}
\lefteqn{\| S(t)(\alpha I+S(T))^{-1}f_1-S(t)(\alpha I+S(T))^{-1}f_2\|}\\
&=& \| S(t)(\alpha I+S(T))^{-1}(f_1-f_2)\| \\
&\leq&\frac 1\alpha \| f_1-f_2\|\,. 
\end{eqnarray*}
Uniqueness follows from the fact that any solution $v$ must satisfy $%
v(0)=(\alpha I+S(T))^{-1}f$ and the uniqueness of solutions to the forward
problem. $\Box$

We make two observations at this point which will be useful later. First,
from the above it is clear that $\| u_\alpha (t)\|
\leq \frac 1\alpha \| f\|$. Secondly, if 
$u=\sum_{i=1}^\infty a_i\phi _i,$ then $(\alpha I+S(T))u=\sum_{i=1}^\infty
(\alpha +{e}^{-T\lambda _i})a_i\phi _i$ and 
$$
(\alpha I+S(T))^{-1}u=\sum_{i=1}^\infty \frac{a_i}{\alpha +{e}^{-T\lambda _i}%
}\phi _i\,. 
$$

\begin{theorem}
For all $f$ in $H,$ $\alpha >0$, and $t$ in $[0,T]$ we have that 
$$
\| u_\alpha (t)\| \leq \alpha ^{\frac{t-T}T}\| f\|\, . 
$$
\end{theorem}

\paragraph{Proof.} If $f=\sum_{i=1}^\infty b_i\phi _i,$ we have 
\begin{eqnarray*}
\| u_\alpha (t)\|^2&=&\sum_{i=1}^\infty e^{-2t\lambda _i}b_i^2
\left( \alpha +e^{-T\lambda _i}\right)^{-2}\\
&\leq& \sum_{i=1}^\infty e^{-2t\lambda _i}b_i^2 
\left[ \left( \alpha +{e}^{-T\lambda _i}\right) ^{\frac tT}
\left( \alpha +e^{-T\lambda _i}\right) ^{1-\frac tT}\right] ^{-2} \\
&\leq& \sum_{i=1}^\infty b_i^2\left( \alpha ^{1-\frac tT}\right)^{-2} \\
&=& \left( \alpha ^{\frac{t-T}T}\right)^2\sum_{i=1}^\infty b_i^2
\end{eqnarray*}
and we are done. $\Box$ 

\begin{theorem}
For all $f$ in $H$, $\left| \left| u_\alpha \left( T\right) -f\right|
\right| $ tends to zero as $\alpha $ tends to zero. That is $u_\alpha \left(
T\right) $ converges to $f$ in $H.$
\end{theorem}

\paragraph{Proof.} If $f=\sum_{i=1}^\infty b_i\phi _i$, then 
\begin{eqnarray*}
\| u_\alpha (T)-f\| ^2&=&\|S(T)(\alpha I+S(T))^{-1}f-f\| ^2 \\ 
&=&\alpha ^2\|(\alpha I+S(T))^{-1}f\| ^2 \\ 
=\sum_{i=1}^\infty \alpha ^2b_i^2\left( \alpha +{e}^{-T\lambda
_i}\right) ^{-2}\,.
\end{eqnarray*}

Fix $\epsilon >0.$ Choose $N$ so that $\sum_{i=N}^\infty b_i^2<
\epsilon /2.$ Thus
\begin{eqnarray*}
\|u_\alpha (T)-f\| ^2&<&\sum_{i=1}^N
\alpha ^2b_i^2\left( \alpha +{e}^{-T\lambda _i}\right) ^{-2}+\frac
\epsilon 2 \\
&\leq& \alpha ^2\sum_{i=1}^Nb_i^2e^{2\lambda _iT}+\frac \epsilon
2\,.
\end{eqnarray*}
Now let $\alpha $ be such that $\alpha ^2< \epsilon
\left(2\sum_{i=1}^Nb_i^2e^{2\lambda _iT}\right)^{-2}$ and we are done. 
 $\Box$ 

\begin{theorem}
For all $f$ in $H,$ (FVP) has a solution $u$ if and only if the sequence $%
u_\alpha (0)$ converges in $H$. Furthermore, we then have that $u_\alpha (t)$
converges to $u(t)$ as $\alpha $ tends to zero uniformly in $t$.
\end{theorem}

\paragraph{Proof.} Assume that $\lim _{\alpha \downarrow 0}u_\alpha
(0)=u_0$ exists. Let $u(t)=S(t)u_0$. Since $\lim _{\alpha \downarrow
0}u_\alpha (T)=f$, 
\begin{eqnarray*}
\lim _{\alpha \downarrow 0}\|u(t)-u_\alpha (t)\|
&=&\|S(t)u_0-u_\alpha (t)\|  \\ 
&=&\lim _{\alpha \downarrow 0}\|S(t)\left( u_0-(\alpha
I+S(T))^{-1}f\right) \|  \\ 
&\leq& \lim _{\alpha \downarrow 0}\|u_0-(\alpha
  I+S(T))^{-1}f\|  \\ 
&=&\lim _{\alpha \downarrow 0}\|u_0-u_\alpha (0)\| =0\,.
\end{eqnarray*}
Thus, $u(T)=f$ and $u(t)=S(t)u_0$ solves (FVP). We also see that $u_\alpha
(t)$ converges to $u(t)$ uniformly in $t$.

Now let us assume that $u(t)$ is the solution to (FVP). Let $\epsilon >0$
and $f=\sum_{i=1}^\infty b_i\phi _i$. From Lemma~1 we have that 
$\| u(0)\| ^2=\sum_{i=1}^\infty b_i^2e^{2T\lambda _i}$.
Choose $N$ so that $\sum_{i=N}^\infty b_i^2{e}^{2T\lambda _i}<\frac \epsilon
2$. Let $\alpha ,\gamma >0$. Then%
\begin{eqnarray*}
\|u_\alpha (0)-u_\gamma (0)\| ^2
&=&\|(\alpha I+S(T))^{-1}f-(\gamma I+S(T))^{-1}f\|  \\ 
&=&\|\sum_{i=1}^\infty \left( \frac 1{\alpha + e^{-T\lambda _i}}
-\frac 1{\gamma +e^{-T\lambda _i}}\right) b_i\phi _i\|  \\
 &=&\sum_{i=1}^\infty  
(\gamma -\alpha )^2\left( \alpha \gamma +(\alpha +\gamma )
e^{-T\lambda _i}+{e}^{-2T\lambda _i}\right)^{-2} b_i^2 \\ 
&=&\sum_{i=1}^N 
(\gamma -\alpha )^2\left( \alpha \gamma +(\alpha +\gamma )
e^{-T\lambda _i}+e^{-2T\lambda _i}\right) ^{-2} b_i^2\\
&&+\sum_{i=N+1}^\infty (\gamma -\alpha )^2\left( \alpha
\gamma +(\alpha +\gamma )e^{-T\lambda _i}+e^{-2T\lambda _i}\right)^{-2}
 b_i^2 \\ 
&\leq& \sum_{i=1}^N(\gamma -\alpha )^2e^{4T\lambda
_i}b_i^2+\sum_{i=N+1}^\infty \left( 
\frac{\gamma -\alpha }{\alpha +\gamma }\right) ^2b_i^2e^{2T\lambda _i} \\ 
&\leq& \sum_{i=1}^N(\gamma -\alpha )^2e^{4T\lambda _i}b_i^2+
\frac \epsilon 2\,. 
\end{eqnarray*}

Now if we choose $\delta >0$ so that $\delta ^2<\epsilon\left(
\sum_{i=1}^Ne^{4T\lambda _i}b_i^2\right)^{-1}$ and require that $\alpha $ and $\gamma $
be less than $\delta $, we have that 
$$
\|u_\alpha (0)-u_\gamma (0)\| ^2<\epsilon\, . 
$$
We therefore have that $\{u_\alpha (0)\}$ is Cauchy and thus converges. From
the first part of the theorem, we have that $u_\alpha (t)$ converges to $u(t)
$ uniformly in $t$. $\Box$

We end this paper with a result that gives explicit convergence rates 
in the case that (FVP) is soluble for some positive final time.

\begin{theorem}
If $f=\sum_{i=1}^\infty b_i\phi _i$ is in $H$ and there exists an $\epsilon
>0$ so that $\sum_{i=1}^\infty b_i^2e^{\epsilon \lambda _iT}$ converges,
then $\|u_\alpha (T)-f\| $ converges to zero with
order $\alpha ^\epsilon \epsilon ^{-2}$.
\end{theorem}

\paragraph{Proof.} Let $\epsilon $ be in $(0,2)$ such that $%
\sum_{i=1}^\infty b_i^2e^{\epsilon \lambda _iT}$ is 
finite and let $k$ be in $(0,2)$. Fix a
natural number $n$. Define 
$$
g_n(\alpha )=\frac{\alpha ^k}{(\alpha +e^{-\lambda _nT})^2}. 
$$

Differentiating with respect to $\alpha $ yields%
$$
g_n'(\alpha )=\alpha ^{k-1}\frac{(k-2)\alpha +ke^{-T\lambda _n}}{%
(\alpha +e^{-\lambda _nT})^3}. 
$$
Thus $g_n'(\alpha )=0$ when either $\alpha =0$ or 
$$
\alpha =\frac k{2-k}e^{-T\lambda _n}. 
$$
Since $g_n(\alpha )>0$, $g_n(0 )=0$, and $%
\lim _{\alpha \rightarrow \infty }g_n(\alpha )=0$ we have that $\alpha
_0=\frac k{2-k}e^{-T\lambda _n}$ is the critical value at which $g_n$
achieves its maximum. Thus we have the inequality%
$$
g_n(\alpha )\leq \frac{\left( \frac k{2-k}\right) ^ke^{-kT\lambda _n}}{%
(\alpha _0+e^{-\lambda _nT})^2}. 
$$

We now calculate
\begin{eqnarray*}
\|u_\alpha (T)-f\| ^2&=&\sum_{n=1}^\infty b_n^2
\alpha ^2(\alpha +e^{-\lambda _nT})^{-2} 
 =\alpha^{2-k}\sum_{n=1}^\infty b_n^2g_n(\alpha ) \\ 
&\leq& \alpha ^{2-k}\sum_{n=1}^\infty 
b_n^2\left( \frac k{2-k}\right) ^ke^{-kT\lambda _n}
(\alpha_0+e^{-\lambda _nT})^{-2} \\ 
&\leq& \alpha ^{2-k}\sum_{n=1}^\infty 
b_n^2\left( \frac k{2-k}\right) ^ke^{(2-k)T\lambda _n}
(\alpha_0^2+2\alpha _0e^{\lambda _nT}+1)^{-1} \\ 
&\leq& \alpha ^{2-k}\sum_{n=1}^\infty
b_n^2\left( \frac k{2-k}\right) ^ke^{(2-k)T\lambda _n} \\ 
&=&\alpha ^{2-k}\left( \frac k{2-k}\right) ^k\sum_{n=1}^\infty
b_n^2e^{(2-k)T\lambda _n}.
\end{eqnarray*}
If we choose $k=2-\epsilon $ we arrive at
$$
\|u_\alpha (T)-f\| ^2\leq \left( \frac 2\epsilon
\right) ^2\alpha ^\epsilon \sum_{n=1}^\infty b_n^2e^{\epsilon T\lambda _n}  
=c\alpha ^\epsilon \epsilon ^{-2}\,.
$$
\hfill $\Box$\newline
If we assume that $\sum_{i=1}^\infty b_i^2e^{(2+\epsilon )\lambda _iT}$
converges, working as above, we have that 
\begin{eqnarray*}
\|u_\alpha (0)-u(0)\| ^2&=&\alpha
^{2-k}\sum_{n=1}^\infty b_n^2g_n(\alpha )e^{2T\lambda _n} \\ 
&\leq& \alpha ^{2-k}\sum_{n=1}^\infty b_n^2\left( \frac k{2-k}\right)
^ke^{(4-k)T\lambda _n}\,.
\end{eqnarray*}
As above, letting $k=2-\epsilon $, we arrive at the following.

\begin{corollary}
If $f=\sum_{i=1}^\infty b_i\phi _i$ is in $H$ and there exists an $\epsilon
>0$ so that $\sum_{i=1}^\infty b_i^2e^{(2+\epsilon )\lambda _iT}$ converges,
then $\|u_\alpha (t)-u(t)\| $ converges to zero
with order $\alpha ^\epsilon\epsilon ^{-2}$ uniformly in $t$.
\end{corollary}

\begin{thebibliography}{9}
\bibitem{1}  Conway, J.B., ``A Course in Functional Analysis,
Springer-Verlag, New York, 1990

\bibitem{2}  Lattes, R. and Lions, J.L., ``Methode de Quasi-Reversibility et
Applications'', Dunod, Paris, 1967 (English translation R. Bellman,
Elsevier, New York, 1969)

\bibitem{3}  Miller, K., {\it Stabilized quasireversibility and other nearly
best possible methods for non-well-posed problems}, ``Symposium on
Non-Well-Posed Problems and Logarithmic Convexity'', Lecture Notes in
Mathematics, Vol. 316, Springer-Verlag, Berlin, 1973, pp 161-176

\bibitem{4}  Payne, L.E., {\it Some general remarks on improperly posed
problems for partial differential equations}, ``Symposium on Non-Well-Posed
Problems and Logarithmic Convexity'', Lecture Notes in Mathematics, Vol.
316, Springer-Verlag, Berlin, 1973, pp 1-30

\bibitem{5}  Pazy, A., ``Semigroups of Linear Operators and Applications to
Partial Differential Equations'', Springer-Verlag, New York, 1983

\bibitem{6}  Showalter, R.E., {\it The Final Value Problem for Evolution
Equations}, J. Math. Anal. Appl. 47, 1974, pp 563-572

\bibitem{7}  Showalter, R.E., {\it Cauchy Problem for Hyper-Parabolic
Partial Differential Equations}, ``Trends in the Theory and Practice of
Non-Linear Analysis'', Elsevier, 1983

\bibitem{8}  Yosida, K.,``Functional Analysis'', Springer-Verlag, Berlin,
1980
\end{thebibliography}

{\sc Gordon W. Clark \\
Department of Mathematics\\
Kennesaw State College\\
P O Box 444\\
Marietta, GA 30061}\\
E-mail address: clark@math.msstate.edu
\smallskip
{\sc Seth F. Oppenheimer\newline 
Department of Mathematics and Statistics\\
Mississippi State University\\
Drawer MA MSU, MS 39762}\\
E-mail address: seth@math.msstate.edu
\end{document}