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Commonfixedpointresultswithapplications...
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andametricd:X×X→[0j∞)byd(xjg)=|x1−g1|+|x1x2−g1g2|.It
canbeverifiedthatXisaconvexmetricspacebutnotanormedspace.
Definition2.LetTjS:X→X.Apointx∈Xiscalled:
(1)afixedpointofTifT(x)=x;
(2)acoincidencepointofthepair{TjS}ifTx=Sx;
(3)acommonfixedpointofthepair{TjS}ifx=Tx=Sx.
F(T)jC(TjS)andF(TjS)denotesetofallfixedpointsofTjthesetofall
coincidencepointsofthepair{TjS}jandthesetofallcommonfixedpoints
ofthepair{TjS}jrespectively.
Definition3.LetEbeaq−starshapedsubsetofaconvexmetricspace
X,q∈F(S),withEisbothTandSinvariantwhere,TjS:X→X.Put
YTx
q
={gλ:gλ=W(TxjqjA)andA∈[0j1]}j
and,foreachxinXjd(SxjYTx
q
)=inf{d(Sxjgλ):A∈[0j1]}.ThemapT
issaidtobe:
(1)anS−contractionifthereexistsk∈(0j1)suchthat
d(TxjTg)≤kd(SxjSg);
(2)asymptoticallyS−nonexpansiveifthereexistsasequence{kn}jknł
1jwithlim
n→∞
kn=1suchthatd(TnxjTng)≤knd(SxjSg)jforeach
xjginEandn∈N.Ifkn=1jforalln∈N,thenTiscalledan
S−nonexpansivemapping.IfS=I(theidentitymap),thenTisan
asymptoticallynonexpansivemapping;
(3)R−weaklycommutingifthereexistsarealnumberR>0suchthat
d(STxjTSx)≤Rd(TxjSx)
forallxinE;
(4)R−subweaklycommutingifthereexistsarealnumberR>0suchthat
d(TSxjSTx)≤Rd(SxjYTx
q
);
forallx∈E;
(5)uniformlyR-subweaklycommutingifthereexistsarealnumberR>0
suchthat
d(TnSxjSTnx)≤Rd(SxjYT
q
nx
);
forallx∈E.
