Variational Analysis, Optimization, and Fixed Point Theory - Hindawi PDF
Preview Variational Analysis, Optimization, and Fixed Point Theory - Hindawi
Abstract and Applied Analysis Variational Analysis, Optimization, and Fixed Point Theory Guest Editors: Qamrul Hasan Ansari, Mohamed Amine Khamsi, Abdul Latif, and Jen-Chih Yao Variational Analysis, Optimization, and Fixed Point Theory Abstract and Applied Analysis Variational Analysis, Optimization, and Fixed Point Theory GuestEditors:QamrulHasanAnsari,MohamedAmineKhamsi, Abdul Latif, and Jen-Chih Yao Copyright©2013HindawiPublishingCorporation.Allrightsreserved. Thisisaspecialissuepublishedin“AbstractandAppliedAnalysis.”AllarticlesareopenaccessarticlesdistributedundertheCreative CommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginal workisproperlycited. Editorial Board RaviP.Agarwal,USA JaeyoungChung,Korea LorenzoGiacomelli,Italy BashirAhmad,SaudiArabia SilviaCingolani,Italy JaumeGine´,Spain M.O.Ahmedou,Germany JeanM.Combes,France ValeryY.Glizer,Israel NicholasD.Alikakos,Greece MonicaConti,Italy LaurentGosse,Italy DeboraAmadori,Italy DiegoCo´rdoba,Spain JeanP.Gossez,Belgium PabloAmster,Argentina JuanCarlosCorte´sLo´pez,Spain JoseL.Gracia,Spain DouglasR.Anderson,USA GrazianoCrasta,Italy MaurizioGrasselli,Italy JanAndres,CzechRepublic B.Dacorogna,Switzerland QianGuo,China GiovanniAnello,Italy VladimirDanilov,Russia YuxiaGuo,China StanislavAntontsev,Portugal MohammadT.Darvishi,Iran ChaitanP.Gupta,USA MohamedKamalAouf,Egypt LuisF.P.deCastro,Portugal UnoHa¨marik,Estonia NarcisaC.Apreutesei,Romania T.Diagana,USA FerencHartung,Hungary NatigM.Atakishiyev,Mexico Jesu´sI.D´ıaz,Spain BehnamHashemi,Iran FerhanM.Atici,USA JosefDibl´ık,CzechRepublic NorimichiHirano,Japan IvanG.Avramidi,USA FasmaDiele,Italy JafariHossein,Iran SoohyunBae,Korea TomasDominguez,Spain JiaxinHu,China ChuanzhiBai,China A.I.Domoshnitsky,Israel ChengmingHuang,China ZhanbingBai,China MarcoDonatelli,Italy ZhongyiHuang,China DumitruBaleanu,Turkey Bo-QingDong,China GennaroInfante,Italy Jo´zefBana´s,Poland OnduˇejDoˇslY´,CzechRepublic IvanIvanov,Bulgaria GerassimosBarbatis,Greece Wei-ShihDu,Taiwan JaanJanno,Estonia MartinoBardi,Italy LuizDuarte,Brazil ArefJeribi,Tunisia RobertoBarrio,Spain RomanDwilewicz,USA UnC.Ji,Korea FeyziBa¸sar,Turkey PaulW.Eloe,USA ZhongxiaoJia,China A.Bellouquid,Morocco AhmedEl-Sayed,Egypt L.Jo´dar,Spain DanieleBertaccini,Italy LucaEsposito,Italy J.S.Jung,RepublicofKorea MichielBertsch,Italy JoseA.Ezquerro,Spain HenrikKalisch,Norway LucioBoccardo,Italy KhalilEzzinbi,Morocco HamidRezaKarimi,Norway IgorBoglaev,NewZealand DashanFan,USA S.Kichenassamy,France MartinJ.Bohner,USA AngeloFavini,Italy TeroKilpela¨inen,Finland JulianF.Bonder,Argentina MarciaFederson,Brazil S.G.Kim,RepublicofKorea GeraldoBotelho,Brazil S.Filippas,EquatorialGuinea LjubisaKocinac,Serbia ElenaBraverman,Canada AlbertoFiorenza,Italy AndreiKorobeinikov,Spain RomeoBrunetti,Italy ToreFla˚tten,Norway PekkaKoskela,Finland JanuszBrzdek,Poland IlariaFragala,Italy VictorKovtunenko,Austria DetlevBuchholz,Germany BrunoFranchi,Italy Ren-JiehKuo,Taiwan Sun-SigByun,Korea XianlongFu,China PavelKurasov,Sweden FabioM.Camilli,Italy MassimoFuri,Italy MiroslawLachowicz,Poland JindeCao,China GiovanniP.Galdi,USA KunquanLan,Canada AnnaCapietto,Italy IsaacGarcia,Spain RuedigerLandes,USA JianqingChen,China Jesu´sGarc´ıaFalset,Spain IrenaLasiecka,USA W.-S.Cheung,HongKong J.A.Garc´ıa-Rodr´ıguez,Spain MattiLassas,Finland MichelChipot,Switzerland LeszekGasinski,Poland Chun-KongLaw,Taiwan ChangbumChun,Korea Gyo¨rgyGa´t,Hungary Ming-YiLee,Taiwan SoonY.Chung,Korea VladimirGeorgiev,Italy GongbaoLi,China ElenaLitsyn,Israel SomyotPlubtieng,Thailand JuanJ.Trujillo,Spain ShengqiangLiu,China MilanPokorny,CzechRepublic CiprianA.Tudor,France YanshengLiu,China SergioPolidoro,Italy GabrielTurinici,France CarlosLizama,Chile ZiemowitPopowicz,Poland MilanTvrdy,CzechRepublic MiltonC.LopesFilho,Brazil MariaM.Porzio,Italy MehmetUnal,Turkey JulianLo´pez-Go´mez,Spain EnricoPriola,Italy CsabaVarga,Romania GuozhenLu,USA VladimirS.Rabinovich,Mexico CarlosVazquez,Spain JinhuLu¨,China I.Rachu˚nkova´,CzechRepublic GianmariaVerzini,Italy GrzegorzLukaszewicz,Poland MariaA.Ragusa,Italy JesusVigo-Aguiar,Spain WanbiaoMa,China SimeonReich,Israel YushunWang,China ShiwangMa,China AbdelazizRhandi,Italy Qing-WenWang,China EberhardMalkowsky,Turkey HassanRiahi,Malaysia JingPingWang,UK SalvatoreA.Marano,Italy JuanP.Rinco´n-Zapatero,Spain ShawnX.Wang,Canada CristinaMarcelli,Italy LuigiRodino,Italy QingWang,USA PaoloMarcellini,Italy YuriyV.Rogovchenko,Norway YouyuWang,China Jesu´sMar´ın-Solano,Spain JulioD.Rossi,Argentina PeixuanWeng,China JoseM.Martell,Spain WolfgangRuess,Germany NoemiWolanski,Argentina M.MastyThlo,Poland BernhardRuf,Italy Ngai-ChingWong,Taiwan MingMei,Canada MarcoSabatini,Italy PatriciaJ.Y.Wong,Singapore TarasMel’nyk,Ukraine SatitSaejung,Thailand ZiliWu,China AnnaMercaldo,Italy StefanG.Samko,Portugal YongHongWu,Australia ChangxingMiao,China MartinSchechter,USA ShanheWu,China StanislawMigorski,Poland JavierSegura,Spain Tie-chengXia,China MihaiMihaˇilescu,Romania SigmundSelberg,Norway XuXian,China FelizMinho´s,Portugal ValerySerov,Finland YanniXiao,China DumitruMotreanu,France NaseerShahzad,SaudiArabia GongnanXie,China RobertaMusina,Italy AndreyShishkov,Ukraine FudingXie,China G.M.N’Gue´re´kata,USA StefanSiegmund,Germany NaihuaXiu,China MariaGraziaNaso,Italy A.-M.A.Soliman,Egypt DaoyiXu,China SylviaNovo,Spain PierpaoloSoravia,Italy XiaodongYan,USA MicahOsilike,Nigeria MarcoSquassina,Italy ZhenyaYan,China M.Oˆtani,Japan S.Staneˇk,CzechRepublic NorioYoshida,Japan TurgutOˆzi¸s,Turkey StevoStevic,Serbia BeongInYun,Korea FilomenaPacella,Italy AntonioSua´rez,Spain VjacheslavYurko,Russia N.S.Papageorgiou,Greece WenchangSun,China AgacikZafer,Turkey SehiePark,Korea RobertSzalai,UK SergeyV.Zelik,UK AlbertoParmeggiani,Italy SanyiTang,China JianmingZhan,China KailashC.Patidar,SouthAfrica Chun-LeiTang,China ChengjianZhang,China KevinR.Payne,Italy YoushanTao,China WeinianZhang,China AdemirFernandoPazoto,Brazil GabriellaTarantello,Italy MeirongZhang,China J.E.Pecˇaric´,Croatia N.-e.Tatar,SaudiArabia ZengqinZhao,China ShuangjiePeng,China SusannaTerracini,Italy SiningZheng,China S.V.Pereverzyev,Austria GerdTeschke,Germany TianshouZhou,China MariaEugeniaPerez,Spain AlbertoTesei,Italy YongZhou,China JosefinaPerles,Spain BevanThompson,Australia Chun-GangZhu,China AllanPeterson,USA SergeyTikhonov,Spain QijiJ.Zhu,USA AndrewPickering,Spain ClaudiaTimofte,Romania MalisaR.Zizovic,Serbia CristinaPignotti,Italy ThanhTran,Australia WenmingZou,China Contents VariationalAnalysis,Optimization,andFixedPointTheory,QamrulHasanAnsari, MohamedAmineKhamsi,AbdulLatif,andJen-ChihYao Volume2013,ArticleID823070,2pages ExistenceandApproximationofAttractivePointsoftheWidelyMoreGeneralizedHybridMappingsin HilbertSpaces,Sy-MingGuuandWataruTakahashi Volume2013,ArticleID904164,10pages ImplicitIshikawaApproximationMethodsforNonexpansiveSemigroupsinCAT(0)Spaces, Zhi-binLiu,Yi-shenChen,Xue-songLi,andYi-binXiao Volume2013,ArticleID503198,8pages ParatingentDerivativeAppliedtotheMeasureoftheSensitivityinMultiobjectiveDifferential Programming,F.Garc´ıaandM.A.MelguizoPadial Volume2013,ArticleID812125,11pages SomeResultsonFixedandBestProximityPointsofMultivaluedCyclicSelf-MappingswithaPartial Order,M.DelaSen Volume2013,ArticleID968492,11pages StrongConvergenceTheoremsforSemigroupsofAsymptoticallyNonexpansiveMappingsinBanach Spaces,D.R.Sahu,Ngai-ChingWong,andJen-ChihYao Volume2013,ArticleID202095,8pages MultistepHybridExtragradientMethodforTripleHierarchicalVariationalInequalities, Zhao-RongKong,Lu-ChuanCeng,QamrulHasanAnsari,andChin-TzongPang Volume2013,ArticleID718624,17pages ImplicitRelaxedandHybridMethodswithRegularizationforMinimizationProblemsand AsymptoticallyStrictPseudocontractiveMappingsintheIntermediateSense,Lu-ChuanCeng, QamrulHasanAnsari,andChing-FengWen Volume2013,ArticleID854297,14pages ImplicitIterativeSchemeforaCountableFamilyofNonexpansiveMappingsin2-UniformlySmooth BanachSpaces,MingTianandXinJin Volume2013,ArticleID264910,9pages ExistenceofSolutionsforaPeriodicBoundaryValueProblemviaGeneralizedWeaklyContractions, SirousMoradi,ErdalKarapınar,andHassenAydi Volume2013,ArticleID704160,7pages Meir-KeelerTypeMultidimensionalFixedPointTheoremsinPartiallyOrderedMetricSpaces, ErdalKarapınar,AntonioRolda´n,JuanMart´ınez-Moreno,andConcepcionRolda´n Volume2013,ArticleID406026,9pages ANon-NP-CompleteAlgorithmforaQuasi-FixedPolynomialProblem, Yi-ChouChenandHang-ChinLai Volume2013,ArticleID893045,10pages ApproximationfortheHierarchicalConstrainedVariationalInequalitiesovertheFixedPointsof NonexpansiveSemigroups,Li-JunZhu Volume2013,ArticleID604369,8pages ApproximateControllabilityofFractionalSobolev-TypeEvolutionEquationsinBanachSpaces, N.I.Mahmudov Volume2013,ArticleID502839,9pages TheMann-TypeExtragradientIterativeAlgorithmswithRegularizationforSolvingVariational InequalityProblems,SplitFeasibility,andFixedPointProblems,Lu-ChuanCeng,HimanshuGupta, andChing-FengWen Volume2013,ArticleID378750,19pages ANewFixedPointTheoremandApplications,MinFangandXiePingDing Volume2013,ArticleID432402,5pages NeuralNetworkDynamicswithoutMinimizingEnergy,Mau-HsiangShihandFeng-ShengTsai Volume2013,ArticleID496217,4pages OrderedVariationalInequalitiesandOrderedComplementarityProblemsinBanachLattices, JinluLiandChing-FengWen Volume2013,ArticleID323126,9pages Berinde-TypeGeneralizedContractionsonPartialMetricSpaces,HassenAydi,SanaHadjAmor, andErdalKarapınar Volume2013,ArticleID312479,10pages GeneralSplitFeasibilityProblemsinHilbertSpaces,MohammadEslamianandAbdulLatif Volume2013,ArticleID805104,6pages ExistenceSolutionsofVectorEquilibriumProblemsandFixedPointofMultivaluedMappings, KanokwanSitthithakerngkietandSomyotPlubtieng Volume2013,ArticleID952021,6pages BestProximityPointResultsforMK-ProximalContractions,MohamedJleli,ErdalKarapınar, andBessemSamet Volume2012,ArticleID193085,14pages AMaximumPrincipleforControlledTime-SymmetricForward-BackwardDoublyStochastic DifferentialEquationwithInitial-TerminalSateConstraints,ShaolinJi,QingmengWei, andXiuminZhang Volume2012,ArticleID537376,29pages TheEquivalenceofConvergenceResultsbetweenMannandMultistepIterationswithErrorsfor UniformlyContinuousGeneralizedWeakΦ-PseudocontractiveMappingsinBanachSpaces, GuiwenLvandHaiyunZhou Volume2012,ArticleID169410,14pages SomeModifiedExtragradientMethodsforSolvingSplitFeasibilityandFixedPointProblems, Zhao-RongKong,Lu-ChuanCeng,andChing-FengWen Volume2012,ArticleID975981,32pages HindawiPublishingCorporation AbstractandAppliedAnalysis Volume2013,ArticleID823070,2pages http://dx.doi.org/10.1155/2013/823070 Editorial Variational Analysis, Optimization, and Fixed Point Theory QamrulHasanAnsari,1MohamedAmineKhamsi,2AbdulLatif,3andJen-ChihYao4 1DepartmentofMathematics,AligarhMuslimUniversity,Aligarh202002,India 2DepartmentofMathematicalSciences,TheUniversityofTexasatElPaso,ElPaso,TX79968,USA 3DepartmentofMathematics,KingAbdulazizUniversity,Jeddah21589,SaudiArabia 4CenterofGeneralEducation,KaohsiungMedicalUniversity,Kaohsiung807,Taiwan CorrespondenceshouldbeaddressedtoQamrulHasanAnsari;[email protected] Received5September2013;Accepted5September2013 Copyright©2013QamrulHasanAnsarietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttribution License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly cited. In the last two decades, the theory of variational analysis theory in all of analysis. This is because the contractive including variational inequalities (VI) emerged as a rapidly condition on the mapping is simple and easy to verify, and growing area of research because of its applications in because it requires only completeness of the metric space. nonlinear analysis, optimization, economics, game theory, Although, the basic idea was known to others earlier, the andsoforth;see,forexample,[1]andthereferencestherein. principlefirstappearedinexplicitforminBanach’s1922thesis In the recent past, many authors devoted their attention whereitwasusedtoestablishtheexistenceofasolutiontoan to study the VI defined on the set of fixed points of a integralequation. mapping, called hierarchical variational inequalities. Very Caristi’s fixed point theorem [10, 11] has found many recently, several iterative methods have been investigated applications in nonlinear analysis. It is shown, for example, to solve VI, hierarchical variational inequalities, and triple thatthistheoremyieldsessentiallyalltheknowninwardness hierarchical variational inequalities. Since the origin of the results of geometric fixed point theory in Banach spaces. VI,ithasbeenusedasatooltostudyoptimizationproblems. Recallthatinwardnessconditionsaretheoneswhichassert Hierarchical variational inequalities are used to study the that, in some sense, points from the domain are mapped bilevel mathematical programming problems. A triple level toward the domain. This theorem is amazing equivalent to mathematicalprogrammingproblemcanbestudiedbyusing Ekeland’svariationalprinciple. triplehierarchicalvariationalinequalities. QamrulHasanAnsari TheEkeland’svariationalprincipleprovidestheexistence MohamedAmineKhamsi ofanapproximateminimizerofaboundedbelowandlower AbdulLatif semicontinuous function. It is one of the most important Jen-ChihYao results from nonlinear analysis and it has applications in different areas of mathematics and mathematical sciences, References namely, fixed point theory, optimization, optimal control theory,gametheory,nonlinearequations,dynamicalsystems, [1] Q. H. Ansari, C. S. Lalitha, and M. Mehta, Generalized and so forth,forexample, [2–9] and thereferences therein. Convexity,NonsmoothVariationalInequalities,andNonsmooth Duringthelastdecade,ithasbeenusedtostudytheexistence Optimization,CRCPress,Taylor&FrancisGroup,BocaRaton, ofsolutionsofequilibriumproblemsinthesettingofmetric Fla,USA,2014. spaces,forexample,[2,3]andthereferencestherein. [2] Q.H.Ansari,MetricSpaces:IncludingFixedPointTheoryand Banach’s contraction principle is remarkable in its sim- Set-ValuedMaps,NarosaPublishingHouse,NewDelhi,India, plicity, yet it is perhaps the most widely applied fixed point 2010. 2 AbstractandAppliedAnalysis [3] M. Bianchi, G. Kassay, and R. Pini, “Existence of equilibria viaEkeland’sprinciple,”JournalofMathematicalAnalysisand Applications,vol.305,no.2,pp.502–512,2005. [4] D. G. de Figueiredo, Lectures on the Ekeland Variational PrinciplewithApplicationsandDetours,vol.81ofTataInstitute ofFundamentalResearchLecturesonMathematicsandPhysics, TataInstituteofFundamentalResearch,Mumbai,India,1989. [5] I.Ekeland,“Surlesproble`mesvariationnels,”ComptesRendus del’Acade´miedesSciences,vol.275,pp.1057–1059,1972. [6] I.Ekeland,“Onthevariationalprinciple,”JournalofMathemat- icalAnalysisandApplications,vol.47,pp.324–353,1974. [7] I.Ekeland,“Nonconvexminimizationproblems,”Bulletinofthe AmericanMathematicalSociety,vol.1,no.3,pp.443–474,1979. [8] O.Kada,T.Suzuki,andW.Takahashi,“Nonconvexminimiza- tion theorems and fixed point theorems in complete metric spaces,”MathematicaJaponica,vol.44,no.2,pp.381–391,1996. [9] M.A.KhamsiandW.A.Kirk,AnIntroductiontoMetricSpaces andFixedPointTheory,PureandAppliedMathematics,Wiley- Interscience,NewYork,NY,USA,2001. [10] J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of the American Mathe- maticalSociety,vol.215,pp.241–251,1976. [11] J. Caristi and W. A. Kirk, “Geometric fixed point theory and inwardnessconditions,”inTheGeometryofMetricandLinear Spaces, vol. 490 of Lecture Notes in Mathematics, pp. 74–83, Springer,Berlin,Germany,1975.
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