Internat. J. Math. & Math. Sci.
VOL. 20 NO. (1997) 13-18 13
COMMON
FIXED
POINTTHEOREMS
AND
APPLICATIONS
HEMANTKUMARPATHAK
DepartmentofMathematics,Kalyan Mahavidyalaya
BhilaiNagar (M.P.)490006,INDIA
SHIH SENCHANG
Department
of Mathematics,SichuanUniversity Chengdu, Sichuan610064,PEOPLE’SREPUBLIC OF CHINAYEOLJECliO
Departmentof Mathematics,
Gyeongsang
NationalUniversity Jinju 660-701, KOREAJONG 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 allzG,
(G-2)
for any,
vG, sup{u,
v}
existsandbelongstoG,
(G-3) foranyu/G,
,
(G-4) for any
,
v,wG,
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 andsupF(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) byF=,u(t
which representthevalueofF=,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)
1andF.z(
1, thenF,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 withTix
T:x,
O<
<j<_
n-1,(2.1)
SexSJ’x,
0_<i’<._<q-1,,(T"
S"
=,ST
=)
<
max supr(TJz,
S =), supr(T:=,
x), supr(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 Xsuchthat
Tx
SP=Tnx
sqx
(2.3)and either
F(S)
orF(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 satisfyingTx
Tx,
k>
n. Putting y T"x and p rn-n, we have T y y and PlistheminimalintegersatisfyingTky
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 andT’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 ofT.
Further, since S and T arecommuting,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 xEF(T),
we have, by terchangingtheroleofSandT,
that y Sqx isacommonfixedpoint ofSandT. Thiscompletes the proofOntheotherhand, 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, yEX,
x-
y, thereexists a positive integerp(x, y) suchthat
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 existx’, y’
A,
xy’,
satisfying the following conditions:(ST)e’z’
u,(ST)’
vand either
F(S)
QP(ST) #
orF(T)
QP(St) #
@.
Ifthese conditions are satisfied, then the pointxistheuniquecommon 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 ofST.
Uniqueness givesSx
x. Similarly,Tx
x. Thiscompletesthe proofThe 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 xX,
there exists a positive integer p(x) suchthat everyyEX,
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 existx’, y’
EA,
x’
=f=
y’,
satisfying the following conditions:(ST)
()z’
u,(ST)e)y
vand either
F(S)
NP(ST)
0orF(T)
NP(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, yX,
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’lP(ST)
t
orF(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 spaceandletS
andTbetwocommuting self-mappings ofX. Assumethat for any x, yX,
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 pointxX
ofST
and either
F(S)
qP(ST)
orF(T)
P(ST)
:/:.
Ifthiscondition is satisfied, then thepointxis 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 andT
betwocommuting self-mappings ofX. Assumethatthere existpositive integersp, q suchthat forany x,y, EX,
xt
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 pointx6X
ofST withperiodic indexkwhich satisfiesthefollowing condition:
where p
=/hk
+
P2,q qlk+
q2,0_<
P2, q2<
k and Pl,ql are non-negative integers and eitherF(S)
glP(ST)
orF(T)
;
P(ST)
-
f}. If this conditionissatisfied, then the pointx isthe unique common fixed point ofSandT.
PROOF. The necessity condition is obvious.
To
prove converse,if weuse Theorem 5 of[6]
byreplacingT
withST,
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 anyF, 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, ye
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 andT
be twocommuting self-mappings ofX. If for anyx
X
and positive integer n,q_>
2 withTixTTx,
O<_i<j<_n-1,S’x
Sx,
O<
<
j<_
q-FT,
s,x.s,z(T)
>
rffm(
rainFr,:
s,,
(t)
rainFT,
(t)
nfinFs,,.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 ifthere existintegersm,p, q,m
>
n_>
0,p, q>_
0,and apointxEXsuch thatTx
Sx
T"x
andeither
F(S)
:
O
orF(T)
:/:
.
Ifthis condition is satisfied, then eitherT"x
orSqx isacommon fixedpoint ofSandT.THEOREM 3.3 Let
(X,Y’)
be a probabilistic quasi-metric space and let S and T be twocommuting self-mappings of
X.
If there exist positive integers p, q such,thatforany.x,
ye
X,
x:
y, and for all t>
0,F(sr),.(sr),()
>
min{F.(),
F.(ST),(t),
F.(ST),(),
F.ST),(),
F.(ST),()
},
then
S
andT
haveacommon fixed pointinX
if andonlyifthereexistsa periodic pointx XofST withperiodic indexk whichsatisfies thecondition(2.5)
inTheorem2.6andeitherF(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 obtainedbysettin
p q 1COROLLARY 3.4 Let
(X,.’)
bea probabilistic quasi-metric space and let S andT betwocommuting 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)
IqP(ST) :fi
0 orF(T)
f3P(ST)
$. Ifthis condition issatisfied, then the pointz isTHEOREM 3.5 Let
(X,.T)
be a probabilistic quasi-metric space and let S and T be two commutingself-mappings of X. If foranyxX,
z#
y, thereexistsapositiveintegerp(z, V)such thatforallt
>
0,F(
sT).),(sr)z.% t>
min{F.v(t), F,(
ST)., tFv.(
ST).% tF.(sr),.%
(t),
F.(ST),,,
(t)
},
thenSand Thaveacommon fixed pointinXifandonlyif there existsaperiodicpointx
X
ofST
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
vandeither
F(S)
n
P(ST)
orF(T)
fqP(ST)
.
If theseconditions aresatisfied, then thepointxis theunique common fixed point of
S
andT.
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 Tbetwocommutingself-mappings ofX. If thereexistsa positive integer
p(x)
such that for every yX,
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 pointxX
ofST with periodic index k such that for any u,vA
{x, (ST)x
(ST)-Ix},
u#
v, there exist:d,y’ e
A,
Z’ #
y,
satisfying thefollowingconditions:(:T)ez
u,(ST)eV’
vandeither
F(S)
fqP(ST) -#-
orF(T)
fP(ST)
#-
t3. If theseconditions aresatisfied, then the pointxisthe 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.CHANG,
S.S.,
OnRhoades’openquestions and some fixed point theorems for a class of mappings, Proc. Amer.Math.Soc. 97(1986),
343-346.[2]
CHANG,S.S.,
On the generalized 2-metric space and probabilistic 1-metric spaces withapplicationsto fixedpointtheory, Math Japonica34(1989),885-900.
[3]
CHANG, S.S., CHO,
Y.J.andKANG,
S.M.,
ProbabilisticMetricSpaces
and NonlinearOperator
Theory,Sichuan University
Press,
P.R. China, 1994.[4]
CHANG, S.S. andHUANG,
N.J.,Fixedpoint theorems forsomemappingsinprobabilisticmetricspaces,J.Natural Sci. 12(1959),474-475.
[5]
CHANG,S.S.,
HUANG, N.J.andCHO,Y.J.,
Fixedpoint theoremsingeneral quasi-metric spaces and applications,toappearinMath. Japonica.[6]
CHANG,
S.S. andCHANG,
Q.C., On Rhoades’ open questions, Prov. Amer. Math. Soc. 109 (990),269-27a.