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Coincidenceandcommonfixedpoint...
9
(01):Obviously.
u+u
u+u
2
2
(02):Let1<k<
}−(1−a)max{u2juuj1
}+(1−a)max{u2juuj1
1
b
,u≤ktand0(tjujujuju+uj0)=t−b[amax{uju,
2u(u+u)}
2u(u+u)}
1
2]≤0.Then,u≤kt≤kb[amax{uju,
1
2].Ifu>0andu≥u,itfollows
thatu≤kbu<uwhichisacontradictionandsou≤hu,whereh=kb<1.
Ifu=0,thereforeu≤hu.Similarly,u≤ktand0(tjujujuj0ju+u)≤0
impliesu≤hu.
Example7.0(t1jt2jt3jt4jt5jt6)=t1−at2−b
t2
t5+t6
5+t2
6
−C(t3+t4),t5+t6/=
0,ajb>0,C≥0anda+2b+2C<1.
Example8.0(t1jt2jt3jt4jt5jt6)=t1−at2−b
t2
t3+t4
3+t2
4
−C(t5+t6),t3+t4/=
0,ajbjC>0anda+2b+2C<1.
TheyfollowasinExample1since
t2
t5+t6
5+t2
6
≤t5+t6and
t2
t3+t4
3+t2
4
≤t3+t4
ift5+t6/=0andt3+t4/=0.
3.Mainresults
Theorem2.Let(Xjd)beametricspace,SjT:X→XandFjG:
X→CB(X)satisfying(3)
(6)
0(H(FxjGy)jd(TxjSy)jD(TxjFx)jD(SyjGy)j
D(TxjGy)jD(SyjFx))≤0
forallxjy∈X,where0∈06,wheneverD(TxjGy)+D(SyjFx)/=0and
H(FxjGy)=0wheneverD(TxjGy)+D(SyjFx)=0.Supposethatoneof
S(X)orT(X)iscomplete.Then
a)thereexistsqjp∈XsuchthatTq∈FqandSp∈Gp.
Further,ifthepair(TjF)isR-weaklycommutingoftype(AT)and(SjG)
isR-weaklycommutingoftype(AS)attheircoincidencepoints,
b)thereexistsz∈XsuchthatTz∈FzandSz∈Gz.
C)Inthecase(b),ifSz=Tz,thenSz=Tz∈FznGz.
d)Inthecase(c),ifSz=Tz=z,thenzisacommonfixedpointof
SjTjFandG.
Proof.First,assumethatthereexistsqjp∈XsuchthatD(SpjFq)+
D(TqjGp)=0.So,D(SpjFq)=0andD(TqjGp)=0whichimpliesthat
Sp∈FqandTq∈Gp.SinceH(FqjGp)=0,itfollowsthatD(TqjFq)≤
H(FqjGp)=0andhenceTq∈Fq.Inasimilarmanner,wegetSp∈Gp.
