数理解析研究所講究録
1071 巻 1998 年 44-58

44

NONLINEAR ERGODIC THEOREMSFOR ALMOST NONEXPANSIVE CURVES
東京工業大学大学院情報理工学研究科厚芝幸子 (Sachiko Atsushiba)

1. INTRODUCTION
Let $H$ be a real Hilbert space with inner productof $H$ . Then, a mapping

$T$

of

$C$

$\langle\cdot, \cdot\rangle$

and norm$||\cdot||$. Let $C$ be a subset

into itself is called nonexpansive if $||Tx-Ty||\leq||x-y||$for all $y\in C$ . We denote by $F(T)$ the set of fixed points of .
The first nonlinear ergodic theorem for nonexpansive mappings in a Hilbert space wasestablished by Baillon [2]: Let be a nonempty closed convex subset of a Hilbert spaceand let be a nonexpansive mapping of into itself. If for some $x_{0}\in C,$
is bounded, then for each $x\in C$ , the Ces\‘aro means
$T$

$x,$

$C$

$T$

$C$

$\{T^{n}x_{0:n}\in \mathbb{N}\}$$S_{n}(x)= \frac{1}{n}\sum_{k=0}^{n-1}T^{k}X$converge weakly to some $y\in F(T)$ . In Baillon’s theorem, putting $y=Px$ for each $x\in C$ ,is a nonexpansive retraction of onto $F(T)$ such that $PT^{n}=T^{n}P=P$ for all positive
$P$

$C$

integers and $Px\in\overline{co}\{T^{n_{X:n}}=1,2, \ldots\}$ for each $x\in C,$ whereis the closure of theconvex hull of . Takahashi $[22, 23]$ proved the existence of such retractions,retractions”, for noncommutative semigroups of nonexpansive mappings in a Hilbert space.
Rod\’e [19] found a sequence of means on the semigroup, generalizing the Ces\‘aro means on
the positive integers, such that the corresponding sequence of mappings converges to an$\overline{Co}A$

$n$

(

$A$

$‘ \mathrm{e}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{C}$ergodic retraction onto the set of common fixed points. Recently Takahashi [25] proved anonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings without
convexity in a Hilbert space. On the other hand, Miyadera and Kobayasi [17] introduced
the notion of almost-orbits of a one-parameter nonexpansive semigroup on
and stud$C$

ied wealc and strong convergence theorems of such almost-orbits (see also [6, 7]). Then,
1991 Mathematics Subject Classification. Primary; Secondary.
Key words and phrases. Nonlinear ergodic theorem, weak convergence, invariant mean, nonexpansive$47\mathrm{A}35$

curve.

$47\mathrm{H}20$

45

Rouhani $[20, 21]$ introduced the notion of almost nonexpansive sequences and curves in a
Hilbert space and proved weak and strong convergence theorems for such sequences andcurves. Kada and Takahashi [12] introduced the notion of almost nonexpansive curves
over a commutative semigroup. They studied the asymptotic behavior of such almost
nonexpansive curves over a commutative semigroup.
In this article, we recall the notion of almost nonexpansive sequences and curves overa commutative semigroup and nonlinear ergodic theorems for such sequences and curves.
Further, we introduce the notion of almost nonexpansive curves over a noncommutative
semigroup and for any almost nonexpansive curve , consider generalized fixed point set$F(u)$ . Then, we prove nonlinear ergodic theorems for almost nonexpansive curves over a$u$right reversible semitopological semigroup.

2. THEOREMS FOR NONEXPANSIVE SEQUENCES AND CURVES
is a nonempty closed convex subset of a
Throughout this article, we assume that
real Hilbert space $H$ . We also assume that is a subspace of $B(S)$ containing constants(or $w- \lim_{n}x_{n}=x$ ) to indicate that the sequence
unless other specified. We write
$C$

$D$

$x_{n}arrow x$) andof vectors converges weakly to . Similarly $x_{n}arrow x$ (or-convergence, respectively.
) will symbolize strong convergence and(orand the set of all real numbers, nonnegative real numbers andWe denote bymean the convex
nonnegative integer, respectively. For a subset of $H,$ $coA$ andhull of and the closure of convex hull of , respectively.
$x_{n}arrow w^{*}x$$x$$\{x_{n}\}$$\lim_{narrow\infty}x_{n}=x$$w^{*}$$w^{*}- \lim_{narrow\infty}X_{n}=x$$\mathbb{R},$$\mathbb{R}^{+}$$\mathbb{N}$$A$$\overline{co}A$

$A$

$A$

The first nonlinear ergodic theorem for nonexpansive mappings in a Hilbert space wasestablished by Baillon [2]:Theorem 2.1 ([2]). Let
$T$

$C$

be a $nonempt8Jclo\mathit{8}ed$ convex subset of a Hilbert space and let

be a nonexpansive mapping of$C$into itself.

If for somebounded, then for each $x\in C$ , the Ces\‘aro means$S_{n}(X)= \frac{1}{n}\sum_{k=0}^{n-1}T^{k}X$converge weakly to a fixed point of .
$T$

$x_{0}\in C,$$\{T^{n}x_{0} :n\in \mathrm{N}\}$

is

46

Many mathematicians obtained generalizations of Baillon’s result [2] (for example, see,[17, 19, 22, 23, 25] . Among other things, by modifying the method used by. Rouhani$)$$\mathrm{B}.\mathrm{D}$and S. Kakutani (” Ergodic theorems for nonexpansive nonlinear operators in a Hilbert
space”, preprint, 1984) and. Rouhani (”Ergodic theorems for nonexpansive sequencesin Hilbert spaces and related problems”, Part I, Thesis, Yale University, and ” A new
proof of the weak convergence theorems for nonexpansive sequence and curves in Hilbert
spaces,” preprint, 1984), Rouhani $[20, 21]$ introduced the notion of almost nonexpansive
sequences and curves in a Hilbert space and studied nonlinear ergodic theorems for suchsequences and curves. Let
be a sequence in $H$ . Then,is called an almost
nonexpansive curve if there exists a nonnegative real-valued function
on
such
that
$\mathrm{B}.\mathrm{D}$$\{x_{n}\}$$\{x_{n}\}$$\epsilon(\cdot, \cdot)$$\mathrm{N}\cross \mathrm{N}$$||x_{i+k}-x_{j}+k||2\leq||_{X_{i}}-X_{j}||^{2}+\epsilon(i,j)$for every $i,j$ and$\mathrm{N},$$\{x_{n}\}$

$k$

in

$\mathrm{N}$and$\lim_{i,j}\epsilon(i,j)=0$. In the case when $\epsilon(s, t)=0$ for every $i,j\in$is called a nonexpansive sequence (see [20]).

Remark 2.2. Let$\{x_{n}\}$be a bounded sequence in $H$ such that$||_{X-X}i+kj+k||\leq||_{X_{i}}-x_{j}||+\epsilon_{1}(_{S}, t)$for every $i,j$ and

$k$

in

$\mathbb{N}$and$\lim_{i,j}\epsilon_{1}(i,j)=0$

nonexpansive sequence curve with

A sequence$\{x_{n}\}$. Then, it is obvious that$\{x_{n}\}$$\epsilon(i,j)=4(\sup_{i\in \mathrm{N}}||X_{i}||)\epsilon_{1}(i,j)+\epsilon_{1}(i,j)^{2}$

in $H$ is called an almost-orbit of$T$

is an almost(see also [20, 21]).

if$\lim_{k}\sup_{0n\geq}||x_{n+}k-T^{n}xk||=0$(see [6]).

Example 2.3. Let$T$be a nonexpansive mapping from a closed convex subset

is a bounded almost-orbit of , from Remark 2.2,
nonexpansive sequence in $H$ . Hence, we also see that for $x\in C,$
into itself. If

nonexpansive curve from$\mathbb{R}^{+}$

to

$C$

of $H$$\{x_{n}\}$is an almost$\{T^{n}x\}$

is an almost$T$$\{x_{n}\}$$C$

if $\{T^{n}x\}$ is bounded (see also [20]).

be a sequence $H$ . Then, we denote the subsets
andof $H$ as follows:$q\in F_{1}$ if and only if $||x_{i+k}-q||\leq||x_{i}-q||$ for every$k\in S$ and $q\in F$ if and onlyLet$\{x_{n}\}$$F_{1}$$F$$i,$

if $\lim_{n}||x_{n}-q||$ exists. We can prove that

$F_{1}$and$F$

are closed convex subset of

$H$

and

47

$F_{1}\subset F$(see [20]). Rouhani [20] obtained the following nonlinear ergodic theorem for analmost nonexpansive sequence which is a generalization of Baillon’s result [2]:Theorem 2.4 ([20]). Let$\{x_{n}\}$converges weakly toasuniformly in. Further,
is a
$H,$
$i.e.,$ $z_{0} \in\{z\in H:\varlimsup-||x_{n}-Z||=\inf_{y\in H}\varlimsup-||Xn-y||\}$ .
-asymptotic center ofin$\{\frac{1}{n}\sum_{i=}^{n-}0^{1}+X_{i}k\}$$\varlimsup$

be a bounded almost nonexpansive sequence in H. Then,$z_{0}\in F$$narrow\infty$$k\in \mathbb{R}^{+}$$z_{0}$$\{x_{n}\}$We do not know whether Theorem 2.4 would hold in the case when $H$ is a Banach
space.
A family $\{T(s) : s\in \mathbb{R}^{+}\}$ of mappings ofnonexpansive semigroup on$C$into itself is called a one-parameter
if it satisfies the following conditions:$C$(a) $s\mapsto T(S)x$ is continuous for all $x\in C$ ;(b) $T(s+t)=T(s)T(t)$ for all 8, $t\in S$ ;(c) $||T(s)X-T(S)y||\leq||x-y||$ for all$x,$ $y\in

and $s\in S$ ;C$(d) $T(0)=I$ .

Baillon [3] proved a nonlinear ergodic theorem for a one-parameter nonexpansive semi-group in a Hilbert space:
Theorem 2.5 ([3]). Let$\{T(t) :t\in \mathbb{R}^{+}\}$

be

$a$

one-parameter nonexpansive semigroup on

C. If for some $x_{0}\in C,$ $\{T(t)X0:t\in \mathbb{R}^{+}\}$ is bounded, then for any $x\in C,$$\{\frac{1}{t}\int_{0}^{t}\tau(S)_{Xd\}}\mathit{8}$converges weakly to a fixed point of .
$T$

Rouhani $[20, 21]$ also introduced the notion of almost nonexpansive curve in a Hilbert
space and studied a nonlinear ergodic theorem for such a curve which is a generalizationof Baillon’s result [3].
Let be a function from. Then, is called an almost nonexpansive curve if
there exists a nonnegative real-valued functiononsuch that $||u(h+s)-$
for every
and inand. In thecase when $\epsilon(s, t)=0$ for every $t\in S,$ is called a nonexpansive curve (see [20]).
$\mathbb{R}^{+}\mathrm{i}\mathrm{n}\mathrm{t}_{\mathrm{O}}H$

$u$

$u$

$\mathbb{R}^{+}\cross \mathbb{R}^{+}$$\epsilon(\cdot, \cdot)$$u(h+t)||^{2}\leq||u(\mathit{8})-u(t)||^{2}+\epsilon(\mathit{8}, t)$$s,$Remark 2.6. Let

$u$

$s,$

$h$

$t$

$\mathbb{R}^{+}$$\lim_{s,t}\epsilon(s, t)=0$$u$

be a bounded function from$\mathbb{R}^{+}$

into $H$ such that$||u(h+s)-u(h+t)||\leq||u(S)-u(t)||+\epsilon_{1}(s, t)$

48

for every

$s,$

$t$

and

$h$

in

and

$\mathbb{R}^{+}$

nonexpansive curve with$\lim_{s,t}\epsilon_{1}(s, t)=0$. Then, it is obvious that$\epsilon(s, t)=4(\sup_{r\in S}||u(r)||)\epsilon 1(s, t)+\epsilon_{1}(s, t)^{2}$

A continuous function from$u$$\mathbb{R}^{+}$

into

$C$

$u$

is an almost(see also [20, 21]).

is called an almost-orbit of$\mathfrak{S}=\{T(t):t\in \mathbb{R}^{+}\}$

if$\lim_{s}\sup_{t}||u(t+s)-\tau(t)u(S)||=0$(see [17]).
Example 2.7. Let$\{T(s) :s\in \mathbb{R}^{+}\}$

be a one-parameter nonexpansive semigroup on

is a bounded almost-orbit of $\{T(s) : s\in \mathbb{R}^{+}\}$ , from Remark 2.6,
to . Hence, we also see that for $x\in C,$
nonexpansive curve from

If

$u$

$C$

$\mathbb{R}^{+}$$u$$\mathbb{R}^{+}$

to

$C$

if

$\{T(t)x :t\in \mathbb{R}^{+}\}$.

is an almost$\{T(t)x :is an almost nonexpansive curve from[20] .

$C$t\in \mathbb{R}^{+}\}$

is bounded (see also

$)$

Let

$u$

be a function from$S$

into $H$ . Then, we denote the subsets$F_{1}(u)$and $F(u)$ of $H$

andas follows: $q\in F_{1}(u)$ if and only if $||u(h+s)-q||\leq||u(S)-q||$ for every$q\in F(u)$ if and only if
and $F(u)$ are closed
exists. We can prove thatconvex subset of and $F_{1}(u)\subset F(u)$ (see [20, 21]). Rouhani [20] proved the following
nonlinear ergodic theorem for an almost nonexpansive curve which is a generalization of$h,$ $s\in \mathbb{R}^{+}$$F_{1}(u)$$\lim_{s}||u(\mathit{8})-q||$$H$

Baillon’s result [3]:Theorem 2.8 ([20]). Let

curve in H. Then,in$k\in \mathbb{R}^{+}$$\{u(s) :s\in \mathbb{R}^{+}\}$

converges weakly to $z_{0}\in F(u)$ as$\{\frac{1}{t}\int_{0}^{t}u(s+k)ds\}$. Further,$z_{0}$

is$a\varlimsup$

be a bounded continuous almost nonexpansive-asymptotic center of$\varlimsup-||u(t)-z||=\inf_{y\in C}\varlimsup-||u(t)-Z||\}$$u(\cdot)$$tarrow\infty$

uniformly

in $H,$ $i.e.,$ $z_{0}\in\{z\in H$ :.

We do not lcnow whether Theorem 2.8 would hold in the case when $H$ is a Banach

space.

3. THEOREMS FOR COMMUTATIVE SEMIGROUPSIn this section, we prove nonlinear ergodic theorems for almost nonexpansive curves

over a commutative semigroup. At first, we state some definitions and notations.
Let be a semitopological semigroup with identity, i.e., a semigroup with a Hausdorff
$S$

49

and
from into
topology such that for each $t\in S$ , the mappings
itself are continuous. Then, is called right reversible if any two closed left ideals of
have non-void intersection. In this case, $(S, \leq)$ is a directed system when the binary$b\in S$ . Right reversible
is defined by $a\leq b$ if and only if
relation
semitopological semigroups include all commutative semigroups (see [11]).
$S$$s\vdasharrow t\cdot s$$\mathit{8}\mapsto s\cdot t$

$S$

$S$

$\zeta‘\leq$” $\mathrm{o}\mathrm{n}S$$\overline{Sa}\supseteq\overline{Sb},$$a,$

Throughout this section, we assume that is a right reversible semitopological semigroup with identity and is a subspace of $B(S)$ containing constants which is and-invariant for each $s\in S$ unless other specified. We introduce the notion of almost nonexpansive curves over a noncommutative semigroup. Let be a function from into $H$ .
$S$

$D$

$r_{s}$$l_{s}$$S$$u$

is called an almost nonexpansive curve if there exists a nonnegative real-valued
on $S\cross S$ such thatfunction

Then,

$u$

$\epsilon(\cdot, \cdot)$$||u(h_{S})-u(ht)||2\leq||u(\mathit{8})-u(t)||^{2}+\epsilon(s, t)$for every

$s,$

$t$

and

$h$

in

$S$

and

$\lim_{s,t}\epsilon(S, t)=0$. In the case when $\epsilon(s, t)=0$ for every$s,$ $t\in

S,$

$u$

is called a nonexpansive curve (see [1, 12]).
Remark 3.1. Let$u$be a bounded function from$S$

into $H$ such that$||u(h_{S})-u(ht)||\leq||u(S)-u(t)||+\epsilon_{1}(s, t)$for every 8,

$t$

and

$h$

in

$S$

nonexpansive curve withand$\lim_{s,t}\epsilon_{1}(s, t)=0$. Then, it is obvious that$\epsilon(\mathit{8}, t)=4(\sup_{r\in S}||u(r)||)\epsilon 1(s, t)+\epsilon_{1}(s,t)^{2}$$u$

is an almost(see also [1, 12]).

A family $\mathfrak{S}=\{T(s) : s\in S\}$ of mappings of into itself is called a nonexpansive
semigroup on if it satisfies the following conditions:
$C$

$C$

(a) 8 $[]arrow T(S)x$ is continuous for all $x\in C$ ;(b) $T(\mathit{8}t)=T(s)T(t)$ for all 8, $t\in S$ ;(c) $||T(s)X-\tau(s)y||\leq||x-y||$ for all$x,$ $y\inC$

and $s\in S$ .

the set of common fixed points of $T(t),t\in S$ , that is,
We denote byis called an almost-orbit of { $T(t)$ :. A continuous function from into$F(\mathfrak{S})=$$F(\mathfrak{S})$

$u$

$S$

$C$

$\bigcap_{t\in S}F(T(t))$$t\in S\}$if$\lim_{s}\sup_{t}||u(t_{S})-\tau(t)u(.S)||=0$(see [26, 27]).

50

Example 3.2. Let $\mathfrak{S}=\{T(s) : s\in S\}$ be a nonexpansive semigroup on

is abounded almost-orbit of , from Remark 3.1, is an almost nonexpansive curve fromto . Hence, we also see that for $x\in C,$ $\{T(t)_{X:}t\in S\}$ is an almost nonexpansive curvefrom to if $\{T(t)x:t\in S\}$ is bounded (see also [12]).
$\mathfrak{S}$

$C$

. If

$u$

$u$

$S$

$C$

$S$

$C$

be a function from into $H$ . Then, we denote the subsets
and $F(u)$ of$H$ as follows: $q\in F_{1}(u)$ if
and only if $||u(hS)-q||\leq||u(S)-q||$ for every $s\in S$ and
Let

$S$

$u$

$F_{1}(u)$$h,$$q\in F(u)$

Let

$S$

if and only if $\lim_{s}||u(S)-q||$ exists (see [1, 12]).
be a semigroup and let $B(S)$ be the Banach space of all bounded real-valued

functions on with supremum norm. Then, for each $s\in S$ and $f\in B(S)$ , we can define
elements $r_{s}f\in B(S)$ and $l_{s}f\in B(S)$ by $(r_{s}f)(t)=f(tS)$ and $(l_{s}f)(t)=f(st)$ for all$t\in S$ , respectively. We also denote byandthe conjugate operators ofand ,respectively. Let be a subspace of $B(S)$ and let be an element of , whereis thedual space of . Then, we denote by
the value of at $f\in D$ . Sometimes, $\mu(f)$ will
be denoted by
or $\int f(t)d\mu(t)$ . When contains constants, a linear functional
on is called a mean on
if $||\mu||=\mu(1)=1$ . We also know that is a mean onif$S$$r_{s}^{*}$$l_{s}^{*}$$l_{s}$$r_{s}$$D$$D$$\mu(f)$$D^{*}$

$\mu$

$D$

$\mu_{t}(f(t))$$D$$D^{*}$

$\mu$

$\mu$

$D$

$D$

$\mu$

and only if$\inf_{s\in S}f(S)\leq\mu(f)\leq\sup_{ss\in}f(s)$for each $f\in D$ . For $s\in S$ , we can define a point evaluation$\delta_{s}$by $\delta_{s}(f)=f(s)$ for every

A convex combination of point evaluations is called a finite mean on . A finite
mean on is also a mean on any subspace of $B(S)$ containing constants. Further,letbe a subspace of $B(S)$ containing constants which is -invariant i.e., $r_{s}D\subset D$ foreach $s\in S$ . Then, a mean on is called right invariant if$f\in B(S)$ .
$\mu$

$S$

$S$

$D$

$D$

$r_{s}$

$\mu$

$D$

$\mu(r_{s}f)=\mu(f)$for all $s\in S$ and $f\in D$ . Similarly, we can define a left invariant mean on an -invariant
subspace of $B(S)$ containing constants. A right and left invariant mean is called an$l_{s}$

invariant mean. We also denote by $C(S)$ the set of all bounded continuous real-valued
functions on .
$S$

The following definition which was introduced by Talcahashi [22] is crucial in nonlinear
ergodic theory for abstract semigroups. Let be a bounded function from into $H$ suchthatfor every $y\in H$ . Let be an element of . Then, there exists a unique
$S$

$u$

$\langle u(\cdot), y\rangle\in D$$\mu$$D^{*}$

51

element
such thatis contained indenoted by $\int u(t)d\mu(t)$ .
$u_{\mu}\in H$$\overline{co}\{u(t) :$u_{\mu}$

Lemma 3.3. Suppose thatcurve from$s\in S.$$S$

to $H$ withfor all $y\in H$ . If is a mean on , then(for example, see [13, 14, 22]). Sometimes,
will be$\langle u_{\mu}, y\rangle=\mu_{S}\langle u(S), y\rangle$$D$$\epsilon(\cdot, \cdot)$t\in S\}$

$D$

$\mu$

$u_{\mu}$

has an invariant mean . Let be an almost nonexpansive
such thatandare in for all $y\in H$ and
$u$

$\mu$

$||u(\cdot)-y||^{2}$$\epsilon(s, \cdot)$$D$

Then, (i), (ii) and (iii) hold.

(i) $F(u)$ and$F_{1}(u)$are closed convex subsets of $H$ ;(ii) $F_{1}(u)\subset F(u)$ ;(iii) $u_{\mu}\in F(u)$ .

Let$\{\mu_{\alpha} :be a net of means on\alpha\in I\}$$D$

. Then,$\{\mu_{\alpha} :\alpha\in I\}$is said to be asymptoticallyinvariant if
and$\mu_{\alpha}(f)-\mu\alpha(r_{s}f)arrow \mathrm{O}$$\mu_{\alpha}(f)-\mu_{\alpha}(l_{s}f)arrow \mathrm{O}$for every $s\in S$ and $f\in D$ (see [19]). Let
be a net of continuous linear
functionals on . Then,is said to be left strongly regular if the following$\{\lambda_{\alpha} :$D$$\{\lambda_{\alpha} :\alpha\in I\}$

\alpha\in I\}$

conditions are satisfied:(a)$\sup||\lambda_{\alpha}||<\infty$;$\alpha$(b)(c)$\lim_{\alpha}\lambda_{\alpha}(1)=1$;$\lim_{\alpha}||\lambda_{\alpha}-l_{S}*\lambda\alpha||=0$

for every$\mathit{8}\in S$.

Right strong regularity is defined similarly. A strongly regular net is a left and rightstrongly regular net (see [10]).
Let be a bounded function from into such that for any $x\in C,$ $||u(\cdot)-x||^{2}\in D$ .
Then, for a mean on , the set $\mu- AC(u, C)$ defined by
$C$

$S$

$u$

$\mu$

$D$

$\mu- AC(u, C)=\{x\in C:\mu_{s}||u(\mathit{8})-X||^{2}=\inf_{y\in C}\mu s||u(S)-y||^{2}\}$is called the -asymptotic center of
$\mu$

$u$

in

$C$

(see also [9, 12, 15, 18]). Similarly, the set

defined by $\varlimsup- AC(u, c)=\{x\in C:\mu_{s}||u(s)-X||^{2}=\inf_{y\in c\mu_{s}}||u(\mathit{8})-y||^{2}\}$is-asymptotic center of in .
theKada and Takahashi [12] proved nonlinear ergodic theorems for almost nonexpansive
curves over a commutative semigroup which are generalizations of Rouhani’s results [20,$\varlimsup_{-}AC(u, C)$$\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\prime \mathrm{d}$21]:$\varlimsup$

$u$

$C$

52

Theorem 3.4 ([12]). Let$S$

subspace of $B(S)$ containing

be a commutative semigroup with
$conStant\mathit{8}$almost nonexpansive curve from$S$

in $D$ for all $y\in H$ and $s\in S.$ If$\{\mu_{\alpha} :

on

$D$

to $H$ with\alpha\in I\}$$r_{s}$

identity and let-invariant for eachLet$\mathit{8}\in S.$

be a

$D$

be an

$u$

are
is an asymptotically invariant net of means$\epsilon(\cdot, \cdot)$such that $||u(\cdot)-y||^{2}$ and$\epsilon(s, \cdot)$, then $\{\int u(t)d\mu\alpha(t)\}$ converges weakly to $y0 \in F(u)\cap\bigcap_{s\in S}\overline{co}\{u(t):t\geq s\}$ . Further,$y_{0}=u_{\mu}$Ifwhich is$a$and $\varlimsup- Ac(u, H)=\mu- Ac(u, H)=\{u_{\mu}\}$ for every invariant mean$\{\mu_{\alpha} :\alpha\in I\}$$s\in S.$$S$

of finite means onLet$continuou\mathit{8}$weakly to

$u$

and

$D$

.

is strongly regular net, the convergence is uniform.

Theorem 3.5 ([12]). Let$\beta\in J\}$

on

$\mu$

$D$

be

$a\mathit{8}$be as in Theorem 3.4. Assume that there exists a net {$SsuCf_{l}$

that

:

for every
be a strongly regular net of$\lim_{\beta}||\lambda_{\beta}-l_{s}^{*}\lambda_{\beta}||=\lim_{\beta}||\lambda_{\beta}-r^{*}\lambda_{\beta}|S|=0$in Theorem 3.4. Let

linear functionals on D. Then,$/ \mathrm{t}0\in F(u)\cap\bigcap_{s\in S}\overline{co}\{u(t):t\geq s\}$$\varlimsup-AC(u, H)=\mu- Ac(u, H)=\{u_{\mu}\}$$\lambda_{\beta}$$\{\mu_{\alpha} :\alpha\in I\}$$\{\int u(th)d\mu_{\alpha}(t)\}$and$\{\int u(ht)d\mu_{\alpha}(t)\}$converge

uniformly in $h\in\Lambda(S)$ . Further, $y0=u_{\mu}$ andfor every invariant mean

$\mu$

on

$D$

.

By using Theorem 3.5, Theorems 2.4 and 2.8 can be proved (see [12]). We do not knowwhether Theorems 3.4 and 3.5 would hold in the case when $H$ is a Banach space.

4. THEOREMS FOR NONCOMMUTATIVE SEMIGROUPSIn this section, we prove nonlinear ergodic theorems for almost nonexpansive curves

over a noncommutative semigroup. Throughout this section, we assume that is a rightreversible semitopological semigroup with identity and is a subspace of $B(S)$ containing
$S$

$D$

constants which is and -invariant for each $s\in S$ unless other specified. We denote by
the algebraic center of , i.e., all $s\in S$ such that $st=ts$ for all $t\in S$ .
$l_{s}$$r_{s}$$S$$\Lambda(S)$Theorem 4.1 ([1]). Let$u$be an almost nonexpansive curve from$S$

to $H$ with$\epsilon(\cdot, \cdot)$such

are in for all $y\in H$ and $s\in S.$ If
is anasymptotically invariant net of means on , thenconverges weakly to. Further,and $\varlimsup- Ac(u, H)=\mu- AC(u, H)=\{u_{\mu}\}$ forthat $||u(\cdot)-y||^{2}$ and$D$$\epsilon(s, \cdot)$$\{\mu_{\alpha} :$D$$y_{0}=u_{\mu}$$F(u)\cap \mathrm{n}\overline{C\mathit{0}}\{u(t)s\in S : t\geq s\}$every invariant mean

$\mu$

on

$D$

.

$\{\int u(t)d\mu\alpha(t)\}$\alpha\in I\}$$y_{0}\in$53

We consider the case when$\{\mu_{\alpha} :\alpha\in I\}$is strongly regular. Then, we obtain thefollowing theorem:
Theorem 4.2 ([1]). Assume that there exists a net$S$such that$\lim_{\beta}||\lambda_{\beta}-l_{S}^{*}\lambda_{\beta}||=\lim_{\beta}||\lambda_{\beta}-r_{s}^{*}\lambda_{\beta}||=0$

most nonexpansive curve fromto $H$ with$S$

in $D$ for all $y\in H$ and $s\in$ S. Let$\{\mu_{\alpha} :tinuous linear functionals on D. Then,weaklyof finite means on\beta\in J\}$

for every $s\in S.$ Let\alpha\in I\}$

be an al-$u$arebe a strongly regular net of conconverge
andsuch that $||u(\cdot)-y||^{2}$ and$\epsilon(\mathit{8}, \cdot)$$\{\int u(ht)d\mu\alpha(t)\}$$\{\int u(th)d\mu_{\alpha}(t)\}$$to/\mathrm{t}0\in F(u)\cap s\in S\cap\overline{CO}\{u(t):t\geq \mathit{8}\}$$\varlimsup-AC(u, H)=\mu- Ac(u, H)=\{u_{\mu}\}$$\epsilon(\cdot, \cdot)$$\{\lambda_{\beta} :uniformly in $h\in\Lambda(S)$ . Further,$y_{0}=u_{\mu}$andfor every invariant mean on .
$D$

$\mu$

To prove Theorems 4.1 and 4.2, we need the following lemmas and theorem.
The following lemma is a modification of [25] (see also [12]).
Lemma 4.3 ([1]). Assume that
pansive curve from$y\in H$and$\mathit{8}\in S$$S$

to $H$ with$D$

has an invariant mean . Let$\epsilon(\cdot, \cdot)$$\mu$such that $||u(\cdot)-y||^{2}$ and$u$be an almost nonex-$\epsilon(\mathit{8}, \cdot)$

are in$D$

for all. Then,$\varlimsup-AC(u, H)=\mu- Ac(u, H)=\{u_{\mu}\}$,where $\varlimsup- Ac(u, H)=\{x\in H:\varlimsup_{s}||u(\mathit{8})-X||^{2}=\inf_{y\in H}\overline{\lim S}||u(S)-y||^{2}\}$ . Consequently, if
$\mu$

and

$\lambda$

are invariant meanso.n , then$D$$u_{\mu}=\mu_{\lambda}$.

The following theorem plays an important role in the proofs of Theorems 4.1 and 4.2(see also [12]).

Theorem 4.4 ([1]). Assume that $D$ has an invariant mean . Let$\mu$expansive curve from$y\in H$and$s\in St_{\}}$$S$

to $H$ with$\epsilon(\cdot, \cdot)$such that $||u(\cdot)-y||^{2}$ and$u$be an almost non-$\epsilon(s, \cdot)$

are in$D$

for all. Then, $F(u) \cap\bigcap_{s\in S}\overline{co}\{u(t) : t\geq s\}=\{u_{\mu}\}$ .

The following lemma is essential to prove Theorem 4.2.

Lemma 4.5 ([1]). Let
Let$\{\mu_{\alpha} :\alpha\in A\}$$u$

be a bounded almost nonexpansive curve frombe a net of finite means on

$S$

$S$

to $H$ with$\epsilon(\cdot, \cdot)$.

such that$\lim_{\alpha}||\mu_{\alpha}-l_{S}^{*}\mu\alpha||=\lim_{\alpha}||\mu_{\alpha}-r_{s}^{*}\mu\alpha||=0$

for every$s\in S.$

$(*)$

54

Then,

$\{\int u(th)d\mu_{\alpha}(t)\}$$h\in\Lambda(S)$

mean

$\mu$

. Further,

on

$D$

converges weakly to$y_{0}\in F(u)\cap s\in S\cap\overline{co}\{u(t):t\geq s\}$uniformly inand $\varlimsup- Ac(u, H)=\mu- AC(u, H)=\{u_{\mu}\}$ for every invariant$y_{0}=u_{\mu}$.

Sketch of the proof of Lemma 4.5. Letmeans on such that$\{\mu_{\alpha} :\alpha\in A\}$and$\{\lambda_{\beta} :\beta\in B\}$be nets of finite

$S$

and

$\lim_{\alpha}||\mu_{\alpha}-l_{S}^{*}\mu\alpha||=\lim_{\alpha}||\mu_{\alpha}-r_{S}^{*}\mu\alpha||=0$

for every $s\in S$ . Define
be a net in .

$(\beta_{1}, \gamma_{1})\leq(\beta_{2}, \gamma_{2})$$\{p_{\beta,\gamma} :$t\geq s\}$$\beta_{1}\leq\beta_{2}$and$\gamma_{1}\leq\gamma_{2}$. Let$\{[\int u(tp\beta,\gamma q)d\lambda\beta(t)d\lambda_{\gamma}(q)\}$converges weakly to$y_{0}\in F(u)\cap s\in S\mathrm{n}\overline{co}\{u(t)$:. From Lemma 4.4, it is sufficient to show that all weak limit points of subnets of

the netSinceof

if and only if$S$(\beta, \gamma)\in B\cross B\}$We show that$\lim_{\beta}||\lambda_{\beta}-l_{s}*\lambda_{\beta}||=\lim_{\beta}||\lambda_{\beta}-r_{S}\lambda_{\beta}*||=0$

are in$\{\iint u(tp_{\beta,\gamma}q)d\lambda\beta(t)d\lambda_{\gamma}(q)\}$$\bigcap_{s\in S}\overline{CO}\{u(t):t\geq \mathit{8}\}\cap F(u)$is bounded, there is a subnet
such that$\{[\int u(tp_{\beta}’,\prime q)\gamma d\lambda_{\beta};(t)d\lambda_{\gamma’}(q)\}$$\{\iint u(tp_{\beta,\gamma}q)d\lambda\beta(t\lambda f\lambda_{\gamma}(q)\}$$\{\iint u(tp_{\beta,\gamma}q)d\lambda\beta(t)\lambda_{\gamma}(q)\}$. Put $M= \sup_{t\in S}||u(t)||$ .

$\iint u(tp_{\beta^{l},\gamma^{;}}q)d\lambda\beta’(t)d\lambda_{\gamma’}(q)arrow y0\in H$.

Then, we have that for any $a\in S$ ,$\iint u(tp_{\beta\gamma}’,\prime qa)d\lambda_{\beta^{\prime(}}t)d\lambda_{\gamma^{i}}(q)arrow y_{0}\in H$

We obtain $y_{0}\in F(u)$ . Indeed, letfor all$t\geq t_{0}$and$s\geq t_{0}$. Let$\epsilon>0$$s\geq t_{0}$(1).

. Then, there exists $t_{0}\in S$ such that$\epsilon(s, t)<\epsilon$

and $h\in S$ . Then, we can show that$||u(h_{S})-y0||^{2}-||u(s)-y_{0||^{2}-2}\langle u(hs)-u(S),$ $\iint u(tp\beta^{;_{\gamma}},\prime qt_{0})d\lambda_{\beta}’(t)d\lambda_{\gamma}’(q)-y_{0}\rangle$$<\epsilon+4M^{2}||\lambda_{\beta^{;-}}l^{*}h\lambda\beta’||\cdot||\lambda_{\gamma}’||$.

So, it follows from (1) that $\lim_{s}||u(S)-y0||$ exists. This implies $y_{0}\in F(u)$ .

From the separation theorem, we obtain$s\in S\mathrm{n}\overline{co}\{u(\iota):i\geq s\}\cap F(u)$$/\mathrm{t}0\in s\in S\mathrm{n}\overline{co}\{u(t) :t\geq s\}$and hence $y0\in$. This implies that all weak limit points of subnets of the net$\{\iint u(tp\beta,\gamma q)d\lambda\beta(t)d\lambda_{\gamma}(q)\}$are in$\bigcap_{s\in S}\overline{co}\{u(t):t\geq s\}\cap F(u)$.

55

Next, we prove thatSince$\{p_{\beta,\gamma} :converges weakly tois arbitrary, we see that$\{\int u(\mathit{8}h)d\mu_{\alpha}(s)\}$(\beta, \gamma)\in B\cross B\}$

verges weakly to$y_{0}$$y_{0}$

uniformly in $h\in\Lambda(S)$ .

con-$\{\iint u(thp\beta,\gamma q)d\lambda\beta(t)d\lambda_{\gamma}(q)\}$uniformly in $h\in S$ . Then, there exists$(\beta_{0\gamma 1},)\in B\cross B$

such that(2)$| \iint\langle u(thp_{\beta,\gamma}q), X\rangle d\lambda_{\beta}(t)d\lambda_{\gamma}(q)-\langle y_{0}, x\rangle|<\frac{\epsilon}{3}$for every

and $h\in S$ . So, sinceconverges weakly to$\beta\geq\beta_{0},$ $\gamma\geq\gamma_{1}$$\{\int u(sh)d\mu_{\alpha}(\mathit{8})\}$$\{\mu_{\alpha}\}$satisfies $(*)$ , we can show thatuniformly in $h\in$$y0 \in F(u)\cap\bigcap_{s\in S}\overline{co}\{u(t) :$\Lambda(S)$t\geq s\}$. From Theorem 4.4 and Lemma 4.3, $y_{0}=u_{\mu}$ and $\varlimsup-AC(u, H)=\mu- Ac(u, H)=$$\{u_{\mu}\}$for every invariant mean

$\mu$

on D.

$\square$We can prove the following lemma as in the proof of Lemma 4.5.
Lemma 4.6 ([1]). Let $S,$ $D,$

converges$y_{0}=u_{\mu}$$weakl\mathrm{c}/$

to

$u$

and

$\{\mu_{\alpha} :\alpha\in A\}$be as in Lemma 4.5. Then, $\{[u(ht)d\mu\alpha(t)\}$uniformly in $h\in\Lambda(S)$ . Further,$y_{0} \in F(u)\cap\bigcap_{s\in S}\overline{co}\{u(t):t\geq s\}$and $\varlimsup- Ac(u, H)=\mu- Ac(u, H)=\{u_{\mu}\}$ for every invariant mean

$\mu$

on

$D$

.

Now, we can prove the nonlinear ergodic theorems (Theorems 4.1 and 4.2).

Sketch of the proof of Theorem 4.1. Letis bounded,on . Since
converges weakly to a point in $H$ . So, let$\{\mu_{\alpha}\}$$D$

be an asymptotically invariant net of means
must contain a subnet which$\{\int u(t)d\mu_{\alpha}(t)\}$$\{\int u(t)d\mu\alpha(\iota)\}$$\{\int u(t)d\mu\alpha\beta(t)\}$be a subnet of$\{\int u(t)d\mu_{\alpha}(t)\}$such that$\int u(t)d\mu_{\alpha}\beta(t)arrow z_{0}$Let$B_{1}(D^{*})$$\{\mu_{\alpha_{\beta_{\gamma}}}\}$

ofbe the closed unit ball of$\{\mu_{\alpha_{\beta}}\}$$D^{*}$. Since(3).

$\{\mu_{\alpha_{\beta}}\}\subset B_{1}(D^{*})$, there exists a subnet

such that$\mu_{\alpha_{\beta_{\gamma}}}arrow\mu w^{*}$

Then, we can show that

$\mu$

.

is an invariant mean on$D$. Since$\mu_{\alpha_{\beta_{\gamma}}}arrow\mu w^{*}$$\int\langle u(t), X\rangle d\mu_{\alpha}\beta\gamma(t)arrow\int\langle u(t), X\rangle d\mu(t)=\langle u_{\mu}, x\rangle$

Then, from (3), we have that$\mu$

are invariant means on$D$$u_{\mu}=u_{\lambda}$.

. From Lemma 4.3, if andand. Therefore, since
is arbitrary,$\int u(t)d\mu\alpha_{\beta}(t)arrow z_{0}$, then, for any $x\in H$ ,$z_{0}=u_{\mu}$$\{\int u(t)d\mu_{\alpha}\beta(t)\}$$\lambda$

56$\{\int u(t)d\mu_{\alpha}(t)\}$converges weakly to$\mu- Ac(u, H)=\varlimsup-AC(u, H)$ .

$u_{\mu}$. Furthermore, $\{u_{\mu}\}=F(u)\cap\bigcap_{ss\in}\overline{CO}\{u(t) : t\geq s\}=$$\square$Sketch of the proof of Theorem 4.2. Let be an invariant mean on
be a net in . From Lemma 4.5, we have thatuniformly in $h\in S$ . We also know
converges weakly to

$D$

$\mu$

$S$

$(\beta, \gamma)\in J\cross J\}$

$F(u) \cap\bigcap_{s\in S}\overline{co}\{u(t) :for every invariant meansuch that$\varlimsup-AC(u, H)=\mu- Ac(u, H)=\{u_{\mu}\}$$(\beta_{0}, \gamma 1)\in J\cross J$$| \langle\iint u(thp\beta,\gamma q)d\lambda_{\beta}(t)d\lambda_{\gamma}(q),$for every$\{\mu_{\alpha}\}$$\beta\geq\beta_{0},$ $\gamma\geq\gamma_{1}$$p_{\beta,\gamma}$:$\{\iint u(thp_{\beta},\gamma q)d\lambda\beta(t)d\lambda_{\gamma}(q)\}$$u_{\mu}$Then, there exists

and let {and $h\in S$ . Put

$\mu$

on

$D$

. Let $x\in H,$

t\geq s\}=$$\epsilon>0$.

$x \rangle-\langle u_{\mu}, x\rangle|<\frac{\epsilon}{\sup_{\alpha}||\mu_{\alpha}||}$$\lambda_{0}=\lambda_{\beta_{0},p_{0}}=p_{\beta 0,\gamma_{1}}$and$\lambda_{1}=\lambda_{\gamma_{1}}$. So, since

is strongly regular, from$| \int\langle u(sh), x\rangle d\mu\alpha(S)-\langle u_{\mu}, x\rangle|$$\leq|\int\langle u(sh), X\rangle d\mu\alpha(s)-\iint\langle u(t\mathit{8}h), X\rangle d\lambda 0(t)d\mu_{\alpha}(s)|$$+| \iint\langle u(tsh), X\rangle d\lambda_{\mathrm{o}(}t)d\mu_{\alpha}(\mathit{8})-\iiint\langle u(tshp0q), X\rangle d\lambda_{0}(t)d\lambda_{1}(q)d\mu_{\alpha}(s)|$$+| \int\langle\iint u(tShp_{0}q\mathrm{I}d\lambda \mathrm{o}(t)d\lambda_{1}(q)-uX\rangle\mu’ d\mu_{\alpha}(s)|+|\int\langle u_{\mu}, x\rangle d\mu_{\alpha}(s)-\langle u_{\mu}, x\rangle|$

we can prove that$\{\int u(sh)d\mu_{\alpha}(s)\}$converges weakly to,$u_{\mu}\in F(u)\cap s\in S\cap\overline{co}\{u(t):t\geq s\}$uniformly in $h\in\Lambda(S)$ .

As in the above argument, we obtain that
$F(u)\cap s\in S\mathrm{n}\overline{co}\{u(t) :t\geq \mathit{8}\}$$\{\int u(h\mathit{8})d\mu\alpha(\mathit{8})\}$

uniformly in $h\in\Lambda(S)$ .

converges weakly to$u_{\mu}\in$$\square$

We do not know whether Theorems in this section would hold in the case when $H$ is a
Banach space.

57

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DEPARTMENT OF MATHEMATICAL AND COMPUTING SCIENCES, TOKYO INSTITUTE ON TECHNOLOGYO-OKAYAMA, MEGURO-KU, TOI$<\mathrm{Y}\mathrm{O}$

$E$

-mail

$addres\mathit{8}$152-8552, JAPAN: atsusiba@is. titech.

$\mathrm{a}\mathrm{c}$. jp