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Maetal.
FixedPointTheoryandApplications(2015)2015:31DOI10.
1186/s13663-015-0281-xRESEARCHOpenAccessConvergencetheoremsforsplitequalitymixedequilibriumproblemswithapplicationsZhaoliMa1,LinWang2*,Shih-senChang2andWenDuan3*Correspondence:WL64mail@aliyun.
com2CollegeofStatisticsandMathematics,YunnanUniversityofFinanceandEconomics,LongquanRoad,Kunming,650221,ChinaFulllistofauthorinformationisavailableattheendofthearticleAbstractInthispaper,weintroduceanewalgorithmforsolvingsplitequalitymixedequilibriumproblemsintheframeworkofinnite-dimensionalrealHilbertspaces.
Thestrongandweakconvergencetheoremsareobtained.
Asapplication,weshallutilizeourresultstostudythesplitequalitymixedvariationalinequalityproblemandthesplitequalityconvexminimizationproblem.
Ourresultspresentedinthispaperimproveandextendsomerecentcorrespondingresults.
MSC:47H09;47J25Keywords:splitequalitymixedequilibriumproblems;splitequalitymixedvariationalinequalityproblem;splitequalityconvexminimizationproblem1IntroductionLetHbearealHilbertspacewiththeinnerproduct·,·andthenorm·.
LetDbeanonemptyclosedconvexsubsetofH.
LetT:D→Dbeanonlinearmapping.
ThexedpointsetofTisdenotedbyF(T),thatis,F(T)={x∈D:Tx=x}.
AmappingTissaidtobenonexpansiveifTx–Ty≤x–y,x,y∈D.
(.
)IfDisaboundednonemptyclosedconvexsubsetofHandTisanonexpansivemappingofDintoitself,thenF(T)isnonempty[].
AmappingTissaidtobequasi-nonexpansiveifF(T)=,andTx–p≤x–pforeachx∈Dandp∈F(T).
(.
)Formodelinginverseproblemswhicharisefromphaseretrievalsandinmedicalimagereconstruction[],inCensorandElfving[]rstlyintroducedthefollowingsplitfeasibilityproblem(SFP)innite-dimensionalHilbertspaces:LetCandQbenonemptyclosedconvexsubsetsoftheHilbertspacesHandH,respec-tively,letA:H→Hbeaboundedlinearoperator.
Thesplitfeasibilityproblem(SFP)isformulatedasndingapointxwiththepropertyx∈CandAx∈Q.
(.
)2015Maetal.
;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-tionLicense(http://creativecommons.
org/licenses/by/4.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycredited.
Maetal.
FixedPointTheoryandApplications(2015)2015:31Page2of18IthasbeenfoundthattheSFPcanbeusedinmanyareassuchasimagerestoration,computertomograph,andradiationtherapytreatmentplaning[–].
Somemethodshavebeenproposedtosolvesplitfeasibilityproblems;see,forinstance,[,–].
AssumingthattheSFPisconsistent(i.
e.
,(.
)hasasolution),itisnothardtoseethatx=PCI+γA(PQ–I)Ax,x∈C,(.
)wherePCandPQarethe(orthogonal)projectionsontoCandQ,respectively,γ>,andAdenotestheadjointofA.
Thatis,xsolvesSFP(.
)ifandonlyifxsolvesxedpointequation(.
)(see[]).
ThisimpliesthatSFPcanbesolvedbyusingxedpointalgo-rithms.
Recently,Mouda[]introducedthefollowingnewsplitfeasibilityproblem,whichisalsocalledgeneralsplitequalityproblem:LetH,H,HberealHilbertspaces,CH,QHbetwononemptyclosedconvexsets,A:H→H,B:H→Hbetwoboundedlinearoperators.
Thenewsplitfeasibilityproblemistondx∈C,y∈QsuchthatAx=By.
(.
)Thisallowsasymmetricandpartialrelationsbetweenthevariablesxandy.
Itiseasytoseethatproblem(.
)reducestoproblem(.
)asH=HandB=I(IstandsfortheidentitymappingfromHtoH)in(.
).
Thereforethenewsplitfeasibilityproblem(.
)proposedbyMoudaisageneralizationofsplitfeasibilityproblem(.
).
Theinterestofthisproblemistocovermanysituations,forinstance,indecompositionmethodsforPDE's,applicationsingametheoryandinintensity-modulatedradiationtherapy.
ManyauthorshaveproposedsomeusefulmethodstosolvesomekindsofgeneralsplitfeasibilityproblemsandgeneralsplitequalityproblemsinrealHilbertspaces,andun-dersuitableconditionssomestrongconvergencetheoremshavebeenproved;see,forin-stance,[–]andthereferencestherein.
Theequilibriumproblem(forshort,EP)istondx∈CsuchthatFx,y≥,y∈C.
(.
)ThesetofsolutionsofEPisdenotedbyEP(F).
GivenamappingT:C→C,letF(x,y)=Tx,y–xforallx,y∈C.
Thenx∈EP(F)ifandonlyifx∈CisasolutionofthevariationalinequalityTx,y–x≥forally∈C,i.
e.
,xisasolutionofthevariationalinequality.
Letφ:C→R∪{+∞}beafunction.
Themixedequilibriumproblem(forshort,MEP)istondx∈CsuchthatFx,y+φ(y)–φx≥,y∈C.
(.
)ThesetofsolutionsofMEPisdenotedbyMEP(F,φ).
Ifφ=,thenthemixedequilibriumproblem(.
)reducesto(.
).
Maetal.
FixedPointTheoryandApplications(2015)2015:31Page3of18IfF=,thenthemixedequilibriumproblem(.
)reducestothefollowingconvexmin-imizationproblem:ndx∈Csuchthatφ(y)≥φx,y∈C.
(.
)Thesetofsolutionsof(.
)isdenotedbyCMP(φ).
Themixedequilibriumproblem(MEP)includesservalimportantproblemsarisinginphysics,engineering,scienceoptimization,economics,transportation,networkandstructuralanalysis,Nashequilibriumproblemsinnoncooperativegamesandothers.
Ithasbeenshownthatvariationalinequalitiesandmathematicalprogrammingproblemscanbeviewedasaspecialrealizationoftheabstractequilibriumproblems(e.
g.
,[–]).
Recently,Bnouhachem[]introducedthefollowingsplitequilibriumproblems:LetF:C*C→RandG:Q*Q→RbenonlinearbifunctionsandA:H→Hbeaboundedlinearoperator,thenthesplitequilibriumproblem(SEP)istondx∈CsuchthatFx,x≥,x∈C,(.
)andsuchthaty=Ax∈QsolvesGy,y≥,y∈Q.
(.
)Inthispaper,weconsiderthefollowingpairofequilibriumproblemscalledsplitequalityequilibriumproblems(SEEP).
Denition.
LetF:C*C→RandG:Q*Q→Rbenonlinearbifunctions,letA:H→HandB:H→Hbetwoboundedlinearoperators,thenthesplitequalityequilibriumproblem(SEEP)istondx∈Candy∈QsuchthatFx,x≥,x∈C,Gy,y≥,y∈QandAx=By.
(.
)Thesetofsolutionsof(.
)isdenotedbySEEP(F,G).
Thesplitequalitymixedequilibriumproblem(SEMEP)isdenedasfollows.
Denition.
LetF:C*C→RandG:Q*Q→Rbenonlinearbifunctions,letφ:C→R∪{+∞}and:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctionssuchthatC∩domφ=andQ∩dom=,andletA:H→HandB:H→Hbetwoboundedlinearoperators,thenthesplitequalitymixedequilibriumproblem(SEMEP)istondx∈Candy∈QsuchthatFx,x+φ(x)–φx≥,x∈C,Gy,y+(y)–y≥,y∈QandAx=By.
(.
)Thesetofsolutionsof(.
)isdenotedbySEMEP(F,G,φ,).
Maetal.
FixedPointTheoryandApplications(2015)2015:31Page4of18Remark.
()In(.
),ifφ=,thenthesplitequalitymixedequilibriumproblem(.
)reducesto(.
).
()IfF=andG=,thenthesplitequalitymixedequilibriumproblem(.
)reducestothefollowingsplitequalityconvexminimizationproblem:ndx∈Candy∈Qsuchthatφ(x)≥φx,x∈C,(y)≥y,y∈QandAx=By.
(.
)Thesetofsolutionsof(.
)isdenotedbySECMP(φ,).
()IfF=,G=,B=Iandy=Ax,thenthesplitequalitymixedequilibriumproblem(.
)reducestothefollowingsplitconvexminimizationproblem:ndx∈Csuchthatφ(x)≥φx,x∈C,andy=Ax∈Q,(y)≥y,y∈Q.
(.
)Thesetofsolutionsof(.
)isdenotedbySCMP(φ,).
Inordertosolvethesplitequalityproblem(.
),MoudaandAl-Shemas[]presentedthefollowingsimultaneousiterativemethodandobtainedweakconvergencetheorem:(SIM–FPP)xk+=U(xk–γkA(Axk–Byk));yk+=T(yk+γkB(Axk–Byk)),(.
)whereH,H,HarerealHilbertspaces,U:H→H,T:H→Haretwormlyquasi-nonexpansivemappings,A:H→H,B:H→Haretwoboundedlinearoperators,AandBaretheadjointofAandB,respectively.
Undersomesuitableconditions,theyobtainedsomeweakconvergencetheorems.
Inthispaper,motivatedbytheaboveworksandrelatedliterature,weintroduceanewalgorithmforsolvingsplitequalitymixedequilibriumproblemsintheframeworkofinnite-dimensionalrealHilbertspaces.
Undersuitableconditionssomestrongandweakconvergencetheoremsareobtained.
Asapplication,weshallutilizeourresultstostudythesplitequalitymixedvariationalinequalityproblemandthesplitequalityconvexmin-imizationproblem.
Ourresultspresentedinthispaperimproveandextendsomerecentcorrespondingresults.
2PreliminariesThroughoutthispaper,wedenotethestrongconvergenceandweakconvergenceofase-quence{xn}toapointx∈Xbyxn→xandxnx,respectively.
LetHbeaHilbertspacewiththeinnerproduct·,·andthenorm·,CbeanonemptyclosedconvexsubsetofH.
Foreverypointx∈H,thereexistsauniquenearestpointofC,denotedbyPCx,suchthatx–PCx≤x–yforally∈C.
ThemappingPCiscalledthemetricprojectionfromHontoC.
ItiswellknownthatPCisarmlynonexpansivemappingfromHtoC,i.
e.
,PCx–PCy≤PCx–PCy,x–y,x,y∈H.
Further,foranyx∈Handz∈C,z=PCxifandonlyifx–z,z–y≥,y∈C.
(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page5of18Forsolvingmixedequilibriumproblems,weassumethatthebifunctionF:C*C→Rsatisesthefollowingconditions:(A)F(x,x)=,x∈C;(A)F(x,y)+F(y,x)≤,x,y∈C;(A)Forallx,y,z∈C,limt↓F(tz+(–t)x,y)≤F(x,y);(A)Foreachx∈C,thefunctiony→F(x,y)isconvexandlowersemi-continuous;(A)Forxedr>andz∈C,thereexistsaboundedsubsetKofHandx∈C∩KsuchthatF(z,x)+ry–x,x–z≥,y∈C\K.
Lemma.
([])LetCbeanonemptyclosedconvexsubsetofaHilbertspaceH.
LetFbeabifunctionfromC*CtoRsatisfying(A)-(A),andletφ:C→R∪{+∞}beaproperlowersemi-continuousandconvexfunctionsuchthatC∩domφ=.
Forr>andx∈H,deneamappingTFr:H→Casfollows:TFr(x)=z∈C:F(z,y)+φ(y)–φ(z)+ry–z,z–x≥,y∈C.
(.
)Then()Foreachx∈H,TFr(x)=;()TFrissingle-valued;()TFrisrmlynonexpansive,thatis,x,y∈H,TFrx–TFry≤TFrx–TFry,x–y;()F(TFr)=MEP(F,φ);()MEP(F,φ)isclosedandconvex.
AssumethatG:Q*QtoRsatisfying(A)-(A),andlet:Q→R∪{+∞}beaproperlowersemi-continuousandconvexfunctionsuchthatQ∩dom=,andfors>andu∈H,deneamappingTGs:H→Qasfollows:TGs(u)=ν∈Q:G(ν,w)+(w)–(ν)+sw–ν,ν–u≥,w∈Q.
(.
)ThenitfollowsfromLemma.
thatTGssatises()-()ofLemma.
,andF(TGs)=MEP(G,).
Denition.
LetHbeaHilbertspace.
()Asingle-valuemappingT:H→Hissaidtobedemiclosedatoriginif,foranysequence{xn}Hwithxnxandxn–Txn→,wehavex=Tx.
()Asingle-valuemappingT:H→Hissaidtobesemi-compactif,foranyboundedsequence{xn}Hwithxn–Txn→,thereexistsasubsequence{xni}{xn}suchthat{xni}convergesstronglytoapointx∈H.
Lemma.
([])LetCbeanonemptyclosedconvexsubsetofaHilbertspaceandTbeanonexpansivemappingfromCintoitself.
IfThasaxedpoint,thenI–TisdemiclosedMaetal.
FixedPointTheoryandApplications(2015)2015:31Page6of18atorigin,whereIistheidentitymappingofH,thatis,whenever{xn}isasequenceinCweaklyconvergingtosomex∈Candthesequence{(I–T)xn}convergesstronglytosome,itfollowsthatTx=x.
Lemma.
([])LetHbeaHilbertspaceand{μn}beasequenceinHsuchthatthereexistsanonemptysetWHsatisfying:(i)Foreveryμ∈W,limn→∞μn–μexists.
(ii)Anyweak-clusterpointofthesequence{μn}belongstoW.
Thenthereexistsμ∈Wsuchthat{μn}weaklyconvergestoμ.
Lemma.
([])LetHbearealHilbertspace,thenforallx,y∈H,wehavex–y=x–y–x–y,y.
(.
)3MainresultsTheorem.
LetH,H,HberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
AssumethatF:C*C→RandG:Q*Q→Rarebifunctionssatisfying(A)-(A),andletφ:C→R∪{+∞}and:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctionssuchthatC∩domφ=andQ∩dom=.
LetT:H→H,S:H→Hbetwononexpansivemap-pings,andA:H→H,B:H→Hbetwoboundedlinearoperators.
Let(x,y)∈C*Qandtheiterationscheme{(xn,yn)}bedenedasfollows:F(un,u)+φ(u)–φ(un)+rnu–un,un–xn≥,u∈C;G(vn,v)+(v)–(vn)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–Bvn));yn+=αnvn+(–αn)S(vn+ρnB(Aun–Bvn)),n≥;(.
)whereλAandλBstandforthespectralradiiofAAandBBrespectively,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA+λB–ε)(forεsmallenough),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
If:=F(T)∩F(S)∩SEMEP(F,G,φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofproblem(.
).
ProofNowweproveconclusion(I).
Taking(x,y)∈,itfollowsfromLemma.
thatx=TFrnxandy=TGrny,wehaveun–x=TFrnxn–TFrnx≤xn–x,(.
)vn–y=TGrnyn–TGrny≤yn–y.
(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page7of18Let(x,y)∈.
Since·isconvexandS,Tarenonexpansivemappings,wehavexn+–x=αnun+(–αn)Tun–ρnA(Aun–Bvn)–x=αnun–x+(–αn)Tun–ρnA(Aun–Bvn)–x+αn(–αn)un–x,Tun–ρnA(Aun–Bvn)–x≤αnun–x+(–αn)un–ρnA(Aun–Bvn)–x+αn(–αn)un–x,Tun–ρnA(Aun–Bvn)–x≤αnun–x+(–αn)un–ρnA(Aun–Bvn)–x+αn(–αn)un–x+un–ρnA(Aun–Bvn)–x=αnun–x+(–αn)un–ρnA(Aun–Bvn)–x≤αnun–x+(–αn)un–x+ρnA(Aun–Bvn)–ρnAun–Ax,Aun–Bvn≤xn–x+(–αn)ρnA(Aun–Bvn)–(–αn)ρnAun–Ax,Aun–Bvn.
(.
)SinceρnA(Aun–Bvn)=ρnA(Aun–Bvn),A(Aun–Bvn)=ρnAun–Bvn,AA(Aun–Bvn)≤λAρnAun–Bvn,Aun–Bvn=λAρnAun–Bvn.
(.
)Combine(.
)and(.
),thenwehavexn+–x≤xn–x+(–αn)λAρnAun–Bvn–(–αn)ρnAun–Ax,Aun–Bvn.
(.
)Similarly,fromthefourthequalityin(.
),wecangetyn+–y=yn–x+(–αn)λBρnAun–Bvn+(–αn)ρnBvn–By,Aun–Bvn.
(.
)Since(x,y)∈,soweknowthatAx=By,andnallywehavexn+–x+yn+–y≤xn–x+yn–y–ρn(–αn)–ρn(λA+λB)Aun–Bvn.
(.
)Letn(x,y):=xn–x+yn–y,thenwehaven+(x,y)≤n(x,y)–ρn(–αn)–ρn(λA+λB)Aun–Bvn.
(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page8of18Obviouslythesequence{n(x,y)}isdecreasingandislowerboundedby,soitcon-vergestosomenitelimit,sayω(x,y).
ThismeansthattherstconditionofLemma.
(Opial'slemma)issatisedwithμn=(xn,yn),μ=(x,y)andW=.
Andbypassingtolimitin(.
),weobtainthatlimn→∞Aun–Bvn=.
(.
)Sincexn–x≤n(x,y),yn–y≤n(x,y)andlimn→∞n(x,y)exists,weknowthat{xn}and{yn}arebounded,andlimsupn→∞xn–xandlimsupn→∞yn–yexist.
From(.
)and(.
),wehavelimsupn→∞un–xandlimsupn→∞vn–yalsoexist.
Letxandyberespectivelyweakclusterpointsofthesequences{xn}and{yn}.
FromLemma.
,wehavexn+–xn=xn+–x–xn+x=xn+–x–xn–x–xn+–xn,xn–x=xn+–x–xn–x–xn+–x,xn–x+xn–x,xn–x.
Solimsupn→∞xn+–xn=.
(.
)Similarly,wehavelimsupn→∞yn+–yn=.
(.
)Thisimpliesthatlimn→∞xn+–xn=,(.
)andlimn→∞yn+–yn=.
(.
)ItfollowsfromLemma.
thatun=TFrnxnandun+=TFrn+xn+,wehaveF(un+,u)+φ(u)–φ(un+)+rn+u–un+,un+–xn+≥,u∈C,andF(un,u)+φ(u)–φ(un)+rnu–un,un–xn≥,u∈C.
Particularly,wehaveF(un+,un)+φ(un)–φ(un+)+rn+un–un+,un+–xn+≥(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page9of18andF(un,un+)+φ(un+)–φ(un)+rnun+–un,un–xn≥.
(.
)Summingup(.
)and(.
)andusing(A),weobtainrn+un–un+,un+–xn++rnun+–un,un–xn≥,thusun+–un,un–xnrn–un–xn+rn+≥,whichimpliesthat≤un+–un,un–xn–rnrn+(un+–xn+)=un+–un,un–un++un+–xn–rnrn+(un+–xn+).
Therefore,un+–un≤un+–un,xn+–xn+–rnrn+(un+–xn+)≤un+–un·xn+–xn+–rnrn+·un+–xn+.
Thus,wehaveun+–un≤xn+–xn+–rnrn+·un+–xn+.
(.
)Sincelimn→∞|rn+–rn|=,{un}and{xn}arebounded,from(.
)wehavelimn→∞un+–un=.
(.
)Usingthesameargumentastheproofoftheabove,wehavelimn→∞vn+–vn=.
(.
)Itfollowsfrom(.
)and(.
)thatxn+–x≤un–x+(–αn)λAρnAun–Bvn–(–αn)ρnAun–Ax,Aun–Bvn(.
)andyn+–y=vn–x+(–αn)λBρnAun–Bvn+(–αn)ρnBvn–By,Aun–Bvn.
(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page10of18ByaddingthelasttwoinequalitiesandbytakingintoaccountthefactthatAx=By,wehavexn+–x+yn+–y≤un–x+vn–y–ρn(–αn)–ρn(λA+λB)Aun–Bvn,(.
)whereun–x=TFrnxn–TFrnx≤xn–x,un–x=xn–x+un–x–xn–un,(.
)andvn–y=TGrnyn–TGrny≤yn–y,vn–y=yn–y+vn–x–yn–vn.
(.
)Itfollowsfrom(.
),(.
)and(.
)thatxn–un+yn–vn≤xn–x–xn+–x+yn–y–yn+–y–ρn(–αn)–ρn(λA+λB)Aun–Bvn.
(.
)By(.
)and(.
),weobtainlimn→∞xn–un=,(.
)limn→∞yn–vn=.
(.
)Itfollowsfrom(.
)and(.
)thatunxandvny,respectively.
SinceTandSarenonexpansivemappings,soun–Tun=un–xn++xn+–Tun≤un–xn++xn+–Tun=un–un+–un+–xn++αnun+(–αn)Tun–ρnA(Aun–Bvn)–Tun≤un–un++un+–xn++αnun–Tun+(–αn)Tun–ρnA(Aun–Bvn)–Tun≤un–un++un+–xn++αnun–Tun+(–αn)–ρnA(Aun–Bvn).
Maetal.
FixedPointTheoryandApplications(2015)2015:31Page11of18Thatis,(–αn)un–Tun≤un–un++un+–xn++(–αn)–ρnA(Aun–Bvn).
(.
)By(.
),(.
)and(.
),wegetlimn→∞Tun–un=.
(.
)Similarly,limn→∞Svn–vn=.
(.
)Sincexn–Txn≤xn–un+un–Tun+Tun–Txn≤xn–un+un–Tun+Tun–Txn≤xn–un+un–Tun.
(.
)Itfollowsfrom(.
)and(.
)thatlimn→∞xn–Txn=.
(.
)Inaddition,sinceyn–Syn≤yn–vn+vn–Svn+Svn–Syn≤yn–vn+vn–Svn,(.
)then,from(.
)and(.
),wehavelimn→∞yn–Syn=.
(.
)Since{xn}and{yn}convergeweaklytoxandy,respectively,thenitfollowsfrom(.
),(.
)andLemma.
thatx∈F(T)andy∈F(S).
SinceeveryHilbertspacesat-isesOpial'scondition,Opial'sconditionguaranteesthattheweaklysubsequentiallimitof{(xn,yn)}isunique.
Wenowprovex∈MEP(F,φ)andy∈MEP(G,).
Sinceun=TFrnxn,wehaveF(un,u)+φ(u)–φ(un)+rnu–un,un–xn≥,u∈C.
(.
)From(A)weobtainφ(u)–φ(un)+rnu–un,un–xn≥–F(un,u)≥F(u,un),u∈C.
(.
)Andhenceφ(u)–φ(unj)+rnju–unj,unj–xnj≥F(u,unj),u∈C.
(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page12of18From(.
)weobtainunjx.
Itfollowsfrom(A)thatlimj→∞unj–xnjrnj=,andfromtheproperlowersemicontinuityofφthatFu,x+φx–φ(u)≤,u∈C.
(.
)Putzt=tu+(–t)xforallt∈(,]andu∈C.
Consequently,wegetzt∈CandhenceF(zt,x)+φ(x)–φ(zt)≤.
Sofrom(A)and(A)wehave=F(zt,zt)–φ(zt)+φ(zt)≤tF(zt,u)+(–t)Gzt,x+tφ(u)+(–t)φx–φ(zt)≤tF(zt,x)+φ(u)–φ(zt).
(.
)Hence,wehaveF(zt,u)+φ(u)–φ(zt)≥,u∈C.
(.
)Lettingt→,from(A)andtheproperlowersemicontinuityofφ,wehaveFx,u+φ(u)–φx≥,u∈C.
(.
)Thisimpliesthatx∈MEP(F,φ).
Followingasimilarargumentastheproofoftheabove,wehavey∈MEP(G,).
Ontheotherhand,sincethesquarednormisweaklylowersemicontinuous,wehaveAx–By≤liminfn→∞Aun–Bvn=,thereforeAx=By.
Thisimpliesthat(x,y)∈SEMEP(F,G,φ,).
Therefore,(x,y)∈.
ThusfromLemma.
weknowthat{(xn,yn)}convergesweaklyto(x,y).
Theproofofconclusion(I)iscompleted.
Next,weproveconclusion(II).
SinceTandSaresemi-compact,{xn}and{yn}areboundedandlimn→∞xn–Txn=,limn→∞yn–Syn=,thenthereexistsubsequences{xnj}and{ynj}of{xn}and{yn}suchthat{xnj}and{ynj}convergestronglytouandv(somepointinHandH,respectively),respectively.
Since{xnj}and{ynj}convergeweaklytoxandy,respectively,thisimpliesthatx=uandy=v.
FromLemma.
,wehavex∈F(T)andy∈F(S).
Usingthesameargumentasintheproofinconclusion(I),wehavex∈MEP(F,φ)andy∈MEP(G,).
Further,sincethenormisweaklylowersemicontinuousandAunj–Bvnj→Ax–By,wehaveAx–By≤liminfj→∞Aunj–Bvnj=,soAx=By.
Thisimpliesthat(x,y)∈.
Ontheotherhand,sincen(x,y)=xn–x+yn–yforany(x,y)∈,weknowthatlimj→∞nj(x,y)=.
Fromconclusion(I),wehavelimn→∞n(x,y)exists,thereforeMaetal.
FixedPointTheoryandApplications(2015)2015:31Page13of18limn→∞n(x,y)=.
Further,wecanobtainthatlimn→∞xn–x=andlimn→∞yn–y=.
Thiscompletestheproofofconclusion(II).
Takingφ=and=inTheorem.
,wealsohavethefollowingresult.
Corollary.
LetH,H,HberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
AssumethatF:C*C→RandG:Q*Q→Rarebifunctionssatisfying(A)-(A).
LetT:H→H,S:H→Hbetwononexpansivemappings,andA:H→H,B:H→Hbetwoboundedlinearoperators.
Let(x,y)∈C*Qandtheiterationscheme{(xn,yn)}bedenedasfollows:F(un,u)+rnu–un,un–xn≥,u∈C;G(vn,v)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–Bvn));yn+=αnvn+(–αn)S(vn+ρnB(Aun–Bvn)),n≥;whereλAandλBstandforthespectralradiiofAAandBB,respectively,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA+λB–ε)(forεsmallenough),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
If:=F(T)∩F(S)∩SEEP(F,G,φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofproblem(.
).
InTheorem.
takingB=IandH=H,fromTheorem.
wecanobtainthefollowingconvergencetheoremforgeneralsplitequilibriumproblem(.
)Corollary.
LetHandHberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
AssumethatF:C*C→RandG:Q*Q→Rarebifunctionssatisfying(A)-(A),andletφ:C→R∪{+∞},:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctionssuchthatC∩domφ=andQ∩dom=.
LetT:H→H,S:H→Hbetwononexpansivemappings,andA:H→Hbeaboundedlinearoperator.
Let(x,y)∈C*Qandtheiterationscheme{(xn,yn)}bedenedasfollows:F(un,u)+φ(u)–φ(un)+rnu–un,un–xn≥,u∈C;G(vn,v)+(v)–(vn)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–vn));yn+=αnvn+(–αn)S(vn+ρn(Aun–vn)),n≥;whereλAstandsforthespectralradiusofAA,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA–ε)(forεsmallenough),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
Maetal.
FixedPointTheoryandApplications(2015)2015:31Page14of18If:=F(T)∩F(S)∩SMEP(F,G,φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofproblem(.
).
4Applications4.
1ApplicationtothesplitequalitymixedvariationalinequalityproblemThevariationalinequalityproblem(VIP)isformulatedastheproblemofndingapointxwithpropertyx∈C,Ax,z–x≥,z∈C.
WewilldenotethesolutionsetofVIPbyVI(A,C).
In[],themixedvariationalinequalityofBrowdertype(VI)isshowntobeequivalenttondingapointu∈CsuchthatAu,y–u+(y)–(u)≥,y∈C.
WewilldenotethesolutionsetofamixedvariationalinequalityofBrowdertypebyVI(A,C,).
AmappingA:C→Hissaidtobeanα-inverse-stronglymonotonemappingifthereexistsaconstantα>suchthatAx–Ay,x–y≥αAx–Ayforanyx,y∈C.
SettingF(x,y)=Ax,y–x,itiseasytoshowthatFsatisesconditions(A)-(A)asAisanα-inverse-stronglymonotonemapping.
In,Censoretal.
[]introducedthesplitvariationalinequalityproblem(SVIP)whichisformulatedasfollows:ndapointx∈Csuchthatfx,x–x≥forallx∈C,andsuchthaty=Ax∈Qsolvesgy,y–y≥forally∈Q.
(.
)Theso-calledsplitequalitymixedvariationalinequalityproblemisshowntobeequiv-alenttondingx∈C,y∈QsuchthatBx,x–x+φ(x)–φx≥forallx∈C,andBy,y–y+(y)–y≥forally∈Q,andsuchthatAx=By.
(.
)WewilldenotethesolutionsetofasplitequalitymixedvariationalinequalityproblembySEMVIP(φ,).
Theso-calledsplitmixedvariationalinequalityproblemisshowntobeequivalenttondingapointx∈CsuchthatBx,x–x+φ(x)–φx≥forallx∈C,andsuchthaty=Ax∈QsolvesBy,y–y+(y)–y≥forally∈Q.
(.
)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page15of18ThesetofsolutionsofasplitmixedvariationalinequalityproblemisdenotedbySMVIP(φ,).
SettingF(x,y)=Bx,y–xandG(x,y)=Bx,y–x,itiseasytoshowthatFandGsatisfyconditions(A)-(A)asBi(i=,)isanηi-inverse-stronglymonotonemapping.
ThenitfollowsfromTheorem.
thatthefollowingresultholds.
Theorem.
LetH,H,HberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
LetBi(i=,)beηi-inversestronglymonotonemappings,andletφ:C→R∪{+∞}and:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctionssuchthatC∩domφ=andQ∩dom=.
LetT:H→H,S:H→Hbetwononexpansivemappings,andA:H→H,B:H→Hbetwoboundedlinearoperators.
Assumethat(x,y)∈C*Qandtheiterationscheme{(xn,yn)}isdenedasfollows:Bun,u–un+φ(u)–φ(un)+rnu–un,un–xn≥,u∈C;B(vn),v–vn+(v)–(vn)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–Bvn));yn+=αnvn+(–αn)S(vn+ρnB(Aun–Bvn)),n≥;whereλAandλBstandforthespectralradiiofAAandBB,respectively,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA+λB–ε)(forεsmallenough),ηi>(i=,),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
If:=F(T)∩F(S)∩SEMVIP(φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofthesplitequalitymixedvariationalinequalityproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofthesplitequalitymixedvariationalinequalityproblem(.
).
InTheorem.
takingB=IandH=H,fromTheorem.
wecanobtainthefollowingconvergencetheoremforsplitmixedvariationalinequalityproblemSMVIP(φ,).
Corollary.
LetHandHberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
LetBi(i=,)beηi-inversestronglymonotonemappings,andletφ:C→R∪{+∞}and:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctionssuchthatC∩domφ=andQ∩dom=.
LetT:H→H,S:H→Hbetwononexpansivemappings,andA:H→Hbeaboundedlinearoperator.
Assumethat(x,y)∈C*Qandtheiterationscheme{(xn,yn)}isdenedasfollows:Bun,u–un+φ(u)–φ(un)+rnu–un,un–xn≥,u∈C;B(vn),v–vn+(v)–(vn)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–vn));yn+=αnvn+(–αn)S(vn+ρn(Aun–vn)),n≥;Maetal.
FixedPointTheoryandApplications(2015)2015:31Page16of18whereλAstandsforthespectralradiusofAA,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA–ε)(forεsmallenough),ηi>(i=,),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
If:=F(T)∩F(S)∩SMVIP(φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofthesplitmixedvariationalinequalityproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofthesplitmixedvariationalinequalityproblem(.
).
4.
2ApplicationtothesplitequalityconvexminimizationproblemItiseasytoseethatthesplitequalitymixedequilibriumproblem(.
)reducestothesplitequalityconvexminimizationproblem(.
)asF=andG=.
Therefore,Theorem.
canbeusedtosolvesplitequalityconvexminimizationproblem(.
),andthefollowingresultcanbedirectlydeducedfromTheorem.
.
Theorem.
LetH,H,HberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
Letφ:C→R∪{+∞}and:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctions.
LetT:H→H,S:H→Hbetwononexpansivemappings,andA:H→H,B:H→Hbetwoboundedlinearoperators.
Assumethat(x,y)∈C*Qandtheiterationscheme{(xn,yn)}isdenedasfollows:φ(u)–φ(un)+rnu–un,un–xn≥,u∈C;(v)–(vn)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–Bvn));yn+=αnvn+(–αn)S(vn+ρnB(Aun–Bvn)),n≥;whereλAandλBstandforthespectralradiiofAAandBB,respectively,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA+λB–ε)(forεsmallenough),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
If:=F(T)∩F(S)∩SECMP(φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofproblem(.
).
InTheorem.
takingB=IandH=H,fromTheorem.
wecanobtainthefollowingconvergencetheoremforsplitconvexminimizationproblem(.
)SCMP(φ,).
Corollary.
LetHandHberealHilbertspaces,CHandQHbenonemptyclosedconvexsubsetsofHilbertspacesHandH,respectively.
Letφ:C→R∪{+∞}and:Q→R∪{+∞}beproperlowersemi-continuousandconvexfunctions.
LetT:H→H,Maetal.
FixedPointTheoryandApplications(2015)2015:31Page17of18S:H→Hbetwononexpansivemappings,andA:H→Hbeaboundedlinearopera-tor.
Assumethat(x,y)∈C*Qandtheiterationscheme{(xn,yn)}isdenedasfollows:φ(u)–φ(un)+rnu–un,un–xn≥,u∈C;(v)–(vn)+rnv–vn,vn–yn≥,v∈Q;xn+=αnun+(–αn)T(un–ρnA(Aun–vn));yn+=αnvn+(–αn)S(vn+ρn(Aun–vn)),n≥;whereλAstandsforthespectralradiusofAA,{ρn}isapositiverealsequencesuchthatρn∈(ε,λA–ε)(forεsmallenough),{αn}isasequencein(,)and{rn}(,∞)satisesthefollowingconditions:()andlimn→∞|rn+–rn|=.
If:=F(T)∩F(S)∩SCMP(φ,)=,then(I)Thesequence{(xn,yn)}convergesweaklytoasolutionofproblem(.
).
(II)Inaddition,ifS,Tarealsosemi-compact,then{(xn,yn)}convergesstronglytoasolutionofproblem(.
).
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsTheauthorscontributedequallytothiswork.
Theauthorsreadandapprovedthenalmanuscript.
Authordetails1SchoolofInformationEngineering,TheCollegeofArtsandSciences,YunnanNormalUniversity,LongquanRoad,Kunming,650222,China.
2CollegeofStatisticsandMathematics,YunnanUniversityofFinanceandEconomics,LongquanRoad,Kunming,650221,China.
3YaoanBranch,ChuxiongStateCo.
,YunnanTobaccoCo.
,Chuxiong,Yunnan675300,China.
AcknowledgementsTheauthorswouldliketoexpresstheirthankstothereviewersandeditorsfortheirhelpfulsuggestionsandadvice.
ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina(GrantNo.
11361070).
Received:19October2014Accepted:4February2015References1.
Takahashi,W:NonlinearFunctionalAnalysis.
YokohamaPublishers,Yokohama(2000)2.
Byne,C:Iterativeobliqueprojectionontoconvexsetsandthesplitfeasibilityproblem.
InverseProbl.
18,441-453(2002)3.
Censor,Y,Elfving,T:AmultiprojectionalgorithmusingBregmanprojectionsinaproductspace.
Numer.
Algorithms8,221-239(1994)4.
Censor,Y,Bortfeld,T,Martin,B,Tromov,A:Auniedapproachforinverseprobleminintensity-modulatedradiationtherapy.
Phys.
Med.
Biol.
51,2352-2365(2006)5.
Censor,Y,Elfving,T,Kopf,N,Bortfeld,T:Themultiple-setssplitfeasibilityproblemanditsapplications.
InverseProbl.
21,2071-2084(2005)6.
Censor,Y,Motova,A,Segal,A:Perturbedprojectionsandsubgradientprojectionsforthemultiple-setssplitfeasibilityproblem.
J.
Math.
Anal.
Appl.
327,1244-1256(2007)7.
Combettes,PL:Hilbertianconvexfeasibilityproblem:convergenceofprojectionmethods.
Appl.
Math.
Optim.
35,311-330(1997)8.
Qu,B,Xiu,N:AnoteontheCQalgorithmforthesplitfeasibilityproblem.
InverseProbl.
21,1655-1665(2005)9.
Aleyner,A,Reich,S:Block-iterativealgorithmsforsolvingconvexfeasibilityproblemsinHilbertandinBanach.
J.
Math.
Anal.
Appl.
343,427-435(2008)10.
Yang,Q:TherelaxedCQalgorithmsolvingthesplitfeasibilityproblem.
InverseProbl.
20,103-120(2004)11.
Ceng,LC,Ansari,QH,Yao,JC:Anextragradientmethodforsolvingsplitfeasibilityandxedpointproblems.
Comput.
Math.
Appl.
64,633-642(2012)12.
Ceng,LC,Ansari,QH,Yao,JC:Relaxedextragradientmethodsforndingminimum-normsolutionsofthesplitfeasibilityproblem.
NonlinearAnal.
75,2116-2125(2012)13.
Ansari,QH,Rehan,A:Splitfeasibilityandxedpointproblems.
In:Ansari,QH(ed.
)NonlinearAnalysis:ApproximationTheory,OptimizationandApplications,pp.
281-322.
Springer,NewDelhi(2014)Maetal.
FixedPointTheoryandApplications(2015)2015:31Page18of1814.
Latif,A,Sahu,DR,Ansari,QH:VariableKM-likealgorithmsforxedpointproblemandsplitfeasibilityproblems.
FixedPointTheoryAppl.
2014,ArticleID211(2014)15.
Xu,HK:Iterativemethodsforsplitfeasibilityproblemininnite-dimensionalHilbertspaces.
InverseProbl.
26,105018(2010)16.
Mouda,A:ArelaxedalternatingCQ-algorithmforconvexfeasibilityproblems.
NonlinearAnal.
79,117-121(2013)17.
Eslamian,M,Latif,A:GeneralsplitfeasibilityproblemsinHilbertspaces.
Abstr.
Appl.
Anal.
2013,ArticleID805104(2013)18.
Chen,RD,Wang,J,Zhang,HW:GeneralsplitequalityproblemsinHilbertspaces.
FixedPointTheoryAppl.
2014,ArticleID35(2014)19.
Chang,SS,Wang,L:Mouda'sopenquestionandsimultaneousiterativealgorithmforgeneralsplitequalityvariationalinclusionproblemsandgeneralsplitequalityoptimizationproblems.
FixedPointTheoryAppl.
2014,ArticleID215(2014)20.
Chang,SS,Lee,HWJ,Chan,CK:Anewmethodforsolvingequilibriumproblemxedpointproblemandvariationalinequalityproblemwithapplicationtooptimization.
NonlinearAnal.
70,3307-3319(2009)21.
Qin,X,Shang,M,Su,Y:AgeneraliterativemethodforequilibriumproblemandxedpointprobleminHilbertspaces.
NonlinearAnal.
69,3897-3909(2008)22.
Blum,E,Oettli,W:Fromoptimizationandvariationalinequalitiestoequilibriumproblems.
Math.
Stud.
63,123-145(1994)23.
Noor,M,Oettli,W:Ongeneralnonlinearcomplementarityproblemsandquasi-equilibria.
Matematiche49,313-346(1994)24.
Bnouhachem,A:Amodiedprojectionmethodforacommonsolutionofasystemofvariationalinequalities,asplitequilibriumproblemandahierarchicalxed-pointproblem.
FixedPointTheoryAppl.
2014,ArticleID22(2014)http://www.
xedpointtheoryandapplications.
com/content/2014/1/2225.
Mouda,A,Al-Shemas,E:Simultaneousiterativemethodsforsplitequalityproblems.
Trans.
Math.
Program.
Appl.
1(2),1-11(2013)26.
Peng,JW,Liou,YC,Yao,JC:Aniterativealgorithmcombiningviscositymethodwithparallelmethodforageneralizedequilibriumproblemandstrictpseudocontractions.
FixedPointTheoryAppl.
2009,ArticleID794178(2009).
doi:10.
1155/2009/79417827.
Geobel,K,Kirk,WA:TopicsinMetricFixedPointTheory.
CambridgeStudiesinAdvancedMathematics,vol.
28.
CambridgeUniversityPress,Cambridge(1990)28.
Marino,G,Xu,HK:WeakandstrongconvergencetheoremsforstrictpseudocontractionsinHilbertspace.
J.
Math.
Anal.
Appl.
329,336-346(2007)29.
Alber,YI:MetricandgeneralizedprojectionoperatorsinBanachspace:propertiesandapplication.
In:Kartosator,AG(ed.
)TheoryandApplicationsofNonlinearOperatorsofAccretiveandMonotoneType,pp.
15-50.
Dekker,NewYork(1996)30.
Censor,Y,Gibali,A,Reich,S:Algorithmsforthesplitvariationalinequalityproblem.
Numer.
Algorithms59,301-323(2012)