\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 45, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2004/45\hfil Continuous descent methods]
{Convergence results for a class of abstract continuous descent methods} 

\author[S. Aizicovici, S. Reich, \& A. J. Zaslavski\hfil EJDE-2004/45\hfilneg]
{Sergiu Aizicovici, Simeon Reich, \& Alexander J. Zaslavski}  

\address{Sergiu Aizicovici \hfill\break
 Department of Mathematics, Ohio University, Athens, OH
45701-2979, USA}
\email{aizicovi@math.ohiou.edu}

\address{Simeon Reich\hfill\break
Department of Mathematics, The Technion-Israel Institute of
Technology, 32000 Haifa, Israel}
\email{sreich@tx.technion.ac.il} 

\address{Alexander J. Zaslavski\hfill\break
Department of Mathematics, The Technion-Israel Institute of
Technology, 32000 Haifa, Israel}
\email{ajzasl@tx.technion.ac.il}

\date{}
\thanks{Submitted January 7, 2004. Published March 30, 2004.}
\subjclass[2000]{37L99, 47J35, 49M99, 54E35, 54E50, 54E52, 90C25}
\keywords{Complete metric space,  descent method,  Lipschitzian function, 
\hfill\break\indent
porous set, regular vector field}


\begin{abstract}
 We study continuous descent methods for the minimization of 
 Lipschitzian functions defined on a general Banach space. 
 We establish convergence theorems for those methods which 
 are generated by approximate solutions to evolution equations 
 governed by regular vector fields.
 Since the complement of the set of regular vector fields is 
 $\sigma$-porous, we conclude that our results apply to most 
 vector fields in the sense of Baire's categories. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

Let $(X,\|\cdot \|)$ be a Banach space, $(X^*,\|\cdot \|_*)$ its dual space,
and let $f: X \to \mathbb{R}^1$ be a function
which is bounded from below and Lipschitzian on bounded subsets of $X$. 
Recall that for each pair of sets $A,B \subset X^*$,
$$
H(A,B)=\max \{\sup_{x \in A} \inf_{y \in B} \|x-y\|_*,\;
\sup_{y \in B} \inf_{x \in A}\|x-y\|_*\}
$$
is the Hausdorff distance between $A$ and $B$.
For each $x \in X$, let
\begin{equation}
f^0(x,h)=\limsup_{t \to 0^+,\;y \to x}[f(y+th)-f(y)]/t,\quad h \in X,
\label{e1.1}
\end{equation}
be the Clarke generalized directional  
derivative of $f$ at the point $x$, let
\begin{equation}
\partial f(x)=\{l \in X^*: f^0(x,h) \ge l(h) \text { for all }
h \in X\} \label{e1.2}
\end{equation}
be Clarke's generalized gradient of $f$ at $x$,
and set
\begin{equation}
\Xi(x)=\inf \{f^0(x,h): h \in X \text { and } \|h\| =1\}. \label{e1.3}
\end{equation}
It is well known that $\partial f(x)$ is nonempty and bounded.
Set
$$
\inf(f) =\inf \{f(x): x \in X\}.
$$
Denote by $\mathcal{A}$ the set of all mappings $V: X \to X$ such that $V$ is
bounded on every bounded subset of $X$, 
and for each $x \in X$, $f^0(x,Vx) \le 0$.
We denote by $\mathcal{A}_c$ the set of all continuous 
$V \in \mathcal{A}$ and by $\mathcal{A}_b$ the set of all $V \in \mathcal{A}$ 
which are bounded on $X$.
Finally, let $\mathcal{A}_{bc}=\mathcal{A}_{b} \cap \mathcal{A}_c$.
Next we endow the set $\mathcal{A}$ with two metrics, $\rho_s$ and $\rho_w$.
To define $\rho_s$, we first set, for each $V_1,V_2 \in \mathcal{A}$,
$$
\tilde \rho_s(V_1,V_2)=\sup\{\|V_1x-V_2x\|: x \in X\} 
$$
and then let
\begin{equation}
\rho_s(V_1,V_2)=\tilde \rho_s(V_1,V_2)(1+\tilde \rho_s(V_1,V_2))^{-1}.
\label{e1.4}
\end{equation}
(Here we use the convention that $\infty/\infty =1$.)
Clearly, $(\mathcal{A},\rho_s)$ is a complete metric space.
To define $\rho_w$, we first set, for each $V_1,V_2 \in \mathcal{A}$
and each integer $i \ge 1$, 
\begin{equation}
\rho_i(V_1,V_2)=\sup\{\|V_1x-V_2x\|: x \in X \text { and } \|x\| \le i\} \label{e1.5}
\end{equation}
and then let
\begin{equation}
\rho_w(V_1,V_2)=\sum_{i=1}^{\infty}2^{-i}[\rho_i(V_1,V_2)
(1+\rho_i(V_1,V_2))^{-1}]. \label{e1.6}
\end{equation}
Clearly, $(\mathcal{A},\rho_w)$ is a complete metric space.
It is also not difficult to see that the collection of the sets
$$
E(N,\epsilon)=\{(V_1,V_2) \in \mathcal{A} \times \mathcal{A}:
\|V_1x-V_2x\| \le \epsilon,\; x \in X,\; \|x\| \le N\},
$$
where $N,\epsilon>0$, is a base 
for the uniformity generated by the metric $\rho_w$. 
It is easy to see that 
$$
\rho_w(V_1,V_2) \le \rho_s(V_1,V_2) \quad \text {for all } V_1,
V_2 \in \mathcal{A}.
$$
 The metric $\rho_w$ induces on $\mathcal{A}$ a topology which 
is called the weak topology and $\rho_s$ induces 
a topology which is called the
strong topology.
Clearly, $\mathcal{A}_c$ is a closed subset of $\mathcal{A}$ with the weak topology 
while
$\mathcal{A}_b$ and $\mathcal{A}_{bc}$ are closed subsets of $\mathcal{A}$ 
with the strong topology. We consider the subspaces $\mathcal{A}_c$, $\mathcal{A}_b$
and $\mathcal{A}_{bc}$ with the metrics $\rho_s$ and $\rho_w$ which induce 
the strong and the weak topologies, respectively. 

The study of steepest descent and other
minimization methods  
is a central topic in optimization theory. 
See, for example, \cite{c1,h1,n1,r1,r2,r3,r4}.
When the function $f$ is convex, one usually looks for a sequence 
$\{x_i\}_{i=1}^{\infty}$ which either tends to a minimum point of $f$ 
(if such a point exists) or at least 
such that $\lim_{i \to \infty}f(x_i)=\inf(f)$. If 
$f$ is not necessarily convex, but
$X$ is finite-dimensional, then we expect to construct a sequence which tends
to a critical point $z$ of $f$, namely a point $z$
for which $0 \in \partial f(z)$.
If $f$ is not necessarily convex and
$X$ is infinite-dimensional, then
the problem is more difficult and  
less understood because 
 we cannot guarantee, in general, the existence of a critical point and
a convergent subsequence.
To partially overcome this difficulty, we have introduced the function
$\Xi: X \to \mathbb{R}^1$. Evidently, a point $z$ is a critical point of $f$
if and only if $\Xi(z) \ge 0$. Therefore we say that $z$ is 
$\epsilon$-critical for a given $\epsilon>0$ if $\Xi(z)  \ge -\epsilon$.
In \cite{r3} we looked for sequences $\{x_i\}_{i=1}^{\infty}$
such that either $\liminf_{i \to \infty} \Xi(x_i) \ge 0$ or
at least
$\limsup_{i \to \infty} \Xi(x_i) \ge 0$. In the first case, given 
$\epsilon >0$, all the points $x_i$, except possibly
a finite number of them, are $\epsilon$-critical, while in the second case 
this holds for a subsequence of $\{x_i\}_{i=1}^{\infty}$.


In \cite{r3} we show, under certain assumptions on $f$, 
that for most (in the sense of Baire's categories)
vector fields $W \in \mathcal{A}$, 
certain discrete iterative processes yield sequences with the 
desirable properties. Moreover, we show that the complement of the set
of ``good'' vector fields is not only of the first category, but also
$\sigma$-porous. Analogous results for convex functions $f$ were
obtained in \cite{r1,r2}.
This approach, when 
a certain property
is investigated not for a single point of a complete metric space, but
for the whole space, 
has also been successfully applied in 
the theory of dynamical systems \cite{d1,d2}, optimization \cite{i1}, and
optimal control \cite{z3}, as well as in approximation theory \cite{d3}.
Before we continue, we briefly recall the concept of porosity \cite{b1,z1,z2}.  
As a matter of fact, several different notions of porosity have been
used in the literature. 
In the present paper we will use porosity with
respect to a pair of metrics, a concept which was introduced in \cite{z3}.

When $(Y,d)$ is a metric space
we denote by $B_d(y,r)$ 
the closed ball of center $y \in Y$ and radius $r>0$.
Assume that $Y$ is a nonempty set
and $d_1,d_2:Y \times Y \to [0,\infty)$
are two metrics which satisfy $d_1(x,y) \le d_2(x,y)$ for all $x,y \in Y$.

A subset $E \subset Y$ is called
porous with respect 
to the pair $(d_1,d_2)$ (or just porous if 
the pair of metrics is fixed)
if there exist $\alpha \in (0,1)$ and $r_0>0$ such that for each
$r \in (0,r_0]$ and each $y \in Y$, there exists $z \in Y$ for which
$d_2(z,y) \le r$
and 
$$
 B_{d_1}(z,\alpha r) \cap E=\emptyset.
$$
 

A subset   of the space $Y$ is called $\sigma$-porous 
with respect 
to $(d_1,d_2)$ 
(or just $\sigma$-porous if the pair of metrics is understood)
if it is a 
 countable union of porous (with respect 
to $(d_1,d_2)$)
subsets of $Y$.

Note that if $d_1=d_2$, then by Proposition 1.1 of \cite{z3}
our definitions reduce to those in \cite{d2,d3,r2}.
We use porosity with respect to a pair of metrics because 
in applications a space is usually endowed with a pair of metrics and one 
of them is weaker than the other.
Note that the porosity of a set with respect to one of these two metrics does
not imply its porosity with respect to the other metric. However, 
it is shown in \cite[Proposition 1.2]{z3}  that if 
a subset $E \subset Y$ is porous with respect to $(d_1,d_2)$, then
$E$ is porous with respect to any metric which is weaker
than $d_2$ and stronger than $d_1$. 
For each set $E \subset X$, we denote by $cl(E)$ the closure of $E$ in the
norm topology.
The results of \cite{r3} were established in any Banach space 
and for those functions which satisfy 
the following two assumptions.

A(i) For each $\epsilon >0$, there exists $\delta \in (0,\epsilon)$ such that
$$
cl(\{x \in X: \Xi(x) <-\epsilon\}) \subset \{x \in X: \Xi(x)<-\delta\};
$$

A(ii) for each $r>0$, the function $f$ 
is Lipschitzian on the ball $\{x \in
X: \|x\| \le r\}$.

We will say that a mapping $V \in \mathcal{A}$ is regular if for any natural number $n$,
there exists a positive number $\delta(n)$ such that for each $x \in X$
satisfying 
$\|x\| \le n $ and  $\Xi(x) < -1/n,$
we have
$f^0(x,Vx)  \le -\delta(n)$.

This concept of regularity is a non-convex analog of the regular 
vector fields introduced in \cite{r3}.
We denote by $\mathcal{F}$ the set of all regular vector fields $V \in \mathcal{A}$.

The following result was established in \cite{r3}.

\begin{theorem} \label{thm1.1} 
Assume that both A(i) and A(ii) hold. Then 
$\mathcal{A} \setminus \mathcal{F}$ (respectively,  
$\mathcal{A}_c \setminus \mathcal{F}$, $\mathcal{A}_{b} \setminus \mathcal{F}$ and
$\mathcal{A}_{bc} \setminus \mathcal{F}$) is a $\sigma$-porous subset of the space
$\mathcal{A}$ (respectively,   
$\mathcal{A}_c$, $\mathcal{A}_{b}$ and
$\mathcal{A}_{bc}$) with respect to the pair $(\rho_w,\rho_s)$.
\end{theorem}

In \cite{r3} two of the authors studied the convergence of 
discrete descent methods generated 
by regular vector fields. In \cite{a1} we obtained analogs of the main 
results of \cite{r3} for continuous descent methods generated by regular vector
fields. Our purpose in the present paper is to study some
continuous descent methods for the minimization of Lipschitzian functions
which are generated by approximate solutions to
evolution equations governed by regular vector fields.
Such methods would be quite useful in practice.
Section 2 contains an auxiliary result.
In Section 3 we state and prove three convergence
theorems. 
An extension of our convergence theory to Lipschitzian functions 
satisfying a Palais-Smale type condition is presented in Section 4.
 In view of Theorem \ref{thm1.1}, our results apply to most vector 
fields in the sense of Baire's categories.   


\section{An auxiliary result}

Throughout this paper we let $x \in W^{1,1}(0,T;X)$, i.e. (see, e.g., \cite{b2}),
$$
x(t)=x_0+\int_0^tu(s)ds, \quad t \in [0,T], 
$$
where $T>0$, $x_0 \in X$ and $u \in L^1(0,T;X)$.
Then $x:[0,T] \to X$ is absolutely continuous
and $x'(t)=u(t)$ for a.e. $t \in [0,T]$.
Recall that the function $f:X \to \mathbb{R}^1$ is Lipschitzian on bounded subsets of $X$.
Thus the restriction of $f$ to the set $\{x(t): t \in [0,T]\}$ is 
Lipschitzian. Hence the function 
$(f\cdot x)(t):=f(x(t)),\; t \in [0,T]$, is absolutely continuous. It follows
that for almost every $t \in [0,T]$, both the derivatives
$x'(t)$ and $(f\cdot x)'(t)$ exist:
$$
x'(t)=\lim_{h \to 0}h^{-1}[x(t+h)-x(t)],
$$
$$
(f\cdot x)'(t)=\lim_{h \to \infty}h^{-1}[f(x(t+h))-f(x(t))].
$$

We need the following result proved in \cite[Proposition 2.1]{a1}.

\begin{proposition} \label{prop2.1} Assume that $t \in [0,T]$ and 
that both the derivatives
$x'(t)$ and $(f\cdot x)'(t)$ exist. Then
\begin{equation}
(f \cdot x)'(t)=\lim_{h \to 0}h^{-1}[f(x(t)+hx'(t))-f(x(t))]. \label{e2.1}
\end{equation}
\end{proposition}

In the sequel we denote by $\mu(E)$ the Lebesgue measure of a Lebesgue 
measurable set 
$E \subset \mathbb{R}^1$.

Assume that $V \in \mathcal{A}$ and that $x \in W^{1,1}(0,T;X)$ satisfies
$$
x'(t)=V(x(t)) \text { a.e. } t \in [0,T].
$$
Then by Proposition \ref{prop2.1}, $(f\cdot x)'(t) \le 0$ for a.e. $t \in [0,T]$
and $f(x(\cdot))$ is decreasing.

\section{Three convergence theorems}


\begin{theorem} \label{thm3.1} 
Let A(ii) hold, let $V \in \mathcal{A}$ be 
regular, and assume that
$$
\lim_{\|x\| \to \infty}f(x)=\infty.
$$
Let $K_0$ and $\epsilon $ be positive numbers. Then there exist $N_0>0$ 
and $\tilde K>0$ such that the following property holds:

For each $T \ge N_0$, there is $\gamma >0$ such that if  
$x \in W^{1,1}(0,T;X)$ satisfies
\begin{equation}
\|x(0)\| \le K_0 \label{e3.1}
\end{equation}
and
\begin{equation}
\|x'(t)-V(x(t))\| \le \gamma \text { for a.e. } t \in [0,T], \label{e3.2}
\end{equation}
then 
\begin{equation}
\|x(t)\| \le \tilde K,\; t \in [0,T] \label{e3.3}
\end{equation}
and 
\begin{equation}
\mu\{t \in [0,T]: \Xi(x(t))<-\epsilon \} \le N_0. \label{e3.4}
\end{equation}
\end{theorem}

\begin{proof} We may assume that $\epsilon <1/4$. Choose
\begin{equation}
K_1>\sup\{|f(x)|: x \in X,\; \|x\| \le K_0+1\}. \label{e3.5}
\end{equation}
Then the set
\begin{equation}
\{x \in X: f(x) \le K_1+|\inf(f)|+4\} \label{e3.6}
\end{equation}
is bounded. Consequently,
there exists a constant $\tilde K>K_0+K_1+2$ such that
\begin{equation}
\text{if } x \in X \text { and } f(x) \le K_1+|\inf(f)|+4, \text { then 
} 
\|x\| \le \tilde K. \label{e3.7}
\end{equation}
There also exists a constant $L>1$ such that
\begin{equation}
|f(y_1)-f(y_2)| \le L\|y_1-y_2\| \label{e3.8}
\end{equation}
for all $y_1,y_2 \in X$ such that
\begin{equation}
\|y_1\|,\|y_2\| \le \tilde K+1. \label{e3.9}
\end{equation}
Since $V$ is regular, there is $\delta_0 \in (0,1)$ such that for each $x 
\in X$ satisfying
\begin{equation}
\|x\| \le \tilde K+1 \text { and } \Xi(x)<-\epsilon, \label{e3.10}
\end{equation}
we have
\begin{equation}
f^0(x,Vx) \le -\delta_0. \label{e3.11}
\end{equation}
Choose 
\begin{equation}
N_0>2\delta_0^{-1}[2+K_1+|\inf(f)|]+1 \label{e3.12}
\end{equation}
and let $T \ge N_0$. Choose a positive number
\begin{equation}
\gamma<(4LT)^{-1}. \label{e3.13}
\end{equation}

Assume that $x \in W^{1,1}(0,T;X)$ satisfies \eqref{e3.1} and \eqref{e3.2}. 
We will show that for each $t \in [0,T]$, 
\begin{equation}
f(x(t)) \le f(x(0))+tL\gamma. \label{e3.14}
\end{equation}
There is $\Delta \in (0,1)$ such that
\begin{equation}
\|x(t)-x(0)\| \le 1/4,\; t \in [0,\Delta]. \label{e3.15}
\end{equation}
Clearly,
\begin{equation}
\|x(t)\| \le \|x(0)\|+1/4 \le K_0+1/4<\tilde K,\; t \in [0,\Delta]. 
\label{e3.16}
\end{equation}
Let $s \in (0,\Delta]$. It follows from Proposition \ref{prop2.1}, the 
relation $V \in \mathcal{A}$, and the
 subadditivity of Clarke's generalized directional derivative that
\begin{equation}
f(x(s))-f(x(0))=\int_0^s(f\cdot x)'(t)dt \le \int_0^sf^0(x(t),x'(t))dt 
\label{e3.17}
\end{equation}
$$
\le \int_0^sf^0(x(t),V(x(t)))dt+\int_0^sf^0(x(t),x'(t)-V(x(t)))dt 
$$
$$
\le \int_0^sf^0(x(t),x'(t)-V(x(t)))dt.
$$
 

By \eqref{e3.16}, \eqref{e3.2}, \eqref{e1.1}, and the definition of $L$ (see
\eqref{e3.8}
and \eqref{e3.9}), we have for 
a.e. $t 
\in [0,s]$,
$$
f^0(x(t),x'(t)-V(x(t))) \le L\|x'(t)-V(x(t))\| \le L\gamma.
$$
When combined with \eqref{e3.17}, this inequality implies that
$$
f(x(s))-f(x(0)) \le L\gamma s.
$$
Thus \eqref{e3.14} holds  for any $t \in [0,\Delta]$.
Set
\begin{equation}
\Omega =\{h \in (0,T]: \text { inequality }
\eqref{e3.14} \text { holds for all } t \in [0,h]\}. 
\label{e3.18}
\end{equation}
Clearly, $\Delta \in \Omega$.  Set $ h_0=\sup \Omega$. It is not difficult 
to see that
\begin{equation}
 f(x(t)) \le f(x(0))+tL\gamma
 \text { for all } t \in [0,h_0]. \label{e3.19}
\end{equation}
We now show that $h_0=T$. Let us assume the converse. Then $h_0<T$. 

By \eqref{e3.19} and \eqref{e3.13},
$$
f(x(h_0)) \le f(x(0))+h_0L\gamma <f(x(0))+TL\gamma<f(x(0))+4^{-1}.
$$
There is $h_1 \in (h_0,T)$ such that for each $t \in [h_0,h_1]$,
\begin{equation}
f(x(t))<f(x(0))+1/4. \label{e3.20}
\end{equation}
Relations \eqref{e3.20}, \eqref{e3.1}, \eqref{e3.5} and \eqref{e3.7} imply that for all $t \in 
[h_0,h_1]$,
\begin{equation}
f(x(t))<K_1+1/4 \text { and } \|x(t)\| \le \tilde K. \label{e3.21}
\end{equation}
Let $s \in (h_0,h_1]$. It follows from Proposition \ref{prop2.1}, the
subadditivity of Clarke's directional derivative,
and the relation $V \in \mathcal{A}$ 
that
\begin{equation}  \label{e3.22}
\begin{aligned}
f(x(s))-f(x(h_0))&=\int_{h_0}^s(f\cdot x)'(t)dt \le \int_{h_0}^s
f^0(x(t),x'(t))dt \\
&\le \int_{h_0}^s f^0(x(t),V(x(t)))dt+\int_{h_0}^sf^0(x(t),x'(t)-V(x(t)))dt\\
&\le \int_{h_0}^sf^0(x(t),x'(t)-V(x(t)))dt.
\end{aligned}
\end{equation}
By \eqref{e3.21}, \eqref{e3.2}, and
the definition of $L$ (see \eqref{e3.8} and \eqref{e3.9}), we have for a.e. $t \in 
[h_0,s]$,
$$
f^0(x(t),x'(t)-V(x(t))) \le L\|x'(t)-V(x(t))\| \le L\gamma.
$$
When combined with \eqref{e3.22}, this inequality implies that 
$$
f(x(s))-f(x(h_0)) \le (s-h_0)L\gamma.
$$
This latter inequality and \eqref{e3.19} lead to
$$
f(x(s)) \le (s-h_0)L\gamma+f(x(h_0)) \le (s-h_0)L\gamma 
+f(x(0))+h_0L\gamma=f(x(0))+sL\gamma.
$$
Thus $f(x(s)) \le f(x(0))+sL\gamma$ for each $s \in (h_0,h_1]$. 
This means that $h_1 
\in \Omega$, a contradiction. The contradiction we have reached proves 
that $h_0=T$ and that \eqref{e3.14} is indeed true for all $t \in [0,T]$.

From \eqref{e3.14}, \eqref{e3.13}, \eqref{e3.1} and \eqref{e3.5} it follows that for all $t \in [0,T]$, 
$$
f(x(t)) \le f(x(0))+tL\gamma \le f(x(0))+TL\gamma <f(x(0))+1/4 
<K_1+1/4.
$$
When combined with \eqref{e3.7}, this inequality implies that
$\|x(t)\| \le \tilde K$, $t \in [0,T]$.
Thus \eqref{e3.3} holds.
Set
\begin{equation}
\Omega_0=\{t \in [0,T]: \Xi(x(t))<-\epsilon\}. \label{e3.23}
\end{equation}
By Proposition \ref{prop2.1} and the subadditivity of Clarke's directional 
derivative,
\begin{equation} \label{e3.24}
\begin{aligned}
&f(x(T))-f(x(0))\\
&=\int_0^T(f\cdot x)'(t)dt \\
&\le \int_0^Tf^0(x(t),x'(t))dt \\
&\le \int_0^Tf^0(x(t),V(x(t)))dt+\int_0^Tf^0(x(t),x'(t)-V(x(t)))dt.
\end{aligned}
\end{equation}
By the relation $V \in \mathcal{A}$, \eqref{e3.23}, \eqref{e3.3}, and the definition of 
$\delta_0$ 
(see \eqref{e3.10} and \eqref{e3.11}), 
\begin{equation}
\int_0^Tf^0(x(t),V(x(t)))dt \le \int_{\Omega_0}f^0(x(t),V(x(t)))dt \le 
-\delta_0 \mu(\Omega_0).\label{e3.25}
\end{equation}
 By \eqref{e3.3}, the definition of $L$ (see \eqref{e3.8} and \eqref{e3.9}),
\eqref{e3.2} and 
\eqref{e3.13}, we have
\begin{align*}
\int_0^Tf^0(x(t),x'(t)-V(x(t)))dt &\le \int_0^TL\|x'(t)-V(x(t))\|dt\\
& \le \int_0^LL\gamma dt =TL\gamma<1/4.
\end{align*}
When combined with \eqref{e3.24}, \eqref{e3.25}, \eqref{e3.1} and \eqref{e3.5}, this inequality implies 
that
$$
\inf(f)-K_1 \le 
f(x(T))-f(x(0)) \le -\delta_0\mu(\Omega_0)+1,
$$
which yields, together with \eqref{e3.12},
$$
\mu(\Omega_0) \le \delta_0^{-1}(1+K_1-\inf(f))<N_0.
$$
Theorem \ref{thm3.1} is proved. \end{proof}

\begin{theorem} \label{thm3.2}
 Let A(ii) hold, let $V \in \mathcal{A}$ be 
regular, and assume that
$$
\lim_{\|x\| \to \infty}f(x)=\infty.
$$
Let $\gamma :[0,\infty) \to [0,1]$ be such that $\lim_{t \to 
\infty}\gamma(t)=0$. If
$x \in W^{1,1}_{\rm loc}([0,\infty);X)$
is  bounded and satisfies
\begin{equation}
\|x'(t)-V(x(t))\| \le \gamma (t) \quad \text {a.e. } t \in [0,\infty),
\label{e3.26}
\end{equation}
then for each $\epsilon >0$, there exists $N_{\epsilon}>0$ such that the 
following property holds:

For each $\Delta \ge N_{\epsilon}$, there is $t_{\Delta}>0$ such that if $s 
\ge t_{\Delta}$, then
$$
\mu\{t \in [s,s+\Delta]: \Xi(x(t))<-\epsilon\} \le N_{\epsilon}.
$$
 \end{theorem}

\begin{proof}
Let $\epsilon >0$. There is $K_0>0$ such that 
\begin{equation}
\|x(t)\| \le K_0,\quad t \in [0,\infty). \label{e3.27}
\end{equation}
By Theorem \ref{thm3.1}, there is $N_{\epsilon}>0$ such that the following property 
holds:

(P1) For each $\Delta \ge N_{\epsilon}$, there is $\gamma_{\Delta}>0$ such 
that if $y \in W^{1,1}(0,\Delta;X)$ satisfies
\begin{equation}
\|y(0)\| \le K_0 \label{e3.28}
\end{equation}
and
\begin{equation}
\|y'(t)-V(y(t))\| \le \gamma_{\Delta} \quad \text {for a.e. } t \in [0,\Delta], 
\label{e3.29}
\end{equation}
then
\begin{equation}
\mu\{t \in [0,\Delta]:\Xi(x(t))<-\epsilon\}\le 
N_{\epsilon}.\label{e3.30}
\end{equation}

Let $\Delta \ge N_{\epsilon}$ and let $\gamma_{\Delta}$ 
be as guaranteed by (P1).
There exists $t_{\Delta}>0$ such that
\begin{equation}
\gamma (t) \le \gamma_{\Delta} \quad\text {for all } t \ge t_{\Delta}. 
\label{e3.31}
\end{equation}
Assume that $s \ge t_{\Delta}$ and set
\begin{equation}
y(t)=x(t+s),\quad t \in [0,\Delta]. \label{e3.32}
\end{equation}
Clearly, by \eqref{e3.27},
$$
\|y(0)\|=\|x(s)\| \le K_0
$$
so that \eqref{e3.28} holds. It follows from \eqref{e3.32}, the relation $s \ge 
t_{\Delta}$, \eqref{e3.26} and 
\eqref{e3.31} that for a.e. $t \in [0,\Delta]$,
$$
\|y'(t)-V(y(t))\|=\|x'(t+s)-V(x(t+s))\| \le \gamma (t+s) \le 
\gamma_{\Delta}.
$$
Thus \eqref{e3.29} holds too.  By property (P1),
$$
N_{\epsilon} \ge \mu\{t \in [0,\Delta]: \Xi(y(t))<-\epsilon\}=\mu\{t 
\in 
[s,s+\Delta]: \Xi(x(t))<-\epsilon\}.
$$
Theorem \ref{thm3.2} is proved. 

\end{proof}

\begin{theorem} \label{thm3.3} 
Let A(ii) hold, let $V \in \mathcal{A}$ be 
regular, and assume that
$$
\lim_{\|x\| \to \infty}f(x)=\infty.
$$
Let a function $\gamma:[0,\infty) \to [0,1]$ satisfy $\lim_{t \to 
\infty} \gamma (t)=0$. If 
$x \in W^{1,1}_{\rm loc}([0,\infty);X)$  
is bounded and satisfies \eqref{e3.26}, then for each $\epsilon >0$,
$$
\lim_{T \to \infty}\mu\{t \in [0,T]: \Xi(x(t))<-\epsilon\}/T=0.
$$
 \end{theorem}

\begin{proof} Let $\epsilon>0$ and $\delta \in (0,1)$. Let $N_{\epsilon}>0$ 
be as guaranteed by Theorem \ref{thm3.2} and choose a number $\Delta$ such that
\begin{equation}
\Delta >4(N_{\epsilon}+1)/\delta. \label{e3.33}
\end{equation}
By Theorem \ref{thm3.2}, there is $t_{\Delta}>0$ such that for each $s \ge 
t_{\Delta}$,
\begin{equation}
\mu\{t \in [s,s+\Delta]: \Xi(x(t))<-\epsilon\} \le N_{\epsilon}. 
\label{e3.34}
\end{equation}
Choose
\begin{equation}
T_0>(t_{\Delta}+2\Delta)(4/\delta). \label{e3.35}
\end{equation}
Let $T \ge T_0$. There is a natural number $n$ such that 
\begin{equation}
T-n\Delta \ge t_{\Delta}>T-(n+1)\Delta. \label{e3.36}
\end{equation}
This implies that 
\begin{equation}
n \le (T-t_{\Delta})/\Delta<n+1\le 2n. \label{e3.37}
\end{equation}
It follows from \eqref{e3.34} that for each integer $i=0,1,\dots,n-1$,
\begin{equation}
\mu\big\{t \in [t_{\Delta}+i\Delta,t_{\Delta}+(i+1)\Delta]: 
\Xi(x(t))<-\epsilon\big\} \le N_{\epsilon}. \label{e3.38}
\end{equation}
Then by \eqref{e3.36}, \eqref{e3.37} and \eqref{e3.38}, 
\begin{align*}
&\mu\big\{t \in [0,T]: \Xi(x(t))<-\epsilon\big\}\\
&=\mu\big\{t \in [0,t_{\Delta}]: \Xi(x(t))<-\epsilon\big\}\\
&+\mu\big\{t \in [t_{\Delta},t_{\Delta}+n\Delta]: \Xi(x(t))<-\epsilon\big\}
+\mu\big\{t \in [t_{\Delta}+n\Delta,T]:\Xi(x(t))<-\epsilon\big\} \\
&\le  t_{\Delta}+\mu\big\{t \in [t_{\Delta},t_{\Delta}+n\Delta]: 
\Xi(x(t))<-\epsilon\big\}+\Delta  \\
&\le t_{\Delta}+\Delta+\sum_{i=0}^{n-1}\mu\big\{t \in 
[t_{\Delta}+i\Delta,t_{\Delta}+(i+1)\Delta]:\Xi(x(t))<-\epsilon\big\} \\
&\le t_{\Delta}+\Delta+nN_{\epsilon}.
\end{align*}
By this inequality, \eqref{e3.35}, \eqref{e3.37} and \eqref{e3.33},
\begin{align*}
\mu\big\{t \in [0,T]: \Xi(x(t))<-\epsilon\big\}/T 
&\le [t_{\Delta}+\Delta+nN_{\epsilon}]/T \\
&\le (t_{\Delta}+\Delta)/T_0+(nN_{\epsilon})/T<\delta 
/4+(nN_{\epsilon})/T\\
&\le \delta /4+(T/\Delta)N_{\epsilon}/T \\
& \le \delta /4+N_{\epsilon}/\Delta<\delta /4+\delta /4<\delta\,.
\end{align*}
Thus
$$
\mu\{t \in [0,T]: \Xi(x(t))<-\epsilon \}/T <\delta 
$$
for all $T \ge T_0$. 
This concludes the proof of Theorem \ref{thm3.3}. \end{proof} 

\section{Lipschitzian functions satisfying  the Palais-Smale condition}

We start this section by recalling several results which were established in our 
previous paper \cite{a1}.

\begin{proposition}[\cite{a1}] \label{prop4.1} 
 For each $\epsilon >0$, there exists 
$x_{\epsilon} \in X$ such that 
$$
f(x_{\epsilon}) \le \inf(f)+\epsilon \text { and } \Xi(x_{\epsilon}) \ge 
-\epsilon.
$$
\end{proposition}

This proposition follows from Ekeland's variational principle \cite{e1}.

We say that the function 
$f$ satisfies the Palais-Smale (P-S) condition if 
each 
sequence 
$\{x_n\}_{n=1}^{\infty} \subset X$ for which
$$
\sup\{|f(x_n)|: n=1,2,\dots\}<\infty
$$
and $\limsup_{n \to \infty}\Xi(x_n) \ge 0$, has a norm convergent 
subsequence.

Denote
$$
\mathop{\rm Cr}(f)=\{x \in X: \Xi(x) \ge 0\}.
$$
\begin{proposition}[\cite{a1}]  \label{prop4.2}  
If $\{x_n\}_{n=1}^{\infty} \subset X$, 
$\lim_{n \to \infty}x_n=x$, and $\liminf_{n \to \infty}\Xi(x_n) \ge 0$, then 
$\Xi(x) \ge 0$. 
\end{proposition} 


Propositions \ref{prop4.1} and \ref{prop4.2} imply the next three propositions 
which can also be found in \cite{a1}.

\begin{proposition}[\cite{a1}] \label{prop4.3}  If $f$ satisfies 
the (P-S) 
condition, then $\mathop{\rm Cr}(f)\not =\emptyset$. 
\end{proposition}

\begin{proposition}[\cite{a1}] \label{prop4.4} 
 If the function $f$ satisfies the (P-S) 
condition, then for each $r>0$, the set
$$
\{x \in X: \|x\| \le r\} \cap \mathop{\rm Cr}(f)
$$
is compact in the norm topology.
\end{proposition}

For each $x \in X$ and $A \subset X$ set 
$$
d(x,A)=\inf\{\|x-y\|: y \in A\}.
$$

\begin{proposition}[\cite{a1}] \label{prop4.5} 
 Let $r,\epsilon >0$, and let $f$ satisfy 
the (P-S) 
condition. Then there is $
\delta >0$ such that if $x \in X$ satisfies $\|x\|\le r$ and $\Xi(x) \ge 
-\delta$, then $d(x,\mathop{\rm Cr}(f)) \le \epsilon$. 
\end{proposition}

We are now ready to present and prove our three convergence results regarding 
functions satisfying the Palais-Smale condition.

\begin{theorem} \label{thm4.1} 
Let A(ii) hold, and let $V \in \mathcal{A}$ be 
regular. Assume that 
$$
\lim_{\|x\| \to \infty} f(x)=\infty,
$$
and that $f$ satisfies the (P-S) condition. Let $K_0$ and $\epsilon$ be positive 
numbers. 
Then 
there exist $N_*,\tilde K >0$ such that the following property holds:

for each $T \ge N_*$, there is $\gamma >0$ such that if  
$x \in W^{1,1}(0,T;X)$ satisfies
\begin{equation}
\|x(0)\| \le K_0 \label{e4.1}
\end{equation}
and
\begin{equation}
\|x'(t)-V(x(t))\| \le \gamma \text { for a.e. } t \in [0,T], \label{e4.2}
\end{equation}
then
\begin{equation}
\|x(t)\| \le \tilde K,\; t \in [0,T], \label{e4.3}
\end{equation}
and
\begin{equation}
\mu\{t \in [0,T]: d(x(t),\text {Cr}(f))>\epsilon\} \le N_*. \label{e4.4}
\end{equation}
\end{theorem}

\begin{proof} By Theorem \ref{thm3.1} (with $\epsilon =1/2$), there are $N_0$, 
$\tilde K>0$ such that the following property holds:

(P2) For each $T \ge N_0$, there is $\gamma >0$ such that if 
$x \in W^{1,1}(0,T;X)$ satisfies \eqref{e4.1} and \eqref{e4.2}, then
\eqref{e4.3} holds.

By Proposition \ref{prop4.5}, there is $\delta >0$ such that
\begin{equation}
\text{if } z \in X,\; \|z\| \le \tilde K \text { and } \Xi(z) \ge -\delta,\; \text 
{ then } d(z,\mathop{\rm Cr}(f)) \le \epsilon. \label{e4.5}
\end{equation}
 
By Theorem \ref{thm3.1} (with 
$\epsilon=\delta)$, there exists $N_1 >0$ such that:

(P3) For each $T \ge N_1$, there is $\gamma >0$ such that if 
$x \in W^{1,1}(0,T;X)$ satisfies \eqref{e4.1} and \eqref{e4.2}, then
\begin{equation}
\mu(\{t \in [0,T]: \Xi(x(t))<-\delta\}\le N_1. \label{e4.6}
\end{equation}
Set  
\begin{equation}
N_*=N_0+N_1.\label{e4.7}
\end{equation}
Let $T \ge N_*$. By virtue of (P2), there is $\gamma_1>0$ such that the 
following property holds:

(P4) If $x \in W^{1,1}(0,T; X)$ satisfies \eqref{e4.1} and 
\begin{equation}
\|x'(t)-V(x(t))\| \le \gamma_1 \text { for a.e. } t \in [0,T], \label{e4.8}
\end{equation}
then \eqref{e4.3} is true.


Also, by (P3) there exists $\gamma_2>0$ such that the following 
property holds:

(P5) If $x \in W^{1,1}(0,T;X)$ satisfies \eqref{e4.1} and
\begin{equation}
\|x'(t)-V(x(t))\| \le \gamma_2 \text { for a.e. } t \in [0,T], \label{e4.9}
\end{equation}
then \eqref{e4.6} holds.

Set
\begin{equation}
\gamma=\min\{\gamma_0,\gamma_1\}. \label{e4.10}
\end{equation}
Assume that $x \in W^{1,1}(0,T;X)$ satisfies \eqref{e4.1} and \eqref{e4.2}. 
Then by property (P4), \eqref{e4.1}, \eqref{e4.2}, and \eqref{e4.10}, inequality
\eqref{e4.3} 
holds as well.
By property (P5), \eqref{e4.1}, \eqref{e4.2}, and \eqref{e4.10}, inequality
\eqref{e4.6} holds.
Let
$$
t \in [0,T] \text { and } d(x(t),\mathop{\rm Cr}(f))>\epsilon.
$$
Then by \eqref{e4.3} and \eqref{e4.5}, 
$\Xi(x(t))<-\delta$, so that
$$
\{t \in [0,T]: d(x(t),\mathop{\rm Cr}(f))>\epsilon \} \subset \{t \in 
[0,T]: \Xi(x(t))<-\delta\},
$$
 while by \eqref{e4.6} and \eqref{e4.7},
$$
\mu\{t \in [0,T]: d(x(t),\text{Cr(f)})>\epsilon\}\le 
\mu\{t \in [0,T]:\Xi(x(t))<-\delta\} \le N_1 \le N_*.
$$
The proof of Theorem \ref{thm4.1} is complete. \end{proof}


\begin{theorem} \label{thm4.2} 
Let A(ii) hold, let
$V \in \mathcal{A}$ be regular. Assume that $f$ satisfies the (P-S) 
condition and that
$$
\lim_{\|x\| \to \infty}f(x)=\infty.
$$
Let $\gamma:[0,\infty) \to [0,\infty)$ be such that
$\lim_{t \to \infty} \gamma (t)=0$. If
$x \in W^{1,1}_{\rm loc}([0,\infty);X)$ is bounded and satisfies 
$$
\|x'(t)-V(x(t))\| \le \gamma (t) \quad\text {for a.e. } t \in [0,\infty),
$$
then 
for each $\delta>0$, there exists $N_0>0$ such that the following 
property holds:

For each $\Delta \ge N_0$, there is $t_{\Delta}>0$ such that if $s 
\ge t_{\Delta}$,  then
$$
\mu\{t \in [s.\,s+\Delta]: d(x(t),\text {Cr}(f))>\delta\}\le 
N_0.
$$
\end{theorem}

\begin{proof} Let $\delta >0$. Since $x$ is assumed to be bounded, there 
is a constant
\begin{equation}
K_0>\sup\{\|x(t)\|: t \in [0,\infty)\}. \label{e4.11}
\end{equation}
By Proposition \ref{prop4.5}, there exists $\epsilon >0$ such that
\begin{equation}
\text{ if } z \in X,\; \|z\| \le K_0,\text { and } \Xi(z) \ge -\epsilon,\; 
\text { then } d(x,\mathop{\rm Cr}(f)) \le \delta. \label{e4.12}
\end{equation}
 
Let $N_{\epsilon}>0$ be as 
guaranteed 
by Theorem \ref{thm3.2}. Put $N_0=N_{\epsilon}$ and let $\Delta \ge 
N_{\epsilon}.$ By the choice of $N_{\epsilon}$ and Theorem \ref{thm3.2}, 
the 
following property holds:

(P6) There is $t_{\Delta}>0$  such that for each $s \ge t_{\Delta}$,
\begin{equation}
\mu\{t \in [s,s+\Delta]: \Xi(x(t))<-\epsilon\} \le N_{\epsilon}. 
\label{e4.13}
\end{equation}

Assume that $s \ge t_{\Delta}$. By \eqref{e4.11} and \eqref{e4.12},
$$
\text{ if } t \in [s,s+\Delta] \text { and }
 d(x(t),\mathop{\rm Cr}(f))>\delta,\; \text 
{ then } \Xi(x(t))<-\epsilon,
$$
and
$$
\{t \in [s,s+\Delta]: d(x(t),\mathop{\rm Cr}(f))>\delta\} \subset 
\{t \in [s,s+\Delta]: \Xi(x(t))<-\epsilon\}.
$$
When combined with \eqref{e4.13}, this inclusion implies that
$$
\mu\{t \in [s,s+\Delta]: d(x(t),\mathop{\rm Cr}(f))>\delta\} \le 
N_{\epsilon}=N_0,
$$
as claimed. Theorem \ref{thm4.2} is proved. \end{proof}

\begin{theorem} \label{thm4.3} 
Let A(ii) hold, let $V \in \mathcal{A}$ be
regular, and assume that
$$
\lim_{\|x\| \to \infty}f(x)=\infty.
$$
 
Let $\gamma
:[0,\infty) \to [0,1]$ be such that
$\lim_{t \to \infty
}\gamma (t)=0$. If 
$x \in W^{1,1}_{\rm loc}([0,\infty);X)$  
is bounded and satisfies 
$$
\|x'(t)-V(x(t))\| \le \gamma (t) \text { for a.e. } t \in [0,\infty),
$$
then for each $\delta >0$,
$$
\lim_{T \to \infty} \mu\{t \in [0,T]: 
d(x(t),\mathop{\rm Cr}(f))>\delta\}/T=0.
$$
\end{theorem}

\begin{proof} Since $x$ is assumed to be bounded,
there is a constant $K_0$ such that
\begin{equation}
K_0>\sup\{\|x(t)\|: t \in [0,\infty)\}. \label{e4.14}
\end{equation}
Let $\delta>0$.
By Proposition \ref{prop4.5}, there exists $\epsilon >0$ such that
\begin{equation}
\text{if } x \in X,\; \|z\| \le K_0, \text { and } \Xi(z) \ge -\epsilon,\; \text { 
then } d(x,\mathop{\rm Cr}(f)) \le \delta. \label{e4.15}
\end{equation}
By Theorem \ref{thm3.3},
\begin{equation}
\lim_{T \to \infty} \mu\{t \in [0,T]: \Xi(x(t))<-\epsilon\}/T=0. 
\label{e4.16}
\end{equation}
By \eqref{e4.14} and \eqref{e4.15}, for each $T>0$,
$$
\{t \in [0,T]: d(x(t),\mathop{\rm Cr}(f))>\delta\} \subset 
\{t \in [0,T]: \Xi(x(t))<-\epsilon\}.
$$
When combined with \eqref{e4.16}, this inclusion implies that
$$
\lim_{T \to \infty} \mu\{t \in [0,T]: 
d(x(t),\mathop{\rm Cr}(f))>\delta\}/T=0.
$$
Theorem \ref{thm4.3} is proved. \end{proof}

\subsection*{Acknowledgments} 
The second author was partially supported by the Israel 
Science Foundation founded by the Israel Academy of Sciences and 
Humanities (Grant 592/00) 
and by the Technion 
VPR Fund - Argentinian Research Fund.
Part of the third author's research
was carried out when
he was visiting Ohio University.


\begin{thebibliography}{00}


\bibitem{a1} S. Aizicovici, S. Reich and A. J. Zaslavski,
\emph{Convergence theorems for continuous descent methods},
 J. Evol. Equ., to appear.

\bibitem{b1} Y. Benyamini and J. Lindenstrauss,
\emph{Geometric Nonlinear Functional Analysis},
Amer. Math. Soc., Providence, RI, 2000. 

\bibitem{b2} H. Brezis, \emph{Op\'erateurs Maximaux Monotones et 
Semi-Groupes de Contractions dans les Espaces de Hilbert},
North Holland, Amsterdam, 1973.

\bibitem{c1} H. B. Curry,  \emph{The method of steepest descent for 
 nonlinear minimization problems}, Quart. Appl. Math., Vol. 2 (1944), 
 258-261.

\bibitem{d1} F. S. De Blasi and J. Myjak,  
\emph{Generic flows generated by continuous vector fields in Banach spaces},
 Adv. Math., Vol. 50 (1983),  266-280.

\bibitem{d2} F. S. De Blasi and J. Myjak, 
\emph{Sur la porosit\'e des contractions sans point fixe},
 C. R. Acad. Sci. Paris, Vol. 308 (1989), 51-54.

\bibitem{d3} F. S. De Blasi, J. Myjak and P. L. Papini, 
\emph{Porous sets in best approximation theory},
 J. London Math. Soc., Vol. 44(1991), 135-142. 

\bibitem{e1} I. Ekeland, \emph{On the variational principle}, 
J. Math. Anal. Appl., Vol. 47 (1974), 324-353.

\bibitem{h1} J.-B. Hiriart-Urruty and C. Lemar\'echal,
\emph{Convex Analysis and Minimization Algorithms}, Springer,
Berlin, 1993.

\bibitem{i1} A. D. Ioffe and A. J. Zaslavski, 
\emph{Variational principles and well-posedness in optimization and
calculus of variations}, SIAM J. Control Optim., Vol. 38 (2000),
566-581.


\bibitem{n1} J.W. Neuberger, 
\emph{Sobolev Gradients and Differential Equations},
Lecture Notes in Math. No. 1670, Springer, Berlin, 1997.


\bibitem{r1}  S. Reich and  A. J. Zaslavski,
\emph{Generic convergence of descent methods in Banach spaces},
 Math. Oper. Research, Vol.  25 (2000), 231-242.

\bibitem{r2}  S. Reich and  A. J. Zaslavski,  
\emph{The set of divergent descent methods in a Banach space is 
$\sigma$-porous},  SIAM J. Optim. Vol. 11 (2001), 1003-1018.

\bibitem{r3}  S. Reich and  A. J. Zaslavski, 
\emph{Porosity of the set of divergent descent methods},
 Nonlinear Anal., Vol. 47 (2001),  3247-3258.

\bibitem{r4} S. Reich and A. J. Zaslavski,
\emph{Two convergence results for continuous descent methods},
 Electron. J. Differential Equations, Vol. 2003 (2003), No. 24, 1-11.

\bibitem{z1} L. Zaj\'\i\v cek  \emph{Porosity and $\sigma$-porosity},
 Real Anal. Exchange, Vol. 13 (1987), 314-350.

\bibitem{z2} L. Zaj\'\i\v cek \emph{Small non-$\sigma$-porous sets
in topologically complete metric spaces},
 Colloq. Math., Vol. 77 (1998), 293-304.
 
\bibitem{z3} A. J. Zaslavski,  
\emph{Well-posedness and porosity in optimal control 
without convexity  assumptions}, Calc. Var., 
Vol. 13 (2001),  265-293.

\end{thebibliography} 

\enddocument