WORDS TWO

Internat. J.

VOL. 17 NO. 4 (1994) 681-686

FIXED POINT THEOREMS FOR A SUM OF TWO MAPPINGS IN LOCALLY CONVEX SPACES

RVIJAYARAJU

Department

^ofMathematics

AnnaUniversity Madras 600 025, India

(Received May

2, 1991 andinrevised form March12,

1992)

ABSTRACT. Cain and Nashed generalized to locally convex spaces a well known fixed point theorem of Krasnoselskii for a sum ofcontraction and compact mappings in Banach spaces. The class of asymptotically nonexpansive mappings includes properly the class of nonexpansive mappings aswellas theclass ofcontraction mappings.

In

this paper, weprove by using thesame method some results concerning the existence of fixed points for a sum of nonexpansive and continuous mappings and also a sumof asymptotically nonexpansive and continuous mappings in locallyconvexspaces. These results extend^aresult of Cain and Nashed.

KEY

WORDS AND PHRASES. Asymptotically nonexpansive and continuous mappings, uniformly asymptoticallyregularwithrespecttoamap.

1991

AMS SUBJECT

CLASSIFICATION

CODES.

47H10,54H25.

1.

INTRODUCTION.

Let K be a nonempty closed convex bounded subset of a Banach space

x. In

1955, Krasnoselskii

[6]

proved first that a sum T

+

^{S of two mappings T and S has a fixed point in} K, when T:K.-X is a contraction and S:K--,X is compact

(that

^{is, a continuousmapping}which maps bounded sets into relatively compact

sets)

^{and satisfies the condition that T;r.}

+

^Sy

.

^{K for all}

:,yeK. Nashed and

Wong [7]

generalized Krasnoselskii’s theorem to sum T

+

^{S of a nonlinear}

contraction mapping T of K into X

(that

is, [[Tz-Ty[[ <([[z-yl[) for all z,Ve K, where ^{is a} real valuedcontinuous functionsatisfyingcertain

condition)

^andacompact mapping SofK into X.

Subsequently, Edmunds

[4],

^Reinermann

[8]

^extended Krasnoselskii’s theorem to a sum T

+

^{S ofa} nonexpansive mappingTandastronglycontinuousmappingS

(that

is, ^acontinuousmapping from the weak topology of

x

to the strong topology ^of

x)

^when

the

underlying spaces

x

^are Hilbert spacesand uniformlyconvexBanachspacesrespectively.

Krasnoselskii’s theorem was further extended by Cain and Nashed

[2]

^{to a sum T}

+

^{S of a}

contractionmappingTofanonempty completeconvexsubset Kofalocallyconvex space X into X and a continuous mappings ^S of K into X. Sehgal and Singh

[9]

generalized the above result of Cain and Nashed

[2]

^{to a sum T}

+

^{S of a} nonlinear contraction mapping T of K into X and a continuousmappingSofK intoX. Thisresultgeneralizes ^theresult of Nashedand

Wong [7].

The study of asymptotically nonexpansive mappings concerning the existence of fixed points have become attractive to the authors working in nonlinear analysis, since the asymptotically

682 P. VIJAYARAJU

nonexpansivemappings include nonexpansiveaswellascontractionmappings. Goebel and Kirk

[5]

introduced the concept of asymptotically nonexpansive mappings in Banach spaces and proved a

theoremonthe existenceoffixedpoints for suchmappingsin uniformlyconvexBanach spaces.

The aim of this paper is to prove fixed point theorems for a sum of nonexpansive and continuous mappings in locally convex spaces. Throughout this paper, let X denote a Hausdorff locally convex linear topological space with a family

(Pa)a

_eJ of seminorms which defines the topologyon

x,

whereJ is any indexset.

Werecall thefollowingdefinition.

DEFINITION 1.1. Let Kbeanonemptysubset of

x.

IfTmaps KintoX, then

(a)

T is called a contraction

[2]

^iffor each ^c J, there is a real number

kc

^{with 0_<ka< such}

that

p(Tx r)_<_{k,p(x- )}

(b)

T iscalledanonexpansive ifk_a in

(a).

(c)

^{T iscalledan}asymptotically nonexpansive

[11]

^if

for all x,v^{in K.}

pa(Tnz

Tny) <knpcr(x y) for all x,Vin K,

foreacha d and forn 2, where{kn}isasequenceof real numbers such thatlira k_n 1.

Itis assumed that

kn

^{> and}

kn

^>

kn ₊

^{forn 1,2}

IVeintroducethefollowingdefinition.

DEFINITION 1.2. If T and ^$ map K into X, then T is called a uniformly aymptotically regularwithrespect to^$if,for eachaindandr/>0, thereexists N N(a,,/) such that

pa(Tnx-

Tⁿ

Ix +

Sx) <r/ foralln>Nand for allxin K.

EXAMPLE

1.3. Let X RandK [0,1].

WedefineamapT:K--,XbyTx

+

^{xfor allx}inK.

Then

T2x

T(1

+

^{x) 2}

+

^z.

^By

^{induction,weprovethat}

Tnx

n

+

^z.

Wedefineamap S: K-XbySz forallxin K.

Therefore

Tnx Tn- lx +

^{Sx 0.}

Hence

T isuniformlyasymptoticallyregularwithrespecttoS.

REMARK

1.4. T isuniformlyasymptoticallyregularwithrespect tothezerooperatormeans that T is uniformly asymptoticallyregular

[11].

^Thefollowingexampleshownin

[11]

that uniform asymptoticregularityisstrongerthan asymptoticregularity.

Let

X

eP,

<^p<^ooandKdenote the closedunitballin

x.

DefineamapT:K-by

Tx

(2,3

^{for allz}

(t[i,2,3

^K.

2.

MAIN RESULTS.

We

state the following TychonoWs theorem and Banach’s contraction principle which will be used toproveourtheorems 2.1 and2.2.

THEOREM A [10].

Let K be a nonempty compact convex subset ofX. If T is continuous mapping ofK intoitself,^thenThasafixed pointin K.

FIXED POINT THEOREMS IN LOCALLY CONVEX

THEOREM B [2].

Let K be a nonempty sequentially complete subset of ^X. If T is a contractionmapping of K intoitself, then Thasauniquefixed point u in K and Tnx--.,u for allrin K.

The following theorem ^{is an} extension of Theorem 3.1 of Cain and Nashed

[2]

^{for a sum of}

contraction and continuous mappings to a sum of certain type of asymptotically nonexpansive mapping Tandcontinuous mappingSinlocallyconvexspacesXbyassuming theconditionsthatT is uniformly asymptoticallyregularwithrespectto Sand

Tnx +

SyEKfor allr,yinKandn 1,2 Thisresult isnewevenin thecaseof normedlinear spaces.

THEOREM

2.1 Let K be a nonempty compact convex subset of X. Let T be an

asymptotically nonexpansive self-mapping of ^K. Let S be a continuous mapping of K into X.

Suppose

^that T isuniformly asymptotically regular self-mappingof K withrespect ^{to the}mapping Sand that

Tnz +

^Sy

_

^{Kfor all z,yeKandn 1,2 ThenT}

+

^{Shas afixedpoint in K.}

PROOF. For^eachfixedyin K,wedefineamap HnfromK to Kby Hn(z

an(Tnz +

SV) for allzeK.

where an (1- 1/n)/kn and {kn} ^{is an} in Definition

1.1(c).

^{SinceT is}asymptoticallynonexpansive, itfollows that

pa(Hn(a)- Hn(b))

anPa(Tna- Tnb)

<(1-1/n)t,a(a-b) ^{for alla,bin K.}

HenceH_nis acontractionon K.

By

Theorem

B,

H_nhasauniquefixedpoint,say,Lnyin K.

Therefore

Lny Hn(LnY

an(Tn(Lny) +

Sy).

Let

u,v

.

^Kbe arbitrary. Thenwehave Therefore

pct(Ln^u

Lnv

^<

anpa(Tn(Lnu) Tn(Lnv)) +

anpa(Su Sv) _<(1 1/n)pa(

Lnu Lnv

^{4-anpa(Su Sv)}

(2.2)

Tnzn

^Tn

^lz

ⁿ

^{+ Szn---,O}

^asn--.oo.

From (2.4)

^and

(2.5)we

^obtain

zn-Tⁿ

-lznO

^as

(2.5)

pct(Lnu-

Lnv

^{<nanpa(Su-Sv).}

SinceS iscontinuous,L_nis continuous.

Using TychonoffsTheorem

A,

we seethatL_nhasafixedpoint, say,

rn

^{in K. Therefore}

xn

Lnx

n

an(Tnr,

n

+

S:n).

(2.3)

Hence zn-Tnr,n-SXn

(an

1)(Tnn +

SZn)--O^{as n-oo,} since an--,l ^{as n--cx} and Kis bounded

and

Tn +

^{Sy Kfor all,yE K.}

(2.4)

Since T is uniform!y asymptoticallyregularwithrespecttoS,itfollows that

684 P. VIJAYARAJU

Now

po(Zn-(T

+

S)zn) < po(Zn-(Tⁿ

+

^S)n)

+

po,((Tⁿ

+

S)Zrn-(T

+

S)zn)

<po(Zn-(Tⁿ

+ S)Zn)+ klPo(Tn- lz

n

Zn).

Using

(2.4)

^and

(2.6)in (2.7)

^weget

z_n (T

+

S)zn--,Oas^n-,.

Since K is compact, thereexistsasubnet

(z/)

of the sequence{zn}such that

u forsomeu K.

SinceTand Sarecontinuous,itfollowsthat

(I-(T

+ S)(fl))

^(I-(T

⁺

^S))(u)

andby

(2.8)

^weget

(z.7)

-

^(T

^{+ S)(,)}

^0.

Since

x

is

Hausdorff,

itfollows_{that (I-(T}

+

S))(u)^=0.

Hence

T

+

^Shasafixed point

For nonexpansive mapping T, the condition that T is uniformly asymptotically regular with respect to the map S is not needed in the following theorem. This theorem is an extension of Theorem 3.1 of Cain and Nashed

[2]

^{for a sumof}contraction and continuous mappings inlocally convexspaces.

THEOREM

2.2. Let Kbeanonemptycompact convexsubset ofX.

Let

Tbeanonexpansive mapping of K into X and S be a continuous mapping of K into X such that Tz

+ St

fiK for all

t K. ThenT

+

^{Shas a}fixed pointin K.

PROOF.

Foreach fixed_tin K,wedefineamap//nfromK to Kby

Hn(.

An(T

+

^{Sy)forall K,}

where

{An}

^{is a}sequenceof real numbers with0<^An< and,nl ^asnc.

Proceeding as inthe above

theorem,

weobtain a sequence

{n}

^{in K such that} SinceK iscompact and

{Xn}

^CK, thereexistsasubset

(x#)

^ofthe sequence

{n}

^{such that}

/ forsome:inK.

Therefore

z/ ,/(Tz/+ Sz/).

Since T and S arecontinuous, it follows that _(T

+

S)z.

Hence

T

+

^Shasafixed point inK.

The following example shows that^theabove theorem cannotbe deduced from Theorem2.1.

EXAMPLE

2.3.

Let

X=space (s), the space of all sequences of complex number whose topologyisdefinedbythe family ofseminormsPn^definedby

pn(X) rna[ for

(1,2

^{Xand n 1,2,....}

LetK={,X:Ijl

<lforj=l,2 }.

ThenK iscompact

[3,

Problem 47,p.

346].

^AlsoK is convex.

Wedefineamap TfromKtoKbyTz (3/4) ^:forallz K. ThenT isnonexpansive.

WedefineamapSfromKtoKby

Su

(1/4)u^{for all K. Then}S is continuous.

Ifa,b K, then^wehave

Pn(Ta

+

Sb)< (3[4)pn(a)

+

(l[4)Pn(b).

FIXED POINT THEOREMS IN LOCALLY CONVEX SPACES 685 ThereforeTa+^Sb

_

^{Kfor alla,b EK.}

Suppose

that (1,0 )E^K. Thenwehave

T(el)

^{(3/4,0,0 ), Tm-}

l(el)

^((3/4)m-

^1,0,0

Tin(el)

((3/4)m,0,0 ^and

S(el)

^(1/4,0

Therefore

Pn(Tm(el)_T

^m-

l(el) ⁺ S(el))=

1(3/4)m_(3/4)m-

+

1/4[

1(1/4)(1-(3/4)^m-

1)1-,1/4

^{as m-oo.}

HenceT is notuniformly asymptoticallyregularwithrespect to S.

Thefollowing exampleshows that if thecondition Tx

+

^{Syin Kfor allx, y K}of Theorem 2.2is dropped,then theconclusionof theoremfails.

EXAMPLE

2.4. LetX RandK [0,1].

We define a map T from K to K by Tx=x/2 for all zeK. Then T is a contraction and hence nonexpansive.

We

defineamapSfromK toKbySy for allyEK. Then S is continuous.

Suppose

that 3/4,b K. Then Ta

+ Sbl

11/8 < ^K. ThereforeTa

+

^Sb

f

^{Kforsome a,b}

.

^K.

If^u is afixed point of^T

+

^Sin K, then ^u Tu

+

Su (u/2)

+

^and therefore ^u 2$ K.

Hence

T

+

^S hasnofixedpointin K.

Toproveof thefollowing Theorems2.5and 2.6,weneed thefollowingextensionofTychonoff’s Theorem

A.

THEOREM C

[1,

p.

169].

^{LetK bea}nonempty closedconvexsubset ofalocally^convexspace X. IfT is acontinuousmapping ofK intoitselfsuch that T(K)^is contained ina compact^subsetof K,thenThasafixedpointin K.

In

Theorems 2.5 and2.6, thecompactnessof the set KofTheorems 2.1 and 2.2is

replaced

by

aweaker condition that the set K is a completeand bounded set, but the mappings T and S are required to satisfy additional conditions that S(K) is contained insome compact subsets of Kand (I T S)(K)isclosed.

THEOREM

2.5.

Let

K be a nonempty complete bounded convex subset ofX.

Let

T bean

asymptoticallynonexpansiveself-mapping ^ofK.

Suppose

that Sis acontinuousmapping ofKinto X such that S(K) is contained in some compact subset M of K. Assume further that T is a uniformly asymptotically regular with respect to S and that

TnX +

^{Sy in K for all x,yeK and}

n 1,2,.... If(I T S)(K)^is

closed,

thenT

+

^{Shasafixedpointin K.}

PROOF. Definea map Unasin theproofof Theorem 2.1. Proceeding^as inTheorem2.1, K and L_n satisfy all hypotheses of Theorem

C,

^where L_n is as in the proof of Theorem 2.1.

By

Theorem

C,

L_n has a fixed point, say, ^x_nin K. Since (I-T-S)(K) is

closed,

^it follows from

(2.8)

that0 (I-T-S)(K).

Hence

theproofiscomplete.

THEOREM 2.6. Let K be a nonempty complete bounded convex subset ofX. Let T be a nonexpansive mapping ofK into X.

Suppose

thatS isacontinuousmapping ofK into Xsuch that S(K) is contained in a compact subset M ofK and Tz+Sy.Kfor all z,V K. If(I-T-S)(K)is closed,thenT

+

^Shasafixed pointin K.

PROOF.

Define a map H_{n as} in the proofof Theorem 2.2. Proceeding as in the proofof Theorem2.2 and using TheoremCinsteadofTychonoff’sTheorem

A,

weobtainasequence

{zn}

ⁱⁿ K such that_zn

An(Tz

n

+

Szn).

Since,nl^{asn--*oandK is}

bounded,

itfollows that

(I-T-S)zn-O

^asn-o.

Since(I-T-S)(K)^isclosed,itfollows that0 (I-T-$)(K).

686 P. VIJAYARAJU

Hencetheproofis complete.

Thefollowing exampleshowsthatif theconditionTr.

+

SVin Kfor allz,vEKof Theorem 2.6is dropped, then theconclusionoftheorem fails.

EXAMPLE

2.7.

Let

X

2

^{and K {xeX:[Ix < 1}. Define a map T from K to K by}

Tz

(0,1,2

^{for all eK. ThenT is an}isometry andhencenonexpansive.

Defineamap Sfrom KtoKbySy=(1-

[1!112,

0

)forallveK.

Then S iscompact and henceS is continuous $(K) iscontainedin ^acompactsubset ofK.

Suppose

thata (1/2,0 ),b (0 eK. Thenwehave Ta

+

^{S’b 2}

+

^(I/4) 5/4.

ThereforeTa

+

^Sb

.

^{It"for somea, 6K.}

^Suppose

^thatzisafixed point ofT

+

^{Sin K. Then}

x (T

+

^S)z (1

2,{1,{

₂ ^).

Therefore {n ²forn 1,2 But

oo{n

^{0. Thus}

moo{n

^1-

=

^{2and hence z 1,}

Therefore (1 ² (0 and so

=

⁰ which contradicts x

z-

^Thus T

+

^$ has no fixedpointinK.

ACKNOWLEDGEMENT.

would liketo thank Professor

T.R.

Dhanapalan forhis guidance and encouragementin the preparation of this paper.

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