AppliedMathematicalSciences,Vol.9,2015,no.86,4291-4297HIKARILtd,www.m- - PDF document

AppliedMathematicalSciences,Vol.9,2015,no.86,4291-4297HIKARILtd,www.m-hikari.comhttp://dx.doi.org/10.12988/ams.2015.54345CommonCoupledCoincidencePointforCoincidentallyCommutingandProperty(E.A.)MapsinIFMSwithoutCompletenessJongSeoParkDepartmentofMathematicsEducationandInstituteofMathematicsEducationChinjuNationalUniversityofEducation,Jinju660-756,SouthKoreaCopyrightc 2015JongSeoPark.ThisarticleisdistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.AbstractInthispaper,wedeneproperty(E.A.)withthecounterpartofthenotionofprop-erty(E.A.)inPark[13],andproveacommoncoupledcoincidencepointtheoremfortwopairsofcoincidentallycommutingandproperty(E.A.)mapsinIFMSwithoutcompleteness.MathematicsSubjectClassication:46S40,47H10,54H25Keywords:Coupledcoincidencepoint,property(E.A.),coincidentallycom-mutingmap1.IntroductionIn2006,Bhaskaret.al.[1]provedtocoupledcontractionmappingtheorem,andthisresultwasgeneralizedtocoupledcoincidencepointtheories([3],[9])undertwoseparatesetsofconditions.Zhuet.al[16]studiedcoupledxedpointtheoremsinfuzzymetricspacesinwhichtheyobtainedafuzzyversionoftheresultof[1].Also,Hu[8]obtainedcommoncoupledxedpointresultsinfuzzymetricspaces.Furthermore,Choudhuryet.al.[2]provedacoupledcoincidencepointtheoremforcoincidentallycommutingmapsinfuzzymetricspaces.Thuscoupledxedpointproblemshavebeenstudiedinstructureswhicharegeneralizationofmetricspaces,inprobabilisticmetricspaces,inG-metricspacesandfuzzymetricspaces. 4292JongSeoParkParket.al.[11]denedanIFMSandprovedaxedpointtheoreminIFMS,Park[12]studiedaxedpointforcommonproperty(E.A.)andweakcompatiblemaps,andobtainedsomecommonxedpointfortheweaklycommutingmaps.Inthispaper,wedeneproperty(E.A.)withthecounterpartofthenotionofproperty(E.A.)inPark[13],andproveacommoncoupledcoincidencepointtheoremfortwopairsofcoincidentallycommutingandproperty(E.A.)mapsinIFMSwithoutcompleteness.2.PreliminariesInthissection,werecallsomedenitions,propertiesandknownresultsintheintuitionisticfuzzysymmetricspaceasfollowing:Letusrecall(see[15])thatacontinuoust�normisaoperation:[0;1][0;1]![0;1]whichsatisesthefollowingconditions:(a)iscommutativeandassociative,(b)iscontinuous,(c)a1=aforalla2[0;1],(d)abcdwheneveracandbd(a;b;c;d2[0;1]).Also,acontinuoust�conormisaoperation:[0;1][0;1]![0;1]whichsatisesthefollowingconditions:(a)iscommutativeandassociative,(b)iscontinuous,(c)a0=aforalla2[0;1],(d)abcdwheneveracandbd(a;b;c;d2[0;1]).Denition2.1.([11])The5�tuple(X;M;N;;)issaidtobeanintuitionis-ticfuzzymetricspace(shortly,IFMS)ifXisanarbitraryset,isacontinuoust�norm,isacontinuoust�conormandM;NarefuzzysetsonX2(0;1)satisfyingthefollowingconditions;forallx;y2Xandt�0,suchthat(a)M(x;y;t)�0,(b)M(x;y;t)=1ifandonlyifx=y,(c)M(x;y;t)=M(y;x;t),(d)M(x;y;t)M(y;z;s)M(x;z;t+s),(e)M(x;y;
):(0;1)!(0;1]iscontinuous,(f)N(x;y;t)�0,(g)N(x;y;t)=0ifandonlyifx=y,(h)N(x;y;t)=N(y;x;t),(i)N(x;y;t)N(y;z;s)N(x;z;t+s),(j)N(x;y;
):(0;1)!(0;1]iscontinuous.Notethat(M;N)iscalledanintuitionisticfuzzymetriconX.ThefunctionsM(x;y;t)andN(x;y;t)denotethedegreeofnearnessandthedegreeofnon-nearnessbetweenxandywithrespecttot,respectively.Inthispaper,weuset-norm=minandt-conorm=max.Denition2.2.([12])LetXbeanIFMS.(a)fxngissaidtobeconvergenttoapointx2Xbylimn!1xn=xiflimn!1M(xn;x;t)=1,limn!1N(xn;x;t)=0forallt�0.(b)fxngiscalledaCauchysequenceifforany�0,thereexistsn02Nsuchthatforallt�0andm;nn0,M(xn;xm;t)�1�;N(xn;xm;t): Commoncoupledcoincidencepoint4293(c)XiscompleteifandonlyifeveryCauchysequenceconvergesinX.Denition2.3.LetXbeanIFMS.Also,letF:XX!X,G:XX!X,h:X!Xandg:X!Xbemaps.(a)Anelement(x;y)2XXiscalledacoupledcoincidencepointofamappingFandhifF(x;y)=hx;F(y;x)=hy:(b)ThemapsFandharecommutingifforallx;y2X,hF(x;y)=F(hx;hy).(c)ThemapsFandharesaidtobecoincidentallycommutingiftheycommuteattheircoupledcoincidencepoints,thatis,ifhx=F(x;y)andhy=F(y;x),forsome(x;y)2XX,thenhF(x;y)=F(hx;hy)andhF(y;x)=F(hy;hx).(d)(x;y)2XXiscalledacommoncoupledcoincidencepointofthepairsof(F;h)and(G;g)ifF(x;y)=hx,F(y;x)=hy,(G(x;y)=gxandG(y;x)=gy.Example2.4.LetXbeanIFMSandletF:XX!Xandh:X!Xbedenedrespectivelyasfollows;F(x;y)=8:1 3ifx�1;0y1;0otherwise;hx=8&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.30; 0 ;&#xTd [;&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.30; 0 ;&#xTd [;&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.30; 0 ;&#xTd [;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;:0ifx=0;100if0x1;1ifx=1;20ifx&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.82; 0 ;&#xTd [;1:ThemapsF;hcommuteattheironlycoupledcoincidencepoint(0;0).There-forethepairofmaps(F;h)iscoincidentallycommuting.Butthepairofmaps(F;h)isnotcommuting.Denition2.5.LetXbeanIFMS.ThemapsF:XX!Xandh:X!Xaresaidtosatisfyproperty(E.A.)ifthereexiststwosequencesfxngandfynginXsuchthatforsomex;y2X,F(xn;yn)=hxn!xasn!1;F(yn;xn)=hyn!yasn!1:Inthispaper,weusethefollowingclassofrealmaps.Dene=f; :[0;1]![0;1]gsatisfyingtheconditions:(i)iscontinuousmonotoneincreasing, iscontinuousmonotonedecreasingon[0,1],(ii)(t)&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.82; 0 ;&#xTd [;tand (t)t)forall0t1with(1)=1and (0)=0.3.MainresultTheorem3.1.LetXbeanIFMS.LetF:XX!X,G:XX!X,h:X!Xandg:X!Xbefourmapsand; 2satisesthefollowingconditions; 4294JongSeoPark(a)forallx;y;u;v2X,s�0and; 2M(F(x;y);G(u;v);s)(minfM(hx;gu;s);M(hy;gv;s);M(hx;F(x;y);s);M(hx;G(u;v);s);M(gu;G(u;v);s);M(gu;F(x;y);s)g);N(F(x;y);G(u;v);s) (maxfN(hx;gu;s);N(hy;gv;s);N(hx;F(x;y);s);N(hx;G(u;v);s);N(gu;G(u;v);s);N(gu;F(x;y);s)g);(b)F(XX)g(X),G(XX)h(X)andh(X),g(X)aretwoclosedsubsetsofX,(c)(h;F)and(g;G)arecoincidentallycommutingpairs.If(h;F)and(g;G)satisfytheproperty(E.A.),thenthereexistx;y2Xsuchthathx=F(x;y),hy=F(y;x),gx=G(x;y)andgy=G(y;x),thatis,thepairsofmaps(h;F)and(g;G)havecommoncoupledcoincidencepointinX.Proof.Since(h;F)and(g;G)satisfytheproperty(E.A.),thereexisttwose-quencesfxngandfynginX,suchthatF(xn;yn)=hxn!xasn!1;F(yn;xn)=hyn!yasn!1:andG(xn;yn)=gxn!xasn!1;G(yn;xn)=gyn!yasn!1:Hencex;y2h(X)\g(X)becauseh(X);g(X)aretwoclosedsubsetsofX.Also,sinceG(XX)h(X),thereexistsu;v2Xsuchthathu=x;hv=y.Thusforalls�0,wehaveM(F(u;v);G(xn;yn);s)(minfM(hu;gxn;s);M(hv;gyn;s);M(hu;F(u;v);s);M(gxn;G(xn;yn);s);M(hu;G(xn;yn);s);M(gxn;F(u;v);s)g);N(F(u;v);G(xn;yn);s) (maxfN(hu;gxn;s);N(hv;gyn;s);N(hu;F(u;v);s);N(gxn;G(xn;yn);s);N(hu;G(xn;yn);s);N(gxn;F(u;v);s)g): Commoncoupledcoincidencepoint4295Asn!1,wehaveforalls�0,M(F(u;v);x;s)(minfM(x;x;s);M(y;y;s);M(x;F(u;v);s);M(x;x;s);M(x;x;s);M(x;F(u;v);s)g)(minf1;1;M(x;F(u;v);s);1;1;M(x;F(u;v);s)g);N(F(u;v);x;s) (maxfN(x;x;s);N(y;y;s);N(x;F(u;v);s);N(x;x;s);N(x;x;s);N(x;F(u;v);s)g) (maxf0;0;N(x;F(u;v);s);0;0;N(x;F(u;v);s)g):Now,ifx6=F(u;v),thenM(F(u;v);x;s)(M(x;F(u;v);s))�M(x;F(u;v);s)andN(F(u;v);x;s) (N(x;F(u;v);s))N(x;F(u;v);s)whichisacontra-dictioninthisinequality.HenceM(x;F(u;v);s)=1andN(x;F(u;v);s)=0whichimpliesthatx=F(u;v).Thereforex=hu=F(u;v).Similarly,wecanprovethaty=hv=F(v;u).SinceF(XX)g(X),thereexistsw;z2Xsuchthatgw=x;gz=y.Thusforalls&#x-278;0,wehaveM(F(xn;yn);G(w;z);s)(minfM(hxn;gw;s);M(hyn;gz;s);M(hxn;F(xn;yn);s);M(gw;G(w;z);s);M(hxn;G(w;z);s);M(gw;F(xn;yn);s)g);N(F(xn;yn);G(w;z);s) (maxfN(hxn;gw;s);N(hyn;gz;s);N(hxn;F(xn;yn);s);N(gw;G(w;z);s);N(hxn;G(w;z);s);N(gw;F(xn;yn);s)g):Asn!1,wehaveforalls&#x-278;0,M(x;G(w;z);s)(minfM(x;x;s);M(y;y;s);M(x;x;s);M(x;G(w;z);s);M(x;G(w;z);s);M(x;x;s)g)(minf1;1;1;M(x;G(w;z);s);M(x;G(w;z);s);1g);N(x;G(w;z);s) (maxfN(x;x;s);N(y;y;s);N(x;x;s);N(x;G(w;z);s);N(x;G(w;z);s);N(x;x;s)g) (maxf0;0;0;N(x;G(w;z);s);N(x;G(w;z);s);0g):Now,ifx6=G(w;z),thenM(x;G(w;z);s)(M(x;G(w;z);s))&#x-278;M(x;G(w;z);s)andN(x;G(w;z);s) (N(x;G(w;z);s))N(x;G(w;z);s)whichisacon-tradictioninthisinequality.HenceM(x;G(w;z);s)=1andN(x;G(w;z);s)=0whichimpliesthatx=G(w;z).Thereforex=gw=G(w;z).Similarly,wecanprovethaty=gz=G(z;w).Thereforex=gw=hu=G(w;z)=F(u;v)andy=gz=hv=F(v;u)=G(z;w).Since(h;F)iscoincidentally 4296JongSeoParkcommuting,hF(u;v)=F(hu;hv)andhF(v;u)=F(hv;hu)whichimplieshx=F(x;y)andhy=F(y;x).Also,since(g;G)iscoincidentallycommut-ing,thereforegG(w;z)=G(gw;gz)andgG(z;w)=G(gz;gw)whichimpliesgx=G(x;y)andgy=G(y;x),thatis,(x;y)isthecommoncoupledcoin-cidencepointofthepairsofmappings(h;F)and(g;G).Thiscompletestheproofofthetheorem.Corollary3.2.LetXbeanIFMS.LetF:XX!Xandh:X!Xbetwomapsand; 2satisesthefollowingconditions;(a)forallx;y;u;v2X,s�0and; 2M(F(x;y);F(u;v);s)(minfM(hx;hu;s);M(hy;hv;s);M(hx;F(x;y);s);M(hu;F(u;v);s);M(hx;F(u;v);s);M(hu;F(x;y);s)g);N(F(x;y);F(u;v);s) (maxfN(hx;hu;s);N(hy;hv;s);N(hx;F(x;y);s);N(hu;F(u;v);s);N(hx;F(u;v);s);N(hu;F(x;y);s)g);(b)F(XX)h(X)andh(X)isaclosedsubsetsofX,(c)(h;F)iscoincidentallycommutingpair.If(h;F)satisfytheproperty(E.A.),thenthereexistx;y2Xsuchthathx=F(x;y)andhy=F(y;x),thatis,(h;F)hascoupledcoincidencepointinX.Proof.TheprooffollowsbyputtingF=G,h=ginTheorem3.1.Corollary3.3.LetXbeanIFMS.LetF:XX!Xbeamapand; 2satisesthefollowingconditions;Forallx;y;u;v2X,s�0and; 2M(F(x;y);F(u;v);s)(minfM(x;u;s);M(y;v;s);M(x;F(x;y);s);M(u;F(u;v);s);M(x;F(u;v);s);M(u;F(x;y);s)g);N(F(x;y);F(u;v);s) (maxfN(x;u;s);N(y;v;s);N(x;F(x;y);s);N(u;F(u;v);s);N(x;F(u;v);s);N(u;F(x;y);s)g):IfFsatisfytheproperty(E.A.),thenthereexistx;y2Xsuchthatx=F(x;y)andy=F(y;x),thatis,FhasxedpointinX.Proof.TheprooffollowsbyputtingF=G,h=g=I(Identityfunction)inTheorem3.1.References[1]T.G.BhaskarandV.Lakshmikantham,Fixedpointtheoremsinpartiallyor-deredmetricspacesandapplications,NonlinearAnal.,65(2006),1379{1393.http://dx.doi.org/10.1016/j.na.2005.10.017 Commoncoupledcoincidencepoint4297[2]B.S.Choudhury,D.GopalandP.Das,Coupledcoincidencepointresultsinfuzzymetricspaceswithoutcompleteness,A.F.M.I.,6(1)(2013),127{133.[3]B.S.ChoudhuryandA.Kundu,Acoupledcoincidencepointresultinpartiallyor-deredmetricspacesforcompatiblemappings,NonlinearAnal.,73(2010),2524{2531.http://dx.doi.org/10.1016/j.na.2010.06.025[4]B.S.ChoudhuryandP.Maity,Coupledxedpointresultsingen-eralizedmetricspaces,Math.Comput.Modeling,54(2011),73{79.http://dx.doi.org/10.1016/j.mcm.2011.01.036[5]A.GeorgeandP.Veeramani,Onsomeresultsinfuzzymetricspaces,FuzzySetsandSystems,64(1994),395{399.http://dx.doi.org/10.1016/0165-0114(94)90162-7[6]D.Gopal,M.ImbadandC.Vetro,Impactofcommonproperty(E.A.)onxedpointtheoremsinfuzzymetricspaces,FixedPointTheoryandAppl.,(2011),ID297360.http://dx.doi.org/10.1155/2011/297360[7]M.Grabiec,Fixedpointinfuzzymetricspaces,FuzzySetsandSystems,27(1988),385{389.http://dx.doi.org/10.1016/0165-0114(88)90064-4[8]X.Q.Hu,Commoncoupledxedpointtheoremsforcontractivemappingsinfuzzymetricspaces,FixedPointTheoryandAppl.,(2011),ID363716.http://dx.doi.org/10.1155/2011/363716[9]V.LakshmikanthamandL.Ciric,Coupledxedpointtheoremsfornonlinearcon-tractionsinpartiallyorderedmetricspaces,NonlinearAnal.,70(2009),4341{4349.http://dx.doi.org/10.1016/j.na.2008.09.020[10]N.V.LuongandN.X.Thuan,Coupledxedpointsinpartiallyor-deredmetricspacesandapplication,NonlinearAnal.,74(2011),983{992.http://dx.doi.org/10.1016/j.na.2010.09.055[11]J.H.Park,J.S.ParkandY.C.Kwun,Acommonxedpointtheoremintheintuition-isticfuzzymetricspace,AdvancesinNaturalComput.DataMining(Proc.2ndICNCand3rdFSKD),(2006),293{300.[12]J.S.Park,Fixedpointtheoremforcommonproperty(E.A.)andweakcompatiblefunctionsinintuitionisticfuzzymetricspace,Int.J.F.I.S.11(3)(2011),149{153.http://dx.doi.org/10.5391/ijs.2011.11.3.149[13]J.S.Park,Somexedpointtheoremusingcommonproperty(E.A.)inintuitionisticfuzzymetricspace,J.KoreanSoc.Math.Educ.Ser.B:PureAppl.Math.,18(3)(2011),255{260.http://dx.doi.org/10.7468/jksmeb.2011.18.3.255[14]J.S.Park,Somecommonxedpointtheoremsfortheweaklycommutingmapsonintuitionisticfuzzymetricspace,F.J.M.S.76(1)(2013),97{104.[15]B.SchweizerandA.Sklar,Statisticalmetricspaces,PacicJ.Math.,10(1960),314{334.http://dx.doi.org/10.2140/pjm.1960.10.313[16]X.H.ZhuandJ.Z.Xiao,Coupledxedpointtheoremsforcontractionsinfuzzymetricspaces,NonlinearAnal.,74(2011),5475{5479.Received:April29,2015;Published:June12,2015