\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 24, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{1cm}}
\begin{document}
\title[\hfilneg EJDE--2003/24\hfil Continuous descent methods]
{Two convergence results for continuous descent methods}
\author[Simeon Reich \& Alexander J. Zaslavski\hfil EJDE--2003/24\hfilneg]
{Simeon Reich and Alexander J. Zaslavski}
\address{Simeon Reich \hfill\break
Department of Mathematics, The Technion-Israel Institute of
Technology,\hfill\break
32000 Haifa, Israel}
\email{sreich@tx.technion.ac.il}
\address{ Alexander J. Zaslavski\hfill\break
Department of Mathematics, The Technion-Israel Institute of
Technology,\hfill\break
32000 Haifa, Israel}
\email{ajzasl@tx.technion.ac.il}
\date{}
\thanks{Submitted August 5, 2002. Published March 10, 2003.}
\subjclass[2000]{37L99, 47J35, 49M99, 54E50, 54E52, 90C25}
\keywords{Complete metric space, convex function, descent method, porous set,
\hfill\break\indent regular vector field}
\begin{abstract}
We consider continuous descent methods for the minimization of
convex functionals defined on general Banach space.
We establish two convergence results for methods which
are generated 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
\newtheorem{theorem}{Theorem}[section] % theorems numbered with section #
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\numberwithin{equation}{section}
\section{Introduction}
The study of discrete and continuous descent methods
is an important topic in optimization theory
and in dynamical systems. See, for example, [3, 7, 9, 10].
Given a continuous convex function $f$ on a Banach space $X$,
we associate with $f$
a complete metric space of vector fields $V:X \to X$
such that $f^0(x,Vx) \le 0$ for all $x \in X$. Here $f^0(x,h)$ is
the right-hand derivative of $f$ at $x$ in the direction $h \in X$.
To each such vector field there correspond two
gradient-like iterative processes.
In our recent papers [10, 11]
we show that for most of the vector fields in this space, both
iterative processes generate sequences $\{x_n\}_{n=1}^{\infty}$ such that
the sequences
$\{f(x_n)\}_{n=1}^{\infty}$ tend to $\inf(f)$ as $n \to \infty$.
In the present paper, we first briefly review these results and then study
the convergence of the values of the function $f$
to its infimum along the trajectories of
an analogous continuous dynamical system governed by such vector
fields.
When we say that most of the elements of a complete metric space $Y$
enjoy a certain property, we mean that the set of points which have this
property contains a $G_{\delta}$ everywhere dense subset of $Y$. In other
words, this property holds generically.
Such an approach, when a certain property is investigated
for the whole space $Y$ and not just for a single point in $Y$,
has already been successfully applied
in many areas of Analysis [1, 4-6, 8, 13, 16].
Now we recall the concept of porosity [2, 5, 6, 11, 14, 17]
which enables us to obtain even more refined results.
Let $(Y,d)$ be a complete metric space.
We denote by $B_d(y,r)$ the closed ball of center $y \in Y$ and radius $r>0$.
We say that a subset $E \subset Y$ is porous in $(Y,d)$
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
$$
B_d(z,\alpha r) \subset B_d(y,r) \setminus E.
$$
A subset of the space $Y$ is called $\sigma$-porous in $(Y,d)$
if it is a countable union of porous subsets in $(Y,d)$.
Other notions of porosity have been used in [2, 14].
We use the rather strong notion which appears in
[5, 6, 11, 17].
Since porous sets are nowhere dense, all $\sigma$-porous sets are of the
first category. If $Y$ is a finite-dimensional Euclidean space
$\mathbb{R}^n$, then
$\sigma$-porous sets are of Lebesgue measure $0$.
The existence of a non-$\sigma$-porous set $P \subset \mathbb{R}^n$,
which is of the first Baire
category and of Lebesgue measure $0$, was established in [14].
It is easy to see
that for any $\sigma$-porous set $A \subset \mathbb{R}^n$, the set $A \cup P
\subset \mathbb{R}^n$ also belongs to the family
$\mathcal{E}$ consisting of
all the non-$\sigma$-porous subsets of $\mathbb{R}^n$
which are
of the first Baire category and have
Lebesgue measure $0$.
Moreover, if $Q \in \mathcal{E}$ is a countable union of
sets $Q_i \subset \mathbb{R}^n$, $i=1,2,\dots $, then there is a natural number
$j$ for which the set $Q_j$ is non-$\sigma$-porous. Evidently, this set
$Q_j$ also belongs to $\mathcal{E}$.
Thus one sees that the family $\mathcal{E}$ is quite large.
Also, every complete metric space
without isolated points contains a closed nowhere dense set which is not
$\sigma$-porous [15].
To point out the difference between porous and nowhere dense sets, note
that if $E \subset Y$ is nowhere dense, $y \in Y$ and $r>0$, then there is
a point $z \in Y$ and a number $s>0$ such that $B_d(z,s) \subset B_d(y,r)
\setminus E$. If, however, $E$ is also porous, then for small enough $r$
we can choose $s=\alpha r$, where $\alpha \in (0,1)$ is a constant which
depends only on $E$.
This paper is organized as follows. In the second section we briefly
review our work on discrete descent methods. In the third section
we present our two theorems (Theorems 3.2 and 3.3)
on continuous descent methods. The first one
is concerned with infinite horizon trajectories, while the second
deals with finite horizon perturbed trajectories. We prove Theorem 3.2
in Section 4. Theorem 3.3 is proved in Section 5.
\section{Discrete descent methods} %sec 2
Let
$(X^*,\|\cdot \|_*)$ be the dual space of the Banach space $(X,\|\cdot\|)$,
and let
$f: X \to \mathbb{R}^1$ be a convex continuous function
which is bounded from below.
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
$$\partial f(x)=\{l \in X^*: f(y)-f(x) \ge l(y-x) \text { for all }
y \in X\}$$
be the subdifferential of $f$ at $x$.
It is well known that the set $\partial f(x)$ is nonempty and
norm-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 $X$
(i.e., for each $K_0>0$ there is $K_1>0$ such that $\|Vx\| \le K_1$
if $\|x\| \le K_0$), and for each $x \in X$ and
each $l \in \partial f(x)$,
$l(Vx)\le 0$. We denote by $\mathcal{A}_c$ the set of all continuous
$V \in \mathcal{A}$, by $\mathcal{A}_u$ the set of all $V \in \mathcal{A}$
which are
uniformly continuous on each bounded subset of $X$, and by $\mathcal{A}_{au}$
the set of all $V \in \mathcal{A}$ which are uniformly continuous on the subsets
$$
\{x \in X: \|x\| \le n \text { and } f(x) \ge \inf(f)+1/n\}
$$
for each integer $ n \ge 1$.
Finally, let $\mathcal{A}_{auc}=\mathcal{A}_{au} \cap \mathcal{A}_c$.
Next we endow the set $\mathcal{A}$ with a metric $\rho$:
For each $V_1,V_2 \in \mathcal{A}$ and each integer $i \ge 1$, we first set
$$
\rho_i(V_1,V_2)=\sup\{\|V_1x-V_2x\|: x \in X \
\text { and } \|x\| \le i\}
$$
and then define
$$
\rho(V_1,V_2)=\sum_{i=1}^{\infty}2^{-i}[\rho_i(V_1,V_2)
(1+\rho_i(V_1,V_2))^{-1}].
$$
Clearly, $(\mathcal{A},\rho)$ 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 $\rho$.
Evidently, ${\mathcal{A}}_c$, $\mathcal{A}_u$, $\mathcal{A}_{au}$ and
$\mathcal{A}_{auc}$ are all
closed subsets of the metric space $(\mathcal{A},\rho)$. In the sequel
we assign to all these spaces the same metric $\rho$.
To compute $\inf(f)$, we associate in [10, 11] with
each vector field $W \in \mathcal{A}$ two
gradient-like iterative processes to be defined below.
Note that the counterexample studied in Section 2.2
of Chapter VIII of [7] shows that, even for two-dimensional problems,
the simplest choice for a descent direction,
namely the normalized steepest descent direction,
$$
V(x)=\mathop{\rm argmin}\big\{\max_{l \in \partial f(x)}: \|d\|=1\big\},
$$
may produce sequences the functional values of
which fail to converge to the infimum of $f$.
This vector field $V$ belongs to $\mathcal{A}$ and the Lipschitzian
function $f$ attains
its infimum. The steepest descent scheme (Algorithm 1.1.7) presented
in Section 1.1 of Chapter VIII of [7] corresponds to any of
the two iterative processes we consider below.
In infinite dimensions the minimization
problem is even more difficult and
less understood. Moreover, positive results usually require
special assumptions on
the space and the functions.
However, as shown in [10]
(under certain assumptions on the function $f$),
for an arbitrary Banach space $X$ and a generic vector field
$V \in \mathcal{A}$, the values of $f$
tend to its infimum for both processes.
In [11]
we introduced the class of regular vector fields $V \in \mathcal{A}$
and established three main results.
The first one, Theorem 2.2 below, shows
(under the two mild assumptions A1 and A2 on $f$
stated below)
that the complement of the set of regular vector fields
is not only of the first
category, but also
$\sigma$-porous
in each of the spaces $\mathcal{A}$, ${\mathcal{A}}_c$, $\mathcal{A}_u$, $\mathcal{A}_{au}$ and
$\mathcal{A}_{auc}$.
We then show (Theorem 2.3) that
for any regular vector field $V \in \mathcal{A}_{au}$,
the values of the function $f$
tend to its infimum for both processes. If, in addition to A1 and A2,
$f$ also satisfies the assumption A3, then this convergence result
is valid for any regular $V \in \mathcal{A}$.
The last result in [11], Theorem 2.4 below, is a stability theorem for regular
vector fields.
These results are valid in any Banach space
and for those convex functions which satisfy
the following two assumptions.
\begin{itemize}
\item[\textbf{A1}] There exists a bounded set $X_0 \subset X$ such that
$$ \inf(f)=\inf\{f(x): x \in X\}=\inf\{f(x): x \in X_0\}; $$
\item[\textbf{A2}] for each $r>0$ the function $f$
is Lipschitzian on the ball $\{x \in X: \|x\| \le r\}$.
\end{itemize}
Note that assumption A1 clearly holds if
$\lim_{\|x\| \to \infty}f(x)=\infty$.
We 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
$$\|x\| \le n \text { and } f(x) \ge \inf(f)+1/n,$$
and each $l \in \partial f(x)$, we have
$$l(Vx) \le -\delta(n).$$
Denote by $\mathcal{F}$ the set of all regular vector fields
$V \in \mathcal{A}$.
It is not difficult to verify the following property of regular
vector fields. It means, in particular, that $\mathcal{A} \setminus \mathcal{F}$
is a face of the convex cone $\mathcal{A}$.
\begin{proposition} \label{prop2.1}
Assume that $V_1,V_2 \in \mathcal{A}$, $V_1$ is
regular, $\phi:X \to [0,1]$, and that for each integer $n \ge 1$,
$$
\inf\{\phi (x): x \in X \text { and } \|x\| \le n\}>0.
$$
Then the mapping $x \to \phi(x)V_1x+(1-\phi(x))V_2x$, $x \in X$,
also belongs to $\mathcal{F}$.
\end{proposition}
The first result of [11]
shows that in a very strong sense most of the vector
fields in $\mathcal{A}$ are regular.
\begin{theorem} \label{thm2.1}
Assume that both A1 and A2 hold. Then
$\mathcal{A} \setminus \mathcal{F}$ (respectively,
$\mathcal{A}_c \setminus \mathcal{F}$, $\mathcal{A}_{au} \setminus \mathcal{F}$ and
$\mathcal{A}_{auc} \setminus \mathcal{F}$) is a $\sigma$-porous subset of the space
$\mathcal{A}$ (respectively,
$\mathcal{A}_c$, $\mathcal{A}_{au}$ and
$\mathcal{A}_{auc}$).
Moreover, if $f$ attains its infimum, then
the set $\mathcal{A}_u \setminus \mathcal{F}$ is also
a $\sigma$-porous subset of the space $\mathcal{A}_u$.
\end{theorem}
Now let $W \in \mathcal{A}$. We associate with $W$ the following
two gradient-like iterative processes.
For $x \in X$ we denote by $P_W(x)$ the set of all
$y \in \{x+\alpha Wx: \alpha \in [0,1]\}$
such that
$$ f(y)=\inf\{f(x+\beta Wx): \beta \in [0,1]\}. $$
Given any initial point
$x_0 \in X$, one can construct a sequence $\{x_i\}_{i=0}^{\infty} \subset X$
such that for $i=0,1,\dots $,
\[
x_{i+1} \in P_W(x_i). \tag{2.1}
\]
This is our first iterative process.
Next we describe the second iterative process.
Given a sequence $\mathbf{a}=\{a_i\}_{i=0}^{\infty} \subset (0,1]$ such that
\[
\lim_{i \to \infty} a_i=0 \quad \text {and}\quad \sum_{i=0}^{\infty} a_i =\infty,
\tag{2.2}
\]
we construct for each initial point
$x_0 \in X$ a sequence $\{x_i\}_{i=0}^{\infty} \subset X$
according to the following rule: For $i=0,1,\dots $,
\[
x_{i+1}=\begin{cases}
x_i+a_iW(x_i) &\text {if } f(x_i+a_iW(x_i))0$ such that
for each $x_1,x_2 \in X$
satisfying
$$\|x_1\|,\|x_2\| \le n,\; f(x_i) \ge \inf (f)+1/n,\; i=1,2,
\text { and } \|x_1-x_2\| \le \delta, $$
we have $H(\partial f(x_1),\partial f(x_2)) \le 1/n$.
\end{itemize}
This assumption is certainly satisfied if $f$ is differentiable and its
derivative is uniformly continuous on those bounded subsets of $X$ over
which the infimum of $f$ is larger than $\inf(f)$.
The next result of [11] is a convergence theorem for
those iterative processes associated with
regular vector fields.
It is of interest to note that we obtain convergence
when either the regular vector field $W$ or the subdifferential
$\partial f$ enjoys a certain uniform continuity property.
\begin{theorem} \label{thm2.2}
Assume that $W \in \mathcal{A}$ is regular, A1, A2 are valid
and that at least one of the following two conditions holds:
1. $W \in \mathcal{A}_{au}$, or
2. A3 is valid. Then the following two assertions are true
\begin{itemize}
\item[(i)] Let $\{x_i\}_{i=0}^{\infty} \subset X$
satisfy (2.1) for all $i=0,1,\dots $.
If $\liminf_{i \to \infty}\|x_i\|<\infty$,
then $\lim_{ i \to \infty}f(x_i)=\inf(f)$.
\item[(ii)] Let the sequence $\mathbf{a}=\{a_i\}_{i=0}^{\infty} \subset
(0,1]$ satisfy (2.2)
and let the sequence
$\{x_i\}_{i=0}^{\infty} \subset X$ satisfy (2.3) for all
$i=0,1,\dots$.
If
$\{x_i\}_{i=0}^{\infty}$ is bounded, then
$\lim_{i \to \infty } f(x_i)=\inf(f)$.
\end{itemize}
\end{theorem}
Finally, we impose an additional coercivity condition on $f$,
and establish the following stability theorem. Note that this
coercivity condition implies A1.
\begin{theorem} \label{thm2.3}
Assume that $ f(x) \to \infty $ as $\|x\| \to \infty$,
$V \in \mathcal{A}$ is regular, A2 is valid
and that at least one of the following two conditions holds:
1. $V \in \mathcal{A}_{au}$, or
2. A3 is valid.
Let $K,\epsilon >0$ be given. Then
there exist a neighborhood $\mathcal{U} $ of $V $ in $\mathcal{A}$ and a natural
number $N_0$ such that the following two assertions are true:
\begin{itemize}
\item[(i)] For each $W \in \mathcal{U} $ and each sequence $\{x_i\}_{i=0}^{N_0} \subset X$
which satisfies
$\|x_0\| \le K$ and (2.1) for all $
i=0,\dots, N_0-1$,
the inequality $f(x_{N_0}) \le \inf(f)+\epsilon$ holds.
\item[(ii)] For each sequence of numbers $
\mathbf{a}=\{a_i\}_{i=0}^{\infty} \subset (0,1]$ satisfying (2.2),
there exists a natural number $N$ such that
for each $W \in \mathcal{U}$ and each sequence $\{x_i\}_{i=0}^N \subset X$
which satisfies
$\|x_0\| \le K$ and
(2.3) for all $ i=0,\dots ,N-1$,
the inequality $f(x_N) \le \inf(f)+\epsilon$ holds.
\end{itemize}
\end{theorem}
A partial extension of the results reviewed in this section to
functions $f:X \to \mathbb{R}$ which are not necessarily convex
can be found in [12].
\section{Continuous descent methods} %sec. 3
Let $T>0$, $x_0 \in X$ and let $u:[0,T] \to X$ be a Bochner integrable function.
Set
$$x(t)=x_0+\int_0^tu(s)ds,\quad t \in [0,T].$$
Then $x:[0,T] \to X$ is differentiable and
$x'(t)=u(t)$ for a.e. $t \in [0,T]$.
Recall that the function $f:X \to \mathbb{R}$ is assumed to be
convex and continuous
and therefore it is, in fact,
locally Lipschitzian. It follows that its restriction to
the set $\{x(t): t \in [0,T]\}$ is Lipschitzian.
Indeed, since the set $\{x(t): t \in [0,T]\}$ is compact, the
closure of its convex hull $C$
is both compact and convex, and so the restriction of $ f$ to $C$ 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:
\begin{gather*}
x'(t)=\lim_{h \to 0}h^{-1}[x(t+h)-x(t)],\\
(f\cdot x)'(t)=\lim_{h \to 0} h^{-1}
[f(x(t+h))-f(x(t))].
\end{gather*}
\begin{proposition} \label{prop3.1}
Assume that $t \in [0,T]$ and that
both the derivatives
$x'(t)$ and $(f\cdot x)'(t)$ exist.
Then
\[
(f\cdot x)'(t)=\lim_{h \to 0}h^{-1}
[f(x(t)+hx'(t))-f(x(t))]. \tag{3.1}
\]
\end{proposition}
\paragraph{Proof.}
There exist a neighborhood $\mathcal{U}$ of $x(t)$ in $X$ and a constant $L>0$
such that
\[
|f(z_1)-f(z_2)| \le L\|z_1-z_2\| \quad \text { for all } z_1,z_2 \in \mathcal{U}.
\tag{3.2}
\]
Let $\epsilon >0$. There exists $\delta >0$ such that
\[
x(t+h),\; x(t)+hx'(t) \in \mathcal{U} \quad \text {for each } h \in [-\delta,
\delta] \cap [-t,T-t], \tag{3.3}
\]
and such that
for each $h \in [(-\delta,\delta) \setminus \{0\}]\cap [-t,T-t]$,
\[
\|x(t+h)-x(t)-hx'(t)\| <\epsilon |h|. \tag{3.4}
\]
Let
\[
h \in [(-\delta,\delta) \setminus \{0\}]\cap [-t,T-t].\tag{3.5}
\]
It follows from (3.3), (3.2) and (3.4)
that
\[
|f(x(t+h))-f(x(t)+hx'(t))| \le L
\|x(t+h)-x(t)-hx'(t)\| 0$
and $\delta>0$ such that for each $T \ge N_0$
and each differentiable mapping $x:[0,T] \to X$ satisfying
$$\|x(0\| \le K_0 \quad \text {and}\quad
\|x'(t)-V(x(t))\| \le \delta\; \text { for a.e. } t \in [0,T],
$$
the following inequality holds for all
$t \in [N_0,T]$:
$$f(x(t)) \le \inf(f)+\epsilon.$$
\end{theorem}
\section{Proof of Theorem 3.2}
Assume the contrary.
Since $f(x(t))$ is decreasing on $[0,\infty)$, this means that
there exists $\epsilon >0$ such that
\[
\lim_{t \to \infty} f(x(t))>\inf(f)+\epsilon. \tag{4.1}
\]
Then by Proposition 3.1 and (3.9), for each $T>0$,
\[
\begin{aligned}
f(x(T))-f(x(0))&=\int_0^T(f\cdot x)'(t)dt \\
&=\int_0^Tf^0(x(t),x'(t))dt\\
&=\int_0^T f^0(x(t),V(x(t)))dt\\
&\le \int_{\Omega_T}f^0(x(t),V(x(t)))dt,
\end{aligned} \tag{4.2}
\]
where
\[
\Omega_T=\{t \in [0,T]: \|x(t)\| \le r\}.\tag{4.3}
\]
Since $V$ is regular, there exists $\delta >0$
such that for each $x \in X$
satisfying
\[
\|x\| \le r+1 \text { and } f(x) \ge \inf(f)+\epsilon/2, \tag{4.4}
\]
and each $l \in \partial f(x)$, we have
\[
l(Vx) \le -\delta. \tag{4.5}
\]
It follows from (4.2), (4.3), (4.1),
the definition of $\delta$ (see (4.4), (4.5))
and (3.10)
that for each $T>0$,
$$
f(x(T))-f(x(0)) \le
\int_{\Omega_T}f^0(x(t),V(x(t)))dt \le -\delta \mu(\Omega_T) \to -\infty
$$
as $T \to \infty$, which is a contradiction completing the poof.
\section{Proof of Theorem 3.3} %sec.5
We assume that $\epsilon <1/2$ and choose
\[
K_1 >\sup\{f(x): x \in X \text { and } \|x\| \le K_0+1\}. \tag{5.1}
\]
The set
\[
\{x \in X: f(x) \le K_1 +|\inf(f)|+4\} \tag{5.2}
\]
is bounded. Therefore there exists $K_2>K_0+K_1$
such that
\[
\text {if} \; f(x) \le K_1+|\inf(f)|+4, \quad \text {then } \|x\| \le K_2.
\tag{5.3}
\]
There exists a number $K_3 >K_2+1$ such that
\[
\sup\{f(x): x \in X \text { and } \|x\| \le K_2+1\}+2
<\inf\{f(x): x \in X \text { and } \|x\| \ge K_3\}. \tag{5.4}
\]
There exists a number $L_0>0$ such that
\[
|f(x_1)-f(x_2)| \le L_0\|x_1-x_2\| \tag{5.5}
\]
for each $x_1,x_2 \in X$ satisfying
\[
\|x_1\|,\|x_2\| \le K_3+1. \tag{5.6}
\]
Fix an integer
\[
n>K_3+8/\epsilon. \tag{5.7}
\]
There exists
a positive number $\delta(n)<1$ such that
\begin{itemize}
\item[(P1)] for each $x \in X$ satisfying
$\|x\| \le n \text { and } f(x) \ge \inf(f)+1/n$,
and each $l \in \partial f(x)$, we have
$l(Vx) \le -\delta(n)$.
\end{itemize}
Choose a natural number $N_0>8$ such that
\[
8^{-1}\delta(n)N_0 >|\inf(f)|+\sup\{|f(z)|: z \in X
\text { and } \|z\| \le K_2\}+4
\tag{5.8}
\]
and a positive number $\delta$ which satisfies
\[
8\delta(N_0+1)(L_0+1)<\epsilon
\quad\text {and} \quad (1+L_0)\delta < \delta(n)/2. \tag{5.9}
\]
Let $T \ge N_0$ and let $x:[0, T] \to X$ be a differentiable
function such that
\[
\|x(0)\| \le K_2 \tag{5.10}
\]
and
\[
\|x'(t)-V(x(t))\| \le \delta \text { for a.e. } t \in [0,T]. \tag{5.11}
\]
We will show that
\[
\|x(t)\| \le K_3,\; t \in [0,\min\{2N_0,T\}]. \tag{5.12}
\]
Assume the contrary. Then there exists $t_0 \in (0,\min\{2N_0,T\}]$
such that
\[
\|x(t)\| \le K_3,\; t \in [0,t_0) \text { and } \|x(t_0)\|=K_3. \tag{5.13}
\]
It follows from Proposition 3.1, the convexity of directional derivatives,
the inequality $f^0(x(t),Vx(t)) \le 0$ which holds for all $t \in [0,T]$,
(5.13), the definition of $L_0$ (see (5.5), (5.6)), and (5.11)
that
\begin{align*}
f(x(t_0))-f(x(0))&=\int_0^{t_0}(f \cdot x)'(t)dt\\
&=\int_0^{t_0}f^0(x(t),x'(t))dt\\
&\leq \int_0^{t_0}f^0(x(t),V(x(t))dt+\int_0^{t_0}f^0(x(t),x'(t)-V(x(t)))dt\\
&\le\int_0^{t_0}f^0(x(t),x'(t)-V(x(t)))dt \\
&\le \int_0^{t_0}L_0\|x'(t)-V(x(t))\|
\le t_0L_0\delta.
\end{align*}
Thus by (5.9),
$$ f(x(t_0)) \le f(x(0))+2N_0L_0\delta \inf(f)+\epsilon/8 \text { and } \|x(t)\| \le K_3, \;
t \in [1,N_0]. \tag{5.16}
\]
It follows from (5.16), Property (P1) and (5.7) that
\[
f^0(x(t),V(x(t))) \le -\delta(n),\; t \in [1,N_0]. \tag{5.17}
\]
By (5.17), (5.16), (5.11), (5.9), the convexity
of the directional derivatives of $f$, and
the definition of $L_0$ (see (5.5), (5.6)), we have,
for a.e. $t \in [1,N_0]$,
\[
\begin{aligned}
f^0(x(t),x'(t)) &\le f^0(x(t),V(x(t)))+f^0(x(t),x'(t)-V(x(t)))\\
&\le -\delta(n)+L_0\|x'(t)-V(x(t))\|\\
&\le -\delta(n)+L_0\delta
\le -\delta(n)/2.
\end{aligned} \tag{5.18}
\]
It follows from the convexity of the
directional derivatives of $f$, the inclusion
$V \in \mathcal{A}$, (5.11), (5.12)
and the definition of $L_0$ (see (5.5), (5.6)),
that for a.e. $t \in [0,1]$,
\[
\begin{aligned}
f^0(x(t),x'(t)) &\le f^0(x(t),V(x(t)))+f^0(x(t),x'(t)-V(x(t)))\\
&\le f^0(x(t),x'(t)-V(x(t))) \\
&\le L_0\|x'(t)-V(x(t))\|\le L_0\delta.
\end{aligned} \tag{5.19}
\]
The inequalities (5.10), (5.18) and (5.19)
imply that
\begin{align*}
&\inf(f)-\sup\big\{f(z): z \in X, \;\|z\| \le K_2\big\} \\
&\le f(x(N_0))-f(x(0))\\
&=\int_0^{N_0}f^0(x(t),x'(t))dt\\
&=\int_0^1f^0(x(t),x'(t))dt+\int_1^{N_0}f^0(x(t),x'(t))dt\\
&\le -2^{-1}\delta(n)N_0/2+1.
\end{align*}
This contradicts (5.8). The contradiction we have obtained
yields the existence of
a point $t_0$ which satisfies both (5.14) and (5.15).
Clearly, $\|x(t_0)| \le K_2$.
Having established (5.12) and the existence of such a point $t_0$
for an arbitrary mapping $x$ satisfying both (5.10) and (5.11),
we now consider the mapping $x_0(t)=x(t+t_0)$, $t \in [0,T-t_0]$.
Evidently, (5.10) and (5.11) hold true with $x$ replaced by
$x_0$ and $T$ replaced by $T-t_0$. Hence, if $T-t_0 \ge N_0$, then
we have
$$
\|x(t)\|=\|x_0(t-t_0)\| \le K_3,\quad t \in [t_0,t_0+\min\{2N_0,T\}],
$$
and there exists
$t_1 \in [t_0+1,t_0+N_0]$
for which
$$f(x(t_1)) \le \inf(f)+\epsilon/8.
$$
Repeating this procedure,
we obtain by induction a finite sequence of points
$\{t_i\}_{i=0}^q$ such that
\begin{gather*}
t_0 \in [1,N_0],\quad t_{i+1}-t_i \in [1,N_0],\quad i=0,\dots, q-1,\;
T-t_q