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6
AbdelkrimAlioucheandValeriuPopa
Definition2.1)fandTaresaidtobecommuting[4]inXifforall
x∈X,fTx=Tfx.
2)fandTaresaidtobeweaklycommutingonX[17,18]ifforall
x∈X,fTx∈CB(X)and
H(fTxjTfx)≤D(fxjTx)
3)fandTaresaidtobecompatible[5,7]ifforallx∈XjfTx∈CB(X)
and
n→∞
lim
H(fTxnjTfxn)=0
whenever{xn}isasequenceinXsuchthatlim
n→∞
fxn=t∈A=lim
n→∞
Txn
forsomet∈XandA∈CB(X).
Commutingimpliesweaklycommutingimpliescompatible,butthecon-
verseisnottrueingeneral.See[7].
LetT:X→Xbeasingle-valuedmappingandF:X→CB(X)bea
multi-valuedmapping.
Definition3([10],[19]).1)TandFaresaidtobeR-weaklycommuting
atx∈XifTFx∈CB(X)andthereexistsanR>0suchthat
(1)
H(TFxjFTx)≤RD(TxjFx).
2)TandFaresaidtobepointwiseR-weaklycommutingonXifforall
x∈X,TFx∈CB(X)and(1)holdsforsomeR>0.
Definition4([6]).TandFaresaidtobeR-weaklycommutingoftype
(AT)atx∈XifthereexistsanR>0suchthat
(2)
D(TTxjFTx)≤RD(TxjFx).
TandFaresaidtobeR-weaklycommutingoftype(AT)onXifforall
x∈X,(2)holds.
Remark1.IfFisasingle-valuedmapping,thenthedefinitionofR-weak
commutativityoftype(AT)reducestothatofPathaket.al[11].
IfTandFarecompatible,thentheyareR-weaklycommutingoftype
(AT),buttheconverseisnottrueingeneral,see[6].
Thefollowingtheoremwasprovedby[8].
Theorem1.Let(Xjd)beacompletemetricspace,SjT:X→Xand
FjG:X→CB(X)satisfying
(3)
F(X)⊂S(X)andG(X)⊂T(X)j
