RESEARCH OpenAccess contractionsinpartiallyorderedmetricspaces HassenAydi 1 ErdalKarapinar 2 andMihaiPostolache 3 Correspondencehassen aydiisimarnutn 1 InstitutSup ID: 448489
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RESEARCH OpenAccess Tripledcoincidencepointtheoremsforweak -contractionsinpartiallyorderedmetricspaces HassenAydi 1* ,ErdalKarapinar 2 andMihaiPostolache 3 *Correspondence:hassen. aydi@isima.rnu.tn 1 InstitutSupérieurd Informatique etdesTechnologiesde CommunicationdeHammam Sousse,UniversitédeSousse,Route GP1-4011,HammamSousse, Tunisie Fulllistofauthorinformationis availableattheendofthearticle Abstract Inthisarticle,wepresenttripledcoincidencepointtheoremsfor F : X 3 ® X and g : X ® X satisfyingweak -contractionsinpartiallyorderedmetricspaces.Wealso providenontrivialexamplestoillustrateourresultsandnewconceptspresented herein.Ourresultsunify,generalizeandcomplementvariousknowncomparable resultsfromthecurrentliterature,BerindeandBorcutandAbbasetal. Fixedpointtheoryhasfascinatedhundredsofresearcherssince1922withthecele- bratedBanach sfixedpointtheorem.Thistheoremprovidesatechniqueforsolvinga varietyofappliedproblemsinmathematicalsciencesandengineering.Thereexistsa lastliteratureonthetopicandthisisaveryactivefieldofresearchatpresent.There aregreatnumberofgeneralizationsoftheBanachcontractionprinciple.Bhaskarand Lakshmikantham[1]introducedthenotio nofcoupledfixedpointandprovedsome coupledfixedpointresultsundercertainconditions,inacompletemetricspace endowedwithapartialorder.Later,Lakshmikanthamand iri [2]extendedthese resultsbydefiningthemixed g -monotoneproperty.Moreaccurately,theyproved coupledcoincidenceandcoupledcommonfixedpointtheoremsforamixed g -mono- tonemappinginacompletemetricspaceendowedwithapartialorder.Karap nar [3,4]generalizedtheseresultsonacompleteconemetricspaceendowedwithapartial order.Forotherresultsoncoupledfixe dpointtheory,weaddressthereadersto [5-13]. Tomakeourexpositionselfcontained,inthissectionwerecallsomepreviousnota- tionsandknownresults. Forsimplicity,wedenotefromnowon X × X ··· X × X k terms by X k ,where k Î N and X beanon-emptyset. Let( X , )beapartiallyorderedset.Accordingto[1],themapping F : X 2 X issaid tohave mixedmonotoneproperty if F ( x,y )ismonotonenon-decreasingin x andis monotonenon-increasingin y ,thatis,forany x,y Î X , x 1 x 2 F ( x 1 , y ) F ( x 2 , y ),for x 1 , x 2 X , y 1 y 2 F ( x , y 2 ) F ( x , y 1 ),for y 1 , y 2 X . Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 ©2012Aydietal;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribution License(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium, providedtheoriginalworkisproperlycited. Anelement( x , y ) Î X 2 issaidtobea coupledfixedpoint ofthemapping F : X 2 ® X if F ( x , y )= x and F ( y , x )= y . Theorem1.1 .([1]) Let ( X , ) beanorderedsetsuchthatthereexistsametricdonX suchthat ( X , d ) iscomplete.LetF : X 2 ® Xbeacontinuousmappinghavingthemixed monotonepropertyonX.Assumethatthereexistsk Î [0,1) with d ( F ( x , y ), F ( u , v )) k 2 [ d ( x , u )+ d ( y , v )], forallu x , y u . (1 : 1) Ifthereexistx 0 , y 0 Î Xsuchthatx 0 F ( x 0 , y 0 ) andF ( y 0 , x 0 ) y 0 , then,thereexistx, y Î Xsuchthatx = F ( x , y ) andy = F ( y , x ). Recently,SametandVetro[14]introducedthenotionoffixedpointof N -orderas naturalextensionofthatofcoupledfixed pointandestablishedsomenewcoupled fixedpointtheoremsincompletemetricspaces,usinganewconceptof F -invariant set.Later,BerindeandBorcut[15]obtained existenceanduniquenessoftriplefixed pointresultsinacompletemetricspace,endowedwithapartialorder. Again,let( X , )beapartiallyorderedset.Inacco rdancewith[15],themapping F : X 3 ® X issaidtohavethe mixedmonotoneproperty ifforany x,y,z Î X x 1 , x 2 X , x 1 x 2 F ( x 1 , y , z ) F ( x 2 , y , z ), y 1 , y 2 X , y 1 y 2 F ( x , y 1 , z ) F ( x , y 2 , z ), z 1 , z 2 X , z 1 z 2 F ( x , y , z 1 ) F ( x , y , z 2 ). Anelement( x , y , z ) Î X 3 iscalleda tripledfixedpoint of F if F ( x , y , z )= x , F ( y , x , y )= y and F ( z , y , x )= z . BerindeandBorcut[15]provedthefollowingtheorem. Theorem1.2 .([15]) Let ( X , ) beapartiallyorderedsetand ( X,d ) beacomplete metricspace.LetF:X 3 ® XbeamappinghavingthemixedmonotonepropertyonX. Assumethatthereexistconstantsa,b,c Î [0,1) suchthata + b + c forwhich d ( F ( x , y , z ), F ( u , v , w )) ad ( x , u )+ bd ( y , v )+ cd ( z , w ) (1 : 2) forallx u,y , z w.Assumeeither ( I ) Fiscontinuous,or ( II ) Xhasthefollowingproperties: ( i ) ifnon-decreasingsequencex n ® x,thenx n xforalln , ( ii ) ifnon-increasingsequencey n ® y,theny n yforalln . Ifthereexistx 0 , y 0 , z 0 Î Xsuchthat x 0 F ( x 0 , y 0 , z 0 ), y 0 F ( y 0 , x 0 , y 0 ), andz 0 F ( x 0 , y 0 , z 0 ) thenthereexist x,y,z Î Xsuchthat F ( x , y , z )= x , F ( y , x , y )= y , andF ( z , y , x )= z . Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page2of12 Inthisarticle,weestablishtripledcoincidencepointtheoremsforandsatisfyingnonlinearcontractiveconditions,inpartiallyorderedmetricspaces.Thepresentedtheoremsextendandimprovesomeresultsinlitterature.2MainresultsWeshallstartthissectionbyrecallingthefollowingbasicnotions,introducedby[Abbas,AydiandKarapnar,Tripledcommonfixedpointinpartiallyorderedmetricspaces,submitted].Inthisrespect,letusconsider()apartiallyorderedset,twomappings.Themappingissaidtohavethemixedg-mono-tonepropertyifforanyx,y,z x1,x2X,gx1gx2F(x1,y,z)F(x2,y,z),y1,y2X,gy1gy2F(x,y1,z)F(x,y2,z),z1,z2X,gz1gz2F(x,y,z1)F(x,y,z2). Anelement()iscalledatripledcoincidencepoint ,and while(gx,gy,gz)issaidatripledpointofcoincidenceofmappings.Moreover,x,y,z)iscalledatripledcommonfixedpoint ,and Atlast,mappingsarecalled ))= Inthesamepaper,theyprovedthefollowingresult.Theorem2.1beapartiallyorderedsetandsupposethereisametricdonXsuchthatX,disacompletemetricspace.Assumethereisafunction:[0,+[0,+suchthattforeacht�0.AlsosupposeF:XXandgXaresuchthatFhasthemixedg-monotonepropertyandsupposethereexistp,q,r[0,1)withpsuchthat d(F(x,y,z),F(u,v,wpdgxgu)+qd(gy,gv)+rd(gzgw), foranyx,y,zXforwhichgxgu,ggyandgzSupposeFgiscontinuousandcommuteswithF.SupposeeitherFiscontinuous,orXhasthefollowingproperties:ifnon-decreasingsequencegxrespectively,gzthengxrespectively,gzforallnifnon-increasingsequencegyy,thengyyforallnIfthereexistxXsuchthatgxandgzthenthereexistx,y,zXsuchthat andF thatis,FandghaveatripledcoincidencepointetalFixedPointTheoryandApplicationshttp://www.fixedpointtheoryandapplications.com/content/2012/1/44Page3of12 Beforestartingtointroduceourresults,letusconsiderthesetoffunctions = \b :[0,+ ) \b [0,+ ) | ( t ) t andlim r \b t + ( r ) t , t � 0 \t . Ourfirstmainresultisthefollowing: Theorem2.2 . Let ( X , ) beapartiallyorderedsetandsupposethereisametricdon Xsuchthat ( X,d ) isacompletemetricspace.SupposeF : X 3 ® Xandg : X ® Xare suchthatFhasthemixedg-monotonepropertyandF ( X 3 ) g ( X ). Assumethereisa function ÎF suchthat d ( F ( x , y , z ), F ( u , v , w ))+ d ( F ( y , x , y ), F ( v , u , v ))+ d ( F ( z , y , x ), F ( w , v , u )) 3 \n d ( gx , gu )+ d ( gy , gv )+ d ( gz , gw ) 3 , (2 : 2) foranyx,y,z,u, ,w Î Xforwhichgx gu,g gyandgz gw.AssumethatFis continuous,giscontinuousandcommuteswithF.Ifthereexistx 0 , y 0 , z 0 Î Xsuchthat gx 0 F ( x 0 , y 0 , z 0 ), gy 0 F ( y 0 , x 0 , y 0 ) andgz 0 F ( z 0 , y 0 , x 0 ), (2 : 3) thenthereexistx,y,z Î Xsuchthat F ( x , y , z )= gx , F ( y , x , y )= gy , andF ( z , y , x )= gz , thatis,Fandghaveatripledcoincidencepoint . Proof .Let x 0 , y 0 , z 0 Î X besuchthat gx 0 F ( x 0 , y 0 , z 0 ), gy 0 F ( y 0 , x 0 , y 0 )and gz 0 F ( z 0 , y 0 , x 0 ).Wecanchoose x 1 , y 1 , z 1 Î X suchthat gx 1 = F ( x 0 , y 0 , z 0 ), gy 1 = F ( y 0 , x 0 , y 0 )and gz 1 = F ( z 0 , y 0 , x 0 ). (2 : 4) Thiscanbedonebecause F ( X 3 ) g ( X ).Continuingthisprocess,weconstruct sequences{ x n },{ y n },and{ z n }in X suchthat gx n +1 = F ( x n , y n , z n ), gy n +1 = F ( y n , x n , z n ),and gz n +1 = F ( z n , y n , x n ). (2 : 5) Byinduction,wewillprovethat gx n gx n +1 , gy n +1 gy n ,and gz n gz n +1 . (2 : 6) Since gx 0 F ( x 0 , y 0 , z 0 ), gy 0 F ( y 0 , x 0 , y 0 ),and gz 0 F ( z 0 , y 0 , x 0 ),thereforeby(2.4) wehave gx 0 gx 1 , gy 1 gy 0 ,and gz 0 gz 1 . Thus(2.6)istruefor n =0.Wesupposethat(2.6)istrueforsome n �0.Since F hasthemixed g -monotoneproperty,by gx n gx n +1 , gy n +1 gy n ,and gz n gz n +1 ,we havethat gx n +1 = F ( x n , y n , z n ) F ( x n +1 , y n , z n ) F ( x n +1 , y n , z n +1 ) F ( x n +1 , y n +1 , z n +1 )= gx n +2 , Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page4of12 gy n +2 = F ( y n +1 , x n +1 , y n +1 ) F ( y n +1 , x n , y n +1 ) F ( y n , x n , y n +1 ) F ( y n , x n , y n )= gy n +1 , and gz n +1 = F ( z n , y n , x n ) F ( z n +1 , y n , x n ) F ( z n +1 , y n +1 , x n ) F ( z n +1 , y n +1 , x n +1 )= gz n +2 . Thatis,(2.6)istrueforany n Î N .Ifforsome k Î N gx k = gx k +1 , gy k = gy k +1 ,and gz k = gz k +1 , then,by(2.5),( x k ,y k ,z k )isatripledcoincidencepointof F and g .Fromnowon,we assumethatatleast gx n \t = gx n +1 or gy n \t = gy n +1 or gz n \t = gz n +1 (2 : 7) forany n Î N .From(2.6)andtheinequality(2.2) d ( gx n +1 , gx n )+ d ( gy n +1 , gy n )+ d ( gz n +1 , gz n ) = d ( F ( x n , y n , z n ), F ( x n 1 , y n 1 , z n 1 ))+ d ( F ( y n , x n , y n ), F ( y n 1 , x n 1 , y n 1 )) + d ( F ( z n , y n , x n ), F ( z n 1 , y n 1 , x n 1 )) 3 \n 1 3 ( d ( gx n , gx n 1 )+ d ( gy n , gy n 1 )+ d ( gz n , gz n 1 ) . Foreach n 1,take n := 1 3 ( d ( gx n , gx n 1 )+ d ( gy n , gy n 1 )+ d ( gz n , gz n 1 )). (2 : 8) Onecanwrite n +1 ( n ) n 1. (2 : 9) By(2.7),wehave n �0.Havinginmind ( t ) t foreach t �0,sowehave ( n ) n . From(2.9),weget n +1 n n 1, thatis,thesequence{ n }isnon-negativeanddecreasing.Therefore,thereexists some 0suchthat lim n \b + n =lim n \b + 1 3 d ( gx n , gx n 1 )+ d ( gy n , gy n 1 )+ d ( gz n , gz n 1 ) = + . (2 : 10) Weshallprovethat =0.Assume,onthecontrary,that �0.Thenbyletting n ® + in(2.9)wehave 0 =lim n \b + n +1 lim n \b + ( n )=lim r \b + ( r ) , whichisacontradiction.Thus, =0,andby(2.10),weget lim n \b + n =0. (2 : 11) Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page5of12 Wenowprovethat{ gx n },{ gy n },and{ gz n }areCauchysequencesin( X,d ). Suppose,onthecontrary,thatatleastoneof{ gx n },{ gy n },and{ gz n }isnotaCauchy sequence.So,thereexists �0forwhichwecanfindsubsequences{ gx n (k) },{ gx m (k) }of { gx n },{ gy n ( k ) },{ gy m (k) }of{ gy n },and{ gz n ( k ) },{ gz m ( k ) }of{ gz n }with n ( k �) m ( k ) k suchthat d ( gx n ( k ) , gx m ( k ) )+ d ( gy n ( k ) , gy m ( k ) )+ d ( gz n ( k ) , gz m ( k ) ) . (2 : 12) Additionally,correspondingto m ( k ),wemaychoose n ( k )suchthatitisthesmallest integersatisfying(2.12)and n ( k �) m ( k ) k .Thus, d ( gx n ( x ) 1 , gx m ( k ) )+ d ( gy n ( k ) 1 , gy m ( k ) )+ d ( dz n ( k ) 1 , gz m ( k ) ) . (2 : 13) Byusingtriangleinequalityandhavinginmind(2.12)and(2.13) t k = d ( gx n ( k ) , gx m ( k ) )+ d ( gy n ( k ) , gy m ( k ) )+ d ( gz n ( k ) , gz m ( k ) ) d ( gx n ( k ) , gx n ( k ) 1 )+ d ( gx n ( k ) 1 , gx m ( k ) )+ d ( gy n ( k ) , gy n ( k ) 1 ) + d ( gy n ( k ) 1 , gy m ( k ) )+ d ( gz n ( k ) , gz n ( k ) 1 )+ d ( gz n ( k ) 1 , gz m ( k ) ) d ( gx n ( k ) , gx n ( k ) 1 )+ d ( gy n ( k ) , gy n ( k ) 1 )+ d ( gz n ( k ) , gz n ( k ) 1 )+ . (2 : 14) Letting k ® in(2.14)andusing(2.11) lim k \b t k =lim k \b d ( gx n ( k ) , gx m ( k ) )+ d ( gy n ( k ) , gy m ( k ) )+ d ( gz n ( k ) , gz m ( k ) )= . (2 : 15) Againbytriangleinequality, t k = d ( gx n ( k ) , gx m ( k ) )+ d ( gy n ( k ) , gy m ( k ) )+ d ( gz n ( k ) , gz m ( k ) ) d ( gx n ( k ) , gx n ( k )+1 )+ d ( gx n ( k )+1 , gx m ( k )+1 )+ d ( gx m ( k )+1 , gx m ( k ) ) + d ( gy n ( k ) , gy n ( k )+1 )+ d ( gy n ( k )+1 , gy m ( k )+1 )+ d ( gy m ( k )+1 , gy m ( k ) ) + d ( gz n ( k ) , gz n ( k )+1 )+ d ( gz n ( k )+1 , gy m ( k )+1 )+ d ( gz m ( k )+1 , gz m ( k ) ) n ( k )+1 + m ( k )+1 + d ( gx n ( k )+1 , gx m ( k )+1 )+ d ( gy n ( k )+1 , gy m ( k )+1 ) + d ( gz n ( k )+1 , gz m ( k )+1 ). (2 : 16) Since n ( k �) m ( k ),then gx n ( k ) gx m ( k ) , gy n ( k ) gy m ( k ) , gz n ( k ) gz m ( k ) . (2 : 17) Take(2.17)in(2.2)toget d ( gx n ( k )+1 , gx m ( k )+1 )+ d ( gy n ( k )+1 , gy m ( k )+1 )+ d ( gz n ( k )+1 , gz m ( k )+1 ) = d ( F ( x n ( k ) , y n ( k ) , z n ( k ) ), F ( x m ( k ), y m ( k ) , z m ( k ) ) + d ( F ( y n ( k ) , x n ( k ) , y n ( k ) ), F ( y m ( k ) , x m ( k ) , y m ( k ) ) + d ( F ( z n ( k ) , y n ( k ) , x n ( k ) ), F ( z m ( k ) , y m ( k ) , x m ( k ) )) 3 \n 1 3 [ d ( gx n ( k ) , gx m ( k ) )+ d ( gy n ( k ) , gy m ( k ) )+ d ( gz n ( k ) , gz m ( k ) )] =3 \n t k 3 . Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page6of12 Combiningthisin(2.16),weobtainthat t k n ( k )+1 + m ( k )+1 + d ( gx n ( k )+1 , gx m ( k )+1 )+ d ( gy n ( k )+1 , gy m ( k )+1 ) + d ( gz n ( k )+1 , gz m ( k )+1 ) n ( k )+1 + m ( k )+1 +3 \n t k 3 . Letting k ® andhavinginmind(2.11)and(2.15),weget 3lim k \b + \n 1 3 t k =3lim r \b \f 1 3 t \r + ( r ) 3 \n 1 3 = , whichisacontradiction.Thisshowsthat{ gx n },{ gy n },and{ gz n }areCauchysequences in(X, d ). Since X iscomplete,thereexist x,y,z Î X suchthat lim n \b + gx n = x ,lim n \b + gy n = y andlim n \b + gz n = z . (2 : 18) From(2.18)andthecontinuityof g . lim n \b + g ( gx n )= gx ,lim n \b + g ( gy n )= gy ,andlim n \b + g ( gz n )= gz . (2 : 19) Fromthecommutativityof F and g ,wehave g ( gx n +1 )= g ( F ( x n , y n , z n ))= F ( gx n , gy n , gz n ), g ( gy n +1 )= g ( F ( y n , x n , y n ))= F ( gy n , gx n , gy n ), g ( gz n +1 )= g ( F ( z n , y n , x n ))= F ( gz n , gy n , gx n ). (2 : 20) Nowweshallshowthat gx = F ( x,y,z ), gy = F ( y,x,y ),and gz = F ( z,y,x ). Supposethat F iscontinuous.Letting n ® + in(2.20),thereforeby(2.18)and (2.19),weobtain gx =lim n \b + g ( gx n +1 )=lim n \b + F ( gx n , gy n , gz n ) = F lim n \b + gx n ,lim n \b + gy n ,lim n \b + gz n = F ( x , y , z ), gy = lim n \b + g ( gy n +1 )=lim n \b + F ( gy n , gx n , gy n ) = F lim n \b + gy n ,lim n \b + gx n ,lim n \b + gy n = F ( y , x , y ), and gz =lim n \b + g ( gz n +1 )=lim n \b + F ( gz n , gy n , gx n ) = F lim n \b + gz n ,lim n \b + gy n ,lim n \b + gx n = F ( z , y , x ). Wehaveprovedthat F and g haveatripledcoincidencepoint. Corollary2.3 . Let ( X , ) beapartiallyorderedsetandsupposethereisametricdon Xsuchthat ( X,d ) isacompletemetricspace.SupposeF : X 3 ® Xandg : X ® Xare suchthatFhasthemixedg-monotonepropertyandF ( X 3 ) g ( X ). Assumethereexists a Î [0,1) suchthat Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page7of12 d ( F ( x , y , z ), F ( u , v , w ))+ d ( F ( y , x , y ), F ( v , u , v ))+ d ( F ( z , y , x ), F ( w , v , u )) ( d ( gx , gu )+ d ( gy , gv )+ d ( gz , gw )), foranyx,y,z,u, ,w Î Xforwhichgx gu,g gy,andgz gw.AssumethatFis continuous,giscontinuousandcommuteswithF.Ifthereexistx 0 , y 0 , z 0 Î Xsuchthat gx 0 F ( x 0 , y 0 , z 0 ), gy 0 F ( y 0 , x 0 , y 0 ),and gz 0 F ( z 0 , y 0 , x 0 ), thenthereexistx,y,z Î Xsuchthat F ( x , y , z )= gx , F ( y , x , y )= gy , andF ( z , y , x )= gz , thatis,Fandghaveatripledcoincidencepoint . Proof .Itfollowsbytaking ( t )= a t inTheorem2.2. Inthefollowingtheorem,weomitthecontinuityhypothesisof F .Weneedthefol- lowingdefinition. Definition2.1 . Let ( X , ) beapartiallyorderedsetanddbeametriconX.Wesay that ( X , d , ) isregularifthefollowingconditionshold : (i) ifanon-decreasingsequence ( x n ) issuchthatx n ® x,thenx n xforalln , (ii) ifanon-increasingsequence ( y n ) issuchthaty n ® y,theny y n foralln . Theorem2.4 . Let ( X , ) beapartiallyorderedsetanddbeametriconXsuchthat ( X,d , ) isregular.Supposethatthereexist ÎF andmappingsF : X 3 ® Xandg : X ® Xsuchthat (2.2) holdsforanyx,y,z,u, ,w Î Xforwhichgx gu,g gyandgz gw.Supposealsothat ( g ( X ), d ) iscomplete,Fhasthemixedg-monotonepropertyand F ( X 3 ) g ( X ). Ifthereexistx 0 , y 0 , z 0 Î Xsuchthatgx 0 F ( x 0 , y 0 , z 0 ), gy 0 F ( y 0 , x 0 , y 0 ), andgz 0 F ( z 0 , y 0 , x 0 ), thenthereexistx,y,z Î Xsuchthat F ( x , y , z )= gx , F ( y , x , y )= gy , andF ( z , y , x )= gz , thatis,Fandghaveatripledcoincidencepoint . Proof .ProceedingexactlyasinTheorem2.2,wehavethat( gx n ),( gy n ),and( gz n )are Cauchysequencesinthecompletemetricspace( g ( X ), d ).Then,thereexist x,y,z Î X suchthat gx n ® gx,gy n ® gy ,and gz n ® gz .Since( gx n )and( gz n )arenon-decreasing and( gy n )isnon-increasing,usingtheregularityof( X,d , ),wehave gx n gx,gz n gz , and gy gy n forall n 0.If gx n = gx,gy n = gy ,and gz n = gz forsome n 0,then gx = gx n gx n +1 gx = gx n ,gz = gz n gz n +1 gz = gz n ,and gy gy n +1 gy n = gy ,which impliesthat gx n = gx n +1 = F ( x n ,y n ,z n ), gy n = gy n +1 = F ( y n ,x n ,y n ),and gz n = gz n +1 = F ( z n ,y n ,x n ),thatis,( x n ,y n ,z n )isatripledcoincidencepointof F and g .Then,wesup- posethat( gx n ,gy n ,gz n ) ( gx,gy,gz )forall n 0.Usingthetriangle inequality,(2.2) andtheproperty ( t ) t forall t �0, d ( gx , F ( x , y , z )) d ( gx , gx n +1 )+ d ( gx n +1 , F ( x , y , z )) = d ( gx , gx n +1 )+ d ( F ( x , y , z ), F ( x n , y n , z n )) d ( gx , gx n +1 )+3 \n 1 3 [ d ( gx n , gx )+ d ( gy n , gy )+ d ( dz n , gz )] d ( gx , gx n +1 )+ d ( gx n , gx )+ d ( gy n , gy )+ d ( gz n , gz ). (2 : 21) Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page8of12 intheaboveinequalityweobtainthatx,y,z))=0,soy,zAnalogously,wefindthat F(y,x,y)=gy,F(z,y,x)=gz thus,wehaveprovedthathaveatripledcoincidencepoint.Corollary2.5beapartiallyorderedsetandsupposethereisametricdonXsuchthatisregular.SupposeF:XXandgXaresuchthatFhasthemixedg-monotonepropertyandFAssumethereexistsists)such ))+))+ foranyx,y,z,u,Xforwhichgxgu,ggy,andgzgw.Supposealsothatiscomplete.IfthereexistxXsuchthat )and thenthereexistx,y,zXsuchthat andF thatis,Fandghaveatripledcoincidencepoint.ItfollowsbytakinginTheorem2.4.Now,weshallprovetheexistenceandtheuniquenessofatripledcommonfixedpointtheorem.Foraproductofapartialorderedset(),wedefineapartialorderinginthefollowingway:Forall(x,y,z),( (x,y,z)(u,v,r)\nxu,yvandzr. Wesaythat(x,y,z)and()arecomparableif )or( Also,wesaythat(x,y,z)isequalto()ifandonlyifu,yTheorem2.6InadditiontohypothesisofTheoremsupposethatforallx,y,zandinXthereexistsa,b,cinXsuchthata,b,cb,a,bc,b,aiscomparabletox,y,zF(y,x,yz,y,x,u,)).Also,assumethatisnon-decreasing.Then,Fandghaveauniquetripledcommonfixedpointx,y,zthatis andz Proof.DuetoTheorem2.2,thesetoftripledcoincidencepointsofandisnotempty.Assumenow,that(x,y,z)and()aretwotripledcoincidencepointsof,thatis, ,and,and Weshallshowthat(gx,gy,gz)and(gu,g,gr)areequal.etalFixedPointTheoryandApplicationshttp://www.fixedpointtheoryandapplications.com/content/2012/1/44Page9of12 Byassumption,thereis( a,b,c )in X 3 suchthat( F ( a,b,c ), F ( b,a,b ), F ( c,b,a ))is comparableto( F ( x,y,z ), F ( y,x,y ), F ( z,y,x ))and( F ( u, ,r ), F ( ,u,v ), F ( r, ,u )). Definethesequences{ ga n },{ gb n },and{ gc n }suchthat a = a 0 , b = b 0 , c = c 0 and ga n = F ( a n 1 , b n 1 , c n 1 ), gb n = F ( b n 1 , a n 1 , b n 1 ), gc n = F ( c n 1 , b n 1 , a n 1 ), forall n .Further,set x 0 = x,y 0 = y,z 0 = z and u 0 = u, 0 = ,r 0 = r ,andsimilar definethesequences{ gx n },{ gy n },{ gz n }and{ gu n },{ g n },{ gr n }.Then, gx n = F ( x , y , z ), gu n = F ( u , v , r ), gy n = F ( y , x , y ,), gv n = F ( v , u , v ), gz n = F ( z , y , x ), gr n = F ( r , v , u ,), (2 : 23) forall n 1.Since( F ( x,y,z ), F ( y,x,y ), F ( z,y,x ))=( gx 1 , gy 1 , gz 1 )=( gx,gy,gz )is comparableto( F ( a,b,c ), F ( b,a,b ), F ( c,b,a ))=( ga 1 , gb 1 , gc 1 ),thenitiseasytoshow that( gx,gy,gz ) ( ga 1 , gb 1 , gc 1 ).Recursively,wegetthat ( gx , gy , gz ) ( ga n , gb n , gc n )forall n 0. (2 : 24) By(2.24)and(2.2),wehave d ( gx , ga n +1 )+ d ( gb n +1 , gy )+ d ( gz , gc n +1 )= d ( F ( x , y , z ), F ( a n , b n , c n )) + d ( F ( b n , a n , b n ), F ( y , x , y ))+ d ( F ( z , y , x ), F ( c n , b n , a n ) 3 \n d ( gx , ga n )+ d ( gy , gb n )+ d ( gz , gc n ) 3 . (2 : 25) Set \b n = d ( gx , ga n )+ d ( gy , gb n )+ d ( gz , gc n ) 3 From(2.25),wededucethat g n +1 ( g n ).Since isnon-decreasing,itfollows \b n n ( \b 0 ). Fromthedefinitionof F ,weget lim n \b + n ( t )=0 .Then,wehave lim n \b + \b n =0 .Thus, lim n \b d ( gx , ga n )=0,lim n \b d ( gy , gb n )=0,lim n \b d ( gz , gc n )=0. (2 : 26) Byanalogy,weshowthat lim n \b d ( gu , ga n )=0,lim n \b d ( gv , gb n )=0,lim n \b d ( gr , gc n )=0. (2 : 27) Combining(2.26)and(2.27)yieldsthat( gx,gy,gz )and( gu,g ,gr )areequal. Since gx = F ( x,y,z ), gy = F ( y,x,y ),and gz = F ( z,y,x ),bythecommutativityof F and g ,wehave g ( gx )= g ( F ( x , y , z ))= F ( gx , gy , gz ), g ( gy )= g ( F ( y , x , y ))= F ( gy , gx , gy ), g ( gz )= g ( F ( z , y , x ))= F ( gz , gy , gx ). Denote gx = x ,gy = y ,and gz = z .Fromtheprecedentidentities, gx = F ( x , y , z ), gy = F ( y , x , y ),and gz = F ( z , y , x ), Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page10of12 thatis,( x ,y ,z )isatripledcoincidencepointof F and g .Consequently,( gx ,gy ,gz ) and( gx,gy,gz )areequal,thatis, gx = gx ,gy = gy ,and gz = gz . Wededuce gx = gx = x ,gy = gy = y ,and gz = gz = z .Therefore,( x ,y ,z )isa tripledcommonfixedof F and g .ItsuniquenessfollowsfromTheorem2.2. 3Examples RemarkthatTheorem2.2ismoregeneralthanTheorem2.1,sincethecontractive condition(2.2)isweakerthan(2.1),afactwhichisclearlyillustratedbythefollowing example. Example3.1 .Let X = with d ( x,y )=| x - y |andnaturalorderingandlet g : X ® X , F : X 3 ® X begivenby g ( x )= n +1 n x , n =1,2, ... , x X ; F ( x , y , z )= x ,( x , y , z ) X 3 . Itisclearthat F iscontinuousandhasthemixed g -monotoneproperty.Wenow take ( t )= n n +1 t .Weshallshowthat(2.2)holdsforall gx gu,gy g ,and gz gw . Let x,y,z,u, ,and w suchthat gx gu,gy g ,and gz gw ,andbydefinitionof g , itmeansthat x u,y and z w ,sowehave d ( F ( x , y , z ), F ( u , v , w ))+ d ( F ( y , x , y ), F ( v , u , v ))+ d ( F ( z , y , x ), F ( w , v , u )) = | x u | + y v + | z w | =3 \n d ( gx , gu )+ d ( gy , gv )+ d ( gz , gw ) 3 . whichisthecontractivecondi tion(2.2).Ontheotherhand, x 0 =0, y 0 =0, z 0 =0 satisfy(2.3).AllthehypothesesofTheorem2.2areverified,and(0,0,0)isatripled coincidencepointof F and g . Ontheotherhand,assumethat(2.1)holds.Then,thereexist p,q,r 0suchthat p + 2 q + r 1and satisfying ( t ) foreach t �0.If x � u,z = w and y = ,wehave 0 | x u | = d ( F ( x , y , z ), F ( u , v , w )) ( pd ( gx , gu )+ qd ( gy , gv )+ rd ( gz , gw )) = \n n +1 n p | x u | n +1 n p | x u | , whichimplies p � n n +1 forany n 1,andletting n ® + ,weget p 1,thatisa contradiction.Thus,Theorem2.1isnotapplicableinthiscase. FollowingexampleshowsthatTheorem2.2ismoregeneralthanTheorem1.2. Example3.2 . LetX = beendowedwiththeusualorderingandtheusualmetric. Considerg : X ® XandF : X 3 ® Xbegivenbytheformulas g ( x )= x , F ( x , y , z )= 3 x 6 y +3 z 16 , forallx , y , z X Take :[0, ) ® [0, ) begivenby ( t )= 3 t 4 forallt Î [0, ). ItisclearthatallconditionsofTheorem2.2aresatisfied.Moreover ,(0,0,0) isa tripledcoincidencepoint(alsoacommonfixedpoint)ofFandg . Aydi etal . FixedPointTheoryandApplications 2012, 2012 :44 http://www.fixedpointtheoryandapplications.com/content/2012/1/44 Page11of12 Now,forx = u,z = wand � y,wehave d ( F ( x , y , z ), F ( u , v , w )= 3 8 ( v y ) � 1 3 ( v y ) k 3 [ d ( x , u )+ d ( y , v ) d ( z , w )], foranyk Î [0,1), thatistheresultofBerindeandBorcut [15] givenbyTheorem1.2is notapplicable ( fora = b = c = k 3 ) . Authordetails 1 InstitutSupérieurd InformatiqueetdesTechnologiesdeCommunicationdeHammamSousse,UniversitédeSousse, RouteGP1-4011,HammamSousse,Tunisie 2 DepartmentofMathematics,At õ l õ mUniversity,Incek,Ankara06836,Turkey 3 UniversityPolitehnicaofBucharest,FacultyofAppliesSciences,313SplaiulIndependen ei,Romania Authors contributions Allauthorscontributedequallyandsignificantlyinwritingthisarticle.Allauthorsreadandapprovethefinal manuscript. Competinginterests Theauthorsdeclarethattheyhavenocompetinginterests. 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