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.

EKELAND'S PRINCIPLE FOR SET-VALUED VECTOR EQUILIBRIUM PROBLEMS

In this paper, we introduce a concept of quasi C-lower semicontinuity for setvalued mapping and provide a vector version of Ekeland＇s theorem related to set-valued vector equilibrium problems. As applications, we derive an existence theorem of weakly efficient solution for set-valued vector equilibrium problems without the assumption of convexity of the constraint set and the assumptions of convexity and monotonicity of the set-valued mapping. We also obtain an existence theorem of ε-approximate solution for set-valued vector equilibrium problems without the assumptions of compactness and convexity of the constraint set.

LEVITIN-POYAK WELL-POSEDNESS OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS WITH FUNCTIONAL CONSTRAINTS

In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.

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.

GENERIC STABILITY AND EXISTENCE OF ESSENTIAL COMPONENTS OF THE SOLUTION SET FOR THE SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS

By using Fort theorem the generic stability result for the system of generalized vector equilibrium problems is established. Further, by proving the existence and connectivity of minimal essential set the existence result of essential components in the solution set is derived.

Existence results for generalized vector equilibrium problems with applications

By a coincidence theorem, some existence theorems of solutions are proved for four types of generalized vector equilibrium problems with moving cones. Applications to the generalized semi-infinite programs with the generalized vector equilibrium constraints under the mild conditions are also given. The results of this paper unify and improve the corresponding results in the previous literature.

Continuity of the Solution Mappings for Parametric Generalized Strong Vector Equilibrium Problems

The stability analysis of the solution mappings for vector equilibrium problems is an important topic in optimization theory and its applications. In this paper, we focus on the continuity of the solution mapping for a parametric generalized strong vector equilibrium problem. By virtue of a nonlinear scalarization technique, a new density result of the solution mapping is obtained. Based on the density result, we give sufficient conditions for the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mapping to the parametric generalized strong vector equilibrium problem. In addition, some examples were given to illustrate that our results improve ones in the literature.

An Equilibrium Version of Set-Valued Ekeland Variational Principle and its Applications to Set-Valued Vector Equilibrium Problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland varia- tional principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of X complete metric spaces （Z, d）, we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X C Z is countably compact in any Hausdorff topology weaker than that induced by d. When （Z, d） is a Fechet space （i.e., a complete metrizable locally convex space）, our existence result only requires that the domain C Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results.

Error Bounds for Generalized Mixed Vector Equilibrium Problems via a Minimax Strategy

In this paper,by using scalarization techniques and a minimax strategy,error bound results in terms of gap functions for a generalized mixed vector equilibrium problem are established,where the solutions for vector problems may be general sets under natural assumptions,but are not limited to singletons.The other essentially equivalent approach via a separation principle is analyzed.Special cases to the classical vector equilibrium problem and vector variational inequality are also discussed.

On Generalized Vector Equilibrium Problems

A new generalized vector equilibrium problem involving set-valued mappings and the proper quasi concavity of set-valued mappings in topological vector spaces are introduced; its existence theorems and the convexity of the solution sets are established.

Continuity of Solution Mappings for Parametric Generalized Set-Valued Weak Vector Equilibrium Problems

This paper obtains some stability results for parametric generalized set-valued weak vector equilibrium problem. Under new assumptions, which do not contain any information about solution mappings, the authors establish the continuity of the solution mapping to a parametric generalized set-valued weak vector equilibrium problem without monotonicity. These results extend and improve some results in the literature. Some examples are given to illustrate the results.

Upper Semicontinuity of Parametric Lexicographic Vector Equilibrium Problems

In this paper, lexicographic vector equilibrium problems are investigated. By using the idea of sequential process, the upper semicontinuity and closedness of the solution set map are established for a parametric lexicographic strong vector equilibrium problem.

Lower Semicontinuity of the Solution Sets to Parametric Generalized Vector Equilibrium Problems

In this paper,by a scalarization method,the lower semicontinuity of the solution mappings to two kinds of parametric generalized vector equilibrium problems involving set-valued mappings is established under new assumptions which are weaker than the C-strict monotonicity.These results extend the corresponding ones.Some examples are given to illustrate our results.

SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS ON PRODUCT FC-SPACES

By applying a maximal element theorem on product FC-space due to author, some new equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in FC-spaces. By using these equilibrium existence theorems, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are established in noncompact product FC-spaces. These results improve and generalize some recent results in literature to product FC-spaces without any convexity structure.

GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMSIN LOCALLY G-CONVEX SPACES

Some classes of generalized vector quasi-equilibrium problems ( in short, GVQEP) are introduced and studied in locally G-convex spaces which includes most of generalized vector equilibrium problems; generalized vector variational inequality problems, quasi-equilibrium problems and quasi-variational inequality problems as special cases. First, an equilibrium existence theorem for one person games is proved in locally G-convex spaces.. As applications, some new existence theorems of solutions for the GVQEP are established in noncompact locally G-convex spaces. These results and argument methods are new and completely different from that in recent literature.

SYSTEM OF VECTOR QUASI-EQUILIBRIUM PROBLEMS AND ITS APPLICATIONS

A new system of vector quasi-equilibrium problems is introduced and its existence of solution is proved. As applications, some existence results of weak Pareto equilibrium for both constrained multicriteria games and multicriteria games without constrained correspondences are also shown.

Upper and Lower Semicontinuity of Solution Sets for Parametric Generalized Vector Quasi-equilibrium Problems

In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.

MATHEMATICAL PROGRAMS WITH SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM CONSTRAINTS IN FC-SPACES

In this article, four new classes of systems of generalized vector quasi-equilibrium problems are introduced and studied in FC-spaces without convexity structure. The notions of Ci（x）-FC-partially diagonally quasiconvex, Ci（x）-FC-quasiconvex, and Ci（x）-FC- quasiconvex-like for set-valued mappings are also introduced in FC-spaces. By applying these notions and a maximal element theorem, the nonemptyness and compactness of solution sets for four classes of systems of generalized vector quasi-equilibrium problems are proved in noncompact FC-spaces. As applications, some new existence theorems of solutions for mathematical programs with system of generalized vector quasi-equilibrium constraints are obtained in FC-spaces. These results improve and generalize some recent known results in literature.

向量平衡问题强有效解集的连通性

本文主要在自反Banach空间中研究向量平衡问题强有效解集的连通性。利用新定义的非线性函数,将1个非凸非闭集合和1个紧致集分离;证明强向量平衡问题的强有效解集可以表示为非线性标量问题解集的并集,从而得到向量平衡问题强有效解集的连通性。

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.