Noncompact infinite optimization and equilibria of constrained games in generalized convex spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＩｂｅａｎｙ (ｆｉｎｉｔｅｏｒｉｎｆｉｎｉｔｅ)ｉｎｄｅｘｓｅｔａｎｄｆｏｒｅａｃｈｉ∈Ｉ,Ｘｉｂｅａｔｏｐｏｌｏｇｉｃａｌｓｐａｃｅ.ＷｅｕｓｅｔｈｅｎｏｔａｔｉｏｎＸ =∏ｉ∈ＩＸｉ　ａｎｄ　Ｘｉ=∏ｊ∈Ｉ,ｊ≠ｉＸｊ.Ｆｏｒｅａｃｈｘ∈Ｘ ,ｘｉｄｅｎｏｔｅｓｉｔｓｉｔｈｃｏｏｒｄｉｎａｔｅａｎｄｘｉｔｈｅｐｒｏｊｅｃｔｉｏｎｏｆｘｏｎＸｉ.Ｗｒｉｔｅｘ=(ｘｉ,ｘｉ) .Ｆｏｒｅａｃｈｉ∈Ｉ,ｌｅｔＦｉ:Ｘｉ → 2 Ｘｉｂｅａｓｅｔ＿ｖａｌｕｅｄｍａｐｐｉｎｇａｎｄｆｉ:Ｘ →Ｒ …     

CONSTRAINED MULTIOBJECTIVE GAMES IN LOCALLY CONVEX H-SPACES

IntroductionTheconceptofanequilibriumpointforann_persongamewasfirstintroducedbyNash[1,2 ],whoestablishedtheexistenceoftheequilibriumpointundercertainassumptions.Sincethen ,theNashequilibriumproblemforn_persongameshavebeenintensivelystudiedandextendedbyanumberofauthorsundervariousassumptionsanddifferentdirections.Theconstrainedgameswithfiniteorinfiniteplayersisanimportantgeneralizationofn_persongameswhichhavebeenwidelystudiedbymanyauthors.Recently ,muchattentionhasbeenfocusedonthegameproblemsw…     

Quasi-equilibrium problems in noncompact generalized convex spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＸａｎｄＹｂｅｎｏｎｅｍｐｔｙｓｅｔｓａｎｄ2ＸｂｅｔｈｅｆａｍｉｌｙｏｆａｌｌｓｕｂｓｅｔｓｏｆＸ.ＬｅｔＴ:Ｘ→Ｙｂｅａｓｉｎｇｌｅ＿ｖａｌｕｅｄｍａｐｐｉｎｇ,Ａ:Ｘ→2Ｘｂｅａｓｅｔ＿ｖａｌｕｅｄｍａｐｐｉｎｇａｎｄｆ:Ｘ×Ｙ→Ｒ∪±∞ｂｅａｆｕｎｃｔｉｏｎ.Ｔｈｅｑｕａｓｉ＿ｅｑｕｉｌｉｂｒｉｕｍｐｒｏｂｌｅｍＱＥＰ(Ｔ,Ａ,ｆ)ｉｓｔｏｆｉｎｄ^ｘ∈Ｘｓｕｃｈｔｈａｔ^ｘ∈Ａ(^ｘ),ｆ(^ｘ,Ｔ^ｘ)≤ｆ(ｙ,Ｔ^ｘ),　　ｙ∈Ａ(^ｘ).(1)ＴｈｅＱＥＰ(Ｔ,Ａ,ｆ)ｗａｓｉｎｔｒｏｄ…     

Constrained multiobjective games in locally convex<Emphasis Type="Italic">H</Emphasis>-spaces

A new class of constrained multiobjective games with infinite players in noncompact locally convex H-spaces without linear structure are introduced and studied. By applying a Fan-Glicksberg type fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values and a maximum theorem, several existence theorems of weighted Nath-equilibria and Pareto equilibria for the constrained multiobjective games are proved in noncompact locally convex H-spaces. These theorems improve, unify and generalize the corresponding results of the multiobjective games in recent literatures. Contributed by Ding Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Ding Xie-ping (1938-)     

Generalized constrained multiobjective games in locally <Emphasis Type="Italic">FC</Emphasis>-uniform spaces

A new class of generalized constrained multiobjective games is introduced and studied in locally FC-uniform spaces without convexity structure where the number of players may be finite or infinite and all payoff functions get their values in an infinite-dimensional space. By using a Himmelberg type fixed point theorem in locally FC-uniform spaces due to author, some existence theorems of weak Paxeto equilibria for the generalized constrained multiobjective games are established in locally FC-uniform spaces. These theorems improve, unify and generalize the corresponding results in recent literatures.     

GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS IN LOCALLY G-CONVEX SPACES

IntroductionLetX ,YbetwotopologicalspacesandZbeanonemptyset.LetF∶X×X→ 2 Z,A∶X→ 2 XandC∶X→ 2 Zbeset_valuedmappingswhere2 ZdenotesthefamilyofallsubsetsofZ .　　Thegeneralizedvectorquasi_equilibriumproblemofType (Ⅰ ) (inshort,GVQEP(Ⅰ ) )istofind ^x∈Xsuchthat^x∈A( ^x)andF( ^x ,z) C( ^x) ,　　 z∈A( ^x) . ( 1 )　　Thegeneralizedvectorquasi_equilibriumproblemofType (Ⅱ ) (inshort,GVQEP(Ⅱ ) )istofind ^x∈Xsuchthat^x∈A( ^x)andF( ^x ,z) C( ^x) ,　　 z∈A( ^x) . ( 2 )　　…     

MAXIMAL ELEMENTS FOR G_B-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅰ)

IntroductionItiswell_knownthattheexistenceofmaximalelementsforset_valuedmappingsintopological(vector)spacesanditsimportantapplicationstomathematicaleconomieshavebeenstudiedextensivelybymanyauthorsinbothmathematicsandeconomics.Forexistenceresultsofmaximalelementsofvariousclassesofset_valuedmappingsandtheirapplicationstomathematicaleconomiesandotherbranchesofmathematics,thereadermayconsultRefs.[1～ 2 7]andthereferencestherein .LetXbeatopologicalspaceandY =∏i∈IYibeaproductofnonemptyconvexsetY…     

Nonempty intersection theorems and system of generalized vector equilibrium problems in product G-convex spaces

By using an existence theorems of maximal elements for a family of set-valued mappings in G-convex spaces due to the author, some new nonempty intersection theorems for a family of set-valued mappings were established in noncompact product G-convex spaces. As applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems were proved in noncompact product G-convex spaces. These theorems unify, improve and generalize some important known results in literature.     

MAXIMAL ELEMENTS FOR GB-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅰ)

A new family of set-valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set-valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature. Contributed by DING Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province of China ([2000]25) Biography: DING Xie-ping (1938∼)     

NEW COLLECTIVELY FIXED POINT THEOREMS AND APPLICATIONS IN G-CONVEX SPACES

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ 1 991 ,Ｔａｒａｆｄａｒ[1]ｆｉｒｓｔｅｓｔａｂｌｉｓｈｅｄｔｈｅｆｏｌｌｏｗｉｎｇｃｏｌｌｅｃｔｉｖｅｌｙｆｉｘｅｄｐｏｉｎｔｔｈｅｏｒｅｍｏｎｔｈｅｐｒｏｄｕｃｔｓｐａｃｅｏｆｎｏｎｅｍｐｔｙｃｏｍｐａｃｔｃｏｎｖｅｘｓｕｂｓｅｔｓｏｆｔｏｐｏｌｏｇｉｃａｌｖｅｃｔｏｒｓｐａｃｅｓａｎｄｇａｖｅｉｔｓａｐｐｌｉｃａｔｉｏｎｓｔｏｔｈｅｅｘｉｓｔｅｎｃｅｏｆｅｑｕｉｌｉｂｒｉｕｍｐｏｉｎｔｓｆｏｒａｂｓｔｒａｃｔｅｃｏｎｏｍｉｅｓ.Ｔｈｅｏｒｅｍ 1　Ｌｅｔ Ｘｉ ｉ∈Ｉ…     

Systems of generalized quasi-variational inclusion （disclusion） problems in FC-spaces

By applying an existence theorem of maximal elements of set-valued mappings in FC-spaces proposed by the author, some new existence theorems of solutions for systems of generalized quasi-variational inclusion （disclusion） problems are proved in FC-spaces without convexity structures. These results improve and generalize some results in recent publications from closed convex subsets of topological vector spaces to FC-spaces under weaker conditions.     

MAXIMAL ELEMENTS FOR G_B-MAJORIZED MAPPINGS IN PRODUCT G_CONVEX SPACES AND APPLICATIONS (Ⅱ)

IntroductionThispaperisacontinuumoftheprecedingpaperofauthor[1] .Fortheconceptsandnotationsofgeneralizedconvex (orG_convex)space ,CG_convexspace,theclassB(Y ,X)ofbetteradmissiblemappings ,GB_mappings,GB_majorizedmappingsandtherelatednotions,thereadermayconsultRef.[1].Themainaimofthispaperistogivesomeapplicationsoftheexistencetheoremsofmaximalelementsobtainedbyauthor[1] .ByapplyingourresultsinRef.[1],wegivesomecomponentversionofcoincidencetheoremandFan_Browdertypefixedpointtheorems .Somee…     

New collectively fixed point theorems and applications inG-convex spaces

By applying the technique of continuous partition of unity and Tychonoff’s fixed point theorem, some new collectively fixed point theorems for a family of set-valued mappings defined on the product space of noncompact G-convex spaces are proved. As applications, some nonempty intersetion theorems of Ky Fan type for a family of subsets of the product space of G-convex spaces are proved; An existence theorem of solutions for a system of nonlinear inequalities is given in G-convex spaces and some equilibrium existence theorems of abstract economies are also obtained in G-convex spaces. Our theorems improve, unify and generalized many important known results in recent literature.     

CONTINUOUS SELECTION THEOREMS FOR FAN-BROWDER MAPPINGS IN TOPOLOGICAL SPACES AND THEIR APPLICATIONS

The concept of Fan-Browder mappings was first introduced in topological spaces without any convex structure. Then a new continuous selection theorem was obtained for the Fan-Browder mapping with range in a topological space without any convex structure and noncompact domain. As applications, some fixed point theorems, coincidence theorems and a nonempty intersection theorem were given. Both the new concepts and results unify and extend many known results in recent literature.     

Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications(Ⅱ)

This paper is a continuum of the preceding paper of author.Some new sys- tems of generalized vector quasi-equilibrium problems involving condensing mappings are introduced and studied in locally FC-uniform spaces.By applying the existence theorem of maximal elements of condensing set-valued mappings in locally FC-uniform spaces obtained by author in the preceding paper,some new existence theorems of solutions for the systems of generalized vector quasi-equilibrium problems are proved in locally FC- uniform spaces.These results improve and generalize some known results in literature to locally FC-uniform spaces.     

Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications (Ⅱ)

This paper is a continuum of the preceding paper of author. Some new systems of generalized vector quasi-equilibrium problems involving condensing mappings are introduced and studied in locally FC-uniform spaces. By applying the existence theorem of maximal elements of condensing set-valued mappings in locally FC-uniform spaces obtained by author in the preceding paper, some new existence theorems of solutions for the systems of generalized vector quasi-equilibrium problems are proved in locally FC-uniform spaces. These results improve and generalize some known results in literature to locally FC-uniform spaces.     

GENERALIZED VECTOR VARIATIONAL-TYPE INEQUALITIES IN FC-SPACES

A class of generalized vector variational-type inequality problems (GVVTIP) are studied in FC-spaces, which includes the most of vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems and generalized vector variational inequality problem as special cases. By using F-KKM theorem, some new existence results for GVVTIP are established in noncompact FC-space. As consequences, some recent known results in literature are obtained under much weaker assumption.     

Systems of generalized vector quasi-variational inclusions and systems of generalized vector quasi-optimization problems in locally FC-uniform spaces

In this paper, we introduce some new systems of generalized vector quasi-variational inclusion problems and system of generalized vector ideal (resp., proper, Pareto, weak) quasi-optimization problems in locally FC-uniform spaces without con-vexity structure. By using the KKM type theorem and Himmelberg type fixed point theorem proposed by the author, some new existence theorems of solutions for the sys-tems of generalized vector quasi-variational inclusion problems are proved. As to its applications, we obtain some existence results of solutions for systems of generalized vec-tor quasi-optimization problems.     

Generalized variational inequalities and generalized quasi-variational inequalities

Under much weaker hypotheses and in a more general setting some existence theorems of solutions to generalized variational inequalities,generalized quasi-varia-tional inequalities and minimax inequalities are established.The results presented in this paper generalize the corresponding results of[3—13]to the noncompact case,thus improving these results.     

MAXIMAL ELEMENTS OF A FAMILY OF GB-MAJORIZED MAPPINGS IN PRODUCT FC-SPACES AND APPLICATIONS

A new family of GB-majorized mappings from a topological space into a finite continuous topological spaces （in short, FC-space） involving a better admissible set-valued mapping is introduced. Some existence theorems of maximal elements for the family of GB-majorized mappings are proved under noncompact setting of product FCspaces. Some applications to fixed point and system of minimax inequalities are given in product FC-spaces. These theorems improve, unify and generalize many important results in recent literature.