NORMAL STRUCTURE AND FIXED POINT PROPERTY FOR NONEXPANSIVE MAPPINGS(Nonlinear Analysis and Convex Analysis)

(1)

NORMAL STRUCTURE

AND

FIXED

POINT

PROPERTY

FOR NONEXPANSIVE MAPPINGS

ANTHONY TO-MING

LAU1

1. Introduction

Let

$E$

be

a

Banach space

and

$X$

be

a

weakly

_conlpact

convex

subset

of

$E$

.

Let

$S=\{T_{s} ; s\in S\}$

be

a

continuous representation

of

a

semitopological

semigroup

$S$

as

non-expansive

self-maps

on

$X$

. In

this paper,

we

shall

report,

among

other things

, on

some

recent

$1^{\cdot}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{s}$

concerning

tlle relation

of

$\mathrm{i}\mathrm{n}\mathrm{V}\mathrm{a}\mathrm{l}\cdot \mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}$

submean

on

the space

of bounded

continuous real-valued functiolls

on

S. nornlal

$\mathrm{s}\mathrm{t}\mathrm{l}\cdot \mathrm{u}\mathrm{C}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$

of

$K$

,

and

the

existence of

a

common

fixed

point

in

$X$

for

$S$

.

We shall also

$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{l}\cdot \mathrm{t}$

on

some

sufficient

or

necessaly

conditions

on a

locally

compact

group

$G$

such

that

every

weak*-compact

convex

subset

of the

$\mathrm{F}\mathrm{o}\mathrm{u}\Gamma \mathrm{i}\mathrm{e}\mathrm{l}$

Stieltjes algebra of

$G$

has normal

structure and hence the fixed point

$\mathrm{p}\mathrm{l}\cdot \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{y}\mathrm{f}\mathrm{o}1$

nonexpansive mappings.

This

paper

contains

palt

of

our

talk

given

$\mathrm{d}\mathrm{u}\mathrm{l}\cdot \mathrm{i}\mathrm{n}\mathrm{g}$

the

1995

RIMS Symposium

on

Noulillcal

$\cdot$

Analysis alld Convex Analysis held at Kyoto University, Kyoto. We thank

the organizers for their kind invitation to

speak and

theil

$\cdot$

warm

hospitality

$\mathrm{d}\mathrm{u}\mathrm{l}\cdot \mathrm{i}\mathrm{n}\mathrm{g}$

the

conference.

2. Normal

Structure

and

Submean

Let

$E$

be

a

Banach space.

$D$

be

a

bounded

subset

of

$E,$

$u\in D$

.

Define

$r_{u}(D)= \sup\{||u-v||:v\in D\}$

.

(2)

Thell

$r_{u}(D)\leq \mathrm{d}\mathrm{i}\mathrm{a}\ln(D)=\mathrm{s}’ \mathrm{u}_{1^{)}\{}||v_{1}-v_{2}||:v_{1},$

_{$v_{2}\in D\}$}

.

A point

$u$

in

$D$

is said to

be

dianletl

$\cdot$

al if

$r_{u}(D)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(D)$

.

Otherwise,

$u$

is said to be non-diametral.

A

convex

subset

$X$

of

$E$

is said to

have normal

structure if each closed

convex

subset

$D$

of

$\cdot$

$X$

with

$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{l}(D)>0$

contains

a

noll-diallletl

$\cdot$

al point

i.e. there exists

$u\in D$

sncll that

$\sup\{||u-v||\backslash v\in D\}<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{u}(D)$

.

As

well known compact

convex

subset of

a

Banach

space

$E$

has nornlal

structure.

Also.

$\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}_{01}\cdot \mathrm{n}11\mathrm{y}$

convex

Banach

spaces

llave normal

$\mathrm{s}\mathrm{t}\mathrm{r}\iota 1\mathrm{c}\mathrm{t}_{\mathrm{U}1}\cdot \mathrm{e}$

(see

[4]).

However it follows

fronl

[1]

that

weakly compact

convex

subset

of

a

Banach space

need

not have

normal

structure.

Let

$S$

be

a

semitopological

semigroup i.e.

$S$

is

a

$\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{g}\mathrm{l}\cdot \mathrm{o}\mathrm{u}\mathrm{p}$

with

a

$\mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{d}_{01}\cdot \mathrm{f}\mathrm{f}$

topology such that for each

$a\in S$

,

the

mappings

$s-\rangle$

as

and

$s\mapsto sa$

from

$S-S$

$\mathrm{a}\mathrm{l}\mathrm{e}$

continuous.

Let

$CB_{r}(S)$

be

the space

of bounded

$1^{\cdot}\mathrm{e}\mathrm{a}\mathrm{l}$

-valued functions

on

_$G$

.

A

sub

$7neanl^{l}$

is

a

real-valued

function

on

$CB_{r}(S)$

satisfying:

(i)

$l^{l},(f$

.

$+g)\leq_{l^{l}}(f)+\mu(_{j}().$

$f..J(\in CB_{r}(S)$

:

(ii)

$l^{\iota}(\alpha f)=\alpha\mu(f)$

.

$\alpha\geq 0$

,

$f$

.

$\in CB_{r}(S)$

;

(iii)

for

$f’,$

_{$g\in CB_{r}(S)$}

,

$f\leq g$

_,

$l^{b}(f)\leq\{l(cJ)$

;

(iv)

$l^{(},(C)=c$

for

every

constant

$\mathrm{f}_{\mathrm{U}11}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}c$

.

Tlle notion of

$\mathrm{s}\mathrm{u}\mathrm{b}_{1}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{n}$

is due to Mizoguchi-Takahashi [12]. A submean

$l^{l}$

is

left

$invar\cdot i\text{ノ}ant$

if

_{$\mu(\ell_{a}f)=_{l^{\iota(f)}}$}

for

all

_{$a\in S,$}

_{$f\in CB_{r}(S)$}

where

_{$(\ell_{a}f)(t)=f$}

(at),

_{$t\in S$}

_.

A

$\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{O}\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}_{\mathrm{C}\mathrm{a}1}$

senligroup

is

left

$reve7$

sible

if

$\overline{\mathrm{c}xS}\cap\overline{bS}\neq\phi$

for

_$a$

.

$l$

₎

_{$\in S$}

(where

$\overline{A}$

(3)

and

submean:

Lemma 1 ([8]).

Let

$S$

be

a

semitopological

$semigr\cdot oup$

.

(a)

_If

$S$

is

left

reversible,

then

$CB,,(S)$

has

a

left

invariant submean.

(b)

_If

$S$

is

normal,

and

$CB_{r}(s)$

has

a

left

invariant submean. then

$S$

is

left

$r\cdot eversible$

.

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{l}\cdot \mathrm{C}$

is

also

a

relation between

$\mathrm{n}\mathrm{o}\mathrm{l}\cdot 11\mathrm{l}\mathrm{a}\mathrm{l}$

structure

and

invariant submean.

Lemma 2

([9]). Let

$X$

be

a

weakly

compact

conve.

$\iota$

subset

of

$\cdot$

a

Banach space.

_If

$X$

has

$no7^{\cdot}7nal$

structure, then

$X$

has the

$foll\mathit{0}wingp7^{\cdot}operty$

:

(P)

whenever

$S$

is

a

semitopological semigroup and

$S=\{T_{S} ; s\in S\}$

is

a

continuous

representation

_of

$S$

as

nonexpansive

self

maps

on

$X$

,

if

$\ell\iota$

is

a

left

invariant submean

on

$CB_{r}(S)$

.

then

the

set

$A_{x}=\{y\in X;l^{l_{t}(1}|T_{t}x-y||)=\rho_{x}\}$

is

a

proper subset

_of

$X$

for

some

$x\in X$

,

where

$\rho_{x}=$

$\inf\{\mu_{t}(||T_{t}X-y||);y\in X\}$

.

$Furthermo\gamma\cdot e$

.

for

each

$x\cdot\in X$

.

the

set

$A_{x}$

is non-empty. closed.

convex

and S-invariant.

$\mathrm{L}\mathrm{e}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{l}1$

and 2

can

be

used

to obtain

the following

$\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{l}\cdot \mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$

of

$\mathrm{L}\mathrm{i}_{111^{\tau_{\mathrm{S}}}}1^{\cdot}\mathrm{e}\mathrm{S}\mathrm{u}\mathrm{l}\mathrm{t}$

:

Theorem

3 ([9]). Let

$S$

be

a

$se7nitop\mathit{0}logical$

se

migroup. and

$X$

be

a

non-empty weakly

compact

convex

subset

_of

a

Banach space with noraml

stmcture.

_If

$CB_{r}(S)$

has

a

left

invariant submean

(

$e.g$

. when

$S$

is

left

$7^{\cdot}eve\gamma sible$

).

then

whenever

_{$S=\{T_{S}: s\in S\}$}

is

a

continuous representation

of

$S$

as

non-expansive

self

maps

on

X. $X$

contains

a

common

fixed

point

in

$X$

.

Problem: Does

Theorenl

3 remain valid when

$X$

is

a

weak*-compact

convex

subset of

a

dual Banach

space

and

$S=\{T_{S} : s\in S\}$

is

a

weak*-continuous

representation of

$S$

?

(4)

The

following

is

a

partial solution

to

this problenl:

Theorem

4 ([8]).

Let

$S$

be

a

semitopological

semigroup.

If

_{$CB_{r}(S)$}

has

a

non-zero

left

invariant continuous

linear

functional, then

whenever

$S=\{T_{6} ; s\in S\}$

is

a

representation

of

$S$

as norm

non-expansive

mappings

on a

_$nor7n$

-separable

weak*-compact

convex

subset

$X$

of

a

dual

Banach space such that the mapping

$S\cross X-,$

$X$

,

$(s, x)\mapsto T_{s}x$

.

is jointly

continuous ulhen

$X$

has the

weak*-topology,

then

$X$

has

a common

fixed

point

$fo7^{\cdot}S$

.

3. Fixed

Point Property

and the

Fourier-Stieltjes Algebra

Let

$G$

be

a

locally

conlpact

group

with

a

fixed left Haar

$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{l}\cdot \mathrm{e}\lambda$

.

The standard

Lebesgue

space

of integrable functions

with respect to

$\lambda$

will be

denoted

by

_$L^{1}(G)$

:

$CB(G)$

will denote

the

space

of

all

bounded

continuous

conlplex-valued

functions

on

$G$

and

$c_{00()}c$

will denote the space

of functiolls in

$CB(G)$

with

conlpact

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{l}\cdot \mathrm{t}$

.

Let

$P(G)\subseteq$

$CB(G)$

be the

set

of continuous

positive

definite functions

on

$G$

,

_$B(G)$

its linear span.

The space

$B(G)$

can

be

identified

with

the

dual of

the

group

$C^{*}- \mathrm{a}\mathrm{I}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}C^{*}(G)$

,

this

lattel being

the colllpletion

of

$L^{1}(G)$

under

its largest

$C^{*}$

-norm.

Indeed.

we

have the

duality

$\langle\phi. f.\rangle=\int_{G}\emptyset f.d\lambda$

,

$(\phi\in B(G), f$

.

$\in L^{1}(G))$

.

With pointwise

nlultiplication

and the

dual

nornl.

$B(G)$

is

a

comnlutative

Banach

algebla

called the Fourier-Stieltjes

$algeb\gamma\cdot a$

of

$G$

.

The

$F_{\mathit{0}urie}7^{\cdot}$

algebra

$A(G)$

of

$G$

is the closed

lineal

$\cdot$

span

of

$P(G)\cap c_{00}(c)$

in

$B(G)$

. When

$G$

is

abelian,

then

$A(G)\cong L^{1}(\hat{C_{\tau}})$

and

$B(G)\cong M(\hat{G})$

whele

$\hat{G}$

is the dual

groul)

of

$G$

(see

[3]).

Let

$E$

be

a

Banach

space aud

$X$

be

a

weakly

colnpact

convex

subset

of

$E$

.

We

say

that

$X$

has’ the fpp

(

$=\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}$

point

$\mathrm{P}^{\mathrm{l}\mathrm{O}}\mathrm{P}\mathrm{C}1^{\cdot}\mathrm{t}\mathrm{y}$

)

if

$\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}\cdot \mathrm{y}$

nonexpansive nlappillg

$T:Xarrow X$

(

$\mathrm{i}.\mathrm{e}$

.

$||Tx-Ty||\leq||x-y||$

for

evely

$x,$

$y\in X$

)

has

a

fixed

point.

The space

$X$

has

the fpp if

every

weakly

compact

convex

subset

$X\subseteq E$

has

$\mathrm{f}\mathrm{p}\mathrm{p}$

.

It

is well known that

(5)

$\mathrm{B}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{l}.’\mathrm{S}$

Theorem,

showed that

a

weakly

compact

convex

subset of

a

Banach

space

with

normal structure

has

.

It

follows from

Alspach’s example

[1]

that

if

$G=(\mathbb{Z}, +)$

,

then

$A(\mathbb{Z})\cong L^{1}(\mathrm{T})$

(hence

$B(\mathbb{Z})$

)

does not

have

the

,

where

$\mathbb{T}=\{\lambda\in \mathbb{C}, |\lambda|=1\}$

.

If

$E$

is

a

dual Banach

space,

$E$

is said to have

weak*

fpp

(

$=$

weak*

fixed

point

$pr\cdot operty)$

if for

_evely

weak*-compact

convex

subset

$X$

of

$E$

has the

.

It follows from

[11] that if

$G=\mathbb{T}$

,

then eacll

weak*-conlpact

convex

subset of

$B(\mathrm{T})=A(\mathrm{T})\cong l_{1}(\mathbb{Z})$

llas normal

$\mathrm{s}\mathrm{t}\mathrm{l}\cdot \mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$

.

In particular.

$B(\mathbb{T})$

has

weak*

.

More

generally:

Theorem

5 ([6]).

Let

$G$

be

a

locally

compact abelian

group.

The following

are

equivalent:

(a)

$G$

is compact

(b)

Each

weak*-compact

convex

subset

_of

$B(G)$

has

$nor^{\sim}mal$

structure.

(c)

$B(G)$

has

$v$

)

$eak^{*}\mathrm{f}\mathrm{p}\mathrm{p}$

.

Problem: Does

$B(G)$

has

_weak*

”

imply

“

_$G$

is

$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{l}$

)

$\mathrm{a}\mathrm{c}\mathrm{t}’$

?

In

general.

“

_$B(G)$

has

’

does not

imply

$‘(G$

is

$\mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{a}}}\mathrm{C}\mathrm{t}’$

’

_Indeed,

_if

_$G$

is

the

Fell

group

(which

is the

natural semi-direct

product

of the

$p$

-adic nulnbers with the

compact

group

of

p–adic

units

for

a

fixed

$\mathrm{p}_{1}\cdot \mathrm{i}\mathrm{m}\mathrm{e}p$

),

then

$G$

is

non-compact

but

$B(G)$

has fpp

as

shown

in

[7].

A locally

conlpact

group

$G$

is called

an

$[\mathrm{I}\mathrm{N}]- \mathrm{g}\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{P}$

if thele is

a

compact

neighborhood

$U$

of the

identity

$e$

such

that

$:r^{-1}\cup x=U$

for all

$x\in G$

.

This

includes all

groups

$G$

such that the left and right uniformities coincide.

Examples

of

$[\mathrm{I}\mathrm{N}]- \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{P}^{\mathrm{S}}$

include abelian

groups.

compact

groups

and discrete

groups.

Theorem

7 ([7]).

_If

$\cdot$

$G$

is

a

connected

$[\mathrm{I}\mathrm{N}]- group$

,

then

$G$

is compact

if

and only

if

(6)

References

[1]

D. Alspach, A

_fixed

point

_free

non expansive map, Proc. Amer. Math.

82 (1981),

423-424.

[2]

F.E.

$\mathrm{B}_{\mathrm{l}\mathrm{O}\mathrm{W}}\mathrm{d}\mathrm{e}1^{\cdot}$

.

Non expansive

$??O?1linear$ ope7

ators

in

$BanaCl\iota$

spaces, Proc. Nat. Acad. Sci. U.S.A. 54

(1965),

1041-1044.

[3]

P.

$\mathrm{E}\mathrm{y}_{111}\mathrm{a}1^{\cdot}\mathrm{d}$

.

L

$o\tau_{g_{\grave{G}b}\prime}e$

de

$F_{oU\prime}?e?$

d un

qr

oupc

$l_{\mathit{0}\prime_{-}a}lG7’\prime c\cdot nt$

_co7)

$lpa(_{\lrcorner}t$

.

$\mathrm{D}\iota 111$

.

Soc.

$\mathrm{M}\mathrm{a}\mathrm{t}1_{1}$

.

France

92 (1964).

181-236.

[4]

K. Goebel and W.A.

Kirk,

Topics

$\dot{\iota}?$

_7}

_{$?etr?cfi\alpha\cdot e(lp_{\mathit{0}??}\iota t$}

th

$eo\tau y,$

$\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{b}_{1}\cdot \mathrm{i}\mathrm{d}\mathrm{g}\mathrm{e}$

Studies in Advanced

Mathenlatics

28 (1990).

[5]

W.A.

Kirk,

A

_fixed

$poi_{7?}t$

theorem

for

$7’ ?appirlg_{S}$

which do not

$i,?cr\cdot eaSedista??ces$

,

_{Amer. Math. Monthly}

72 (1965),

1004-1006.

[6]

A.T. Lau and P.F.

Mah,

$N_{\mathit{0}77\gamma}alst_{7u}Cturei?\mathit{1}$

dual

Banacl,

spaces associated to locally

conipact

groups,

$\mathrm{T}1^{\cdot}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}_{0}11\mathrm{s}\mathrm{A}_{11}1\mathrm{e}1^{\cdot}$

.

Math.

Soc.

310 (1988).

341-353.

[7]

A.T.

Lau.

P.F. Mali and A.

$\mathrm{U}1_{\mathrm{b}}$

,er,

Fixed

$poi??tp_{7}ope\gamma ty$

and

$?\prime O77’\prime alst\gamma uctu7e$

for

$Bo\mathit{7}?acl$

_?

spaces

associated

to locally compact groups, preprint.

[8]

A.T. Lau and W. Takahashi, Invoria

$??tmea?\mathit{1}Sa\mathit{7}?d$

fixed

$poi?\iota t$

properties

$fo7^{\cdot}$

non-expansive

represen-tations

_of

_{topological semigroups, Topological Methods in Nonlinear Analysis 5}

(1995),

39-57.

[9]

A.T. Lau and W.

Takahashi,

Invariant

$sub_{\mathit{7}l}?eansa\mathit{7}id$

seniigroups

of

_,?

onexpansive

_{$n?appi’\iota gso7?$}

Ba-$?\downarrow ach$

_6paces

with

$??or7$

}

al structure,

$\mathrm{p}_{1}\cdot \mathrm{e}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t}$

.

[10]

T.C.

Lim.

$Ch_{\mathit{0}7\mathit{0}}CtC\Gamma?Z\mathit{0}ti_{on}$

of

nomtal

structure.

$\mathrm{P}_{1}\cdot \mathrm{o}\mathrm{c}$

.

Alller. Math.

Soc.

43 (1974),

313-319.

[11] T.

C. Lim,

$Asy7??\mathrm{P}totic$

centers

$a?ld$

₇₁

on

_expa?)

sive m appin

_gs

on

$con_{J}$

ugate

Banac

$l\iota$

spa ces,

Pacific J.

Math.

90 (1980).

135-143.

[12] N. Mizoguchi and W.

Takahashi,

On

the

existence

_{of fixed}

points and ergodic

$7et\gamma$

actions

for

Lip-sch

itzian semigroups in Hilbert spaces, Nonlineal. Analysis 14

(1990),

69-80.

$\mathrm{D}\mathrm{c}\mathrm{p}\mathrm{a}\mathrm{l}\cdot \mathrm{t}\mathrm{n})\mathrm{e}\mathrm{n}\mathrm{t}$

of

$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}_{\mathrm{C}1}11\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}1$

Sciences

University of Albel.ta

$\mathrm{E}\mathrm{d}\mathrm{n}\mathrm{z}\mathrm{o}\mathrm{n}\mathrm{t}_{\mathrm{o}\mathrm{n}}$

,

Alberta