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FASCICULI
MATHEMATICI
Nr41
2009
AbdelkrimAlioucheandValeriuPopa
COINCIDENCEANDCOMMONFIXEDPOINT
THEOREMSVIAR-WEAKCOMMUTATIVITY
OFTYPE(AT)
Abstract.Weprovecommonfixedpointtheoremsfortwopairs
ofhybridmappingssatisfyingimplicitrelationsincompletemetric
spacesusingtheconceptofR−weakcommutativityoftypeAT
andwecorrecterrorsof[1],[3]and[8].Ourtheoremsgeneralize
resultsof[1-3],[8],[12-16]and[21].
Keywords:hybridmappings,commonfixedpoint,R−weakly
commutingoftypeAT,metricspace.
AMSMathematicsSubjectClassification:54H25,47H10.
1.Introductionandpreliminaries
Let(Xjd)beametricspace.Forx∈XandA⊂X,D(xjA)=inf{d(xjy)j
y∈A}.
LetCB(X)bethesetofallnonemptyclosedandboundedsubsetsofX.
LetHbetheHausdorffmetricwithrespecttoddefinedby
H(AjB)=max{sup
a∈A
D(ajB)jsup
b∈B
D(Ajb)}forallAjB∈CB(X).
Itiswellknownthat(CB(X)jH)isametricspaceandif(Xjd)iscom-
plete,then(CB(X)jH)isalsocomplete
Lemma1([9]).IfAjB∈CB(X)andk>1,thenforeacha∈A,there
existsb∈Bsuchthatd(ajb)≤kH(AjB).
Letf:X→Xbeasingle-valuedmappingandT:X→CB(X)bea
multi-valuedmapping.
Definition1.1)Apointx∈Xissaidtobeacoincidencepointoff
andTiffx∈Tx.WedenotebyC(fjT)thesetofallcoincidencepoints
offandT.
2)Apointx∈XisafixedpointofTifx∈Tx.
