Advances in Mathematical Economics 21 Shigeo Kusuoka Toru Maruyama Editors Advances in Mathematical Economics Volume 21 Advances in MATHEMATICAL ECONOMICS Managing Editors ShigeoKusuoka Toru Maruyama TheUniversity ofTokyo KeioUniversity Tokyo,Japan Tokyo,Japan Editors Robert Anderson Jean-Michel Grandmont KunioKawamata University of California, CREST-CNRS KeioUniversity Berkeley Malakoff,France Tokyo,Japan Berkeley, USA Norimichi Hirano HiroshiMatano Charles Castaing Yokohama National TheUniversityofTokyo UniversitéMontpellierII University Tokyo,Japan Montpellier, France Yokohama, Japan KazuoNishimura Francis H.Clarke Tatsuro Ichiishi KyotoUniversity UniversitédeLyon I The OhioState Kyoto,Japan Villeurbanne, France University YoichiroTakahashi Ohio, USA Egbert Dierker TheUniversityofTokyo University of Vienna Alexander D.Ioffe Tokyo,Japan Vienna,Austria Israel Institute of AkiraYamazaki Technology Darrell Duffie Haifa, Israel Hitotsubashi University StanfordUniversity Tokyo,Japan Stanford, USA Seiichi Iwamoto MakotoYano Lawrence C.Evans Kyushu University KyotoUniversity Fukuoka,Japan University of California, Kyoto,Japan Berkeley Kazuya Kamiya Berkeley, USA TheUniversityofTokyo Takao Fujimoto Tokyo,Japan FukuokaUniversity Fukuoka,Japan Aims and Scope. The project is to publish Advances in Mathematical Economics onceayearundertheauspicesoftheResearchCenterforMathematicalEconomics. Itisdesignedtobringtogetherthosemathematicianswhoareseriouslyinterestedin obtaining new challenging stimuli from economic theories and those economists who are seeking effective mathematical tools for their research. The scope of Advances in Mathematical Economics includes, but is not limited to, the following fields: – Economic theories in various fields based on rigorous mathematical reasoning. – Mathematicalmethods(e.g.,analysis,algebra,geometry,probability)motivated by economic theories. – Mathematical results of potential relevance to economic theory. – Historical study of mathematical economics. Authorsareaskedtodeveloptheiroriginalresultsasfullyaspossibleandalsoto giveaclear-cutexpositoryoverviewoftheproblemunderdiscussion.Consequently, we will also invite articles which might be considered too long for publication in journals. More information about this series at http://www.springer.com/series/4129 Shigeo Kusuoka Toru Maruyama (cid:129) Editors Advances in Mathematical Economics Volume 21 123 Editors ShigeoKusuoka ToruMaruyama ProfessorEmeritus ProfessorEmeritus TheUniversity of Tokyo KeioUniversity Tokyo,Japan Tokyo,Japan ISSN 1866-2226 ISSN 1866-2234 (electronic) Advances in Mathematical Economics ISBN978-981-10-4144-0 ISBN978-981-10-4145-7 (eBook) DOI 10.1007/978-981-10-4145-7 LibraryofCongressControlNumber:2017935013 ©SpringerNatureSingaporePteLtd.2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore Contents Some Problems in Second Order Evolution Inclusions with Boundary Condition: A Variational Approach.... .... ..... .... 1 Charles Castaing, Truong Le Xuan, Paul Raynaud de Fitte and Anna Salvadori On Sufficiently-Diffused Information in Bayesian Games: A Dialectical Formalization .. ..... .... .... .... .... .... ..... .... 47 M. Ali Khan and Yongchao Zhang On Supermartingale Problems..... .... .... .... .... .... ..... .... 75 Shigeo Kusuoka Bolza Optimal Control Problems with Linear Equations and Periodic Convex Integrands on Large Intervals ... .... ..... .... 99 Alexander J. Zaslavski Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 161 v Some Problems in Second Order Evolution Inclusions with Boundary Condition: A Variational Approach CharlesCastaing,TruongLeXuan,PaulRaynauddeFitte andAnnaSalvadori Abstract Weprove,underappropriateassumptions,theexistenceofsolutionsfora secondorderevolutioninclusionwithboundaryconditionsviaavariationalapproach. · · · · Keywords Boundedvariation Epiconvergence BitingLemma Subdifferential Youngmeasures Articletype:ResearchArticle Received:September12,2016 Revised:January6,2017 JELClassification:C61,C73. MathematicsSubjectClassifications(2010):34A60,34B15. B C.Castaing( ) DépartementdeMathématiques,UniversitéMontpellierII,Case051,34095 MontpellierCedex5,France e-mail:[email protected] T.LeXuan DepartmentofMathematicsandStatistics,UniversityofEconomicsofHo ChiMinhCity,59CNguyenDinhChieuStr.Dist.3,HoChiMinhCity,Vietnam e-mail:[email protected] P.RaynauddeFitte LaboratoireRaphaëlSalem,UMRCNRS6085,NormandieUniversité, Rouen,France e-mail:[email protected] A.Salvadori DipartimentodiMatematica,Università‘diPerugia,viaVanvitelli1, 06123Perugia,Italy e-mail:[email protected] ©SpringerNatureSingaporePteLtd.2017 1 S.KusuokaandT.Maruyama(eds.),AdvancesinMathematical Economics,AdvancesinMathematicalEconomics21, DOI10.1007/978-981-10-4145-7_1 2 C.Castaingetal. 1 Introduction In the present paper, we prove, under appropriate assumptions, the existence of solutionsforasecondorderevolutioninclusionwithboundaryconditionsgoverned bysubdifferentialoperatorsoftheform f(t)∈u¨(t)+Mu˙(t)+∂ϕ(u(t)),t ∈[0,T]. (I) Here, M is positive, ϕ is a lower semicontinuous convex proper function defined on Rd and ∂ϕ(u(t)) is the subdifferential of the function ϕ at the point u(t) and the perturbation f belongs to L2 ([0,T]). It is well known that this problem is Rd difficult and needs a specific treatment via the Moreau-Yosida approximation or epiconvergenceapproach.SeeAttouch–Cabot–Redon[4]andSchatzmann[24]fora deepstudyoftheseproblems,Castaing–RaynauddeFitte–Salvadori[11],Castaing– Le Xuan Truong [8] dealing with second order evolution with m-point boundary conditionsviatheepiconvergenceapproach.Theseconsiderationsleadustoconsider thevariationallimitsofafairlygeneralapproximatingproblem fn(t)∈u¨n(t)+Mu˙n(t)+∂ϕ (un(t)),t ∈[0,T] (II) n whereunisaW2,1([0,T])-solution, fnweaklyconverginginL2 ([0,T])to f∞,ϕ Rd Rd n isaconvexLipschitzfunctionwhichepiconvergestoalowersemicontinuousconvex properfunctionϕ∞.Thisapproximatingproblemcoversvarioustypeofproblemsof practicalinterestinseveraldynamicsystems,evolutioninclusion,controltheoryetc. HerewefocusonseveralvariationallimitsofsolutionsviatheBitingLemmaand Youngmeasuresandothertoolsoccurringinthisapproachbyshowingundersuit- ablelimitassumptionontheboundaryconditionsthat(u¨n)isL1 ([0,T])-bounded. Rd Thismainfactallowstostudythevariationallimitofsolutionsinthisproblem,in particular,thetraditionalestimatedenergyforthevariationallimitsolutionsiscon- servedalmosteverywhere.Theapplicabilityofourabstractframeworkgiventherein (Proposition 3.3) will be exemplified in considering the existence of solution for secondorderdifferentialinclusions f(t)∈u¨(t)+Mu˙(t)+∂ϕ(u(t)),t ∈[0,T] under m-point boundary condition or anti-periodic conditions and further related secondorderevolutioninclusionsintheliterature.Thiswillbedonebyapplyingour abstractresulttothesinglevaluedapproximatingproblem fn(t)=u¨n(t)+Mu˙n(t)+∇ϕ (un(t)),t ∈[0,T] (III) n where∇ϕ isthegradientoftheC1,Lipschitz,convexfunctionϕ thatepi-converges n n toaproperconvexlowersemicontinuousfunctionϕ∞ and fn weaklyconvergesin L2 ([0,T])to f∞ sothatthevariationallimitsolutionsu∞ to(III)aregeneralized Rd solutionstotheinclusion SomeProblemsinSecondOrderEvolutionInclusions… 3 f∞(t)∈u¨∞(t)+Mu˙∞(t)+∂ϕ∞(u∞(t)),t ∈[0,T] with appropriate properties, namely, the solution limit u∞ is W1,1([0,T]), that is, BV u∞ iscontinuousanditsderivativeu˙∞ isboundedvariation(BVforshort)andthe estimatedenergyholdsalmosteverywhere 1 1 ϕ∞(u∞(t))+ ||u˙∞(t)||2 =ϕ∞(u0)+ ||u˙0)||2 2 2 (cid:2) (cid:2) t t −M ||u˙∞(s)||2ds+ (cid:5)f∞(s),u˙∞(s)(cid:6)ds 0 0 withfurtherrelatedvariationalinclusion,inparticular, f∞(t)∈ζ∞(t)+Mu∞(t)+∂ϕ∞(u∞(t)),t ∈[0,T] almosteverywhere,ζ∞ beingthebitinglimitofthe L1 ([0,T])-boundedsequence Rd (u¨n). Section3 is devoted to second order evolution inclusion with boundary con- ditions.Wepresentthevariationallimitsofthegeneralapproximatingproblem(II) andtheapplicationsofvariationallimitsoftheapproximatingproblem(III)tothe existenceproblemofsecondorderevolutioninclusion(I)involvingvariationaltech- niques,theBitingLemma,thecharacterizationoftheseconddualofL1 andYoung Rd measures. It is worth to mention that the approximation (III) occurs in practical casesofsecondorderevolutioninclusiongovernedbysubdifferentialoperators.For instance,Attouch–Cabot–Redon[4]consideredtheapproximatingproblem 0=u¨n(t)+γu˙n(t)+∇ϕ (un(t)),t ∈[0,T] n un(0)=un,u˙n(0)=u˙n 0 1 whereγ ispositive,∇ϕ isthegradientofaC1,smoothfunction.Schatzmann[24] n consideredtheapproximatingproblem f(t)=u¨ (t)+∂ϕ (u (t)),t ∈[0,T] λ λ λ u (0)=u ,u˙ (0)=u λ 0 λ 1 where f ∈ L2 ([0,T])and∂ϕ istheMoreau-Yosidaapproximationtothelower Rd λ semicontinuousconvexproperfunctionϕ.M.Mabrouk[19]continuedtheworkof M.Schatzmann[24]byconsideringtheapproximatingproblem f (t)=u¨ (t)+∇ϕ (u (t)),t ∈[0,T] λ λ λ λ u (0)=u ,u˙ (0)=u , λ 0 λ 1 4 C.Castaingetal. with f ∈ L1 ([0,T]).InSect.4,weapplyourtechniquestothestudyofbothfirst λ Rd order and second order evolution equations with anti-periodic boundary condition usingtheapproximatingproblem fn(t)=u¨n(t)+ Mu˙n(t)+∇ϕ (un(t)),t ∈[0,T] n un(0)=−un(T), whereun ∈ W2,2([0,T])and fn ∈ L2 ([0,T]),seeH.Okochi[22],A.Haraux[17], Rd Rd Aftabizadeh,AizicoviciandPavel[1,2],AizicoviciandPavel[3]andthereferences therein. AgeneralanalysisofsomerelatedproblemsinHilbertspaceisavailable,c.fK. Maruo[19]andM.Schatzmann[24]. 2 SomeExistenceTheoremsinSecondOrderEvolution Inclusionswith m-PointBoundaryCondition Wewillusethefollowingdefinitionsandnotationsandsummarizesomebasicresults. • Let E beaseparableBanachspace, B (0,1)istheclosedunitballof E. E • c(E) (resp. cc(E)) (resp. ck(E))(resp. cwk(E)) is the collection of nonempty closed(resp.closedconvex)(resp.compactconvex)(resp.weaklycompactcon- vex)subsetsof E. • If Aisasubsetof E,δ∗(.,A)isthesupportfunctionof A. • L([0,T])istheσ-algebraofLebesguemeasurablesubsetsof[0,T]. • If X isatopologicalspace,B(X)istheBoreltribeof X. • L1([0,T],dt) (shortly L1([0,T])) is the Banach space of Lebesgue–Bochner E E integrablefunctions f :[0,T]→ E. • A mapping u :[0,T]→ E is absol(cid:3)utelycontinuousif there is a function u˙ ∈ L1([0,T])suchthatu(t)=u(0)+ tu˙(s)ds, ∀t ∈[0,T]. E 0 • IfXisatopologicalspace,C (X)isthespaceofcontinuousmappingsu : X → E E equippedwiththenormofuniformconvergence. • A set-valued mapping F :[0,T]⇒ E is measurable if its graph belongs to L([0,T])⊗B(E). • A convex weakly compact valued mapping F : X →ck(E) defined on a topo- logicalspace X isscalarlyuppersemicontinuousifforeveryx∗ ∈ E∗,thescalar functionδ∗(x∗,F(.))isuppersemicontinuouson X. Wereferto[13]formeasurablemultifunctionsandConvexAnalysis. Forthesakeofcompleteness,werecallandsummarizesomeresultsdeveloped in[9].By W2,1([0,T])wedenotethesetofallcontinuousfunctionsinC ([0,T]) E E suchthattheirfirstderivativesarecontinuousandtheirsecondderivativesbelongto L1([0,T]). E