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Commonfixedpointresultswithapplications...
2.Commonfixedpointresults
9
Inthissection,theexistenceofcommonfixedpointsofuniformlyCq−
commuting,Cq−commuting,anduniformlyR−subweaklycommutingmap-
pingsisestablishedinaconvexmetricspace.
Theorem1.LetEbeanonemptyq−starshapedcompletesubsetof
convexmetricspace,andT,fandgbeselfmappingsonX.Supposeq∈
F(f)∩F(g),Tiscontinuous,fandgarecontinuousandaffineonE,
cl(T(E))iscompactandT(E)⊂f(E)=g(E).Ifthepairs{Tjf}and
{Tjg}areCq−commutingandsatisfy,forallxjg∈E,
(1)
d(TxjTg)≤max{d(fxjgg)jd(fxjYTx
q
)jd(ggjYTy
q
)j
1
2
[d(fxjYTy
q
)+d(ggjYTx
q
)]}j
thenTjfandghaveacommonfixedpointinE.
Proof.DefineTn:E→Eby
Tnx=W(TxjqjAn)j
whereAn∈(0j1)withlim
n→∞
An=1.SinceEisq−starshaped,Tnistheself
mappingonEforeachnł1.AsfandTareCq−commutingandfis
affineonEwithfq=qjif,x∈C(fjTn)⊂Cq(fjT)jthen
fTnx=f(W(TxjqjAn))=W(fTxjqjAn)=W(TfxjqjAn)=Tnfx.
ThusfandTnareweaklycompatibleforalln.AlsosincegandTareCq−
commutingandgisaffineonEwithgq=q,gandTnareweaklycompatible
foralln.Also,
d(TnxjTng)=d(W(TxjqjAn)jW(TgjqjAn))≤And(TxjTg)
≤Anmax{d(fxjgg)jd(fxjYTx
q
)jd(ggjYTy
q
)j
1
2
[d(fxjYTy
q
)+d(ggjYTx
q
)]}
≤Anmax{d(fxjgg)jd(fxjTnx)jd(ggjTng)j
1
2
[d(fxjTng)+d(ggjTnx)]}.
ByCorollary3.1of[7],foreachnł1jthereexistxninEsuchthatxnis
acommonfixedpointoffjgjandTn.Thecompactnessofcl(T(E))implies
thatthereexistsasubsequence{Txk}of{Txn}suchthatTxk→gas
k→∞.ThedefinitionsofTkxkandconvexitystructureonXgivexk→g.
FromthecontinuityofTjfandgjwehaveg∈F(T)∩F(f)∩F(g).
.
