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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 120, pp. 1--10.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/120\hfil A stability theorem for convergence]
{A stability theorem for convergence of a lyapounov
function along trajectories of nonexpansive semigroups}

\author[R. Choudhary\hfil EJDE-2006/120\hfilneg]
{Renu Choudhary}

\address{Renu Choudhary \newline
 Department of Mathematics \\
University of Auckland \\
Private Bag 92019 \\
Auckland, New Zealand}
\email{renu@math.auckland.ac.nz}

\date{}
\thanks{Submitted July 28, 2005. Published October 2, 2006.}
\subjclass[2000]{47H05, 47H20}
\keywords{Monotone; semigroup; Lyapounov function}

\begin{abstract}
 It is known that a regularly Lyapounov function for a
 semigroup of contractions on a Hilbert space  converges
 to its minimum along the trajectories of the semigroup.
 In this paper we show that this Lyapounov function nearly
 converges to its minimum  along trajectories of the
 semigroup generated by a small bounded perturbation of
 the semigroup generator.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

 Let $ K $ be a closed convex subset of a real Hilbert space
 $H$ and let $\{ S(t) \} _{t \geq 0 }$ be a semigroup of
 contractions on $K $ generated by a maximal monotone operator
 $ A $ on $ H $. The study of convergence of a trajectory
$ S(t)x $ as $ t \to  \infty $ has called  the attention of several
mathematicians; see, for example,
\cite{bail,bbr,brez1,brez2,bruc1,slem, nr,pazy1,pazy2,reich4}.
In general $ S(t)x $ does not converge strongly or even
weakly as $ t \to \infty $, and convergence requires additional conditions.

The fact that  $ S(t)x $ does not converge even weakly, in general,
as $ t \to \infty $, together with the use of Lyapounov functions
to determine the asymptotic behaviour  of semigroups, inspired us
to consider a  Lyapounov  function $ f : H \to \mathbb{R} \cup \{ \infty \} $
 and then study convergence of  $ f(S(t)x) $ as $ t \to \infty $.

In \cite{chou}, we show that, in general in infinite dimensional space,
$f(S(t)x) \not \to \min(f) $ as  $ t \to \infty $, even if $ f $
strictly decreases along the trajectories of  $ \{ S(t)\}_{t\geq 0} $.
If for a semigroup $ \{ S(t)\}_{t\geq 0} $,  $ f $ decreases  along the
trajectories at a particular rate, then we call  $ f $ regularly Lyapounov
for the semigroup $ \{ S(t)\}_{t\geq 0} $, and we have
$f(S(t)x) \to \min(f) $ as $ t \to \infty $.  Further, under some mild
conditions on $ A $ and $ f $, we  construct a complete metric space
($ \mathcal{A}^1, d $) of  the bounded perturbations of the generator
$ A $ such that $ f $ is Lyapounov for all the semigroups generated by
these perturbations. We show that there is a very large subset
$ \mathcal{F}^1 $ of $\mathcal{A}^1 $ such that  $ f $ is
regularly Lyapounov for all the semigroups generated by the maximal
monotone operators in $\mathcal{F}^1 $. In particular, the Lyapounov
function $ f $  for $ \{ S(t) \}_{t\geq 0 } $, generated by $ A $,
converges along the trajectories of a class of semigroups generated
by most bounded perturbations of $ A $ in $ \mathcal{A}^1 $.

Since a regularly Lyapounov function $ f $ for a nonexpansive semigroup
on a Hilbert space converges to its minimum along the trajectories of
the semigroup a natural question arises about the behaviour of $ f $
along the trajectories of the semigroup  $ \{ S^{1}(t)\}_{t\geq 0} $
generated by a small bounded perturbation of the semigroup generator,
 not assumed to be in $ \mathcal{A}^1 $. So the main question becomes:
is  $ f (S^{1}(t)x ) - \min (f ) $ small for large $ t $ if the perturbation
is small? Theorem 1 and 2 of this paper give an answer to this question.
The idea came  from the work of
Reich and Zaslavski; in  \cite{reich2} they investigated a similar problem
for a continuous convex function $f $ on a Banach space along the trajectories
given by an everywhere defined bounded vector field on a Banach space.
For more recent results on continuous descent methods see
\cite{arz1,arz2,arz3}. In  \cite{reich3,reich1,reich2} one considers
the semigroup generators to be bounded vector fields, whereas here
and in \cite{chou}  we consider them to be maximal  monotone operators.
Hence we are not in a position to use
\cite{reich3,reich1,reich2} as their suppositions do not hold.
On the other hand, you might hope that this paper would
generalise \cite{reich2}. However, it does not because in \cite{reich2}
 one considers single trajectories and does not assume the semigroup
to be nonexpansive or even Lipschitz continuous. Moreover, because of
the properties of maximal monotone operators in Hilbert space, one needs
less stringent conditions on $ f $ to obtain a positive result.

Reich and Zaslavski, in  \cite{reich1,reich2},  considered a bounded below,
convex, continuous function $ f $ on a Banach space $ X $. With $ f $,
they associated a complete metric space of bounded vector fields
$ V : X \to X $ such that the right hand directional derivative of
$ f $ at  $ x $ in the direction of $ Vx $ is non-positive.
In \cite{reich3} and \cite{reich1}, they considered two gradient like
iterative processes defined by these vector fields. Under some assumptions on $ V $ and $ f $, they showed that for most of the vector fields in this complete metric space, both iterative processes  generate a sequence
$ \langle x_{n} \rangle_{n=1}^{\infty} $ such that $ f(x_n) \to \inf(f) $ as
$ n \to \infty $.
In \cite{reich2}, they gave the continuous version of the same problem.
 They showed that if there are trajectories  given by these vector fields
 then $ f $ converges to its infimum along these trajectories for most
vector fields. They defined the notion of regular vector fields and
showed that the function $ f $ converges to its infimum along the
sequences/trajectories given by these regular vector fields.
Under the assumptions that $ f $ is coercive and Lipschitz on bounded
subsets of the Banach space,  they showed that for regular vector fields
 the function $ f $ remains close to its infimum along finite horizon
perturbed trajectories.

\section{Preliminaries and Notation}

   Throughout this paper $ H $ stands for a real Hilbert space. Let $ A $
 be a maximal  monotone operator on  $ H $ such that
$ A^{-1}\{0\} \neq \emptyset  $ and let  $ \{ S(t) \} _{t \geq 0 } $
be the semigroup of contractions generated by  $ A $ on
$ K = \overline{D(A)} $. Usually $ -A $ is called the generator of
   $  \{ S(t) \} _{t \geq 0 } $ but we find it more convenient to say
$ A $ is the generator
   of $  \{ S(t) \} _{t \geq 0 } $ in the same sense as Pazy \cite{pazy}.
 Since in a Hilbert space  there is one to one correspondence between
the maximal monotone operators and the semigroups of contractions
\cite{pazy} we will switch frequently between semigroups and the maximal
monotone operators generating them.


 \begin{definition} \label{defA} \rm
Suppose $A$ is a maximal monotone operator and $\{S(t)\}_{t\geq 0}$ the
semigroup it generates. Assume $A^{-1}\{0\} $ is nonempty.
Let $ f $ be a proper  l.s.c.  function from $ H $ to
$ \mathbb{R} \cup  \{ +\infty \} $ such that
 $ K = \overline{D(A)} \subseteq \mathop{\rm Dom}f) $ and suppose
 there exists $ x_0 \in  A^{-1}\{0\} $ such that
 $ f(x_0 ) = \min(f) :=  \min\{ f(x) : x \in H \} $.
    We say
$ f $ is Lyapounov for $ \{ S(t) \} _{t \geq 0 } $  if
 \[
f(S(t)x) \leq f(x) \quad \forall  \, x \in K, \quad  t \geq 0 ,
\]
and strictly Lyapounov if
\[
f(S(t)x) <  f(x) \quad \forall  \, x \in D(A) \setminus A^{-1} \{0
\}, \quad  t >  0 .
\]
\end{definition}

\begin{definition}\label{def1} \rm
A  Lyapounov function $ f $ for a semigroup $ \{S(t) \}_{t \geq 0} $ is
called regularly  Lyapounov  for $ \{S(t) \}_{t \geq 0} $ if for each
positive integer $ n $
there exists a positive number $\delta (n) $ (depending on $n$) such that
for  every $x$ in $ D_n $, where
 $ D_{n} = \{ x \in D(A) : \|x\| \leq n, \, f(x) > \min(f) + \frac{1}{n} \} $,
there exists $\alpha (x)>0 $ such that
\[
f(x) - f(S(t)x)  \geq  t\delta (n) \quad \forall \, t \in [0, \alpha (x))\, .
 \]
\end{definition}

     The idea of regularity that we use was essentially already given
in \cite[page 4]{reich2}, and it had  previously been given in
\cite[page 1005]{reich1}, in the study of discrete descent methods.
In \cite{reich1,reich2}, they considered a bounded below, convex,
continuous function $f$ on a Banach space $X$, and  with $f$ they
associated a bounded vector field $V : X \to X $. By a regular vector
field they mean, for any natural number $ n $ there exists a positive
number $\delta (n) $ such that  for each $ x \in X$ satisfying
$ \| x\| \leq n $ and $ f(x) \geq \inf (f) + \frac{1}{n}$,
     \begin{equation}\label{a}
      f^{\circ}(x, Vx) \leq  -\delta (n) ,
      \end{equation}
where $f^{\circ}(x, Vx)$ denotes the right hand directional derivative
of $f$ at $x$ in the direction of $Vx$. That means if there are
trajectories governed by a regular vector field then the function
$f $ decreases along these trajectories at a particular rate.
In Definition 2, the function $ f $ need not be convex and continuous
and we are considering the decrease of $ f $ along trajectories of a
semigroup generated by a maximal monotone operator.
In  \cite[Proposition 4]{chou} we assumed $f $ to be convex and continuous
on $ D(A)$ and obtained an analogue of (\ref{a}) from Definition \ref{def1}.

  Through out this paper, for any real $ r > 0 $ by $ D_{r} $ we mean
$D_{r} = \{ x \in D(A): \| x \| \leq r, f(x) > \min(f) + \frac{1}{r} \} $.
 Set $ D_{\epsilon^{-1}}(x_0) = \{ x \in D(A): \| x -x_0 \| < \epsilon^{-1},
f(x) > \min ( f) + \frac{\epsilon}{2} \}$.

\section{Stability Theorems}

We begin with a simple geometrical result that  for a small perturbation
of a closed convex set the elements of minimal norm in perturbed and
unperturbed sets are  not very far from each other.

\begin{lemma}\label{l1}
Let  $K$ be a closed convex subset of $H$ and $e $ be a given vector in $H$.
Let $ K' = K + e $. Let  $y^{\circ} $ and $(y')^{\circ}$ be the elements
of minimal norm in $K$ and $K'$ respectively. Then
 \[
\| (y')^{\circ} - y^{\circ} \| \leq 2 \| e \| \, .
 \]
\end{lemma}

\begin{proof} Let $(y')^{\circ}  = y + e $, for some $y $ in $ K $.
We note that the nearest point projection $P_{K} $ of origin and $-e$
in $K$ are $y^{\circ} $  and $y $. Since $P_K$ is a contraction
\[
\| y^\circ - y \| = \| P_K(0) - P_{K}(-e) \| \leq \| e \| \, .
\]
Hence
\[
\| (y')^{\circ} - y^\circ \| = \| y + e - y^{\circ} \|
\leq  \| y - y^{\circ} \| + \| e \| \leq 2 \| e \| \,  .
\]
\end{proof}

The following hypotheses will be assumed when specified.
\begin{itemize}
\item[(A1)] $A $ is a maximal monotone operator on $H$ and
 $A^{-1}\{ 0\}$ is nonempty.
\item[(A2)] $f : H \to  \mathbb{R} $ is bounded below and
 Lipschitzian on bounded subsets of $ H $,  $ x_{0} \in A^{-1}\{ 0 \} $
 satisfies $ f (x_{0}) = \min\{ f(x) : x \in H \}  = \min (f)$, and
 $ f $ is regularly Lyapounov for the semigroup $ \{ S(t) \}_{t\geq 0 } $
 generated by $ A $.
\item[(A3)] $ D(A) $ is a convex subset of $ H $.
\end{itemize}

Our next result tells that, under the assumption (A1) and (A2),
 the function $ f $ behaves almost like a regularly Lyapounov function
for the semigroup  $ \{ S^1(t) \}_ { t \geq 0 }  $ generated by a small
perturbation of $ A $. Note $ f $ is not even a Lyapounov function for
$ \{ S^1(t) \}_ { t \geq 0 }  $.

\begin{lemma}\label{thmS}
 Let $ A $ and $ f $ satisfy (A1) and (A2). For all $ \epsilon > 0 $,
there exist positive numbers $\overline{\delta} $, $ \delta $ such that
for all $ A' $ satisfying:
\begin{enumerate}
\item $ A' $ is single valued,
\item $ D(A) \subseteq D(A') $,
\item $A' $ is bounded on bounded subsets of $ D(A) $,
\item $ A'x_{0} = 0 $,
\item $ A^1 = A + A' $ is  maximal monotone,
\item $\displaystyle \sup_{ \{ x \in D(A) : \| x - x_0 \| \leq \epsilon^{-1} \} } \| A'x \| < \delta $,
\end{enumerate}
and for every $ x \in D_{\epsilon^{-1}}(x_0)$, where recall
 $ D_{\epsilon^{-1}}(x_0) = \{ x \in D(A) : \| x - x_0 \|
 < \epsilon^{-1}, f(x) > \min (f) + \frac{\epsilon}{2} \} $,
 there exists $ \overline{\alpha} > 0 $ such that
\[
f(x) - f(S^1 (t)x) \geq t\overline{\delta} \quad \forall \,
 t \in [0, \overline{\alpha} ) \, ,
\]
 where  $ \{ S^1(t) \}_ { t \geq 0 }  $ is the semigroup generated
by $A^{1}$.
\end{lemma}

\begin{proof}
 Let $ \epsilon > 0 $ be given, and set $ M$ to be a positive integer greater than $ \max\{ \epsilon^{-1} +  \|x_{0} \| , 2\epsilon^{-1} \} $.
 Since $ f $ is Lipschitzian on bounded subsets of $H$, there exists $ L > 0 $ such that
\begin{equation}\label{S2}
| f(x) - f(y) | \leq L \| x - y \| \quad
\forall \, x, y \in \{ x \in H : \| x \| \leq M \} \, .
\end{equation}
Since $ f $ is a regularly Lyapounov function for the semigroup
$ \{ S(t)\}_ { t \geq 0 }  $, there exists $ \delta_1 > 0 $ such that for
every $ x \in D_M $ there exists $ \alpha_1 (x)> 0 $ such that
\begin{equation}\label{S3a}
f(S(t)x) - f(x) \leq -t \delta_1 \quad \forall \, t \in [0, \alpha_1(x)) \, .
\end{equation}
Let
\begin{equation}\label{S6}
 \overline{\delta} = \frac{\delta_1}{2},
\end{equation}
and
\begin{equation}\label{S7}
\delta = \frac{ \delta_1}{8L}.
\end{equation}
Let $ A' $ satisfy 1-5 and
\begin{equation}\label{1a}
 \sup_{ \{ x \in D(A) : \| x  - x_0 \| \leq \epsilon^{-1} \} } \| A'x \|
< \delta \, ,
\end{equation}
and let $ \{ S^1(t) \}_ { t \geq 0 }  $ be the semigroup generated by $A^{1}$.
Let $ x \in D_{\epsilon^{-1}}(x_0) $ be given.
 We note that  $ \|S(t)x \| $   is  bounded  for all $ t \geq 0 $ as
\begin{equation}\label{S1}
 \|S(t)x \| \leq   \| S(t)x - x_{0 } \| + \| x_{0} \| \leq \|x - x_{0} \| + \|x_{0} \|
  \leq  \epsilon^{-1} +  \| x_{0} \| \leq M  \, ,
\end{equation}
and similarly
\begin{equation}\label{S1.1}
 \| S^1 (t)x \| \leq  M  \quad \forall \, t \geq 0 ,
\forall \, x \in D_{\epsilon^{-1}}(x_0) \, .
 \end{equation}
Also, we note that $ x \in D_M $. Therefore, by (\ref{S3a}) there exists
$ \alpha_1 (x)> 0 $ such that
\begin{equation}\label{S3}
f(S(t)x) - f(x) \leq -t \delta_1 \quad \forall \, t \in [0, \alpha_1(x)) \, .
\end{equation}
Since $ \lim_{t \to 0+ } \frac{S(t)x - x }{t} =- A^{\circ}x $, there exists
$\alpha_2 (x) > 0 $ such that
\begin{equation}\label{S4}
\| \frac{S(t)x - x }{t}  + A^{\circ}x \| < \frac{\delta_1}{8L} \quad \forall \, t \in (0, \alpha_2 (x))\, .
\end{equation}
Similarly,  since
$ \lim_{t \to 0+ } \frac{S^1(t)x - x }{t} = -(A^1)^{\circ}x $,
 there exists $\alpha_3 (x) > 0 $ such that
\begin{equation}
\label{S5}
\| \frac{S^1 (t)x - x }{t}  +  (A^1)^{\circ}x \|
< \frac{\delta_1}{8L} \quad \forall \, t \in (0, \alpha_3 (x)) \, .
\end{equation}
Choose $\overline{\alpha}  = \min (\alpha_1 (x), \alpha_2 (x), \alpha_3 (x)) $.
Then for $ t \in (0, \overline{\alpha} )$,
\begin{align*}
&f(S^1(t)x) - f(x) \\
& =  f(S^1(t)x) - f (S(t)x) + f(S(t)x ) - f(x)\\
& \leq   L\| S^1(t)x - S(t)x \| +  f(S(t)x ) - f(x) \quad
 (\mbox{ by  (\ref{S1}), (\ref{S1.1}) and (\ref{S2}) })\\
& \leq  L\| S^1(t)x - S(t)x \|  -t \delta_1 \quad
 \mbox{(by (\ref{S3})) }\\
& =  Lt \| \frac{S^1 (t)x - x }{t}  +  (A^1)^{\circ}x - (A^1)^{\circ}x + A^{\circ}x - A^{\circ}x
      -  \frac{S(t)x - x }{t} \| - t \delta_1\\
& \leq  Lt \left(\| \frac{S^1 (t)x - x }{t}  + (A^1)^{\circ}x \| + \|(A^1)^{\circ}x - A^{\circ}x \|+ \|A^{\circ}x
      +  \frac{S(t)x - x }{t} \|\right) - t \delta_1\\
& \leq  tL\frac{\delta_1}{8L }+ tL \|(A^1)^{\circ}x - A^{\circ}x \| +tL\frac{\delta_1}{8L }
      -t\delta_1\quad \mbox{(by (\ref{S4}) and (\ref{S5}))}\\
& =  tL \|(A^1)^{\circ}x - A^{\circ}x \| - \frac{3}{4}t \delta_1\\
& \leq  tL2 \| A'x\| - \frac{3}{4}t \delta_1 \quad \mbox{ (by  Lemma \ref{l1} )}\\
& <  tL2\delta - \frac{3}{4}t \delta_1 \quad \mbox{ (by (\ref{1a})) } \\
& =  tL2 \frac{\delta_1}{8L} -\frac{3}{4}t \delta_1 \quad \mbox{ (by (\ref{S7})) }\\
& =  -\frac{t \delta_1}{2} = -t \overline{\delta} \quad
 \mbox{ (by (\ref{S6})).}
\end{align*}
\end{proof}

By using Lemma \ref{thmS}, in the next result, we show that the function
$ f $ may not converge  along the perturbed trajectory  $ S^1(t)x  $
but stays close to $ \min (f) $ if the perturbation is small,
assuming $ x \in D(A) $.

\begin{lemma}\label{thmS'}
Let $ A $ and $ f $ satisfy (A1) and (A2). For all  $ \epsilon > 0 $, there exists
   $ \delta>0 $ such that for all $ A' $ satisfying:
\begin{enumerate}
\item $ A' $ is single valued,
\item $ D(A) \subseteq D(A') $,
\item $A' $ is bounded on bounded subsets of $ D(A) $,
\item $ A'x_{0} = 0 $,
\item $ A^1 = A + A' $ is maximal monotone,
\item  $ \displaystyle \sup_{ \{ x \in D(A) : \| x - x_0 \| \leq \epsilon^{-1}   \}} \| A'x \| < \delta $,
\end{enumerate}
 and for every $ x \in \{ x \in D(A) : \| x - x_0  \| \leq \epsilon^{-1} \} $ there exists $ T \geq 0 $ such that
\begin{equation*}
 f(S^1 (t)x) \leq \min (f) + \frac{\epsilon}{2}  \quad \forall \, t \geq T \, ,
\end{equation*}
where $ \{ S^1(t) \}_ { t \geq 0 }  $ is the semigroup generated by $ A^{1}$.
\end{lemma}

\begin{proof} Let $\epsilon > 0$ be given. For convenience we write
$ M = \epsilon^{-1} + \|x_{0} \|  $.
By Lemma \ref{thmS} there exist positive numbers  $ \delta $,
$ \overline{\delta}  $ such that for all $ A' $ satisfying 1 -  5 and
     \[
\sup_{ \{ x \in D(A) : \| x  - x_0 \| \leq \epsilon^{-1} \} } \| A'x \|
< \delta \, ,
\]
and for every $ x \in D_{\epsilon^{-1}}(x_0) $ there exists
$ \overline{\alpha} > 0 $ such that
\begin{equation}\label{S'1}
f(x) - f(S^1(t)x) \geq t\overline{\delta} \quad \forall \,
t \in [0, \overline{\alpha}) \, ,
\end{equation}
where $ \{ S^1(t) \}_ { t \geq 0 }  $ is the semigroup generated by
 $ A^{1}$.
Let $ x \in   \{ x \in D(A) : \| x - x_0  \| \leq \epsilon^{-1} \} $.
Note $ \|S^1(t)x \| $  is  bounded, as
\begin{equation}\label{S'1.1}
 \|S^1(t)x \| \leq   \| S^1(t)x - x_{0 } \| + \| x_{0} \|
\leq \|x - x_{0} \| + \|x_{0} \|
  \leq \epsilon^{-1} +  \| x_{0} \| = M \, .
\end{equation}
Also,\begin{equation}\label{S'1.1a}
\| S^1(t)x - x_{0 } \| \leq \|x - x_{0} \|  \leq \epsilon^{-1} \, .
\end{equation}

Since $ f $ is Lipschitzian on bounded subsets of $H$, there exists
 $ L > 0 $ such that
\begin{equation}\label{S'1.2}
| f(x) - f(y) | \leq L \| x - y \| \quad \forall \, x, y
\in \{ x \in H : \| x \| \leq M \} \, .
\end{equation}
We note that $   t \to f(S^1(t)x)$ is Lipschitz continuous on $[0, \infty )$
as for every $ t_1, t_2 \in [0, \infty)$ we have
\begin{align*}
| f(S^1(t_1)x) - f(S^1(t_2)x) |
& \leq L \| S^1(t_1)x - S^1(t_2)x \| \quad
 \mbox{(by   (\ref{S'1.1}) and (\ref{S'1.2}))}\\
& \leq L | t_1 - t_2 | \| (A^1)^{\circ}x\| \, .
\end{align*}
Firstly, we claim that there exists $ T \geq 0 $ such that
\[
f(S^1 (T)x) \leq  \min (f) + \frac{\epsilon}{2} \, .
\]
Assume the contrary. Then
 \[
f(S^1 (t)x) > \min (f) + \frac{\epsilon}{2} \quad \forall \, t \geq 0 \, .
 \]
 Therefore by (\ref{S'1.1a}),
$ S^1(t)x \in  D_{\epsilon^{-1}}(x_0) \, \forall \, t \geq 0 $.
   Let $ V = \{ T: f(x) - f(S^1(\tau)x) \geq \tau\overline{ \delta} \;
 \forall \, \tau \in [0,T] \} $. Then  $ V $ is a nonempty subinterval of
$ [0,\infty) $. We claim $ V $ is open and closed in $[0,\infty)$.
To see $ V $ is open in $ [0,\infty) $ let $ T \in V $ be given.
Since $ S^1(T)x \in D_{\epsilon^{-1}}(x_0) $ there exists $ \alpha' > 0 $
such that
  \begin{equation}\label{S'2}
    f(S^1(T)x) - f(S^1(t)S^1(T)x) \geq t \overline{\delta}
\quad \forall \, t \in [0,\alpha'] \, .
    \end{equation}
Also $ T \in V $ implies
   \begin{equation}\label{S'3}
   f(x) - f(S^1(T)x) \geq T\overline{\delta} \, .
   \end{equation}
Adding the inequalities (\ref{S'2}) and (\ref{S'3}) we get
   \begin{equation*}
    f(x) - f(S^1(t+T)x) \geq (t+T) \overline{\delta} \quad
   \forall \, t \in [0,\alpha'] \, .
    \end{equation*}
   Thus $ [ 0, T+ \alpha' ] \subseteq V $, and $ V $ is open.
   To see that $ V $ is closed in $ [0, \infty )  $,
let $ \langle t_n \rangle _{n=1}^{\infty}   $ be a sequence in $ V $ and,
let $t_n \nearrow t $ as $n \to \infty$, $t > 0$.  Since $ t_n \in V $,
for every $n $,  we have
\begin{equation*}
 f(x) - f(S^1(t_n)x)   \geq   t_n \overline{\delta} \, .
 \end{equation*}
 Since  $  t \to f(S^1(t)x)$ is continuous on $[0, \infty )$,
letting $ n $ go to infinity we get
 \begin{equation*}
 f(x) - f(S^1(t)x) \geq    t \overline{\delta}  \, .
\end{equation*}
Hence $t \in V $. Now $ V $ is a nonempty open and closed
subinterval of $ [0, \infty) $, and therefore $ V = [0, \infty) $.
Hence for every $ t \in [0,\infty) $, $ f(x) - f(S^1(t)x) \geq t
\overline{\delta} $. Therefore by taking the limit as
$ t \to \infty $, we get
 $\lim_{t \to \infty } f(S^1(t)x) =  -\infty  $,
 contradicting the fact that $ f $ is bounded below.
 Hence our assumption is wrong, proving the claim.

 Secondly, we show that for every $ t \geq T $
 \[
f(S^1 (t)x) \leq  \min (f) +\frac{ \epsilon }{2} \, .
\]
 Assume, to obtain a contradiction that there exists $ T_1 > T $ such that
 \[
f(S^1 (T_1)x) >  \min (f) + \frac{\epsilon}{2} \, .
 \]
 Since  $  t \to f(S^1(t)x)$ is continuous on $[0, \infty )$ there exists
 $ T_2  \in (T, T_1)$ such that
 \begin{equation}\label{S'4}
 f(S^1 (T_2)x)  =\min (f) + \frac{\epsilon}{2} \, ,
\end{equation}
and
\begin{equation}\label{S'5}
 f(S^1 (\tau)x) >  \min (f) + \frac{\epsilon}{2}  \quad
\forall \tau \in (T_2, T_1] \, .
\end{equation}
Since $  t \to f(S^1(t)x)$ is Lipschitz continuous on $[0, \infty )$,
it is differentiable a.e. and
 \begin{equation}\label{S'6}
f(S^1(T_1)x) = f(S^1(T_2)x ) + \int_{T_2}^{T_1} \frac{d}{dt} f(S^1(t)x) dt \, .
\end{equation}
Let $ T_3 \in (T_2, T_1) $ be such that $ f(S^1(t)x) $ is differentiable
at $ T_3 $.
By (\ref{S'1.1a}) and (\ref{S'5}), $ S^1(T_3)x \in D_{\epsilon^{-1}}(x_0) $.
Therefore by (\ref{S'1}) there exists $ \alpha_3 > 0 $ such that
\begin{equation*}
f(S^1(T_3)x) - f(S^1(T_3 + t )x ) \geq t \overline{\delta} \quad
\forall t \in (0, \alpha_3 ) \, ,
\end{equation*}
which in turn implies
\begin{equation}\label{S'7}
\frac{d}{dt} f(S^1(T_3)x ) \leq -\overline{\delta} \, .
\end{equation}
Using (\ref{S'7}) in (\ref{S'6}) we get
\begin{equation*}
f(S^1(T_1)x) \leq f(S^1(T_2)x ) - \overline{\delta} ( T_1 - T_2)
= \min (f) + \frac{\epsilon}{2}  - \overline{\delta} ( T_1 - T_2) \, ,
\end{equation*}
which contradicts our assumption.
\end{proof}

Let us recall from \cite{pazy} that an operator $ A $ is maximal
$\mathcal{M}(\omega) $ if and only if $ A + \omega I $ is a maximal
monotone operator.

\begin{remark}\label{r1} \rm
We note that if we assume $ f $ to be Lipschitzian  on $ K $, rather
than on bounded sets, and $ A $ to be a maximal $ \mathcal{M}(\omega)$
operator, in (A1) and (A2), then the conclusion of Lemma \ref{thmS} holds.
We are unable to extend  the conclusion of Lemma \ref{thmS'} for a maximal
$\mathcal{M}( \omega)$ operator even if we assume perturbed trajectories
to be bounded.
\end{remark}

Finally we establish the first stability theorem, extending Lemma \ref{thmS'}
by not assuming $ x \in D(A) $.

\begin{theorem}\label{thmS''}
 Let $ A $ and $ f $ satisfy (A1) and (A2). For all $ \epsilon > 0 $,
there exists    $ \delta>0 $ such that for all $ A' $ satisfying:
\begin{enumerate}
\item $ A' $ is single valued,
\item $ D(A) \subseteq D(A') $,
\item $A' $ is bounded on bounded subsets of $ D(A) $,
\item $ A'x_{0} = 0 $,
\item $ A^1 = A + A' $ is maximal monotone,
\item  $ \displaystyle \sup_{ \{ x \in D(A) : \| x  - x_0 \| \leq \epsilon^{-1}   \}} \| A'x \| < \delta $,
\end{enumerate}
 and for every $ x \in \{ x \in \overline{D(A)} : \| x  - x_0 \|
< \epsilon^{-1}  \} $, there exists $ T \geq 0 $ such that
\[
f(S^1 (t)x) \leq \min (f) + \epsilon \quad \forall \, t \geq T \, ,
\]
where $ \{ S^1(t) \}_ { t \geq 0 }  $ is the semigroup generated by $ A^{1}$.
\end{theorem}

\begin{proof} Let $ \epsilon > 0 $ be given, and set
$ M = \epsilon^{-1}  + \| x_{0} \| $. Let
$x \in \{ x \in \overline{D(A)} : \| x  - x_0 \| < \epsilon^{-1}  \} $
then there exists a sequence
$\langle x_k \rangle _{k =1}^{\infty}$ in
$ \{ x \in {D(A)} : \| x  - x_0 \| \leq \epsilon^{-1}  \}$ such that
$x_k \to x$  as $k \to \infty$.
 Since $ f $ is Lipschitzian on bounded subsets of $H$, there exists
$ L > 0 $ such that
\begin{equation}\label{S''3}
| f(x) - f(y) | \leq L \| x - y \| \quad \forall \, x, y \in
\{ x \in H : \| x \| \leq M \} \, .
\end{equation}
Since $x_k \to x $ there exists a positive integer $ k' $ such that
\begin{equation}\label{S''4}
\| x_{k'} - x \| < \frac{1}{L+1} .\frac{ \epsilon}{2}  \, .
\end{equation}
By Lemma \ref{thmS'}, noting $ x_{k'} \in D (A)$, and
$ \| x_{k'}  - x_0 \|  \leq \epsilon^{-1}  $,  there exists $\delta > 0 $
such that for all $ A' $ satisfying 1-6 of this theorem,  there exists
$ T \geq 0 $ such that
\begin{equation}\label{S''6}
f(S^1(t)x_{k'}) \leq \min (f) + \frac{\epsilon}{2} \quad \forall \,
t \geq T \, ,
\end{equation}
where $ \{ S^1(t) \}_ { t \geq 0 }  $ is the semigroup generated by $ A^{1}$.
 Note for every $ k $, and all $ t \geq 0 $,  $ \|S^1(t)x_k \| $ and
$ \|S^1(t)x \| $ are  bounded as
\begin{equation}\label{S''1}
 \|S^1(t)x_k \| \leq   \| S^1(t)x_k - x_{0 } \| + \| x_{0} \|
 \leq \|x_k - x_{0} \| + \|x_{0} \|
  \leq \epsilon^{-1}  + \| x_{0} \| = M  \, ,
\end{equation}
and
\begin{equation}\label{S''2}
 \|S^1(t)x \| \leq   \| S^1(t)x - x_{0 } \| + \| x_{0} \| \leq \|x - x_{0} \| + \|x_{0} \|
  \leq \epsilon^{-1}  +  \| x_{0} \| =  M \, .
\end{equation}
Thus by (\ref{S''1}), (\ref{S''2}), (\ref{S''3}) and (\ref{S''4}), for all
$t \geq 0 $,
\begin{equation}
\begin{aligned}
| f(S^1(t)x_{k'}) - f(S^1(t)x) |
& <  L \| S^1(t)x_{k'} - S^1(t)x \| \\
& \leq  L \| x_{k'} - x \|\quad \mbox{(as  $S^1 $ is a contraction)} \\
& < \frac{L}{L+1} .\frac{\epsilon}{2}  \\
& < \frac{\epsilon} {2} \label{S''5}
\end{aligned}
\end{equation}
Combining (\ref{S''5}) and (\ref{S''6}) we get $ \forall \, t \geq T $
\begin{align*}
f(S^1(t)x) & \leq  | f(S^1(t)x) - f(S^1(t)x_{k'}) | + f(S^1(t)x_{k'})\\
    & <  \frac{\epsilon}{2}  +\min (f)+ \frac{\epsilon}{2} \\
    & =  \min (f)+ \epsilon \, .
\end{align*}
\end{proof}

 Instead of assuming $ A + A' $ to be maximal monotone, one can choose
$ A' $  satisfying   additional conditions such that $ A + A' $
is maximal monotone operator. In Theorem 2, we use our perturbation
result  of \cite{chou} to replace (5) of Theorem \ref{thmS''}
by some suitable conditions.

\begin{theorem} \label{thm2}
Let $ A $ and $ f $ satisfy (A1)-(A3).  Then for all $ \epsilon > 0 $,
there exists    $ \delta>0 $ such that for all $ A' $ satisfying:
\begin{enumerate}
\item $ A' $ is single valued, hemicontinuous and monotone on $ D (A) $,
\item $ D(A) \subseteq D(A') $,
\item $A' $ is bounded on bounded subsets of $ D(A) $,
\item $ A'x_{0} = 0 $,
\item  $ \displaystyle \sup_{ \{ x \in D(A) : \| x - x_0 \| \leq \epsilon^{-1}   \}} \| A'x \| < \delta $,
\end{enumerate}
we have $ A^{1} = A + A' $ maximal monotone, and the semigroup
$ \{ S^1(t) \}_ { t \geq 0 }  $  generated by $ A^{1}$ has the property that
  for every $ x \in \{ x \in \overline{D(A)} : \| x  - x_0 \| < \epsilon^{-1}  \} $ there exists $ T \geq 0 $ such that
\[
f(S^1 (t)x) \leq  \min (f) + \epsilon \quad \forall \, t \geq T \, .
\]
\end{theorem}

\begin{proof}
By  \cite[Proposition 4]{chou}, 1,2,3 and (A3), imply that $ A+ A' $
is maximal monotone. By  Theorem \ref{thmS''} we have our conclusion.
\end{proof}

\subsection*{Acknowledgments} The author is indebted to  B. Calvert for
his assistance and guidance.

\begin{thebibliography}{99}

 \bibitem{arz1} S. Aizicovici, S. Reich and A. J. Zaslavski;
\emph{Convergence theorems for continuous descent methods},
J. Evol. Equ. {\bf 4} (2004), 139-156.

\bibitem{arz2} S. Aizicovici, S. Reich and A. J. Zaslavski;
\emph{Convergence results for a class of abstract continuous descent methods},
Electron. J. Differential Equations {\bf 2004} no. 45 (2004), 1-13.

 \bibitem{arz3} S. Aizicovici, S. Reich and A. J. Zaslavski;
\emph{Most continuous descent methods converge},
Arch. Math. (Basel) {\bf 85} (2005), 268-277.

 \bibitem{bail} J. B. Baillon;
\emph{Un exemple concernant le comportement asymptotique de la solution
 du probl\`{e}me  $ du/dt + \partial\phi(u) \ni 0 $},
J. Functional Anal. {\bf28} (1978), 369-376.

  \bibitem{bbr} J. B. Baillon, R. E. Bruck and S. Reich;
\emph{On the asymptotic behavior of nonexpansive mappings and semigroups
in Banach spaces}, Houston J. Math. {\bf 4} (1978), 1-9.

 \bibitem{brez1}  H. Brezis;
\emph{Op\'{e}rateurs Maximaux Monotones
et Semi-groupes de Contractions dans les Espaces de Hilbert},
North-Holland, Amsterdam, 1973.

  \bibitem{brez2} H. Brezis;
\emph{Monotonicity methods in Hilbert spaces and some applications
to nonlinear partial differential equations},
Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.),
Academic Press, New York, 1971, 101-156.

  \bibitem{bruc1} R. E. Bruck;
\emph{Asymptotic convergence of nonlinear contraction semigroups in
Hilbert space}, J. Functional Analysis {\bf18} (1975), 15-26.

\bibitem{chou} R. Choudhary;
\emph{Generic convergence of a convex Lyapounov function along trajectories
 of nonexpansive semigroups in Hilbert space},
J. Nonlinear Convex Anal. {\bf 7} (2006), no. 2, 245-268.

\bibitem{slem} C. M. Dafermos and M. Slemrod;
\emph{Asymptotic behavior of nonlinear contraction semigroups},
J. Functional Analysis {\bf13} (1973), 97-106.

\bibitem{nr} O. Nevanlinna and S. Reich;
\emph{Strong convergence of contraction semigroups and of iterative methods
for accretive operators in Banach spaces},
Israel J. Math. {\bf 32} (1979), 44-58.

 \bibitem{pazy} A. Pazy;
\emph{Semigroups of nonlinear contractions in Hilbert spaces},
C.I.M.E. Varenna, 1970, Ed. Cremonese (1971), 343-430.

 \bibitem{pazy1} A. Pazy;
\emph{On the asymptotic behavior of semigroups of nonlinear contractions
in Hilbert space}, J. Functional Analysis {\bf 27} (1978), 292-307.

 \bibitem{pazy2} A. Pazy;
\emph{Strong convergence of semigroups of nonlinear contractions
in Hilbert space}, J. Analyse Math. {\bf 34} (1978), 1-35.

  \bibitem{reich4} S. Reich;
\emph{Almost convergence and nonlinear ergodic theorems},
J. Approx. Theory {\bf 24} (1978), 269-272.

 \bibitem{reich3} S. Reich and A. J. Zaslavski;
\emph{Generic convergence of descent methods in Banach spaces},
Math. Oper. Res. {\bf25} (2000), 231-242.

 \bibitem{reich1} S. Reich and A. J. Zaslavski;
\emph{The set  of divergent descent methods in a Banach space is
$ \sigma $-porous}, SIAM J. Optim. {\bf11} (2001), 1003-1018.

 \bibitem{reich2} S. Reich and A. J. Zaslavski;
\emph{Two convergence results for continuous descent methods},
Electron. J. Differential Equations {\bf 2003} no. 24 (2003), 1-11.

\end{thebibliography}

\end{document}