Common fixed point theorems and applications

Internat. J. Math. & Math. Sci.

VOL. 20 NO. (1997) 13-18 13

COMMON

FIXED

POINT

THEOREMS

AND

APPLICATIONS

HEMANTKUMARPATHAK

DepartmentofMathematics,Kalyan Mahavidyalaya

BhilaiNagar (M.P.)490006,INDIA

SHIH SENCHANG

Department

of Mathematics,SichuanUniversity Chengdu, Sichuan610064,PEOPLE’SREPUBLIC OF CHINA

YEOLJECliO

Departmentof Mathematics,

Gyeongsang

NationalUniversity Jinju 660-701, KOREA

JONG SOO JUNG

Department

of Mathematics,

Dong-A

University Pusan 604-714,KOREA

(ReceivedNovember 22, 1994and in revisedform

January

18,

1995)

ABSTRACT. The purposeof thispaperisto discuss the existenceofcommon fixedpoints for mappings in general quasi-metric spaces. As applications, some common fixed point theorems for mappings in probabilistic quasi-metric spacesaregiven. Theresultspresented in thispaper generalize

somerecentresults.

KEY WORDS AND PHRASES. General quasi-metric space, probabilistic quasi-metric space, fixed

point, periodic point, periodic index.

1991AMS SUBJECTCLASSHCICATIONCODES. 54H25, 54A40,54C60.

1. INTRODUCTION

Inthispaper, weshowthe existenceofcommon fixedpoints forcommuting mappingsingeneral quasi-metric spaces. As applications, we givesomefixed pointtheorems for commuting mappings in

probabilistic quasi-metric spaces. Ourmainresults generalize and improvesome recentresultsin 1], [4],

[]

and

[6].

Let

(G, _<,

<

beapartialordersetsatisfyingthe following conditions: (G-l) 0 isthe minimal element inG,i.e.,0

<

ufor allz

G,

(G-2)

for any

,

v

G, sup{u,

v}

existsandbelongsto

G,

(G-3) foranyu/G,

,

(G-4) for any

,

v,w

G,

u

<

wandv

<

w

=

sup{z,

v} <

w, and

<

v,v

_

w

=

u

<

w.

DEFINITION 1.1 Let

X

beanonempty set.

(X,

r)

is calledageneralquasi-metricspace if r-Xx X--,_G

(G,

_,

<

satisfiesthe followingconditions:

(QM-1)

r(x, !/)

0if andonlyifx !/,

(QM-2)

r(x,l)

Itfollows from the definition that every general quasi-metric space includes a metricspace asits

DEFINITION 1.2 LetXbeanonemptysetandletTbeaself-mapping ofX. Apointx EXis called aperiodicpointofT ifthereexists apositive integerksuch that

Tkx

x. Theleastpositive integersatisfyingthis condition iscalledtheperiodicindexofx.

DEFINITION 1.3 A mappingF (-oo,oo) [0, oo) is calleda distribution

function

if it is nondecreasingandleft-continuouswithinfF(t) 0 andsup

F(t)

1.

Inwhat follows we always denote by

F(T)

theset of all fixed points ofT,

P(T)

the set of all periodic points of T and /) the set of all distribution functions, respectively, and let l)+

{F

1):

F(t)

0for all t<

0}.

DEFINITION1.4

(X, .’)

iscalledaprobabilisticquasi-metricspaceifXisanonemptyset,

"

is amapping fromXxXinto 1)+ (we shall denote the distribution function.’(x,y) by

F=,u(t

which representthevalueof

F=,u

att oo,oo))satisfyingthefollowing conditions:

(PQM-I)

F=,u(0)

0,

(PQM-2)

F=,u

(t)

Ifor all t

>

0ifand onlyif x y,

(PQM-3)

F,(t)

Fux(t

for all oo,oo).

DEFINrrION1.5

(X,

’)iscalledaprobabilisticmetricspaceif(X, ’)is aprobabilistic quasi-metricspaceand thefollowingcondition is satisfied:

(PQM-4) if

F=,(tl)

1and

F.z(

1, then

F,z(tl

+

z)

1.

Formoredetails onprobabilistic metricspaces,referto[3]and[7].

2. COMMONFIXED POINT TIOREMS Now,wegiveour maintheorems.

TitEOREM 2.1 Let

(X, r)

bageneral quasi-metricspace and let

’

andTbetwocommuting self-mappings ofX. If for anyx Xand anytwopositive integersn,q_>2 with

Tix

T:x,

O

<

<j

<_

n-1,

(2.1)

SexSJ’x,

0_<i’<._<q-1,

,(T"

S"

=,

ST

=)

<

max sup

r(TJz,

S =), sup

r(T:=,

x), sup

r(S

=, x)

1 <_j<_n, 1

<_] <_q

1 <_j<_n 1

<_] <_q

(2.2)

for 1,2, n 1andj 1,2, q 1, thenSandThaveacommonfixedpointinXifandonlyif there existintegersm, n,p,q,m

>

n

>

0, p

>

q

>

0, andapointx Xsuch

that

Tx

SP=

Tnx

sqx

_(2.3)

and either

F(S)

or

F(T)

7

O.

Ifthis conditionis satisfied, theneitherT"xorSqxisacommon fixedpoint ofSandT.

PROOF. Letx* EXbeacommonfixedpointofT,i.e., x* Sx* Tx*. Then(2.3)is true in casern p land n q O.

Conversely, suppose that there exist a point x X and four integers rn, n, p, q,rn

>

n

>_

0, p

>

q

>

0,suchthat(2.3)issatisfied. Withoutlossof generality,we canassumethat x

_

F(S)

and m is the minimal integer satisfying

Tx

Tx,

k

>

n. Putting y T"x and p rn-n, we have T y _{y and Pl}istheminimalintegersatisfying

Tky

y, k

>

1.

By (2.2),itfollowsthat

r(Tx,

Tix)

<

max{

l <_j< n

sup

{r(Tx’x)}’

l

<_j<_nSUp

{r(Tx,x},O},

r(T"x,

Tix)

<

sup

{r(Tix,

x)}.

(2 4)

COMMONFIXEDPOOREMS ANDAPPLICATIONS 15

Next,weprovethat y isacommonfixedpoint ofSandT. Supposethe contrary. Then V isnot afixed pointof T. Also,p

_>

2 and

T’yeTy,

O<_i<j<_pl-1.

By (2.4),itfollows that, for 1, 2, pl 1,

r(y,

TV)

r(Ty,

T/y)

<

sup

{r(TJy,

y)} _<

1<_j<p

sup

{r(TJY,

V)}.

It follows from(G-4)that

sup

{r(y,T:y)}

<

sup

{r(T:y,y)},

l<j<p-1 l<j<_p-1

which is a contradiction. Therefore, y

Tx

is a fixed point of

T.

Further, since S and T are

commuting,wehave

V

T"x

T"Sx

ST"x

Sy,

i.e., y

Tx

is a common fixed point ofS and T. In this case, when xE

F(T),

we have, by terchangingtheroleofSand

T,

that y Sqx isacommonfixedpoint ofSandT. Thiscompletes the proof

Ontheotherhand, by using Theorem3of[6],wehavethe following:

THEOREM 2.2 Let

(X, r)

beageneral quasi-metric spaceand let SandTbetwo commuting self-mappings ofX. Assumethat for any x, yE

X,

x

-

y, thereexists a positive integerp(x, y) such

that

r((ST)’u)x,

(ST)’U)y)

< sup{r(x,

y),

r(x,

(ST)

’’)x),

r(y,

(ST)’)y),

r(x,

(ST)")y),

r(y,

(ST)

’’u)

x)}

ThenSandThave a common fixedpointinXif andonlyifthere exists aperiodic poimx XofST with periodic index k such that for any u,v

A

{x,

STx,...,(ST)k-x},

u v, there exist

x’, y’

A,

x

y’,

satisfying the following conditions:

(ST)e’z’

u,

(ST)’

v

and either

F(S)

Q

P(ST) #

_or

F(T)

Q

P(St) #

@.

_{Ifthese conditions are satisfied, then the pointx}

istheuniquecommon fixedpointof

S

andT. PROOF. The necessitycondition is obvious.

The sufficiencyconditionfollowsfrom Theorem3of[6] asfollows: InTheorem3 of[6], ifwe

replace T by ST, we can conclude that ST has a unique fixed point x in X. Since S and T are

commuting,

Sx

S(STx)

ST(Sx)

and so

Sx

isalso a fixed point of

ST.

Uniqueness gives

Sx

x. Similarly,

Tx

x. Thiscompletesthe proof

The followingis aspecial case of Theorem2.2by setting

p(x,

y)

p(x):

COROLLARY2.3 Let

(X,

r)

be ageneralquasi-metricspaceand letSandTbetwocommuting self-mappings ofX. Assume that for any x

X,

there exists a positive integer p(x) suchthat every

yEX,

xy,

ThenS andThasacommon fixedpointinXifandonlyifthere existsaperiodic pointx EXofST with periodic index k such that for any u,v EA

{x,

STx,...,(ST)k-lx},

u

:fi

v, there exist

x’, y’

E

A,

x’

=f=

y’,

satisfying the following conditions:

(ST)

()z’

u,

(ST)e)y

v

and either

F(S)

N

P(ST)

0or

F(T)

N

P(ST)

O.

If these conditionsaresatisfied, then thepointx istheuniquecommon fixedpointofS andT.

Thefollowingis obtainedfrom Corollary2.3by settingp(x) p:

COROLLARY2.4 Let

(X, r)

be ageneral quasi-metric space andletSandTbetwocommuting self-mappings ofX. Assumethat there exists apositive integerpsuch that for any x, y

X,

x

-

y,

r((ST)’x,

(ST)y) < sup{r(x,

y),

r(x,

(ST)"x),

r(y,

(ST)y),

r(x,

(ST)y),

r(y,

(ST)x)}

ThenSandThaveacommonfixedpointinXifandonlyif there existsaperiodic pointx XofST andeither

F(S)

f’l

P(ST)

t

or

F(T)

P(ST)

:

.

Ifthisconditionissatisfied, then the pointx is theuniquecommonfixed point ofSandT.

By

settingp 1inCorollary 2.4,wehave the following:

COROLLARY2.5 Let

(X, r)

beageneral quasi-metric spaceandlet

S

andTbetwocommuting self-mappings ofX. Assumethat for any x, y

X,

x

:f-

y,

r(STx,

STy)

<

sup[r(x,

y),

r(x, STx), r(y,

STy),

r(x,

STy),r(y,

STx)}.

ThenSandThave acommon fixed pointin

X

ifandonlyifthere existsaperiodic pointx

X

of

ST

and either

F(S)

q

P(ST)

_or

F(T)

P(ST)

_:/:

.

_{Ifthiscondition is satisfied, then thepoint}xis the unique common fixed point ofSandT.

By

usingTheorem 5of[6],wehave the following:

THEOREM

2.6 Let

(X, r)

beageneral quasi-metric space and letS and

T

betwocommuting self-mappings ofX. Assumethatthere existpositive integersp, q suchthat forany x,y, E

X,

x

t

y,

r((ST)x,

(ST)qy) <

sup(v(x,

y),

r(x,

(ST)"x),

r(y,

(ST)qy), r(x, (ST)qy),

r(y,

Then Sand

T

have a fixedpoint inXifand onlyifthere exists aperiodic pointx6

X

ofST with

periodic indexkwhich satisfiesthefollowing condition:

where p

=/hk

+

P2,q qlk

+

q2,0

_<

P2, q2

<

k and Pl,ql are non-negative integers and either

F(S)

gl

P(ST)

or

F(T)

;

P(ST)

-

f}. If this conditionissatisfied, then the pointx isthe unique common fixed point ofSand

T.

PROOF. The necessity condition is obvious.

To

prove converse,if weuse Theorem 5 of

[6]

byreplacing

T

with

ST,

thenSThasaunique fixed pointxinX. Therefore,employing the same argument asintheproofof Theorem 2.2,itfollows that the pointxisthe unique fixed point ofSandT. Thiscompletestheproof.

REMARK. Theorems 2.1 2.6generalize somemainresults in ], [2]and[4]. 3. APPLICATIONS TO PROBABILISTICQUASI-METRIC SPACES

First ofall, we definepartial orders

<

and

<

"

onT+ asfollows, respectively: For any F1,

F2

EI)+ andt

>

0,

F

<

F

F

() > F(),

F

<

F

F

(t) > F().

COMMONFIXEDPOOREMSANDAPPLICATIONS 17

(G-1) there exists a minimalelement0d&f: HE

G,

where g(t)

{

,

t>0, 0, t

<

0, (G-2) foranyF,

F

E G,

sup{Fl,

Y2}(t)

%r.

rnin{Y (t),

F(t)}

(G-3) foranyF G, F

F,

(G-4)

for any

F, F,

Fs

G,

F

<

Fs,

F:

<

Vs

sup{F,

F) <

Fs,

FI

<

F2,

F2

<_

F3

=

FI

< F3.

THEOREM3.1 (Embedding

Theorem)

Let

(X,

’)

beaprobabilistic quasi-metric space. Then

(X, .’)

is ageneralquasi-metricspace, whereG

(D

+,

_<, <

isthe partial ordersetinducedby the wayasabove.

PROOF. Let

r(x,

y)

Fx,y

for all x, y

e

X. Itiseasytosee thatrsatisfies the conditions

(QM-2)

and(QM-2)of Definition1.1.

The following resultsare obtainedfrom Theorems2.1 2.6andTheorem 3.1immediately: THEOREM 3.2 Let

(X,.T’)

be a probabilistic quasi-metric space and let S and

T

be two

commuting self-mappings ofX. If for anyx

X

and positive integer n,q

_>

2 with

TixTTx,

O<_i<j<_n-1,

S’x

Sx,

O

<

<

j

<_

q-FT,

s,x.s,z

(T)

>

rffm

(

rain

Fr,:

s,,

(t)

rain

FT,

(t)

nfin

Fs,,.x

(t)

)

1 <_j<_n, 1

<_j’

<_q

’1

<_j<_n _{1 <_3}

<_q

for 1,2, n 1andj 1, 2, q 1, thenSand

T

haveacommon fixed pointinXifand only if

there existintegersm,p, q,m

>

n

_>

0,p, q

>_

0,and apointxEXsuch that

Tx

Sx

T"x

andeither

F(S)

:

O

or

F(T)

:/:

.

Ifthis condition is satisfied, then either

T"x

orSqx isacommon fixedpoint ofSandT.

THEOREM 3.3 Let

(X,Y’)

be a probabilistic quasi-metric space and let S and T be two

commuting self-mappings of

X.

If there exist positive integers p, q such,thatfor

any.x,

y

e

X,

x

:

y, and for all t

>

0,

F(sr),.(sr),()

>

min{F.(),

F.(ST),(t),

F.(ST),(),

F.ST),(),

F.(ST),()

},

then

S

and

T

haveacommon fixed pointin

X

if andonlyifthereexistsa periodic pointx XofST withperiodic indexk whichsatisfies thecondition

(2.5)

inTheorem2.6andeither

F(S)

P(ST)

7

1

or

F(T)

P(ST)

7k

$. If this condition is satisfied, then thepointxis theuniquecommon fixedpoint ofSandT.

The

followin

is aspecialcaseof Theorem3.3 obtainedby

settin

p q 1

COROLLARY 3.4 Let

(X,.’)

bea probabilistic quasi-metric space and let S andT betwo

commuting self-mappings ofX. If foranyx,y

X,

x y, and t

>

0,

FST,STu(t)

>

min{F,u(t), F,ST(t),

Fu,ST(t),

F,STu(t), Fu,

sr(t) }

thenSand Thaveacommon fixedpointinXifandonlyifthereexists aperiodic pointx GXofST andeither

F(S)

Iq

P(ST) :fi

_{0 or}

F(T)

f3

P(ST)

$. Ifthis condition issatisfied, then the pointz is

THEOREM 3.5 Let

(X,.T)

be a probabilistic quasi-metric space and let S and T be two commutingself-mappings of X. If foranyx

X,

z

#

y, thereexistsapositiveintegerp(z, V)such that

forallt

>

0,

F(

sT).),(sr)z.% t

>

min{F.v(t), F,(

ST)., t

Fv.(

ST).% t

F.(sr),.%

(t),

F.(ST),,,

(t)

},

thenSand Thaveacommon fixed pointinXifandonlyif there existsaperiodicpointx

X

of

ST

with periodic index k such that for any u,v

A

{x, (ST)x,...,

(ST)k-I},

u v, there exist x

,

y

A,

x’

://:

y,

satisfying thefollowing conditions:

(St)

(e’/)x’

u,

(ST)e’V’)y

v

andeither

F(S)

n

P(ST)

or

F(T)

fq

P(ST)

.

If theseconditions aresatisfied, then thepointx

is theunique common fixed point of

S

and

T.

Bysetting

p(x,

y)

p(x)

inTheorem 3.5,wehave the following:

COROLLARY :}.6 Let

(X,.T’)

bea probabilistic quasi-metric space and let S and Tbetwo

commutingself-mappings ofX. If thereexistsa positive integer

p(x)

such that for every y

X,

x

#:

y, andfor all t

>

0,

F(ST),)z,(ST)Z)v(t

>

min{F,v(t),

Fz,ST),)z(t),

thenSand

T

haveacommonfixed pointinXif andonlyif there existsaperiodic pointx

X

ofST with periodic index k such that for any u,v

A

{x, (ST)x

(ST)-Ix},

u

#

v, there exist:d,

y’ e

A,

Z’ #

y,

satisfying thefollowingconditions:

(:T)ez

u,

(ST)eV’

v

andeither

F(S)

fq

P(ST) -#-

or

F(T)

f

P(ST)

#-

t3. If theseconditions aresatisfied, then the pointx

isthe unique commonfixedpoint ofT.

REMARK.

Theorems3.3 3.6includeTheorems 5 in

[4]

and Theorems3.3 3.6 in[5]as special cases.

ACKNOWLEDGMENT.

The authorswish toexpressthe deepest appreciationtotherefereeforhis helpful comments on this paper. The Present Studies were supported in part by the Basic Silence Research InstituteProgram,Ministry of Educatiort,Korea, 1994, ProjectNo.BSRI-94-1405.

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S.S.,

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S.S.,

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Y.J.and

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Y.J.,

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