ASYMPTOTIC BEHAVIOR

Internat. J. Math. & Math. Sci.

VOL. 20 NO. (1997) 51-60

51

ASYMPTOTIC BEHAVIOR

OF

ALMOST-ORBITS

^OF

REVERSIBLE SEMIGROUPS

^OF

NON-LIPSCHITZIAN ^MAPPINGS

^IN

BANACH SPACES

JONG SOO JUNG

DepartmentofMathematics

Dong-AUniversity Pusan 607-714,KOREA

E-mailaddress jungjs@seunghak dongaackr JONG YEOULPARK

DepartmentofMathematics PusanNational University

Pusan 609-735, KOREA JONGSEOPARK DepartmentofMathematics GraduateSchool,Dong-AUniversity

Pusan607-714,KOREA

(Received February 15, 1994andinrevisedformOctober25, 1995)

ABSTRACT. LetCbea nonemptyclosedconvex subset of auniformly convex BanachspaceEwitha Fr6chet differentiable norm, G a rightreversible semitopological semigroup, andS

{S(t)

:t E

G}

a continuousrepresentation ofG asmappings of asymptotically nonexpansivetypeof C into itself The weak convergence of an almost-orbit

{u(t)

:t E

G}

^{of S}

{S(t)

^:t6

G}

on C is established.

Furthermore,it isshownthat ifPisthe metricprojection ofEontoset

F(S)

^ofall common fixed points ofS

{S(t)

^t6

G},

then the strong limit ofthenet

{Pu(t)

^t6

G}

^exists.

KEY WORDS AND PItRASES: Almost-orbit, fixed point, reversible semitopological semigroup, semigroup of asymptotically nonexpansivetype,uniformlyconvex Banachspace

1991AMSSUBJECTCLASSIFICATION CODES: 47H20, 47H10,^47H09 1. INTRODUCTION

Let Cbeanonempty closedconvex subsetofareal BanachspaceEandlet,5

{

S

(t) > 0}

bea family ofmappings fromCintoitselfsuch that

S(0)

I,

S(t + s) S(t)S(s)

^{for allt,s}

[0, c)

^and

S(t)z

is continuous in t E

[0, x)

^foreachx C. Sis saidtobe

(a) nonexpansive semigroup^onCif

IIS(t)z ’(t)ull _< IIz ull

^{for allz, y Cand}

^>

^0,

(b) asymptotically nonexpansive semigroup^onC [1]ifthereisafunction k

[0,

)

[0, )

^with limsupt-ook(t)

<

1 suchthat

IIS(t)x

S(t)yll

< (t)llx

yll forall x,y ECand t

>

O,

(c) semigroup of asymptoticallynonexpansive type onC iffor eachx C,

limsup{sup[llS(t)z-S(t)yl[-t- vec Ilx- ylI]} ^-<;

see

[2]

for mappings of asymptotically nonexpansivetype. Itis easilyseen that(a)

=

^{(b) =, (c) and}

that both the inclusions areproper(of. [1,p. 112]).

In [3],Myadera and Kobayashi^introducedthenotionofalmost-orbitsof nonexpansive semigroups on C and providedthe weak and strong almost convergences ofsuch an almost-orbit ^{in a}uniformly convexBanach space; ^{see also}[4] foralmost-orbitsof nonexpansive mappings. Recently, Tan ^andXu [5]^extendedthis notionto semigroups of asymptotic nonexpansivetype in Hilbertspaces The case of

JS JUNG,JY PARKANDJS PARK

general commutative nonexpansive semigroups in uniformly convex Banach spaces was studied by Takahashi and Park

[11].

Oka [6] gave the results for the case of commutative asymptotically nonexpansive semigroups inuniformlyconvex Banach spaces. In particular, Takahashi and Zhang [7]

established the convergences of almost-orbits of noncommutative asymptotically nonexpansive semigroupsin the sameBanachspaces,see[8]for the caseofHilbertspaces

Thepurpose ofthispaperistogeneralizetheirresultstothe case of noncommutative semigroups of asymptotically nonexpansive type Section 2 is a preliminary part In Section 3, we prove several lemmaswhich arecrucialforourdiscussion. Mainresults aregivenin Section 4 First,weestablishthe weakconvergence(Theorem 1)ofanalmost-orbit

{u()

E

(7}

ofa semigroup^,S

{S(t)

^E

G}

of asymptoticallynonexpansive type onC’in auniformlyconvexBanachspacewith aFr6chet differentiable norm, whereG isafight reversible semitopological semigroup. Next,we show that ifP isthe metric projection ofEonto set

F(,S)

of all common fixed points of ,S

{S(t) " ^(7},

then the strong limit of the net

(Pu()’ (7}

^exists (Theorem 2). Our proofs employ the methods of Hirano and Takahashi [9], ^Ishihara and Takahashi [10], Takahashi and park [11], and Takahashi and Zhang [7,8]

Theresultsaregeneralizationsof the corresponding resultsⁱⁿ

[5],

[7],[8],[11 ], [12]and[13].

2. PRELEMINARIES

LetEbe areal Banachspaceand letE* beitsdual. Thevalueof

f

^{E* atx Ewillbe denoted}

by

(, f)

^Witheach E

E,

weassociatetheset

J() {f m"- (, f)= IIII

Using the Hahn-Banach theorem, it is readily verified that

J(z)

The multivalued mapping J"

E

E* is calledthe duality mapping ofE. Let

U {z ^e

^E

[[zl[ 1}

bethe unitsphere of

E

Then a BanachspaceEis saidtobe smooth providedthe limit lira

]]:r + t/[]- ]]z][

(2 1)

t0

existsforeachx, Y

U.

Inthiscase, thenormofEissaidtobeG.teauxdifferentiable. Itis saidtobe Frchet differentiable iffor each x

U,

the limit (2.1) is attained uniformly for U It is also knownthatifEissmooth,then theduality mapping Jissinglevalued. Itiseasyto see that the norm of

E

is Frchet differentiable if and only if for any bounded set BC E and any z E

E,

limt_,0(2t)^-1

([[:r + tF[[

²

[[z[[ 2) (/, J(z))

uniformlyinY^C

E;

^see[14].

ABanachspaceEiscalled uniformlyconvex if themodulusof convexity

ispositivein itsdomainofdefinition

{e

⁰

<

^e

_< 2}.

^{Fortheproperties}

off(e),

^see

[15].

Forasubset

D

of

E, D

denotestheclosure of

E,

codtheconvexhull of

D,

and -d-dD theclosed convex hull of

E,

respectively.

Let Gbeasemitopological semigroup, i.e., ⁽⁷isasemigroup with aHausdorff topologysuchthat foreacha (7themappings₉

.

^9and9

--

^{9-afrom to(7}are continuous. ⁽⁷issaidtoberight reversibleifanytwoclosed lett idealsofGhavenonemptyintersection. If⁽⁷isright reversible,

((7,

is a directed system when the binary relation

"

^{on (7 is defined by b if and only if}

Let 6’be anonempty closed convex subset ofaBanach space

E

andlet(7 be asemitopological semigroup. Afamily

, _{S(t)

t

(7}

of mappings from 6’ intoitself issaidtobe a(continuous) representation ofGonC’^if,satisfiesthefollowing:

(i)

s(ts), S(t)S()z

forallt,^s

G

andz 6’

(ii) forevery:c 6’,themappings^s

S(s):c

^{from into6’}is continuous.

ASYMPTOTIC BEHAVIOR OF ALMOST-ORBITS 5 3

DEFINITION 1. Arepresentation^,S

{S(t)

:t E

G}

ofG on C is said to be asemigroup of asymptotically nonexpansivetypeon (7ifforeachz (7,

inf sup sup

(llS(t)z-

s(t)yll-

Ilz-

^yll)

^<

^{0. (2 2)}

sG _{s_t} yC

Let G be right reversibleand let S

{5’(t)

"t E

G}

^bearepresentation ofG onC G Ciscalledan almost-orbitof,5

S(t)

^{"t E}

G}

^if

lim

(sup[lu(ts)- S(t)u(s)ll)

^0.

seG _\teG

A function

(2 3)

w(u)

^{denotes the set} of all weak limit points of subnets of the net

{u(t):t

e

G},

^and

F(S) I’]tecF(S(t))

^thesetofall common fixedpoints of mappings

S(t),

GinC

3. LEMMAS

Inthis section,weproveseveral lemmas whicharecrucialinconvergence ofalmost-orbits.

LEMMA1. Let Cbeanonempty closedconvexsubset ofauniformlyconvexBanachspaceEand let S

{S(t):t G}

^{be a} semigroup of asymptotically nonexpansive type of a right ^reversible semitopological semigroupGonC. Then

F(S)

^{is a}closed andconvexsubset ofC.

PROOF. The closedness of

F(S)

is obvious. To show convexity, ^{it is} sufficient to show that z 6

F(S)

forallx y6

F(S)

Let x,y6

F(S),

^x

:fi

Y IflimecS(t)z z,thenfor anys 6G,

S(s)z

lim

S(s)S(t)z

^lim

S(st)z

^lim

S(t)z

^z,

tG teG teG

i.e.,z

e F(S).

^Henceit sufficestoprove thatlimtecS(t)z ^z. If not, there existse

>

0suchthat for any t6

G,

thereis

t’

6 Gwith

t’ h

^tand

41lS(t’)z- zll 112(S(t’)z- z)

2(y-

S(t’)z)l ^>

^e.

Choose d

>

^{0 sosmallthat}

((’))

(R/d)

^1-6

R+d

^<R,

where

R llx- vii ^>

^{0 and 6 is the} modulus of convexity of E asymptotically nonexpansivetype onC,there is

to

^EGsuchthat

Since S

{S(t)’t e G}

^is

sup sup

(llS(t)z s(t)wll- IIz ^wll) ..

d tot weC

Put u

2(S(fo)Z x),

^v 2(y-

S(fo)Z).

^Then

Ilu vii ^411S(t)z- zll >

^e. Furthermore, since

to _ ^fo,

^wehave

Ilull 2]ls(t’o)Z- zll

2(llS(t)z s(t)zll- IIz ^xll)

^/

211z xll

<_

2sup sup

(llS(t)z s(t)zoll- IIz ^{wll) +} IIx ull ^<

^R

+

^d

to-<t weC and

Ilvll 211u- ^s(t)zll

2(llS(t)z s(tS)ull- IIz

^vii)

+ 211z vii

<

2 sup sup

(llS(t)z s(t)ll I1= ¹¹⁾ + I1= ull ^< + a.

to_-<tweC

Sowehave

((’))

< (R+d)

^1-6 R+d andhence

5/ J.S JUNG, YPARKAND J.S PARK

I1 11

-- ^{<_ ( + e/( (}

This isacontraction. Therefore,limevS($)z z,whichcompletestheproof

LEMMA 2. Let C bea nonempty closed convex subset of Banach space E Let G be afight reversible semitopological semigroup and let ‘9

{S(t)’t

^E

G}

^{be a} semigroup of asymptotically nonexpansivetype onC. If

{u(t)

^t

e G}

^and

{v(t)

6

G}

are almost-orbitsof‘9

{S(t) G},

thenlimteG[[u(t)

v(t)[[

^exists.Inparticular, for everyz 6

F(‘9),

limtec[[u(t)

z[[

^exists PROOF. Put

()

sup

Ilu(=)- s()u()ll, (=)

sup

IIv()- s()v(=)ll

teG

for s6G Then

limsec(s)

limsee(s) 0. Let e

>

0. Since ,9

{S(t)

^t6

G}

is of asymptotically nonexpansive typeon

C,

there exists

to

^6Gsuchthat

o

_

^weC

for alls 6G. On the other hand, since, for anys,^$6G,

Ilu(es) -(es)ll <- ^(s) + ⁽⁼⁾ + (llS()u(s) s(),(s)ll -II,(s) ,(s)ll) + ^{II(s) ,()ll}

< () + (s) +

^sup

(llS()u(s) s(e)wll-

we have

inf"sup

Ilu() v()ll _< (s) +

^g.,(s)

+

^supsup

(llS()u(s) S()wll Ilu(s) wll) + ll,(s)

teG t-< to_tweC

_< () + ^(s) + + ^[I()

and then

infev

^sup_-<

[IU(T) V(r)[[ ^<

SUpeC

inf ^{flu(s) v(s)ll}

^Thus

limtec[[u(t) v(t)[l

^exists.

Letz E

F(,9)

and put

v(t)

^{z. Then}

v(t)

^isanalmost-orbitand hencelimec[[u(t)

z][

^exists

LEMMA3. Let C bea nonempty closed convex subset of Banachspace E. Let Gbe aright reversible semitopological semigroup and let ^,9

{S(t)"

t E

G}

be a scrnigroup of asymptotically nonexpansivetype onC. Let

{u(t)

t

G}

be an almost-orbit of

thenthere exists

to

^Gsuchthat

(u(t)

-t

to}

^isbounded.

PROOF. Let z

F(‘9).

Then, since

limtecl[u(t) zll

^{existsby Lemma}2, there ist0 G such that

{[lu(t)- zl[’t - ^to}

^{isbounded. Hence}

^{u(t)" _ ^to}

^isbounded.

LEMMA 4. LetCbe anonempty closed^convexsubset ofauniformly convex Banach spaceE Let G be a fight reversible semitopological semigroup and let‘9

{S(t)

t E

G)

bca semigroup of asymptotically nonexpansivetypeonC. Let

{u(t)’t G)

beanalmost-orbit of

Supposethat

F(,9) -

^{{. Let y}

^F(,9)

^and0

^<

^c_</

^<

^{1. Thenfor any}

^>

0, there is

to

^Gsuch

that

forall t,s

_ ^to

^andA

^{[c, ].}

PROOF. ByLemma 2,

limec[[u(t) Y[I

exists. Let

>

0 and r lim

flu(t)

If r 0,since,9=

{S(t)"

^t

G}

^isofasymptotically nonexpansivetype on

C,

there exists

to

^EGsuch

that

s.p sup

(IIs()(A() + ^{( A))} s()ll- IIA() +

to^"<t

and

ASYMPTOTICBEHAVIOR OF ALMOST-ORBITS

fort_t0,0<A<landsEG. Hence fors, t_t0,0<A<landsEG,

IIS(t)(A() + (1

A)y)

(AS(t)u(s) + (1 a)u)ll

AIIS ^(t)(Au(s) + (1 A)) S(t)u(s)[[ + (1 A)l]S(t)(Au(s) + ^{(1 A))} wll

A

(sup

ktot ^sup

(llS(t)(() + (1 ))- s(t)wll-

lieu(s)

+ (1 A)-

^w

))

+ AIIA() + (1 A)- ()11 + (x A)(sup

^sup

(II S(t)(A(s)+ (1

A)W)-

ktotwC

<A ^+(1-A) +2A(1-A)l[u(s)-

55

Now,letr

>

^{O. Thenwecan}choosed

>

^0sosmallthat

(r+d>(1-c6(

^r+de

)) ^=ro

^<r,

where 6 is the modulus of convexity ofEand

c

min{2A(1- A)’a <_

^A

<_

Let^a

>

^{0 with 2a}

+ ro <

^r. Then there is

to

^{e Gsuch that}

r a

< Ilu()

yll <

r+

d ^{for s to,}

IIS()u(t)- u(st)ll <

^{a for}

>-_ to

^{and s G,}

and

sup sup

(llS(t)a s(e)ll -IIz ^{11) <}

^{d for}

to^_tweC

zC,

c.

sup sup

(llS(t)u(s) s(t)ll- Ilu(s) wll) <

^d for a

e

G.

to^_tweC

Suppose that

IIS(t)(Au(a) + (1

)y)

(AS(t)u(a) + (1

^{)y)JJ e for some s,} t0 A

[, ]

Put z

Au(s) + (1

A)y,^u

(1 A)(S(t)z-

y) d v

A(S(t)u(s) S(t)z)

wehave

I111 ^{(1 )(llS(t), s(t)ll- II; ull) + (1 )11, ull}

(1 )su su ^{(llS(); s()ll- II; 11) + (1 )11() + (1 )u-} 11

tot weC

< (1 )

^d

+ (1 )() 1

(

5 (1 ) (1 ) +

^r

+ ^{< ( )(} + ^d)

and

Wealso have that

and Then

I1, .11 IIS(t)z- (s(t)() + (

A)y)ll

_>

e

PARK and

Au

+ (1

A)v

A(X A)(S(t)z

y)

+ (1 A)A(S(t)u(s) A(1 A)(S()u(s)-

^y).

BytheLemmain 16],wehave

A(1 A)llS(t)u()

yll

IIA ^{+ ( A)vll}

((’))

<_A(1-A)(r+d) 1-2a(1-A)6

r

+

^d

^,X(1

^,)ro

andhence

[[S(t;)u(s)

y[[

_< ro

^{Thisimpliesthat}

This contradictsthe fact

Ilu()

yll

>

^{r afors}

___ ^to.

^{Theproofiscomplete}

LEMMA 5. LetCbeanonemptyclosedconvex subset ofauniformlyconvexBanach space E Let G be a right reversiblesemitopological semigroup and let S

{S(t)

^-t

e G}

^bea semigroup of asymptoticallynonexpansive typeonC. Let

{u(t)

^"t

e G}

be an almost-orbit ofS

{S(t)

^"t

G}.

Supposethat

F(S) .

^Then

limtecllAu(t) + ^{(1 A)x} ull

^{exists.forevery x,y}

^F(S)

PROOF. LetA

(0,1)

and x,y

F(S).

By (2.2), (2.3),and Lemma 4, forany^e

>

0,thereexists

to e

Gsuch that

IIS<t)<u()

^/

(1 A)x) (AS(t)u(s) + (1 )z)ll <_

for t,s h^to, sup

II(ts)- S(t)u()ll <

^{for s}

___

^to,

teG

supsup

(llS(t)(,x()/ (1 ,X)x) S(t)wll- II,Xu()

^/

(x ,)x wll) <

^{for s G.}

tohtweC Since

II,Xu(ts) + ( ,x)x ull

< ll(ts)- s(t)()ll + IIS(t)() + ( )x s(t)(a(s) + (1 )x)ll +

^sup

(llS(t)(,xu(s) + (1 A)x) s(t)wll II,u(s) + (1 ,x)x wll)

wfC

for allt,^s 6

G,

we have

foralls to,and then

infsup

II,X() + (1 ,X)x

yll

_<

sup

i ^{II,Xu() + ( ,x)x !1.}

teG _{t_z teG}

Thus

limllAu(t)

^/

( A)x Yll

^exists.

LEMMA 6. LetC beanonempty closed ^convex subset ofauniformly convex Banach spaceE with a Frfchet differentiable norm. Let G be a fight reversible.semitopological semigroup and let ,5

{S(t)

^t

G}

^beasemigroup of asymptotically nonexpansivetype onC. Let

{u(t) G}

^{be an}

almost-orbit ofS

{S(t)

^"t

G}.

^Then

F(S)

isat mostasingleton.

ASYMPTOTIC BEHAVIOR OF ALMOST-ORBITS 5 7 PROOF. Notethat

nsc--6{u(t): ___ ^{s} -6-6w(u),}

^{see[17] Letz,V 6}

^F(S)

^{Since Ehasa}

Fr6chet differentiable norm, there existsan increasingfunction _7:R^/ R such that

7(t)/t

^{0 as}

0

+,

^and

1 1

IIx

^yll2

^{+ (h, J(x}

^y))

^<_

-

¹

^Ilx

^y

^{+ hll}

forall h 6E. Takeh

A(u(t) x)

Then

1 1

IIx vii + m(u(t)

x,y(x y))

< Ilmu(t) + (1 A)z vii

1

Using Lemma5,wehave

1

IIx

^{yll /}

,

infsup

(U(r)-

x,J

(x

y))

tEC

<

1lim 2

< IIx

^yll

+ ,

supinf

(U(r)

^X,J

(x

y))

+ "/(AM),

2 _t6G

wheresupeGllu<)

exists. Of course r

(, J(z /))

^{for all 6}

w(u)

and hence for all 6 -6

w(u)

Therefore

F(S) w(u)

^isatmosasingleton.

4. SULTS

Intssection,westudytheconvergenceof most-orbit

{u(t)

^t6

^G}

^ofS

^{S(t)

^t

^{e G}}

EOM 1. LetEbeauifoyconvexBach spacewith aFrchet differemiablenod letCbea

nonemp

closed convex subset ofE LetFbeasubsetofCd letGbeaghtreversible setopoloc

segroup.

LetS

{S(t)

6

G}

bea

segroup

of asptoticly nonexpsive

te

onCd let

{u(t)

^t6

G}

be most-orbitofS

{S(t)

^t6

G}.

^Assumethat (a) F

c F(8).

Assume

so

that

) ifasubnet

{u(to)}

^ofthenet

{u(t)

t6

G}

converges

wyto

z, thenz6 F.

Then either(i) F and

Ilu(t)l[

^or(ii) F ^{d thenet}

{u(t) e G}

converges wetlyto somez6

F(S).

PROOF.

Suppose

thatsomesubnet

{u(to)}

^of

{u(t)

^t6

G}

isbounded. SinceEisrefleve,a subnet of

{u(to)}

^mustconverge

wey

to elememz6

E,

wchisinFbyif). ThusF 0implies If,on the otherhd,F

,

^{thenby Lena3,}

{u(t)

6

G}

^isbounded. So

{u(t)

⁶

G}

^must

comn

a subnet

{u(to)}

^{wch converges to some z}

w(u) ev ^{u(t):

^t6

^G},

^wehave

Therefore follows

om

Lena6hat

Asadreconsequence, wehavehe

follonE

^oroll,wch

s

^a

enerzaton

^ofa[esul

n

[5], COROLLARY

.

^{Le E beauifoy}convex Bachspace h^aPrithee deremmbleno d le C be a

nonempy

closed convex subse of E Le G be a

dEh

^{[versbJe seopoloc}

JS JUNG,JY PARKANDJS PARK

semigroup and let ,.q

{S(t)’t

E

G}

be a semigroup of asymptotically nonexpansive type on C Suppose that

F(S)0

^{and let}

{u(t)"t

E

G}

^{be an} almost-orbit of S=

{S(t)t

E

G}

^If

co(u)

C

F(.5),

then thenet

{u(t)

tE

G}

converges weaklytosomez

F(S)

PROOF. Theresult follows by puttingF

co(u)

^inTheorem

Thefollowingtheorem is also a generalizationof[7,^Theorem4]

TI:IEOIM2. LetCbeanonemptyclosedconvexsubset ofauniformlyconvexBanachspaceE Let G be aright reversible semitopological semigroup and let S

{S(t)

^t

G}

be a semigroup ^of asymptotically nonexpansive type on C Supposethat

F(S)

and let

{u(t)’t G}

be an almost- orbitofS

{S(t) G}

LetPdenotethemetricprojectionofEontoF(,.q) Then the stronglimit ofthenet

{Pu(t)

t

El}

exists andlimtcPu(t) z0,where

zo

^{is auniqueelement ofF(,.q)such that}

lim

,,u()-z.oll =min{lim,lu(t)-z[l’zc= F(S)}.

tG tG

PROOF. Since

F(,S) :/: ,

^{weknow that}

(u(t) G)

^isboundedand

limtoJlu(t) zJl ^(z)

existsforeach z

F(S).

Let R

inf((z)

^z

F(,S))

^{and M}

(

^E

F(,S) (u) R)

Then, since

(z)

^{is convex} and continuous on

F(,S)

^and

(z)

^{oo as}

I1,11

oo, M is a nonemptyclosed convexbounded subset of

F(S).

^Fixz0 Mwith

t/(z0)

^{R. SincePis}the metric projectionofE

onto F(,S),

^wehave

Ilu(t) Pu(t)ll <_ Ilu(t) vii

^{forall t G} and V

F(S),

and hence

infsupIi,()-

P()II <_ .

tG _{t_s}

Supposethat

inftec

supt

I{u(a) Pu(a){I <

R. Then wemay choose

>

^0and

to

^6Gsuchthat

supsup

(llS(t)u(a) S(t)wll- Ilu(t) wll) < ,

to_twC

and

sup

Ilu(ta) S(t)u()ll <

foralls

>- to.

^Since

-I1() P()II

^/

i1() P()II

< ₍₎ +

^sup

(llS(t)u() S(t)wll -Ilu() 11)

^/

II(s) P()II

w_C

for alls, tEGand

lim b(a)

^0,where

(s)

^suptaa

Ilu(ts) S(t)u()ll,

^wehave

for

to

^{and allt} G. Therefore,we obtain

lim

II(e) P()II

infsup

ll() Pu(a)ll <

R

<

^R.

This isacontradiction. Soweconclude that

infsup

]lu(a)- Pu(a)ll

^R.

teG

Now we claim that

limto Pu(t)= zo.

^Ifnot, then there exists

>

^{0 such that for any G,}

[[Pu(t’) zol[ >

^{for some}

t’

t. Choosea

>

^{0sosmall that}

(R+)

^1-

R+e

where isthe modulus of convexity ofthenormof E Wehave

llu(t’) P(t’)ll <_

R

+

^{a and}

II(t’) z011 ^<

^R

+

for large enough

t’.

Therefore

ASYMPTOTIC BEHAVIOR OF ALMOST-ORBITS 59

(t’)

^<_

(R+a)

1-6

Since, byLemma1, the pointwt,

Pu(t;)+zo

^belongsto

^F(S),

^{as intheabove,}

Ilu(tt’) w,ll _< (t’) +

^sup

(llS(t)u(t’) S(t)wll- Ilu(t’) wll) + Ilu(t’) w,,ll.

wC Since

limsec ()

^{0, thereist’ EGsuch that}

and

andhence

(t’) <

sup sup

(I s(t)u(t’) s(t)wll- Ilu(t’) wll) <

^R

R

t’twC 4

R-

R

^R

+ R

lim

Ilu(t)

w, sup

Ilu()

w,

< + R <

R.

tG _t-r 2 2

This contradictsthefact

R

inf{g(z) z E

F(,S)}

Thus wehavelimtecPu(t) zo Consequently,it follows that the element

zo F(S)

^with g(zo)= rriln{g(z)’z

F(,S)}

^is unique. The proof is complete

By

Corollary and Theorem2, we havethe following,which isanimprovement of[8,Theorem

3]

and[5,Theorem3.3].

COROLLARY2. Let Cbeanonempty closed convex subset ofareal HilbertspaceH LetGbe afight reversiblesemitopological semigroup andS

{S(t)’t G}

beasemigroup of asymptotically nonexpansive type on C. Suppose that

F(S):/: .

^Let

^{u(t)’t

^E

^G}

be an almost-orbit of S

{S(t)

^t

G}.

^Then

{u(t)

E

G}

convergesweakly^to somez Cifand onlyif

u(ht) u(t)

convergesweaklyto0forall h G. Inthiscase,z

F(S)

and

limtec Pu(t)

^z

PROOF. Weneedonly provethe "if’ part. ByCorollary 1, it sufficestoshow that

w(u)

C

F(S)

Let

{u(to)}

be a subnet of

{u(t)’t e G}

converging weaklyto y C Given e

>

⁰ SinceSisof asymptotically nonexpansivetype and

{u(to)}

^isbounded,there exists

to

Gsuch thatfor anya,

supsup(ilS(t)u(t,,)-

S(t)wll- Ilu(to) wll) <

e.

to^_twC

Sowehave,for

>-_ to

^{andany a,}

IIS(t)u(to)- S(t)ull -Ilu(to)- yll

(llS(t)u(to)

S(t)yll-

II(to)-

yll)(llS(t)(to)- s(t)yll

+ Ilu(to)-

yll)

_<

sup sup

(llS(t)u(to) s(t)wll- Ilu(to)- wll)(

^supsup

(llS(t)u(to)- s(t)wll

totweC to^_tvoeC

-Ilu(to) wll) + 211u(to) yll)

< (e + ^2M),

whereM sup

Ilu(to) YlI-

^Letu

^F(,S)

^ande’

^e(e + 2M).

^Thenwehave,^{for t}

___ ^to

^{and all a,}

e’ _< [[,(to) y2[[ II-

s(t)u(to)yll

II=(to) =11 + ^2(u(to)

^{u,u y)}

+ I1 Yll

IIS(t)u(to) ull = _2(s(t)u(t)

_u,u S(t)y)

Ilu

^s(t)yll

lu(to) ull

²

-IIS(t)u(t) ull

²

+ IJu Yll -llu

^S(t)yll

+ 2(u(to)

u,S(t)y-

) + 2(u(to) S(t)u(to),

^u S(t)y).

Since

{u(t)

^q

G}is

^an almost-orbit^of,9

{S(t)

^"t

G}

and

u(hs) u(s)

converges weaklyto0 forall h

G,

itfollowsthat

lim

IIS(t)u(to) ull

^lim

^Ilu(tto) ull

^lim

^[[u(to) ull

60 and

Thuswehave

JS JUNG,JY. PARKANDJS PARK

u(t,)

S

(t)u(t,) u(to) u(tt,)

^{0 weakly.}

for

__

^{to,andhencelim}suptecllS(t)t

t[[ _<

^{d. Since}

e’

isarbitrary,wehavelimtcS(t)y / Now, fors EG,

S(s)t

limS(s)S(t)j ^limS(s)y lira

S()t

t,

tG teG teG

e, y E

F(,S)

^and hence

w(u)

^C

F(,S)

By Corollary 1, the ^net

{u(t) G}

converges weaklyto somez

F(,S)

Ontheother hand,sincePisthemetricprojectionofHonto

F(,S),

^weknowthat

(u(t) Pu(t), Pu(t) !1) >

⁰

for all //e

F(S).

So, if

Pu(t)-,

^{u by Theorem} 2, we have

(z-

u,u-y)

>_

0for all _fl

e F(S)

Puttingz=//,we obtain

[[z u[[2 >

0 and hencez u.

Asadirectconsequence,wehave thefollowing

COROLLARY3. LetCbe a nonemptyclosedconvexsubsetofareal HilbertspaceH LetGbe a fight reversible semitopological semigroup and let S=

{S(t)-t

E

G}

be a semigroup of asymptotically nonexpansive typeonC. Supposethat

F(,S) =

^Let

^{{u(t) e G)}

^bean almost-orbit of 5

{S(t)

t

G}.

^If

limteallu(ht) u(t)l[

0 for all h

e

G, then the net

{u(t) e G)

converges weaklytosome z

F(S).

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KIRK,

W.A. and

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W.A.,

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R.E.,

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Math.Anal.Appl. 142(1989),^242-249.

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A.T. and T

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