Approximating fixed points of nonexpansive and generalized nonexpansive mappings

VOL. 16 NO. (1993) 81-86

APPROXIMATING

FIXED

POINTS

OF NONEXPANSIVE

AND

GENERALIZED

NONEXPANSIVE

MAPPINGS

M.MAITI and B. SAHA

Department

ofMathematics

Indian InstituteofTechnology,

Kharagpur,

India

(Received May

20,

1991and in revised formNovember

15,

1991)

In

this

paper

weconsider amapping

S

of theform

$

-aol

+aT

+a2

T

+

+aT

,

where

as

0. al>0with ai 1,andshowthat in auniformly convex Banach

space

the Picard iterates

i-0

of

S converge

toa fixedpointof

T

when

T

isnonexpansiveorgeneralized nonexpansiveoreven quasi-nonexpansive.

KEY

WORDS

AND PHRASES:

Fixedpoints, nonexpansive

mappings,

uniformlyconvexBanach

spaces.

1991

AMS

SUBJECr

CLASSIFICATION CODES.

Primary 47H10;

Secondary

54H25.

1.

INTRODUCTION.

Let

B

be aBanach

space

and

C

aconvex subset

orB.

A

mapping

T: C

C

is saidtobenonexpansive

if

r

Tyl[

II

yll

forall

x, y

C.

A

mapping

T: C

C

is said tobequasi-nonexpansiveifThas

afixedpointpsuch that

Tx

-p[[

[Ix -P]I

foralma:

C.

Theconcept of quasi-nonexpansivenessismore

generalthan thatofnonexpansiveness. Indeed,anonexpansive mappingwithat least one fixedpointis

quasi-nonexpansive,but there existsquasi-nonexpansive mappingswhicharenotnonexpansive.

See,

for

example, Petryshyn

andWilliamson

[7].

If

T

isnonexpansive,then the Picard iterates of

T

may

not

converge

and,even if

they

do

converge,

they

may

not

converge

to a fixedpointof

T.

However,

to circumvent thedifficulty,onemayconsider the

mapping

Tx-

(1- .)//

kT,

(1.1)

where

I

isthe

identity

mapping and 0<

.

<

1,

and show that thePicard iteratesof

T

converge

toa fixed pointofTundercertainrestrictions,see

[3,5,10].

Generalizingthe ideaKirk

[6]

has introduced amapping

S

given by

S

-ctol

+alT

+0.2

T2

+...

+tt

(1.2)

where

a

z

0,

_al>0with _ai 1,and has shown that the Picard iteratesof3

converge

toafixedpoint

i-0

of

T

under conditions similartothoseimposedin connection withthe

convergence

of Picard iteratesof

T,.

Our purpose

here istwo-fold. Firstweshow that the Picard iteratesof

S converge

toafixedpoint of

T

underconditionsweaker than those imposed

by

Kirk

[6].

Secondly,

we establishthat thePicard iterates

general

mapping resultingingeneralization ofsomeresults obtained

by

Ray

and Rhoades

[9].

2.

CONVERGENCE TO

FIXED

POINTS

It

hasbeen establishedbyKirk

[6]

that

S

and

T

havecommonfixedpointsif

T

isnonexpansive.

Let

the common fixedpointsetbedenoted

by F. Further,

the set

F

isclosed when

T

isnonexpansiveoreven when

T

isquasi-nonexpansive

(see

Dotson

[2]).

We

nowstatethe

following

conditions:

CONDITION-A.

A

mapping

T: C

C

witha nonempty fixed point set

F

is said to satisfy

Condition-A if there is a nondecreasingfunction

f:

[0, oo).- [0, oo)

with

]’(0)-

0and

f(r)>

0for all r 1

(0,

oo)

such that

I[x

-Sxll

..f(d(x,F))

forallx 1

C,

where

d(x,F)-

inf

I[x

-p[[.

pEF

CONDITION-B.

A

mapping

T: C

C

with a nonempty fixed point set

F

is said to satisfy

Condition-B if there exists a numberct>0such that

IIx

-Sx[I

cut(x,F)

forallx

t

C.

It may

beremarked that themappingswhichsatisfyCondition-Balsosatisfy Condition-A.

However,

Condition-Bmaybe verifiedeasily by giving

examples. It may

be further remarked that Conditions and

H

of

Senter

and

Dotson

[11]

are identical withConditions

A

and

B

when( (h ctk 0.

We

nowrecall thefollowinglemma due to

Dotson

1].

This willbe used later to establish our results.

LEMMA.

Ifthe

sequences

{s,}

and

{t,}

are in theclosedunitballofauniformlyconvexBanach

space

and

{z,, }

{(1

ct,,)s,

+ct,t,

}

satisfieslim

z,

"1, where 0<affi b<1,then lira

s,

t.II

0.

THEOREM

1.

Let C

be anonempty, closed,convex andbounded subset of auniformlyconvex Banach

space

B

and

T: C

C

be anonexpansive mapping. If

T

satisfiesCondition-A, where

F

isthe fixedpointsetof

T

in

C,

thenfor anarbitrary

x01

C,

the Picard iterates

($"xo) converge

toamemberof

F.

PROOF.

Ifxo

1

F,

thenthe result is trivial.

We

assumethat x0(

C

F.

Then, setting

x,

S’xo,

we havefor an arbitrary

p

1

F

IIx.

/,-pll

-IIs"

/’xo-pll

-ilSx.

-pll

-II

,x.

+

,Tx.

/

thT2x,,

+ +ch,

Tx,,

-pll

-II

o<X.

-P

+

Ctl(Tx,

-e

+

ct.z(

r-x.

-p

+ +

ct,(

Tx.

-vii

oll

1.

-ell

/,ll

Tx.

-ell

/,ll

Tx.

-ell

/

/,ll

(r’x.

-p)ll

IIx.

-ell.

Thisimpliesthat

d(x.

I,F).: d(x. ,F)

andhence that the

sequence

{d(x.,F)}

isnonincreasing. Then lira

d(x.,F)

exists.

In

the

sequel

we shall showthat this limit is zero.

Suppose

thatlira

d(x.,F)

b>0.

Then,

fora

p

EF,

lira

II/.

-ell

b’. b>

o.

Choo a

positive

integer

N

such that

IIx.

-pll

2b’for n

N. Set

yi.,

.(Tix._p)l

x.

-Pl]

for all n and

all/-0,1,2

,k

withTx,,-x,,

rhnlly,’ll

1.

Frth,tz.-

.+(-o)t,,wht.- Y.(cy,)/(1-ao)withllt,

1.

i-1

(2.1)

implying

z,,ll

1asn oo.

But,

forn

N,

wehave x.-p

y.0_

t.II

x.

p

(I

o)

x.

-vii

.f(d(x,,,F))

f(b

O,

(2.2)

Cl-,,.o)llx.-ll

2b’(1- %)

implyinglim

y.0_

t.II

-

0,whichcontradicts thelemma.

Hence

lim

d(x,,V)

O.

Thisimpliesthat

{x,}

converges

to amemberof

F,

since

F

isclosed.

1.

It

is obviousthat the above theorem holds ifCondition-Bis satisfied insteadof

Condition-A.

REMARK

2. Condition-A is moregeneralthan the conditionimposed

by

Kirk

[6]

inestablishing

the

convergence

of Picard iterates

{S’x0},

see

Senter

and

Dotson

[11].

REMARK

3. Inthe above theorem the existenceof anonemptyfixedpointset

F

is not assumed and isensuredbythe conditions assumed therein.

However,

if we assumethat

T

has anonemptyfixedpoint

set

F,

then

T

need not be assumed to benonexpansiveand it is

enough

for

T

tobe

quasi-nonexpansive.

Further,

C

neednot be bounded. Indeed, the following resultholds.

THEOREM

2.

Let C

be anonempty,closed and convex subset of auniformlyconvexBanach

space

B

and

T: C

C

be aquasi-nonexpansive mapping. If

T

satisfiesCondition-A,where

F

isthe fixedpoint setof

T

in

C,

then,for anarbitraryx0(E

C,

the Picard iterates

{S’x0}

converge

to amember of

F.

The

proof

maybe establishedexactlyin thesame

way

as in Theorem1.

It only

remains to be shown here that

S

and

T

have common fixedpoints.

A

fixedpoint of

T

is

obviouslly

a fixed

point

of

S. We

now show that theconverseisalso so.

Let p

be a fixedpointof

S.

Then fromCondition-A it is obvious that

d(p,F)

0, implying

p E F.

Since

T

isquasi-nonexpansive,

F

isclosed andhence

p E

F,

i.e.,

p

is a fixed

pointof

T.

Hence

the result.

Next,

weshow that the Picard iteratesof

S converge

toa fixedpoint of

T

evenwhen

T

is

generalized

nonexpansive.

However,

oneneed not assumeCondition-AorCondition-Bin this case. Theseconditions areautomaticallysatisfied.

THEOREM

3.

Let C

be anonempty, bounded, closedand convex subset of auniformlyconvex Banach

space

B

and

T: C

C

be a continuousmapping such that

IITx-Tyll

allx-Yll

+bTIIx-Txll

+IIy-TylI}+cTIII-TYll +lly-Txll}

(2.3)

for all

x, y

(E

C,

wherea, c 0and b>0with a+ 2b + 2ca1. Thenforanarbitrary

Xo

(E

C,

the Picard iterates

{S’xo}

converges

totheuniquefixedpointof

T.

implying

a +b+ c

IIx

-pll

IIx

-ell

(2.4)

Tx

-ell

-b -c

sincea+ 2b +2c 1. Thus

T

isquasi-nonexpansive. Further,it is

easy

toverifythat

IlSx

-p[[

Tx

-eli

II1

-ell,

<2.5)

implying

S

isalsoquasi-nonexpansive.

It

isobviousthatpisalsoafixedpointofS.

We

now show that

S

cannothave a fixedpointother than

p.

If possible,let

q(, p)

be a fixedpoint of

S.

Then

q

Tq

-II

Sq

Tq

-II

aoq

+

aTq

+

oY’q

+ +

aq

Tell

.o(q

Tq)

+

(T’q

Tq)

+...+

a,,(Tq

Tq

a011

q

Tall

+

11T-q Tall

+...+

tll

rq

Tall

a0{llq

-Pll

+IITq-Pll

}

+’{Tq

-t’ll

+IITq-Pll

+--.+

t,{ll

r’,

-Pll

+

Tq

-Pll

}

Z

+,

+

+,x,

q

-Pll

ZX

t)

q

-PlI.

(2.6)

Since

T

isgeneralized nonexpansive,wehave

Tq

-p

-II

Tq

Tpll

"

a

q

P

+b

Tq

q +

{

Tq P

,(a

+

Zc)ll

q

-Pll

+blTq-qll"

(2.7)

Substituting from

(2.6)

into

(2.7)

andnotingthata+2c-:1 2b we obtain

Tq

-Pll

"(

Zb)ll

q

-pll

+

Zb(1

a)ll

q

-pll

q

-pll

Zb,dl

q

-pll

-

q

P

Zb,dl

T

p

II,

implying

Now

from

(2.8)

wehave

1

Tq

-Pll

+

2bt

q

-Pll

(2.8)

1

IIq

-pll -IlSq -pll

"=

Tq

-Pll

+

2bt

q

-PlI,

whichimplies

q

p,

since

b,

ctl>0. Thus

S

and

T

have auniquefixedpoint

p.

Next,

weshow that

T

satisfiesCondition-B.

For

x

E C

wehave

Tx

-Pll

Tx

Tpll

allx

-pll

+

bllx -Txll

+

{llx

-Pll

+

Tx

-Pll

},

(2.9)

implying

a+c.. b

Tx

-Pll

1

Now,

Alsoweobserve that and that

IISx-pll

ollx-pll

/(

//...

/)llTx-pll

Sx

-xll

IIx

-ell

Sx

-ell.

Now,

substitutingfrom

(2.12)

into

(2.13)

wederive

IlSx-xll

IIx-pll

ollx-pll

-(,

//...

+,)llTx-pll

(1

ct0){llx

-pll

-II

Tx

-ell

},

whence, using

(2.10),

weobtain

a+c.. b

Ilax

-xll

(1

-cq)

IIx

-pll

l_--llx

-pll

l_--llx

Txll

-(1-,){

1-a-2Cllx-plll-c

1--c

}-

x

rxll

(1

-cto)

_--llx

-pll

IIx

rxll

b(1

-%)

l:---c

{211x-Pll-IIx-Txll}.

Now,

substitutingfrom

(2.11)

into

(2.14)we

get

b(1

-%)

IlSx-xll

_1-c

{211x-pll-2(1-q)llx-pll-IlSx-xll

(2.11)

(2.12)

(2.13)

(2.14)

implying

where

b(1

1--C

{2tllx

-Pll

-IlSx

-xll

},

IlSx-xll

llx

-pll,

(2.15)

2b(x(1

-)

>0,

1-c

+b(1-%)

since_{b, 1}>0. Thus

T

satisfiesCondition-B.

Hence,

by

Theorem2,the resultfollows.

REMARK

4.

It

maybe noted that thestipulation

q

>0in

S

is

necessary

torule out thepossibility

that fixedpointof

S

is apointatwhich

T

maybeperiodic.

REMARK

5. Ifwe do not restrictb>0in Theorem3,then the fixedpintsetof

T

is not asingleton,

andCondition-A is to beimposedtoensure the

convergence

of

{S’x0}.

Thepresent analysiscanbe extendedtoamore

general

mapping

T

which satisfies

[[Tx-TYl[

max

{[Ix-yll,[llx-Txll

+lly-Tyll]/2,[l]y-Txll +[[x-Tyl[]/2}

(2.16)

forallx, tE

C.

Thismappingincludesnonexpansiveandgeneralized nonexpansive mappings

(see

Rhoades

[8]).

Itiseasytoverifythat

T

isquasi-nonexpansive. Ithas beenproved by

Ray

andRhoades

[9]

that

S

subsets of

C

setsof

B,

then,for each x0

E C,

thesequence

{S’xo

converges fixed

pointof

T

in

C.

However,

thefact that

I-S maps

boundedclosed subsets of

C

intoclosed setsimpliesCondition-A

(see

Senter

and

Dotson

11]).

Thus Condition-A is moregeneraland

incorporating

thiscondition we

may

obtainthefollowingasgeneralizationsofTheorems 2and3of

Ray

andRhoades

[9].

THEOREM

5.

Let C

be anonempty,closed andconvex subsetofauniformlyconvexBanachspace

B

and

T

aself-mappingof

C

which satisfies

(2.16).

If

T

satisfiesCondition-A,where

F

isthenonempty

fixedpointsetof

T

in

C,

then,for anarbitraryx0

C,

the Picard iterates

{S"x0}

converge

toamember of

F.

We maynotethat

C

need not be bounded in Theorem5.

Because

wehave assumed the existence of

nonemptyfixedpointsetof

T

andCondition-A

(see [11]).

But

theboundedness of

C

cannot be omitted from the statementof Theorem 4.

ACKNOWLEDGEMENT.

The authors are indebted to Professor

B.

E.

Rhoadesfor his valuable

sug-gestionsfor theimprovementof the

paper.

[1]

[Z]

[3]

[4]

[5]

[61

[7]

[8]

[9]

[lO]

[11]

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