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MoosaeiFixedPointTheoryandApplications2014,2014:98http://www.
fixedpointtheoryandapplications.
com/content/2014/1/98RESEARCHOpenAccessCommonxedpointsforsomegeneralizedcontractionpairsinconvexmetricspacesMohammadMoosaei**Correspondence:m.
moosaei@basu.
ac.
irDepartmentofMathematics,Bu-AliSinaUniversity,Hamedan,IranAbstractThepresentstudyfocusesonprovingtheexistenceofcoincidencepointsforself-mappingssatisfyingageneralizedcontractiveconditionwithintheframeworkofconvexmetricspaces.
Theexistenceofcommonxedpointsforweaklycompatibleself-mappingsaswellasBanachoperatorpairsundercertaingeneralizedcontractionsinaconvexmetricspaceisalsoestablished.
MSC:47H09;47H10;47H19;54H25Keywords:Banachoperatorpairs;coincidencepoints;commonxedpoints;compatiblemappings;convexmetricspaces;xedpoints;weaklycompatiblepair1IntroductionandpreliminariesIn,Takahashi[]introducedthenotionofconvexityinmetricspacesandprovedthatallnormedspacesandtheirconvexsubsetsareconvexmetricspaces.
Healsogavesomeexamplesoftheconvexmetricspaceswhicharenotembeddedinanynormed/Banachspaces.
AfterwardGuay,SinghandWhittield[],BegandAzam[],Beg,Azam,AliandMinhas[],ShimizuandTakahashi[],Ciric[],Beg[,],BegandAbbas[],andmanyotherauthorshavestudiedxedpointtheoremsinconvexmetricspaces.
Inthispaper,weintroduce(α,β,γ,η)-generalizedcontractionpairsandstudytheexis-tenceofacoincidencepointforsuchpairsinaconvexmetricspaceundercertaincondi-tions(seeTheorem.
).
Consequently,weprovetheexistenceofacommonxedpointforweaklycompatiblemappingsandalsoBanachoperatorpairswhichare(α,β,γ,η)-generalizedcontractionpairs(seeTheorem.
andTheorem.
).
Wenowreviewnotationsanddenitionsneeded.
WedenotebyNandRthesetofnaturalnumbersandthesetofrealnumbers,respectively.
WealsodenotebyItheidentitymapping.
Inwhatfollows,(X,d)isametricspace,andCisanonemptysubsetofX.
Denition.
LetSandTbetwoself-mappingsofC.
ApointxofCiscalled(i)axedpointofTifTx=x,(ii)acommonxedpointofthepair(S,T)ifSx=Tx=x,and(iii)acoincidencepointofthepair(S,T)ifSx=Tx.
ThesetofxedpointsofTisdenotedbyF(T).
Thesetofcommonxedpoints(respec-tively,coincidencepoints)ofthepair(S,T)isdenotedbyF(S,T)(respectively,C(S,T)).
NotethatC(I,T)=F(T).
Denition.
LetSandTbetwoself-mappingsofC.
ThemappingTiscalled2014Moosaei;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-tionLicense(http://creativecommons.
org/licenses/by/2.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
MoosaeiFixedPointTheoryandApplications2014,2014:98Page2of8http://www.
fixedpointtheoryandapplications.
com/content/2014/1/98(i)acontractionifthereexistsk∈[,)suchthatd(Tx,Ty)≤kd(x,y)forallx,y∈C,(ii)anS-contractionifthereexistsk∈[,)suchthatd(Tx,Ty)≤kd(Sx,Sy)forallx,y∈C,(iii)nonexpansiveifd(Tx,Ty)≤d(x,y)forallx,y∈C,and(iv)S-nonexpansiveifd(Tx,Ty)≤d(Sx,Sy)forallx,y∈C.
Denition.
LetSandTbetwoself-mappingsofC.
Thepair(S,T)issaidtobe(i)commutingifSTx=TSxforallx∈C,(ii)R-weaklycommuting[]ifthereexistsR>suchthatd(STx,TSx)≤Rd(Sx,Tx)forallx∈C.
IfR=,thenthemappingsarecalledweaklycommuting[],(iii)compatible[]iflimn→∞d(STxn,TSxn)=,whenever{xn}∞n=isasequenceinCsuchthatlimn→∞Sxn=limn→∞Txn=xforsomex∈C,and(iv)weaklycompatibleiftheycommuteonC(S,T)i.
e.
STx=TSxforallx∈C(S,T)(see[,]formoredetails).
Itiswellknownthatcommutingmappingsareweaklycommuting,andweaklycommut-ingmappingsareR-weaklymappings.
Moreover,R-weaklymappingsarecompatible,andcompatiblemappingsareweaklycompatible.
Thefollowingexampleshowsthattheconversesoftheaboveresultsarenottrueingeneral.
Example.
LetX=Rwiththeusualmetricd(x,y)=|x–y|forallx,y∈X,wehave:()LetC=[,].
LetSx=xandTx=xforallx∈C.
ItistrivialthatSandTareweaklycommutingbutarenotcommuting.
()LetC=[,∞].
ConsiderSx=x–andTx=xforallx∈C.
ThenSandTare-weaklycommutingbutarenotweaklycommuting(see[]).
()LetC=X,Sx=x,Tx=x,x∈C.
ThenSandTarecompatiblebutarenotR-weaklycommuting(see[,,]formoredetails).
()LetC=[,],anddeneself-mappingsSandTofCbyS()=,S(x)=iforηorη,thenηorηorη<α+γholds.
CompetinginterestsTheauthordeclaresthattheyhavenocompetinginterests.
AcknowledgementsTheauthorisgratefultothereviewersfortheirvaluablecommentswhichimprovedthecontentsofthemanuscript.
Received:28November2013Accepted:27March2014Published:16Apr2014MoosaeiFixedPointTheoryandApplications2014,2014:98Page8of8http://www.
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1186/1687-1812-2014-98Citethisarticleas:Moosaei:Commonxedpointsforsomegeneralizedcontractionpairsinconvexmetricspaces.
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