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Coincidenceandcommonfixedpoint...
13
forallxjy∈X,whereajC>0anda+2C<1,wheneverD(TxjGy)+
D(SyjFx)/=0andH(FxjGy)=0wheneverD(TxjGy)+D(SyjFx)=0.
Then,(a)holds.Further,ifthepair(TjF)isR-weaklycommutingoftype
(AT)and(SjG)isR-weaklycommutingoftype(AS)attheircoincidence
points,thereforetheconclusions(b),(c)and(d)ofTheorem2hold.
Proof.Itfollowsfromthefactthat
D2(SyjFx)+D2(TxjGy)
D(SyjFx)+D(TxjGy)
≤D(Sy,
Fx)+D(TxjGy)ifD(TxjGy)+D(SyjFx)/=0andCorollary1.
.
Remark3.In[16]Remark3and[8]Remark5,wehave:”theconditions
inthehypothesisofTheorem3.1of[1]andTheorem1.7of[8],x/=yjFx/=
FyandGx/=GyarenecessarysincethetheoremfailsforFandGtaken
asconstantmappings”.Thisisdemonstratedbythefollowingexample.
Example11.LetX={0j1}jTx=1−xandFx=Gx={0j1}for
allx∈X.Itiseasytoverifythatthemappingssatisfyallthehypothesis
exceptx/=yjFx/=Fy.
Remark4.1)InExample11,wehaveT(0)∈F(0)andT(1)∈F(1);
i.e.,TandFhavecoincidencepoints.SinceT2(0)/=T(0)andT2(1)/=T(1),
TandFhavenocommonfixedpoint
2)Intheoremsof[1],[3]and[8],x/=yjFx/=FyandGx/=Gyarenot
necessaryasitisshownbythefollowingexample.
3)InTheorem1of[21],Sandgarecompatibleshouldbethepairs(Sjf)
and(TjG)arecompatibleandinCorollary2,gshouldbereplacedbyfand
thepair(Sjf)iscompatibleshouldbeadded.
4)In[16],theauthorsmadethefollowingremark.Itisnotyetknown
whethertheirtheoremremainstrueifoneofthemappingsfandTisnot
continuousandTheorem2of[20]yieldsthattheanswerisaffirmative.
Example12.LetX={0j1j
1
2
},Tx=1−xandFx=Gx={0j
1
2
j1}
forallx∈X.Itiseasytoverifythatthemappingssatisfytheconditionsof
theoremsof[1],[3]and[8]exceptx/=y,Fx/=Fy,butT(
andso
1
2
isacommonfixedpointofTandF.
1
2
)=
1
2
∈F(
1
2
)
Asx/=yjFx/=FyandGx/=Gyarenotnecessary,itfollowsthat
theoremof[1]andTheorems3.2and3.3of[3]part(a)arefalse,itsuffices
totakeExample3.8for[1]andX={0j1}jTx=1−xjSx=Ix=Jx=x
andFx=Gx={0j1}forallx∈Xfor[3].
WecanalsoprovethefollowingtheoremwhichgeneralizesTheorems3.2
and3.3of[3].
