Fixed point theorems for compatible multi valued and single valued mappings

(1)

VOL. 12 NO 2. (1989) 257-262

FIXED

POINT THEOREMS FOR

COMPATIBLE

MULTI-VALUED

AND

SINGLE-VALUED

MAPPINGS

HIDEAKI KANEKO

Department of Mathematics and Statistics Old Dominion University

Norfolk, VA 23529 U.S.A. AND

SALVATORE SESSA

Universita di Napoli; Facolta di Architettura Instituto Mathematics; Via Monteoliveto 3,

80134 Napoli, Italy

(Received July I, 1987 and in revised form October I, 1987)

ABSTRACT

The notion of compatibility for point-to-point mappings recently defined by Jungckisgeneralized to include multi-valued mappings. This ideais used to establish afixed point theorem forageneralizedcontractivemulti-valued mapping andasingle-valuednapping.

KEY

WORDS AND

PHRASES: Compatible Multi-Valued Mappings, Fixed Point. 1980

AMS

Subject Classification: 47H10 54H25

1.

INTRODUCTION

Jungck

[2]

provedthe following theorem forf-contractivepoint-to-point

mappings.

THEOREM

1.

A

continuousself-mappingf ofacompletemetricspace

(X,d)

hasafixed point iff thereexistsamappingg

X

whichcommutes withf andsuchthat

9(X)

C_

f(X),

d(gx, gy) <

hd(fx,

fy)

for all x,

yeX,

where 0

<

h

<

1. Futhermore, fand g have a unique commonfixed point.

The present authors generalized thistheoremin two different directions.

Sessa

[8],

gen-eralizing the notion of commutativity for point-to- point mappings, established the idea of weak commutativity for two self- mappingsfand g ofametric space

(X,d),

i.e.

d(fgx, gfx) <

d(fx,

gx)

for all x in X. Under this concept, he extended theorem 1. On the other hand,

(2)

DEFINITION

1.

Two

self-mappings fand g ofmetric space

(X,d)

are compatible iff

d(fgxn,gfxn)

0whenever

xn

isasequence in

X

suchthat

fxn

t,gx,, forsomepoint tinX.

It

can beseen that twoweakly commuting mappings are compatiblebut theconverse is false. Examples supporting this factcan befound in

[3].

The purpose of this paper is to extend the definition ofcompatibility to include

multi-valuedmappings.

2.

RESULTS

Let

(X,d)

beametric space and

CB(X)

thefamily ofallclosedbounded subsetsof

X. Let H

be theHausdorffmetricon

CB(X)

inducedby

d,

i.e.

H(A,B)

max

{supd(x,B)

xeA,supd(x,A)

xeB}

for all

A,B

in

CB(X),

where

d(x,A)

inf{d(x,y):

yeA}.

It iswell known

[10]

that

(CB(X),H)

isametric space. Itisindeedacompletemetric space in the eventthat

(X,d)

isalsoacomplete

metric space.

DEFINITION

2. The mappings

f

X

and

T

X

CB(X)

are compatible iff

fTxeCB(X)

for all

xeX

and

H(Tfx,,

fTxn)

0whenever x, is asequence in

X

such that

Txn

MeCB(X)

and

f

xn

teM.

Definition 2extends definition 1 above.

We

are nowin aposition to proveourresults.

Theorem 2.

Let

(X,d)

beacompletemetric space,

f"

X

and

T"

X

CB(X)

be compatiblecontinuousmappings such that

T(X)

C_

f(X)

and

{

l[d(fx,

Ty)+d(fy

Tx)]}

(1)

H(Tx,

Ty) <_

hmax

d(fx,

fy),d(fx, Tx),d(fy,

Ty),-for allx,y in

X,

where0

_<

h

<

1. Thenthereexists apoint

teX

such that

fteTt.

Proof Wedraw inspiration from

[6].

Let

xoeX

be arbitraryandchoose x

leX

such that

f

xleTxo.

This ispossiblesince

Txo

C_

f(X).

If h

O,

then

d(fxTx)

<_

H(Txo, Tx)

O,i.e.

/xeTxl

since

Txl

is closed.

Now

assumethat h

>

0 andfork

h

>

1, bythe definition of

H,

there existsa point

yeTx

such that

d(y,fxl) <_ kg(Tx,Txo) (this

is notgenerally

true ifk

_<

1 asit is seenin

[7,

remark at p.

480]).

Since

Tx

C_

f(X),

let

xeX

be such that y

fx2.

In

general, if x, has been

selected,

choose

xn+leX

sothat yn

fx,+leTxn

and

d(yn,fxn) <_

kH(Txn, Txn_)

foreach n

_>

1. Using

(1),

wehave that

d(f

xn,

f

xn+l)

_

kH(Txn_I,TX,)

<_

V/-max

d(fxn-l,fxn),-[d(fxn-,fx,)

+

d(fxn, fxn+)]

(3)

i.e.

d(fx,,fx,+,)

<_

v/-ftd(fx,,fx,_,)

for eachn. Since

<

1, this shows that

{fx,}

is a Cauchysequence, hence it coverges tosomepoint

rex

using thecompleteness ofX. Further-more, the above inequalities also show that

H(Tx,,_I,TX,)

<

hd(fx,_,,fx,,).

Since is Cauchy, this must imply that

{Tx,}

is a Cauchy sequence in the complete metric space

(CB(X),H).

Nowlet

Tz,

MeCB(X).

Thus

d(t,M)

<

d(t, fz,)

+

d(fz,,M)

<

d(t, fxn)

+

H(Txn_I,M)

0 as n 03.

Since

M

isclosed, teM and the compatibility offand

T

implies that

H(Tfx,, fTz,,)

0 as n 03. This alongwith the continuities offand

T

imply that

d(ft,Tt)

<

d(ft, ffx,+l)

+

d(ffxn+x,Tt)

<

d(ft, ffx,,+l)+

H(fTx,,Tt)

< d(ft, ffx,+l)

+

H(fTx,,Tfx,)

+

H(Tfx,,Tt)

0

as n 03,i.e.

fteTt

since

Tt

isclosed.

Q.E.D.

Remark 1.

A

nonemptysubset Sof

X

isproximinal if,for each

xeX,

thereexistsapoint

yes

suchthat

d(x,

y)

d(x, S).

Let

PB(X)

bethefamily of allbounded proximinal subsets of

X.

If

T

X

PB(X),

then the interative process

{y,,}

in the aboveproofcanbesimplified

in the following way: if x, has been

selected,

let

x,+leX

be such that y,t

fx,+leTxn

and

d(fx,,y,)

d(fx,Tx,).

Note that an interationscheme of Smithson

[9],

where

Wx

is

compact

(hence

proximinal)

for all

xeX,

isincludedhere. Since aproximinalset is

closed,

we have

PB(X)

C_

CB(X).

Theresults of

[4]

and

[5]

followascorollaries.

COROLLARY 1. Let

(X,d)

be a complete metric space,

f

X

and

T

X

PB(X)

be continuousmappings such that

fTxePB(X)

and

H(Tfx,

fTx)

<

d(fx, Tx)

for all

xeX.

If

(1)

is satisfied for all x,y in

X,

0

<_

h

<

1, and

T(X)

C_

f(X),

thenthere exists

reX

suchthat

fteTt.

COROLLARY

2. Let

(X,d)

bea completemetric space,

T

X

CB(X)

and fbe a continuousself-mapping of

X

such that

H(Tx,

Ty) <

hd(fx,

fy)

andfor allx,y in

X,

where 0

<

h

<

1and

Tfx

fTx.

If

T(X)

C_

f(X),

then there exists

teX

such that

fteTt.

Notethat the continuity of implies the continuity of

T

incorollary2. The followingexample

shows theorem 2 isindeedaproper generalizationovercorollaries 1and 2.

EXAMPLE

Let

X

[1, 03)

beendowedwith the Euclideanmetricd. Let

fx

2x 4-and

Tx

[1,

x

2]

for each x

>

1.

W

andf are clearly continuous and

T(X)

f(X)

X.

(4)

[1,(2z4-1) 2]

forallz

>_

1, fand

T

arecompatible. Since

H(Tz,

Ty)

=l z2-Y

<-

21z2-Y

II

but the above

z2

+

y2

/2

d(fz,

fy)/2,

all the conditions of theorem 2 hold with h

,

corollaries arenot applicable sincef and

T

do not weaklycommute

(for

x

2),

hence donot commute either.

In

thesequel, weuse the following lemmawhich is aslight generalizationof proposition 2.2

(part

1)

of

[3].

LEMMA. Let T" X

CB(X)

and

f

X

be compatible. If

fweTw

forsome

weX,

then

f

Tw

T

f

w.

PROOF. Let

z, w foreachn. Then

f

x,

fw

f

w and

Tzn

M

Tw. Hence

if

fweTw,

then

H(fTw, Tfw)

H(fTx,,Tfxn)

0 by the compatibility.

Hence

wemust

have

fTw

T

f

w.

Q.E.D.

In

order to obtainafixed point

result,

weneed

additional

assumptionsas those given in

[4]

and

[5].

THEOREM3.

Let

fand

T

have thesamemeaningsasintheorem2. Assumealso that for each

xeX

either

(i)

f

x

-#

f2x

implies

f

x

Tx

or

(ii)

f

xeTx

impliesthat

fnx

zforsome

zeX.

Thenfand

T

haveacommonfixed point in

X.

PROOF.

By

Theorem2,

fteTt

forsome

teX

and

fTt

Tft

by

Lemma.

Assuming

(i),

wededucethat

ft

f2tefTt

T

ft.

Assuming

(ii),

itisclearthat

fz

zbythe continuity of f.

We

claimthat

fnteTfn-lt

for eachn.

To

seethis,wehave that

fzt

fftefTt

Tft.

By

lemma

(w

ft),

f3t

f

f2tefTft

Tf2t.

Repeating thisargument, weobtain

fnteTfn-t

and the continuity of

T

impliesthat

d(z, Tz)

<

d(z, fnt)

+

d(fnt, Tz)

<

d(z,

fnt)

+ H(Tfn-,Tz)

O,

i.e.

zeTz

since

Tz

isclosed.

Hence

z is a commonfixed pointof fand

T.

Q.E.D.

REMARK

2. Simple examples prove that the conditions

"T(X)

C_

f(X)"

and the compatibility off and

T

are necessary in theorem 2. Unfortunately, it is not yet known if the continuity ofbothmappingsfand

T

is necessary in theorem2.

However

in thecasethat

fand

T

are single-valued mappings, it suffices only the continuity of at least one of them.

Moreover,

the inequality

(1)

can be weakened as it is proved in the following

results,

which extends theorem2.1of

[1].

THEOREM

4.

Let

(X,d)

be a complete metric space and

f,T

X

be two compatible mappings

(def.1)

such that

T(X)

C_

f(X)

and

(2)

d(Tx,

Ty)

< Hmax{d(fx,

fy),d(fx, Tx),d(fy, Ty),d(fx, Ty),d(fy,

Tx)}

(5)

PROOF.

As in

[1],

it is seenthat thesequence

{Txn}

where

Tx,.,

fxn+l

foreachn,is aCauchysequence.

Hence

it converges tosomepoint

zeX.

If

T

iscontinuous, then

T

2xn

Tz

and

Tfx=

Tz.

Since

d(fTx,, Tfx,)

0 by compatibility,

fTx,

Tz,

hence using

(2),

d(T2x,,,, Tx,)

<

hmax{d(fTx,,fx,.,),d(fTx,.,,T2x,),d(fxn,

Txn),

d(fTx,, Tx,.,), d(f

x,,

T

x,)}

which implies, as n

,

that

d(z,Tz) < hd(z,Tz),

i.e.

Tz=z.

Since

T(X)

C_

f(X),

there

existsapoint

z’

such that z

Tz

fz’

andusing

(2)

again,

d(T2x,.,,Tz

’)

<

hmax

{d(fTx,.,,z),d(fTx,.,,T2x),d(z,

Tz’),d(fTx,.,,Tz’),d(z,

T2x,.,))

As n oo, we deduce that

d(z,

Tz’)

<_ hd(z, Tz’),

i.e. z

Tz’

fz’

and by

Lemma,

f

z

fTz’

Tf

z’

Tz

z.

A

similarproofcan bemade ifweassumethe continuity offinstead ofT.The uniqueness follows easily from

(2).

Q.E.D.

REMARK

3.

Note

thatourproofis differentfrom thatof

[1],

in which it isproved that

fz

Tz

is the unique fixed point

(only

iffis

continuous).

In

this paper, it is just thecase that

f

z

Tz

z.

REMARK

4. Ifneither

T

norfis continuous,e.g.

X

[0,11,

TO

1/4,

f0

1/2,Tx

fx

forx

#

0, theorem3fails. Similarremarkscanbemadeon the results of

[8]

ACKNOWLEDGEMENT:

The authors would like to thank a referee for suggesting the formulation of definition 2 in itspresentform.

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1.

K.M. Das

and

K.V.

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261-263.

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Math. Sci.

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Kaneko,

Single-valued and multi-valuedf-contractions,Boll.

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(6)

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