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12
AbdelkrimAlioucheandValeriuPopa
Corollary1.Let(Xjd)beametricspace,SjT:X→XandFjG:
X→CB(X)satisfying(3)and
H(FxjGy)≤ad(TxjSy)+b(D(TxjFx)+D(SyjGy))
+C(D(TxjGy)+D(SyjFx))
forallxjy∈X,whereajC>0,b≥0anda+2b+2C<1.Supposethatone
ofS(X)orT(X)iscomplete.Then,(a)holds.Further,ifthepair(TjF)
isR-weaklycommutingoftype(AT)and(SjG)isR-weaklycommutingof
type(AS)attheircoincidencepoints,thereforetheconclusions(b),(c)and
(d)ofTheorem2hold.
Proof.ItfollowsfromTheorem2andExample1.
.
Remark2.InTheoremsof[1]and[8],toprovethatz=Tz,the
authorsused:”Tx2n∈Gx2n11andTz∈Fzimpliesthatd(Tx2njTz)≤
H(Gx2n11jFz)”whichisfalsebecause”a∈Aandb∈Bimpliesd(ajb)≤
H(AjB)”isnottrueingeneralasitshownbythefollowingexample.
0∈Aand1∈B,butd(0j1)=1>H(AjB)=
Example9.Letd(xjy)=|x−y|,A=[0j
1
2
1
2
]andB=[
.Therefore,Theorem1.7
1
4
j1].Wehave
of[8]isfalseasitisprovedbythefollowingexample.
Example10.Let(Xjd)=([1j∞)j|.|),Sx=Tx=x2+1andFx=
Gx=[2jx+3]forallx∈X.Itiseasytoverifythatforallxjy∈X
d(SxjSy)=|
|x2−y2|
|≥2|x−y|=H(FxjFy)
andhence
H(FxjFy)≤
1
2
d(TxjTy)
≤
1
2
d(TxjTy)+
1
8
D2(TyjFx)+D2(TxjFy)
D(TyjFx)+D(TxjFy)
ifD(TyjFx)+D(TxjFy)/=0andtheotherconditionsofTheorem1.7of
[8]aresatisfied,butSandFhavenocommonfixedpoint.
ThefollowingcorollaryisthecorrectformofTheorem1.7of[8].
Corollary2.Let(Xjd)beacompletemetricspace,TjS:X→Xand
FjG:X→CB(X)satisfying(3)and
H(FxjGy)≤ad(TxjSy)+C
D2(SyjFx)+D2(TxjGy)
D(SyjFx)+D(TxjGy)
