Fixed Point Theorems and Nonlinear Ergodic Theorems for Nonexpansive Mappings(Information and mathematics of non-additivity and non-extensivity : from the viewpoint of functional analysis)

Fixed

Point Theorems and Nonlinear

Ergodic

Theorems

for Nonexpansive Mappings

東京工業大学・大学院情報理工学研究科

高橋渉 (Wataru Takahashi)

Department ofMathematical and Computing _Sciences

Tokyo Institute ofTechnology

1

Introduction

Let $E$ be a $Ban\mathfrak{X}h$ space and let $C$ be anonempty $cl\propto ed$

convex

suket of B. Then,

a

mappin$gT:Carrow C$ is said to be $none\varphi ansive$if $\Vert Tx-Ty\Vert\leq||x-y\Vert$ for all _{$x,y\in C$}

.

The

$fir8tfixd$$\backslash$ point thmrem for anonexpansive mapping whose domain

$C$ is not compact $v\mathfrak{s}ae$

$establi_{8}hed$in 1965 byBrowder [11]. Heprovedthat if$C_{\dot{\mathfrak{B}}}$ aboundd closd

convex

subset

of

aHilbert space$E$ and$T$ is anonexpansive mapping of$C$into itself, $th\dot{e}nTha\epsilon$ afixed point

inC. Almost immdiately, both Browder [12] and G\"ohde [21] provdthat the

same

is true if

$E$is aunifomly

convex

Banach$spa\iota e$

.

Kirk [27] $ako$proved the$f_{0}nowing$theorem; Let$E$be

$a$.reflexive Banach$sp\epsilon ce\bm{t}d$ let $C$ beanonempty bounded cloeed

convex

subset of$E$ whii

$hu$ normal structure. Let $T$ be anonexpansive mapping of$C$ into itself. Then the set _$F(T)$

offixd point8 of$Ti_{8}$ nonempty.

$A\hslash er$ Kirk’s theorem, mry fixed point thmrems concerning nonexpansive

$mapp_{\dot{i}}gs$ have

been proved in aHilbert space or aBanach space. $\bm{t}$ particular, Baillon and Sch\"oneberg

[9] introduced the concept of$a\epsilon ymptotic$ nomalstructure and generahzed Kirk’s fixedpoint

theorem$\epsilon s$foUows: Let $E$beareflexiveBanachspac6andlet $C$beanonempty boundd$cl\propto ed$

convex

subset of$E$whichhas asymptoticnormalstructure. Let$T$beanonexpansive mapping

of$C$ into itself. Then_$F(T)$ is nonempty. Aflxed point thmrem forafanily of_nonexptsive

mappings $wa\epsilon$ first proved by DeMarr [18] by $\mathfrak{B}8Uming$ that the famPy is commutative and

$C$ is compact. Later, $\ovalbox{\tt\small REJECT} hi[56]$ extendd thi8 thmrem to anoncommutative $\epsilon emigoup$

of nonexpansive mappings whii is called amenable. Since then, many Axed point thmrems

for

a

$non\alpha pansive$ mapping

or

affimily ofnonexptsive mappin$gs$ have been $\infty tablished$ by

many authors.

On the other hand, in 1975, BaiUon [6] originaUy proved thefirst nonhnear ergodicthmrem

in the&ameworkof Hilbert $spac\infty$:Let $C$ be aclosed and

convex

subset ofaEilbert space

and let $T$ be

a

$nonexpan\epsilon ive$ mapping of $C$ into itself. $\bm{i}F(T)i\epsilon$ nonempty, then for eai

$x\in C$, the Coe\‘aro

means

$S_{n}(x)= \frac{1}{n}\sum_{k=0}^{n-1}T^{k}x$

converge weakly to

some

$y\in F(T)$

.

In this case, _{putting $y=Px$ for each} $x\in C,$ $P$ is a

noneypansive retraction of $C$ onto _$F(T)$ such that $PT=TP=P$ and _$Px$ is contained in

${}^{t}an$ ergodic $retraction’$

.

In 1981, Taihashi [58] proved the existence ofergodic $retr\epsilon ctions$

for amenable semigroups of nonexpaoive mappings on Hilbert $spac\infty.$ Rod\’e [49] ako found

asequence of

means

on asemigroup, generalizingthe Cae\‘aro means, and extended $BaiUon’ s$

thmrem. These results were extended to auniforiy

convex

Banach space whose norm i8

R\’echet _{differentiable in the case of commutative} $8emigroups$ of nonexpaoive mappings by

Hirano, KidoandTaihashi [23]. In 1999, Lau, Shioji and$Ta\bm{L}hashi[33]$extendedIbhhaehi$s$

result and Rod\’e’s $re\epsilon ult$ to amenable semigroup8 of nonexpaoive mapping8 in the Banaaeh

space.

Inthi8 article, we first discuss fixed point theorems for nonexptsive mappings

or

familiae

of nonexptsive mappings in Banach spaces. In particular,

we

state afixed point thmrem

for amenablesemigroupsofnonexpansivemappings in Banach spaces whii generahzae Kirk’s

theorem andTaihaehi’s theorem,simultanmusly. _{Thi8 theorem}

_answers

affirmatively

aProk

$lem$ posed during the Conference

on

Fixed Point Thmry and Applications held

at

$cmM$,

$Marseille- Lum\dot{i}y$, 1989 (see [29]). Then

we

$8how$ generahzd nonlinear ergodic thmrems for

nonexpaoive semigroup8 $\bm{i}$ Banach sPacae. In particular, we $d\dot oeCU8S$ generahzed nonlinear

ergodic theorems for $nonexpan\epsilon ive$ semigroups in unifomly

convex

Banach _Ipac\’e

or

general

Banach spaces. $U_{S\dot{i}}g$ these results, we obtain

some

$non1_{\dot{i}}$

ear

ergodic thmrems in

casae

of

$d\dot oe$crete and one-parameter semigroups of nonexpmsive mapping8. Finally, we discuss two

iterative method8 for approximation offixed point8 of nonexpansive mappings whichare

dif-ferent $kom$ the

mean

_ergodic method.

2 Preliminaries

Let $E$ be

a

realBanach space with

norm

$\Vert\cdot\Vert$ and let $E^{*}$ denote the topological dualof$E$:

We denote the valueof$y^{*}\in E$ “ at$x\in E$ by $\langle x,y^{*}\rangle$

.

When $\{x_{n}\}$ is

a

sequencein$E$, wedenote

the 8trong convergence of $\{x_{n}\}$ to $x\in E$ by _{$x_{n}arrow x$} and the weak convergence by $x_{n}arrow x$

.

The mdulus

_of

convexity$\delta$ of_$E$ is defined by

$\delta(\epsilon)=\inf\{1-\frac{||x+y\Vert}{2}$ : $||x||\leq 1,$$\Vert y||\leq 1,$$||x-y||\geq\epsilon\}$

for every $\epsilon$ with $0\leq\epsilon\leq 2$

.

A Banach space $E$ is said to be uniformly

convex

if _{$\delta(\epsilon)>0$} for

every$\epsilon>0$

.

Let $\epsilon$ and$r$ be real numbers with$r>0$ and $0\leq\epsilon\leq 2r$

.

If$E$is uniformly convex,

then $\delta$ satisfies that _{$\delta(\epsilon/r)>0$}and

$\Vert\frac{x+y}{2}\Vert\leq r(1-\delta(\frac{\epsilon}{r}))$

for every$x,y\in E$with $||x\Vert\leq r,$ $\Vert y||\leq r$and $||x-y||\geq\epsilon$

.

Let$C$ beanonemptyclosed

convex

subset ofa uniformly

convex

Banach space $E$

.

Then weknow that forany$x\in E$, there eXists

a

uniqueelement $z\in C$ suchthat $\Vert x-z\Vert\leq\Vert x-y||$ for all$y\in C$

.

Putting $z=P_{C}(x)$,

we

call

$P_{C}$ the metric prvjectionof$E$ onto $C$

.

The dudity mapping$J$ from$E$into $2^{B^{*}}$ is defined by

$Jx=\{x^{r}\in E^{*} : (x,x^{*}\rangle=||x\Vert^{2}=||x^{*}||^{2}\}$

for every $x\in E$

.

Let $U=\{x\in E : \Vert x\Vert=1\}$

.

The

nom

of $E$ is said to be G\^atenur

differentiable

if foreach $x,y\in U$, the limit

exists. In the case, $E$ is called smooth. The

norm

of $E$ is said to be unifomly G\^ateaux

differentiable

iffor each $y\in U$, the limit (2.1) is attained uniformly for $x\in U$

.

It is also said

tobe FV\’echet

_{differentiable}

if for each$x\in U$, the limit (2.1) isattained uniformlyfor$y\in U$

.

A

Banachspace$E$iscalled uniforrnly smooth ifthelimit (2.1) is attained uniformly for$x,y\in U$

.

We know that if $E$ is smooth, then the duality mapping $J$ is single valued. Further, if the

nom

of$E$ is uniformlyG\^ateaux differentiable, then $J$ is uniformly

norm

to weak’ continuous

on each bounded subset of $E$

.

We know the folowing result: Let $E$ be a uniformly

convex

Banachspace withaG\^ateaux differentiable nom. Let $C$ beanonempty closed

convex

subset

of$E$ and $x\in E$

.

Then, $x_{0}=P_{C}(x)$ if and only if

$\langle x_{0}-y,$$J(x-x_{0}))\geq 0$

for all $y\in C$

,

where $J$is the duality mapping of$E$;

see

_{$[64, 65]$} for

more

details.

A Banach space $E$ is said to satisfy Opial’8 condition [46] if for any sequence $\{x_{\mathfrak{n}}\}\subset E$,

$x_{n}\cdot\wedge yimpli\infty$

$\lim_{narrow}\inf_{\infty}\Vert x_{n}-y\Vert<\lim_{narrow}\inf_{\infty}||x_{n}-z\Vert$

for all $z\in E$ with $z\neq y$

.

A Hilbert space satisfies Opial’s condition.

Let $C$ be aclosed

convex

subset of$E$

.

A mapping $T:Carrow C$ is said to be $none\varphi ansive$ if

11

$Tx-Ty\Vert\leq\Vert x-y\Vert$ for all $x,$$y\in C$

.

We denote the set of all fixed point8 of $T$ by $F(T)$

.

A closed

convex

subset $C$ of a Banach space $E$ is said to have normal

structure

if for each

$bo$unded closed

convex

subset of $K$ of$C$ which contains at least two points, there exists

an

element $x$ of$K$which is

not

a

diametral point of$K$

,

i.e.,

$\sup\{\Vert x-y\Vert : y\in K\}<\delta(K)$

,

where$\delta(K)$ is the diameter of$K$

.

It is well known that aclosed

convex

subset ofaunifomly

convex

Banach space has normal structure and a compact

convex

subset ofa Banach space

hasnomal structure;

see

[64] for

more

details. Let$D$be

a

subsetof$C$andlet $P$be

a

mapping

of$C$ into $D$. Then $P$ is said to be sunnyif

$P(Px+t(x-Px))=Px$

whenever $Px+t(x-Px)\in C$ for $x\in\cdot C$ and $t\geq 0.$ $A$ mapping $P$ of$C$ into $C$is said to be

a

retraction if$P^{2}=P$

.

_We denote by$\overline{D}$

and $\varpi D$ the closure of$D$ and the

convex

_$hul$ of$D$

,

respectively.

Let $I$ denote the identity operator on $E$

.

An operator A C $ExE$ with domain $D(A)=$

$\{z\in E:\cdot Az\neq 1\}$ and range $R(A)=\cup\{Az : z\in D(A)\}$ is said to be accfetive if for each $x_{i}\in D(A)$ and$y_{i}\in Ax_{i)}i=1,2$, there exists$j\in J(x_{1}-x_{2})$ such that ($y_{1}-y_{2},j\rangle$ $\geq 0$

.

If$A$

is accretive, then we have

$||x_{1}-x_{2}\Vert\leq\Vert x_{1}-x_{2}+r(y_{1}-y_{2})\Vert$

for all $r>0$

.

An accretive operator $A$ is said to satisfy the range condition if $\overline{D(A)}\subset$

$\bigcap_{r>0}R(I+rA)$

.

If$A$ is accretive, then we can define, for each $r>0$,

a

nonexpansive single

valued mapping $J_{r}$: $R(I+rA)arrow D(A)$ by $J_{r}=(I+rA)^{-1}$

.

It is called the resolvent of$A$

.

We aiso define the Yosida approvimation $A_{r}$ by $A,$ $=(I-J_{r})/r$

.

We know that $A_{r}x\in AJ_{r}x$

for all $x\in R(I+rA)$ and $\Vert A_{r}x\Vert\leq\inf\{\Vert y\Vert : y\in Ax\}$for

au

$x\in D(A)\cap R(I+rA)$

.

We also

know that for

an

accretive oper.atorA satisfying the range condition, $A^{-1}0=F(J_{r})$ for all

[48] proved the following result: Let $E$ be a uniformly convex and uniformly smooth Banach

space and let $A\subset ExE$ _be an m-accretive operator such that $A^{-1}0$ is nonempty. Then,

for any $x\in E$, the strong limit $\lim_{tarrow\infty}J_{t}x$ exists and belongs to $A^{-1}0$

.

In this case, putting

$Px= \lim_{tarrow\infty}J_{t}x$,

we

have that $P$ is a sunny nonexpansive retraction of$E$ onto $A^{-1}0$

.

We

aJso know that $x_{0}=Px$ if and only if

$\langle x-x_{0},J(x_{0}-z)\rangle\geq 0$

for all $z\in A^{-1}0$

.

Let $S$ be

a

semitopological semigroup, i.e.,

a

semigroup with

Hausdorff

topology such that

for each $s\in S$

,

the $mapp\dot{g}$gs $t\mapsto ts$ and $t-\rangle$ $st$ of$S$ into itself

are

continuous. Let $B(S)$ be

the Banach space ofallbounded real valued functions

on

$S$ with supremum

norm

and let $X$

be asubspaceof$B(S)$containing constants. Then,

an

element $\mu$ of$X^{*}$ (the dual spaceof$X$)

is called a mean

on

$X$ if _{$||\mu\Vert=\mu(1)=1$}

.

Weknow that $\mu\in X^{*}$ is amean on $X$ ifandonly if

$\inf\{f(s) : s\in S\}\leq\mu(f)\leq\sup\{f(s) : \epsilon\in S\}$

for every $f\in X$

.

A real valued function $\mu$

on

$X$ is called

a

submean

on

$X$ if the following

properties

are

satisfied:

(1) $\mu(f+g)\leq\mu(f)+\mu(g)$ for every $f,g\in X$;

(2) $\mu(\alpha f)=\alpha\mu(f)$ for every _{$f\in X$} and $\alpha\geq 0$; (3) for $f,g\in X,$ $f\leq g$ implies $\mu(f)\leq\mu(g)$;

(4) $\mu(c)=c$ for every constant function $c$

.

Clearly every

mean on

$X$ is a submean. The notion of submeans

vas

first introduced by

Mizoguchi andTakahashi [45]. For asubmean $\mu$

on

$X$ and $f\in X$

,

sometimes

we

use$\mu(f(t))$

instead of $\mu(f)$

.

For each $s\in S$ and $f\in B(S)$

,

we define elements $l_{\epsilon}f$ and $r_{\epsilon}f$ of $B(S)$

given by $(\ell_{o}f)(t)=f(st)$ and $(r_{\epsilon}f)(t)=f(t_{8})$ for all $t\in S$

.

Let $X$ be

a

subspace of_$B(S)$

$conta\dot{i}\dot{i}g$ constants which is invariant under $\ell_{l},$ $s\in S$ (resp. _{$r_{l},$} $r\in S$). Then a

mean

$\mu$

on

$X$ is said to be

left

invanant (resp. right invariant) if$\mu(f)=\mu(\ell_{\epsilon}f)$ (resp. _{$\mu(f)=\mu(r.f)$})

for all $f\in X$ and $s\in S$

.

An invariant

mean

is

a

left and right invariant

mean.

A

submean

$\mu$

on

$X$ is said to be

left

subinvariant if$\mu(f)\leq\mu(\ell_{\iota}f)$ for all $f\in X$ and $s\in S$

.

Let $S$ be

a

semitopological semigroup. Then$S$is called

left

(resp. right) reversible if anytwoclosed right

(resp. left) ideals of $S$ have non-void intersection. If $S$ is left reversible, $(S, \leq)$ is

a

directed

system when the binary relation $\leq$ on$S$is definedby$a\leq b$if andonlyif$\{a\}\cup\overline{Sa}\supset\{b\}\cup\overline{Sb}$

,

$a,b\in S$

.

Similarly, we

can

deflnethe binary relation $u\leq$’

on

a

rightreversible semitopological

semigroup $S$

.

3

Fixed

Point Theorems

In this section,

we

discuss fixed point theorems for

a

nonexpansive mapping

or

afa血血y

of nonexpansive maPPings. The first fixed point theorem for nonexpansive $mapp\dot{u}$lffi

was

established in 1965 byBrowder [11]. Heprovedthat if$C$ is

a

bounded closed

convex

subsetof

a

Hilbert space $H$ and $T$is

a

nonexpansive mapping of$C$ into itseff, then $T$ has

a

_丘組edPo 血地

in $C$

.

Almost immediately, both Browder [12] andG\"ohde [21] provedthat the

same

is trueif

$E$ is

a

uniformly

convex

Banch space. Kirk [27] aJso

Proved

the followingtheorem:

$Th\infty rem3.1$ _{([27]). Let} $E$ be

a

reflestve

Banach space and let$C$ be

a

nonempty beunded

closed

convex

subset

_of

$E$ which has nonnalstru

cture.

Let$T$ be a

none

zpansive mapping

of

$C$

After Kirk’s theorem, many fixed point theorems concerning nonexpansive mappings have

been proved in a Hilbert space or a Banach space. In particular, Baillon and Sch\"oneberg

[9] introduced the concept ofasymptotic normal structure and generalized Kirk’s fixed point

theorem

as

follows:

Theorem 3.2 ([9]). Let $E$ be

a

reflenive

Banach space and let $C$ be

a

non-empty bounded

closed

convex

subset

_of

$E$ which has asymptotic normal structure. Let $T$ be

a

none2pansive

mapping

_of

$C$ into

itself.

Then _$F(T)$

.is nonempty.

On the otherhand, DeMarr [18] provedthe followingfixedpointtheorem for

a

commutative

family of nonexpansive mappings.

Theorem 3.3 ([18]). Let $C$ be

a

compact

convex

subset

of

a

Banach space $E$ and let$S$ be

a

commutative famay

_of

$none\varphi ansive$ maPpings

of

$C$ into

itsdf.

Then $S$ has

a

common

fixed

point in$C,$ $i.e.$, there exists $z\in C$ such that_$Tz=z$

for

every$T\in S$

.

Browder [12] proved the following fixedpoint theorem without compactness:

Theorem3.4 ([12]). Let$C_{1}$be a bounded closed

convex

subset

of

a

unifomly

convex

Banach

space

$E$ and let $S$ be

a

commutative family

of

noneapansive

maPpings.of

$C$ into

itsdf.

$n_{en}$

$S$ has

a

common

fixed

point in$C$

.

Further, let

us

consider to extend these theorems to

a

noncommutative semigroup of

non-expansive mappings. Let $S$ be a semitopological semigroup and let $C$ be a nonempty closed

convex

subset of

a

Banach space $E$

.

Then

a

family _{$S=\{T_{f} : s\in S\}$} of$mapp_{\dot{i}}$g8 of $C$ into

itself is called a $none\varphi ansive$ semigroup

on

$C$ if it satisfies the following:

(1) $T_{\epsilon t}x=T_{e}T_{t}x$ for all $s,t\in S$ and $x\in C$;

(2) for each$x\in C$, the mapping $s\mapsto T_{\epsilon}x$ is continuous;

(3) for each $s\in S,$ $T_{l}$ is

a

nonexpansive mapping of$C$into itself.

For anonexpansive semigroup$S=\{T_{l} : s\in S\}$

on

$C$,

we

denote by _$F(S)$ the set of

common

fixed points of$T_{\epsilon},$$s\in S$

.

Let $S$be

a

semitopological semigroup, let$C(S)$ bethe Banach spwe

of all bounded continuous functions

on

$S$ and let $RUC(S)$ bethe space of all bounded right

uniformly continuous functions on $S$

,

i.e., all _{$f\in C(S)$} such that the mapping $s\mapsto r_{\epsilon}f$ is

continuous. Then$RUC(S)$ is

a

closed subalgebra of$C(S)$ containin

constants

and invariant

under $\ell_{\epsilon}$ and$r_{\epsilon},$ $s\in S$; see [40] for

more

details.

In1969, Ibhhashi [56] provedthe firstfixed pointtheoremfora noncommutativesemigroup

ofnonexptsivemappingswhich generalizesDeMarr’s fixed point theorem,that is, heproved

that any discrete leftamenablesemigrouphasa

common

fixedpoint. Mitchel [41] generalized

Takahashi’s result by showin$g$that any discrete left reversible semigrouphas a

common

fixed

point. Lau proved the following theoremin [28]:

Theorem 3.5 ([28]). Let $S$ be a semitopological semigroup and let $A(S)$ be the _$mac6$

of

all

$f\in C(S)$ such that $\{\ell_{\epsilon}f : s\in S\}$ is relatively compact in the

norm

topology

of

$C(S)$

.

Let

$S=\{T : s\in S\}$ be a $none\varphi ansive$ semigroup on a compact convex subset $C$

of

a Banach

space E. Then $A(S)$ has a

left

invariant

mean

if

and only

if

$S$ has a

common

fixed

point in $C$

.

Lim [38] generahized Kirk’s result [27], Browder’s result [12] and Mitchen$s$ result [41]‘by

showing the $foUowing$thmrem;

Theorem 3.6 ([38]). Let $S$ be

a

left

$r\epsilon$versible semitopological semigroup. Let$C$ be

a

weakly

compact

convex

subset

_of

a

Banach $\epsilon paceE$ which has nomalstructure and let$S=\{T_{l}$ : $s\in$

Takahashi and Jeong [66] als$0$ generalized Browder’s result [12] by using the concept of submeans.

Theorem 3.7 ([66]). Let $S$ be a semitopological semigroup. Let $S=\{T_{\epsilon} : s\in S\}$ be

a

none

cpansive semigroup

on

a bounded closed

convex

subset $C$

of

a unifomly convex Banach

space E. Suppose that$RUC(S)$ _has

a

left

_{subinvariant submean. Then}$S$ has

a

common

fixed

point in $C$

.

To prove Theorm 3.7, we need the folowing lemma [72]:

Lemma 3.8 ([72]). Let $p>1$ and $b>0$ be two $\mu_{ed}$ numbers. Then a Banach space $E$

is unifomly

convex

_if

and only

_if

there exists

a

continuous, $str\dot{r}ctly$ increasing, and

convex

flnction

(depending

on

$p$ andb) $g:[0, \infty$) $arrow[0, \infty$) such that$g(O)=0$ and

$\Vert\lambda x+(1-\lambda)y\Vert^{p}\leq\lambda\Vert x\Vert^{p}+(1-\lambda)\Vert y\Vert^{p}-W_{p}(\lambda)g(\Vert x-y\Vert)$

for

all $x,y\in B_{b}$ and $0\leq\lambda\leq 1$, where $W_{p}(\lambda)=\lambda(1-\lambda)^{p}+\lambda^{p}(1-\lambda)$ and $B_{b}$ is the closed

ball with radius $b$ and centered at the oriptn.

We may comment

on

the relationship between $RUC(S)$_. has $an$ invariant mean” and $S$

is left reversible”. As well known, they do not imply each other in general. But if$RUC(S)$

has sufficientlymany functions to separateclosedsets,then $RUC(S)$ has aninvariant mean”

would imply $S$ is left and right reversible”. Lau and Takahashi [36] generalized Lim’s result

[38] andTakahashi and Jeong’s result [66].

Theorem 3.9 ([36]). Let $S$ be a semitopological semigrvup, let $C$ be a nonempty weaely

compact

convex

subset

_of

a Banach space$E$ which hasnomal structure and let$S=\{T_{\epsilon}$ : $s\in$

$S\}$ be

a none

zpansive semigroup

on

C. Suppose $RUC(S)$ has

a

left

subinvariant submean.

Then $S$ has a

common

fixed

point in$C$

.

To prove Theorem 3.9, we need two lemmas.

Lemma 3.10 ([37]). A dosed

convex

subset $C$

of

a Banach space has nomal

structure

if

and only

_if

it does not contain a sequence $\{x_{n}\}$ such that

for

some $c>0$

,

$\Vert x_{n}-x_{m}\Vert\leq c$ and $\Vert x_{n+1}-\overline{x_{n}}\Vert\geq c-\frac{1}{n^{2}}$

for

aZl$n\geq 1$ and$m\geq 1,$ where $\overline{x_{n}}=\frac{1}{\mathfrak{n}}\sum_{1=1}^{n}x_{i}$

.

Lemma3.11 ([20]). Let$X$ be

a

compact

convex

subset

of

a separated topologicdvector spaoe

$E$, let $f_{1},$$f_{2},$

$\ldots,$$f_{n}$ be

a

finite

family

of

lowersemicontinuous

convex

hnctions

ffom

$X$ into

$R$ and let $c\in R$

,

where $R$ denotes the set

of

real numbers. Then thefollouying conditions (1)

and (2)

are

equivalent:

(1) There exists $x_{0}\in X$ such that $f_{1}(x_{0})\leq c$

for

all$i=1,2,$$.$

.:’$n$;

(2)

_for

any

_finite

non-negative real numbers $\{\alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}\}$ with

$\sum_{1=1}^{n}\alpha_{i}=1$

,

there nists$y\in X$ such that $\sum_{1=1}^{n}\alpha:f_{i}(y)\leq c$

.

Theorm 3.9

answers

affirmatively a problem posed during the Conference on Fixed Point

Theory and Applications held at CIRM, Marseille-Luminy, 1989 (see [29]), whether Lim’s

result and Takahashi and Jeong’s result can be fuly extended to such Banach spaces for

amenable semigroups.

Problem. Would “normal structure “in Theorem 3.9 be replaced by “asymptotic normal

4 Nonlinear

Ergodic

Theorems

Inthis section,

we

discuss nonlinear ergodic theorems. The first nonlinear ergodic theorem

for nonexpansive mappings

was established

in1975byBaiUon [6] in theframeworkof

a

Hilbert

space.

Theorem 4.1 ([6]). Let $C$ be a closed

convex

subset

of

a Hilbert space $H$ and let $T$ be

a

$none\varphi ansive$ maPping

of

$C$ into

itself. If

the

set

$F(T)$

of

fxed

points

of

$T$ is nonempty, then

for

each $x\in C$, the Ces\‘aro

means

$S_{n}(x)= \frac{1}{n}\sum_{k=0}^{n-1}T^{k}x$

converye

weakly

to

some

$y\in F(T)$

.

This theorem

was

extended to a uniformly

convex

Banach space whose

nom

is R&het

differentiable by Bruck [15].

Theorem 4.2 ([15]). Let $C$ be a closed

convex

subset

of

a

unifomly

convex

Banach $s\mu oe$

$E$ Utth a Pr\’echet

differentiable

nom.

If

$T:Carrow C$ is a nonespansive mapping with a

ffld

point, then the Ces\‘arv

means

_of

$\{T^{\iota}x\}$

converge

weakly to

a

fixed

point

of

$T$

.

In their theorems, putting $y=Px$ for each $x\in C$, we have that $P$ is

a

$none_{W^{ansi_{V}e}}$

retraction of$C$ onto $F(T)$ such that $PT^{n}=T^{\mathfrak{n}}P=P$ forall $n=1,2,$$\ldots$ and $Px\in\varpi\{T^{n}x$ :

$n=0,1,2,$$\ldots$

}

for each$x\in C$

,

where

$\overline{co}A$is the closure of the

convex

hul of$A$

.

Takahashi [58]

called such

a

retraction”ergodic retraction”. In general, let $S$be

a

semitopologcalsemiyoup,

let $C$ be

a

closed

convex

subset of $E$

,

and let $S=\{T_{t} : t\in S\}$ be

a

nonexpansive semigroup

on

$C$

.

Then,

a

mapping $P$ of$C$ onto $F(S)$ is caled

a

$none\varphi ansive$ ergodic retraction if it

satisfiesthe following conditions:

(1) $\Vert Px-Py\Vert\leq\Vert x-y\Vert$ for all$x,y\in C$;

(2) $P^{2}=P$;

(3) $PT_{t}=T_{t}P=P$ for all $t\in S$;

(4) $Px\in\overline{co}\{T_{t}x:t\in S\}$ for all$x\in C$

.

Let $\{\mu_{\alpha} : \alpha\in A\}$beanet ofmeans

on

$RUC(S)$

.

Then $\{\mu_{\alpha}\in A\}$is saidtobe asymptotically

invariantiffor each $f\in RUC(S)$ and $s\in S$

,

$\mu_{\alpha}(f)-\mu_{\alpha}(\ell_{\epsilon}f)arrow 0$ and $\mu_{\alpha}(f)-\mu_{\alpha}(r_{\epsilon}f)arrow 0$

.

Let us give an example of asymptotically invariant nets. Let $S=\{0,1,2, \ldots\}$ and let $N$ be

the set of positive integers. Then for $f=(x_{0},x_{1}, \ldots)\in B(S)$ and $n\in N$, the real valued

function$\mu_{\mathfrak{n}}$ defined by

$\mu_{n}(f)=\frac{1}{n}\sum_{k=0}^{n-1}x_{k}$

is

a mean.

Further since for $f=(x_{0},x_{1}, \ldots)\in B(S)$ and$m\in N$ $| \mu_{n}(f)-\mu_{n}(r_{m}f)|=|\frac{1}{n}\sum_{k=0}^{n-1}x_{k}-\frac{1}{n}\sum_{k=0}^{n-1}x_{k+m1}$

as $narrow\infty,$ $\{\mu_{n}\}$ is an asymptotically invariant net of

means.

If $C$ is a nonempty closed convex subset ofa Hilbert space $H$ and $S=\{T_{s} : s\in S\}$ is a

nonexpansive semigroup on $C$ such that $\{T_{\epsilon}x : s\in S\}$ is bounded for

some

$x\in C$

,

then

we

know that for each $u\in C$ and $v\in H$, the functions $f(t)=\Vert T_{t}u-v\Vert^{2}$ and $g(t)=(T_{t}u,v)$

are in $RUC(S)$

.

Let $\mu$ be a

mean on

$RUC(S)$

.

Then since for each $x\in C$ and $y\in H$, the

real valued function$t\mapsto(T_{t}x,y)$ is in$RUC(S)$, we

can

define thevalue $\mu_{t}(T_{t}x,y)$ of$\mu$ at this

function. By linearity of$\mu$ and of theinner product, this is linear in $y$; moreover, since

$| \mu_{t}(T_{t}x, y)|\leq\Vert\mu\Vert\cdot\sup_{t}|(T_{t}x,y)|\leq(\sup_{t}\Vert T_{t}x\Vert)\cdot\Vert y\Vert$ ,

it is continuous in$y$

.

So, by the Riesz theorem, there exists

an

$x_{0}\in H$ such that

$\mu_{t}(T_{t}x,y)=(x_{0},y)$

for every $y\cdot\in H$

.

We write suchan $x_{0}$ by $T_{\mu}x$;

see

$[58,64]$ for

more

details.

In 1981, Takahashi [58] proved the first nonlinear ergodic theorem for noncommutative.

semigroups ofnonexpansivemapping8 in

a

Hilbert space.

Theorem 4.3 ([58]). Let$C$ bea nonemptydosed

convex

subset

of

a

Hilbert space and let$S$ be

a

semitopological semigroup such that $RUC(S)$ has

an

invariant

mean.

Let$S=\{T_{t} : t\in S\}$

be $a$ one-parameter nonerpansive semigroup

on

$C$ such that $\{T_{t}x:t\in S\}\dot{u}$ bounded

for

some

$x\in C$

.

Then, there nists

a

uniqu$e$ nonlinear ergodic retraction $P$

from

$C$ onto $F(S)$ such

that$PT_{t}=T_{t}P=P$

for

each$t\in S$ and$Px\in\overline{co}\{T_{t}x:t\in S\}$

for

each $x\in C$

.

Further, Rod\’e [49] provethe folowing theorem.

Theorem4.4 ([49]). Let$C$ be

a

nonemptyclosed

convex

subset

of

aHilbert space$H$ andlet$S$

bea semitopological$\epsilon emigroup$ suchthat$RUC(S)$ has

an

invariantmean. Let$S=\{T_{t} : t\in S\}$

be a nonespansive semigroup on$C$ such that $\{T_{t}x:t\in S\}$ is bounded

for

some

$x\in C$

.

Then,

for

an$as_{\Psi^{nptotioelly}}$invariant net$\{\mu_{\alpha} : \alpha\in A\}$

of

means

on$RUC(S)$, thenet$\{T_{\mu_{\alpha}}x:\alpha\in A\}$

converges weakly to an element $x_{0}\in F(S)$.

Using Theorem 4.4, we have Theorem 4.1. Bythe

same

method,

we

can

prove thefollowing

nonhnear ergodic theorems:

Theorem 4.5. Let $C$ be a closed

convex

subset

of

a

Hilbert space $H$ and let $T$ be

a

$non\varpi$

pansive $mapp|ng$

of

$C$ into

itself. If

_$F(T)$ is nonempty, then

for

each $x\in C$,

$S_{r}(x)=(1-r) \sum_{k=0}^{\infty}r^{k}T^{k}x$,

as $r\uparrow 1$, converges weakly to an element$y\in F(T)$

.

Theorem 4.6. Let$C$ be a dosed

convex

subset

of

a Hilbert spaoe $H$ and let_$S=\{S(t)$ : $t\in$

$[0, \infty)\}$ be $a$ one-parameternonempansive semigroup on C.

If

$F(S)$ is nonempty, then

for

each

$x\in C$,

$S_{\lambda}(x)= \frac{1}{\lambda}\int_{0}^{\lambda}S(t)xdt$

,

as

$\lambdaarrow\infty,$ _{$conve\eta es$} weakly to

an

element$y\in F(S)$

.

Next, let

us

state a nonlinear ergodic theorem for nonexpansive semigroups in

a

Banach

space. Before stating it, we give

a

definition. A net $\{\mu_{\alpha}\}$ ofcontinuous linear functionals

on

(1) $sup\Vert\mu_{\alpha}\Vert<$_十科科;

$\alpha$

(2) $\lim_{\alpha}\mu_{\alpha}(1)=1$;

(3) hm$\Vert\mu_{\alpha}-r_{\epsilon}^{*}\mu_{\alpha}\Vert=0$ forevery $s\in S$

.

Theorem 4.7 ([23]). Let $S$ be a commutative $smitopolo\dot{\wp}cal$ semigrouP and let $E$ be

a

uni omly

convex

Banach space with

a

$P\dagger\cdot\acute{e}chet$

differentiable

nom.

Let $C$ be a nonempty

closed

convex

subset

_of

$E$ and let$S=\{T_{t} : t\in S\}$ be

a

nonespansive semigroup

on

$C$ such

that$F(S)$ is nonempty. Then there

extsts

a

unique nonempansive retmction$P$

of

$C$

onto

$F(S)$

such that$PT_{t}=T_{t}P=P$

for

every$t\in S$ and$Px\in\overline{\infty}\{T_{t}x:t\in S\}$

for

every$x\in C$

.

Purther,

if

$\{\mu_{\alpha}\}$ is

a

strongly regular net

of

continuous linear

functionals

on

$RUC(S)$

,

then

for

each

$x\in C,$ $T_{\mu_{\alpha}}T_{t}x$ converges weakly to $Px$ unifomly in$t\in S$

.

We have not known whether Theorem

4.7

vould hold in the

case

when$S$is noncommutative

(cf. [62]). Lau, Shioji and Takahashi [33] solved the problm

as

follows:

Theorem 4.8 ([33]). Let$C$ be a dosed

convex

subset

of

auniformly

convex

Banach$\epsilon paoeE$,

let$S$ be a semitopological semigroup which $RUC(S)$ has

an

invariant mean, and let$S=\{T_{t}$ :

$t\in S\}$ be

a

$none\varphi ansive$semigrouP

on

C Utth $F(S)\neq\phi$

.

Then there extSts $a$ $none\varpi amive$

ergodic febuction $P$

ffom

$C$ onto $F(S)$ such that $PT_{t}=T_{t}P=P$

for

each $t\in S$ and

$Px\in\varpi\{T_{t}x:t\in S\}$

for

each$x\in C$

.

This is

a

generalizationof$Ta\bm{L}hflshi’ s$result [58]for

an

amenablesemigroup of nonexpansive

mapping8

on

a

Hilbert

space.

Furtherthey extended Rod\’e’s result [49] to

an

amenable semi-group of$nonexpan8ive$ mapp血$gs$

on

aun迂omly

convex

$B$anach _叩acewhose

nom

色 R&het

differentiable.

$Th\infty rem4.\theta$ ([33]). Let$E$ be

a

unifomdy

convex

Banach space with

a

R\’echet

diffmntiable

nom

and let $S$ be

a

semitopological $\epsilon emi\varphi oup$

.

Let$C$ be a closed

convex

subset

of

$\dot{E}$ and let

$S=\{T_{t} : t\in S\}$ be a nonespansive semigroup

on

$C$ with $F(S)\neq\phi$

.

Suppose that$RUC(S)$

has

an

invariant

mean.

Then there enist\epsilon _{a unique nonempansive retraction $PnmC$ onto}

$F(S)$ such that $PT_{t}=T_{t}P=P$

for

each $t\in S$ and $Px\in\overline{co}\{T_{t}x:t\in S\}$

for

each $x\in C$

.

FUrther,

_if

$\{\mu_{\alpha}\}$ is

an

aswnptoticdly invariant net

of

means

on

$X$

,

then

for

each $x\in C$,

$\{T_{\mu_{\alpha}}x\}$

converges

weakly

to

$Px$

.

To prove Theorem 4.9, they used Thmrem

4.8

and the following lemma which has been

proved in Lau, Nishiura and Takahashi [31].

Lemma 4.10 ([31]). Let$E$ be

a

$un|fomdy$

convex

Banach spaoe with $a$I\succ \’echet

differentiable

nom

and let $S$ be

a

semitopological semigroup. Let $C$ be

a

dosed

convex

subset

of

$E$ and let

$S=\{T_{t} : t\in S\}$ be

a

$none\varphi an\theta ive$ semigroup

on

C Utth $F(S)\neq\phi$

.

Then,

for

each $x\in C$,

$F(S) \cap\bigcap_{\in}s\overline{co}\{T_{t\epsilon}x : t\in S\}$ consists

of

at most

one

point.

Thefollowing$th\infty rem$has beenproved in$\ovalbox{\tt\small REJECT} hi[60]$ and Lau, Nishiura and$\ovalbox{\tt\small REJECT}$

[31] when$E$ is a Hilbert

space.

Theorem4.11 ([33]). Let$E$ beaunifomly

convex

Banach

wace

with

a

FVichet

differentiable

nom

and let $S$ be

a

semitopologicalsemigroup. Let $C$ be

a

closed

convex

subset

of

$E$ and let

$S=\{T_{t} : t\in S\}$ be

a

$none\varphi ansive$ semigrvup

on

$C$ with $F(S)\neq\phi$

.

Suppose that $fa$

.

each

$x\in C,$ $F(S) \cap\bigcap_{\partial\in S}\overline{co}\{T_{\ell\epsilon}x:t\in S\}\dot{u}$ nonempty. Then there exists

a

$none\varphi ansiv\epsilon$ retraction

$P$

from

$C$ onto $F(S)such$ that $PT_{t}=T_{t}P=P$

for

each $t\in S$ and $Px\in\varpi\{T_{t}x:t\in S\}$

for

Recently, Miyake and Takahashi [44] proved nonlinear ergodic theorems for nonexpansive

mappingswith compact domains in general Banachspaces.

Theorem 4.12 ([44]). Let $C$ be a compact and

convex

subset

of

a Banach spaoe $E$

,

let $S$

be

a

semigroup, let $S=\{T_{s} : s\in S\}$ be

a

$none\varphi ansive$ semigroup on $C$ into itself, let $X$

be a subspace

_of

$B(S)$ containing 1 such that $\ell_{\theta}XCX$

for

each $s\in S$ and the jfmctions

$s-\rangle(T_{\epsilon}x,x^{*}$

}

and $s\vdash\star\Vert T_{\epsilon}x-y\Vert$

are

contained in $X$

for

each $x,y\in C$ and $x^{*}\in E^{*}$ and

let $\{\mu_{\alpha}\}$ be

an

asymptotically invariant net

of

means on

X. Then,

for

each $x\in C,$ $T_{r_{\dot{h}}\mu_{\alpha}}x$

convefges unifomly in $h\in S$

.

Next, applying Theorem 4.12, we obtain a nonlinear ergodic theorem for nonexpansive

semigroups

on

acompact and

convex

subset ofastrictly

convex

Banach space.

Theorem 4.13 ([44]). Let $C$ be a compact and

convex

subset

of

a

strictly

convex

Banach

space $E$, let $S$ be a semigrvup, let $S=\{T_{\epsilon} : s\in S\}$ be

a

$none\eta ansive$ semigmuP

on

$C$

, into

itself, let $X$ be

a

subspace

of

$B(S)$ containing 1 such that$\ell_{\delta}X\subset X$

for

each $s\in S$ and the

functions

$srightarrow\langle T_{l}x,x^{*}\rangle$ and $s\mapsto\Vert T_{\epsilon}x-y\Vert$

are

contained in $X$

for

each$x,y\in C$ and$x^{*}\in E^{*}$

andlet $\{\mu_{\alpha}\}$ be

an

asymptohcally invariant net

of

means on

X. Then,

for

each$x\in C,$ $T_{r_{\dot{h}}\mu_{\alpha}}x$

convergesstrongly to

a common

jfixed point

_of

$S$ unifomly in $h\in S$

.

UsingTheorem4.12 and Theorem 4.13,

we

obtain

some

nonhnear ergodic theorems in

cases

of discrete and oneparametersemigroups of nonexpansive $mapp_{\dot{i}}$gs.

Theorem 4.14. Let $C$ be a compact and

convex

subset

of

a

Banachspace $E$ and let$T$ be $a$

nonescpansive mapping

_of

$C$ into

itself.

Then,

for

each $x\in C$,

$\frac{1}{n}\sum_{i=0}^{n-1}\dot{\Gamma}^{+h_{X}}$

converges

uniformly in $h\in N$

.

Theorem 4.15. Let $C$ be

a

compact and

convex

subset

of

a

strictly

convex

Banach opace $E$

and let $T$ be

a

nonespansive mapping

of

$C$ into

itself.

Then,

for

each $x\in C$,

$\frac{1}{n}\sum_{i=0}^{n-1}\dot{\Gamma}^{+\hslash_{X}}$

converges to a

_fixed

point

_of

$T$ unifomly in $h\in N$

.

Theorem 4.16. Let $C$ be a compact and

convex

subset

of

a Banach opace $E$ and let $S=$

$\{T(t) : t\in R\}$ be $a$ one-parameter

none

zpansive semigmup

on

C. Then,

for

each $x\in C$, $\frac{1}{t}\int_{0}^{t}T(\epsilon+h)xd\epsilon$

convergesunifomly in $h\in R$

.

Theorem 4.17. Let $C$ be a compact and

convex

subset

of

a strictly

convec

Banach qace$E$

and let$S=\{T(t) : t\in R\}$ be $a$ one-parameter nonespansiveseml’gmup

on

C. Then,

for

each

$x\in C$,

$\frac{1}{t}\int_{0}^{t}T(s+h)xds$

5

Approximation

of fixed

points

There

are

two iterative methodsfor approximationof fixedpoint8ofnonexpansive mappings

in

a

Hilbert space which

are

different from the Ces\‘axo

means.

Mann [39] introduced the followingiterative scheme for finding afixed point ofa

nonexpan-sive mapping. For the proof,

see

Takahashi [65].

Theorem 5.1 ([39]). Let $C$ be

a

closed

convex

subset

of

a Hilbert spaoe and let $T$ be $a$

$none\varphi ansive$ mapping

of

$C$ into

itself

such that $F(T)$ is nonempty. Let $P$ be the metric

projection

_of

$H$ onto _$F(T)$

.

Let $x\in C$ and let $\{x_{n}\}$ be a sequenoe

defined

by _{$x_{1}=x$} and

$x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})Tx_{n}$, $n=1,2,$$\ldots$

,

where $\{x_{n}\}\subset[0,1]$

saMfies

$0\leq\alpha_{\mathfrak{n}}<1$ and $\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$

.

Then, $\{x_{n}\}$ converges weakly to $z\in F(T)$, where $z=\dot{n}$) $m_{narrow\infty}Px_{\mathfrak{n}}$

.

Wittmann [71] dealt with the folowing iterative scheme to approximate a flxed point of

a

nonexpansive mapping in

a

Hilbert

space;

8ee originally Halpem [22]. For the proof,

see

Tahhashi [65].

Theorem 5.2 ([71]). Let $C$ be a dosed

convex

subset

of

$\cdot$a Hilbert spaoe $H$ and let $T$ be

a $none\varphi an\epsilon ive$ mapping

of

$C$ into

itsdf

$such$ that $F(T)$ is nonempty. Let $P$ be the metric

$p\dot{\eta}eetion$

of

$H$ onto$F(T)$

.

Let$x\in C$ and let $\{x_{n}\}$ be a sequenoe

defined

by _{$x_{1}=x$} and

$x_{n+1}=\alpha_{n}x+(1-\alpha_{n})Tx_{n}$, $n=1,2,$$\ldots$,

where $\{\alpha_{\mathfrak{n}}\}\subset[0,1]$

satisfies

$\mathfrak{n}n\ovalbox{\tt\small REJECT} n\alpha_{n}=0,\sum_{\mathfrak{n}=1}^{\infty}\alpha_{n}=\infty$ and $\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

.

Then, $\{x_{n}\}$

converges

strongly to $Px\in F(T)$

.

Shimizu and Takahashi [50] introduced the first iterative schemes for finding

common

fixed

points of families of nonexpansive mappinga and proved strong convergence theorems for

discrete and one-parameter nonexpansive semigroups in Hilbert spaces. Atsushiba, Shioji

and Tahhashi [2] established aweak convergencetheorem ofMann’s type for

a

nonexpansive

semigroup in a Banach space.

Theorem 5.3 ([2]). Let $E$ be

a

unifomly

convex

Banach spaoe Utth aPr\’echet

differentiable

nom.

Let $C$ be

a

nonempty dosed

convex

subset

of

$E$ and let _{$S=\{T_{t} : t\in S\}$} be $a$

nonexpansiv$esem|group$

on

$C$ such that$F(S)\neq\phi$

.

Let$\{\mu_{n}\}$ be

a

sequenoe

of

means on

$C(S)$

such that $\Vert\mu_{n}-\ell_{\epsilon}^{*}\mu_{n}||=0$

for

every$s\in S$

.

Suppose that$x_{1}=x\in C$ and $\{x_{n}\}$ is given by

$x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})T_{\mu_{n}}x_{n}$, $n=1,2,$$\ldots$

,

where $\{\alpha_{n}\}$ is

a

sequ

enoe

in $[0,1]$

.

If

$\{\alpha_{n}\}$ is chosen

so

that $\alpha_{n}\in[0, a]$

for

some

$a$ with

Using Theorem 5.3, we pan prove aweak convergence theorem of Mann’s type for

a one

parameter nonexpansive semigroup.

Theorem 5.4. Let$E$ be

a

unifomly

convex

Banachspaoe with a $P\vdash\acute{e}chet$

differentiable

nom

and let $C$ be a dosed convex subset

of

E. Let $S=\{S(t) : t\in[0, \infty)\}$ be $a$ one-parameter

$none\eta ansive$ semigroup on $C$ such that $F(S)\neq\phi$

.

Suppose that $x_{1}=x\in C$ and $\{x_{n}\}\dot{u}$

$\dot{\mu}ven$ by

$x_{\mathfrak{n}+1}= \alpha_{n}x_{n}+(1-\alpha_{\mathfrak{n}})\frac{1}{s_{n}}\int_{0}^{\epsilon_{n}}S(t)x_{n}dt$, $n=1,2,$_$\ldots$,

where$s_{\mathfrak{n}}arrow\infty$

as

$narrow\infty$ and$\{\alpha_{\mathfrak{n}}\}$ is

a

sequenoe in $[0,1]$

.

If

$\{\alpha_{\mathfrak{n}}\}$ is chosen

so

that$\alpha_{n}\in[0,a]$

for some

$a$ with$0<a<1$ , then $\{x_{n}\}$ converges

weauy

to a

common

ffid

point$z\in F(S)$

.

Shioji and $\ovalbox{\tt\small REJECT} hi[53]$ also established the following strong

convergence

theorem for

a

nonexpansive $semi_{\Psi}oup$ of Halpern’s type in

a

Banach space.

Theorem 5.5 ([53]). Let$E$ be a unifomly

convex

Banach spaoe with a uniformly G\^ateaux

$diffeoen\hslash able$

nom.

Let $C$ be

a

nonempty closed

convex

subset

of

$E$ and let_{$S=\{T_{t} : t\in S\}$}

be a nonezpansive semigroup

on

$C$ such that$F(S)\neq\phi$

.

Let $\{\mu_{\mathfrak{n}}\}$ be a sequenoe

of

means on

$C(S)$ such that $\Vert\mu_{n}-\ell_{\epsilon}^{*}\mu_{n}||=0$

for

every$s\in S$

.

Suppose that$x,y_{1}\in C$ and $\{y_{n}\}$ is given by

$y_{\mathfrak{n}+1}=\beta_{n}x+(1-\beta_{n})T_{\mu},y_{\dot{n}}$, $n=1,2,$$\ldots$,

where $\{\beta_{n}\}$ is in $[0,1]$

.

If

$\{\beta_{n}\}$ is chosen

so

that$hm_{narrow\infty}\beta_{n}=0$ and$\Sigma_{n=1}^{\infty}\beta_{n}=\infty$, then $\{y_{n}\}$

converges strongly to an element

_of

$F(S)$

.

Suzuki and $\ovalbox{\tt\small REJECT} hi[55]$ established

a

strong convergence theorem ofMann’s type for

a

one-parameter nonexpansive semigroup in

a

Banach space without strict convexity. $\cdot$

Theorem 5.6 ([55]). Let $C$ be a compact

convex

subset

of

a

Banach opace $E$ and let $S=$

$\{S(t) : t\in \mathbb{R}_{+}\}$ be $a$ one-parameter $none\eta ansive$ semigrvup

on

C. Let$x_{1}\in C$ and

define

a

sequenoe in $C$ by

$x_{\mathfrak{n}+1}= \frac{\alpha_{n}}{t_{n}}\int_{0}^{t}S(s)x_{\mathfrak{n}}ds+(1-\alpha_{n})x_{n}$

for

every$n\in N$, where $\{\alpha_{n}\}\subset[0,1]$ and $\{t_{n}\}\subset(0,\infty)sati\epsilon\hslash$the

follo

wing $conditio|w$: $0< Minf\alpha_{\mathfrak{n}}narrow\infty\leq\lim_{narrow}\sup_{\infty}\alpha_{n}<1$, $\lim_{narrow\infty}t_{n}=\infty$ and $\lim_{narrow\infty}\frac{t_{n+1}}{t_{n}}=1$

.

Then $\{x_{n}\}$ converges strongly to

a

common

fixed

point

of

$S$

.

Recently, Miyake and TaJdhashi [42] extended Suzuki and Takabashi’s result to

a

general

commutative nonexpansive semigroupin a Banach space.

Theorem 5.7 ([42]). Let $C$ be

a

compact

convex

subset

of

a

Banach space $E$ and let$S$ be $a$

commutative semigroup with identity $0$

.

Let _{$S=\{T_{t} : t\in S\}$} be

a

$none\varphi ansive$ semig up

on C. Let $X$ be a subspace

of

$B(S)$ _{containing 1} such that $\ell_{l}X\subset X$

for

each $s\in S$ and

the

_hnctions

$srightarrow(T_{l}x,x^{u})$ and $s\mapsto\Vert T_{\epsilon}x-y||$ are contained in $X$

for

each $x,y\in C$ and

$x^{*}\in E^{*}$ and let $\{\mu_{\mathfrak{n}}\}$ be an uymptotically invariant sequenoe

of

means

on

$X$ such that

$hm_{narrow\infty}\Vert\mu_{n}-\mu_{n+1}||=0$

.

Let $\{\alpha_{n}\}$ be a sequenoe in $[0,1]$ such that $0< \lim_{\mathfrak{n}arrow}\inf_{\infty}\alpha_{n}\leq\lim_{narrow}\sup_{\infty}\alpha_{n}<1$

.

Let $x_{1}\in C$ and let$\{x_{n}\}$ be the sequence

defined

by

$x_{n+1}=\alpha_{n}T_{\mu_{n}}x_{n}+(1-\alpha_{n})x_{n}$

for

every $n=1,2,$$\ldots$

.

Then $\{x_{n}\}$ convefges strongly to a

common

fixed

$po|nt$

of

$S$

.

Miyake and Takahashi [43] also obtained

a

strong convergence theorem of Halpern’s type

fora general commutative nonexpansive semigroup in a Banach space. See alsoLau, Miyake

and Takahashi [30] for amenablesemigroups.

Theorem 5.8 ([43]). Let $C$ be a compact

convex

subset

of

a

smooth and strictly

convex

Banach spaoe$E$, let$S$ be

a

commutative semigroup with identity$0$

.

Let_{$S=\{T_{t} : t\in S\}$} be

a

$none\varphi amive$ semigrouP

on

$C$, let$X$ be

a

subsPaoe

of

$B(S)$ containing 1 such that$\ell_{\iota}X\subset X$

for

each $s\in S$ and the

functions

$srightarrow(T_{\iota}x,x^{*})$ and $\epsilon\vdasharrow||T_{l}x-y\Vert$

are

contained in $X$

for

each $x,y\in C$ and $x^{*}\in E^{*}$ and let $\{\mu_{n}\}$ be

a

strongly

mular

sequenoe

of

means

on

X. Let

$\{\alpha_{n}\}$ be a sequenoe in $[0,1]$ such that $\sum_{\mathfrak{n}=1}^{\infty}\alpha_{\mathfrak{n}}=\infty$ and $\lim_{\mathfrak{n}arrow\infty}\alpha_{n}=0$

.

Let $x\in C$ and let

$\{x_{n}\}$ be the sequenoe

defined

by

$x_{n+1}=\alpha_{\mathfrak{n}}x+(1-\alpha_{n})T_{\mu_{\hslash}}x_{n}$

for

every $n=1,2,3,$$\ldots$

.

Then $\{x_{n}\}$ converges strongly to $Px$, where $P$ is a unique sunny

$none\varphi ansive$ retmction

of

$C$ onto $F(S)$

.

Using Theorem 5.8,

we

can

obtain the fobowin$g$ strong convergence theorem for

a

one

parameter nonexpansive semigroup.

Theorem 5.9. Let $C$ be

a

compact

convex

subset

of

a

smooth and \epsilon trictly

convex

Banach

spaoe $E$ and let$S=\{S(t) : t\in \mathbb{R}_{+}\}$ be $a$ one-parameter nonezpansive semigroup

on

C. Let

$x_{1}=x\in C$ and let $\{x_{n}\}$ be

a

sequenoe

defined

by

$x_{n+1}= \alpha_{n}x+(1-\alpha_{n})\frac{1}{t_{n}}\int_{0}^{t_{n}}S(s)x_{\mathfrak{n}}d\epsilon$

for

every $n=1,2,3,$$\ldots,$ whert $\{\alpha_{\mathfrak{n}}\}$ is a sequneoe in $[0,1]$ such that $\sum_{n=1}^{\infty}\alpha_{n}=\infty$ and

%\rightarrow \infty$\alpha_{n}=0$ and $\{t_{n}\}$ is

an

increasing sequence in $(0, \infty)$ such that _{血噺=\infty tn} $=\infty$

and $11m_{\mathfrak{n}arrow\infty_{t_{n+1}}^{t}}=1$

.

Then $\{x_{n}\}$ converges strongly to $Px$, where $P$ is

a

unique sunny

$none\varphi ansive$ retracton

of

$C$ onto _$F(S)$

.

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