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10
MujahidAbbas
Corollary1.LetEbeanonemptyq−starshapedcompletesubsetof
aconvexmetricspaceX,andT,fandgbeselfmappingsonX.Suppose
thatq∈F(f)∩F(g),Tiscontinuous,fandgarecontinuousandaffine
onE,cl(T(E))iscompactandT(E)⊂f(E)=g(E).Ifthepairs{Tjf}
and{Tjg}areR−subweaklycommutingmappingssatisfying(1),thenT,
fandghaveacommonfixedpointinE.
Corollary2.LetEbeanonemptyclosedq−starshapedsubsetof
convexmetricspaceX,andTandSbeR−subweaklycommutingmappings
onEsuchthatT(E)⊂S(E),cl(T(E))iscompactwhereq∈F(S).IfT
iscontinuousS−nonexpansiveandSisaffineonEjthenF(T)∩F(S)is
nonempty.
Theorem2([15]).LetEbeasubsetofametricspace(Xjd),andS
andTbeweaklycompatibleself-mapsofE.AssumethatclT(E)⊂S(E),
clT(E)iscomplete,andTandSsatisfy,forallxjg∈Eand0≤h<1,
(2)
d(TxjTg)≤hmax{d(SxjSg)jd(SxjTx)j
d(SgjTg)jd(SxjTg)jd(SgjTx)}.
ThenE∩F(S)∩F(T)isasingleton.
Theorem3.LetEbeanonemptyclosedq−starshapedsubsetofaconvex
completemetricspaceXjandTandSbeuniformlyCq−commutingmap-
pingsonE−{q}suchthatS(E)=EandT(E−{q})⊂S(E−{q}),where
q∈F(S).SupposethatTiscontinuousasymptoticallyS−nonexpansive
withsequence{kn}andSisaffineonE.Foreachnł1,defineamapping
TnonEbyTnx=W(Tnxjqj0n),where0n=
An
kn
and{An}isasequencein
(0j1)withlim
n→∞
An=1.Thenforeachn∈N,F(Tn)∩F(S)isasingleton.
Proof.Forallxjg∈Ejwehave
d(Tn(x)jTn(g))=d(W(Tnxjqj0n)jW(Tngjqj0n))
≤0nd(TnxjTng)≤And(SxjSg).
Moreover,asTandSareuniformlyCq−commutingandSisaffineon
EwithSq=qjforeach,x∈C(SjTn)⊆Cq(SjT)j
STnx=S(W(TnxjqjAn))=W(STnxjqjAn)
=W(TnSxjqjAn)=TnSx.
HenceSandTnareweaklycompatibleforalln.Theresultnowfollows
fromTheorem2.
.
