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Commonfixedpointresultswithapplications...
11
Corollary3.LetEbeanonemptyclosedq−starshapedsubsetofconvex
completemetricspaceXandTandSbeCq−commutingmappingson
E−{q}suchthatS(E)=EandT(E−{q})⊂S(E−{q}),whereq∈F(S).
SupposethatTiscontinuousasymptoticallyS−nonexpansivewithsequence
{kn}andSisaffineonE.Foreachnł1,defineamappingTnonEby
Tnx=W(Tnxjqj0n),where0n=
An
kn
and{An}isasequencein(0j1)with
n→∞
lim
An=1.Thenforeachn∈N,F(Tn)∩F(S)issingleton.
Theorem4.LetEbeanonemptyclosedq−starshapedsubsetofconvex
metricspaceX,andTandSbecontinuousselfmappingsonEsuchthat
S(E)=EandT(E−{q})⊂S(E−{q}),q∈F(S).SupposeTisuniformly
asymptoticallyregular,asymptoticallyS-nonexpansive,andSisaffineon
E.Ifcl(E−{q})iscompactandSandTareuniformlyCq−commuting
mappingsonE−{q}.ThenF(T)∩F(S)isasingletoninE.
Proof.FromTheorem3,foreachn∈NjF(Tn)∩F(S)issingletonin
E.Thus,
Sxn=xn=W(Tnxnjqj0n).
Also,
d(xnjT
nxn)=d(W(Tnxnjqj0n)jTnxn))
≤(1−0n)d(qjTnxn)≤(1−0n)d(qjTnxn).
SinceT(E−{q})isbounded,d(xnjTnxn)→0asn→∞.Now,
d(xnjTxn)≤d(xnjTnxn)+d(TnxnjTn+1xn)+d(Tn+1xnjTxn)
≤d(xnjTnxn)+d(TnxnjTn+1xn)+k1d(STnxnjSxn)
≤d(xnjTnxn)+d(TnxnjTn+1xn)+k1d(STnxnjSW(Tnxnjqj0n))
≤d(xnjTnxn)+d(TnxnjTn+1xn)+k1d(STnxnjW(STnxnjqj0n))
≤d(xnjTnxn)+d(TnxnjTn+1xn)+k1(1−0n)d(STnxnjSq)
≤d(xnjTnxn)+d(TnxnjTn+1xn)+k1(1−0n)d(STnxnjSq)j
whichimpliesthat,d(xnjTxn)→0,asn→∞.Ascl(E−{q})iscompact
andEisclosed,thereexistsasubsequence{xn
i}of{xn}suchthatxn
i→
xo∈Easi→∞.ThecontinuityofTimpliesthatT(xo)=xo.Since
T(E−{q})⊂S(E−{q}),itfollowsthatxo=T(xo)=Sgforsomeg∈E.
Moreover,
d(Txn
ijTg)≤k1d(Sxn
ijSg)=k1d(xn
ijxo).
Takingthelimitasi→∞,wegetTxo=Tg.ThusTxo=Sg=Tg=xo.
SinceSandTareuniformlyCq−commutingonE−{q},andg∈C(SjT),
d(TxojSxo)=d(TSgjSTg)=0.
