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8
MujahidAbbas
(ł)Cq−commutingifSTx=TSxforallx∈Cq(SjT)jwhereCq(SjT)=
U{C(SjTk):0≤k≤1}jandTkx=W(Txjqjk).
ClearlyCq−commutingmapsareweaklycompatiblebutconverseisnot
trueingeneral(seeforexample[2]).
AselfmappingTonaconvexmetricspaceXissaidtobe
(7)affineonEif
T(W(xjgjA))=W(TxjTgjA)j
forallxjg∈EandA∈(0j1);
(8)uniformlyasymptoticallyregularonEif,foreach5>0jthereexists
apositiveintegerNsuchthatd(TnxjTng)<5forallnłNandforall
xinE.
Definition4.LetEbeaq-starshapedsubsetofaconvexmetricspace
XjandTjS:E→Ebemapswithq∈F(S).ThenTandSaresaidtobe
uniformlyCq−commutingonEifSTnx=TnSxforallx∈Cq(SjT)and
n∈N.
Clearly,uniformlyCq-commutingmapsonEareCq-commutingbutnot
converselyingeneral,asthefollowingexampleshows.
Example3.LetXbesetofallrealnumberswithusualmetricand
E=[1j∞).Define,Tx=2x−1andSx=x2,forallx∈E.Take,q=1.
ThenEisq−starshapedwithSq=qandCq(SjT)={1}.NotethatS
andTareCq-commutingmapsbutnotuniformlyCq-commuting,because
ST21/=T2S1.
UniformlyR−subweaklycommutingmapsareuniformlyCq-commuting
buttheconversedoesnotholdingeneral,forthis,weconsiderafollowing
example.
Example4.LetXbesetofallrealnumberswithusualmetric,and
E=[0j∞).If,
Sx={x
2
x
if
if
0≤x<1j
xł1
and
Tx={1
2
x2
if
if
0≤x<1j
xł1j
thenEis1−starshapedwithS1=1andCq(SjT)=[1j∞].NotethatS
andTareuniformlyCq−commutingbutnotR−weaklycommutingforall
R>0.ThusSandTareneitherR−subweaklycommutingnoruniformly
R−subweaklycommutingmaps.
