Equilibrium Problems under Generalized Convexity and Generalized Monotonicity

Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.     

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 the Solution Set to Parametric Weak Vector Equilibrium Problems

In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems. This research was partially supported by the National Natural Science Foundation of China (Grant 10561007) and the Natural Science Foundation of Jiangxi Province, China.     

向量均衡问题

本首先引进了集值映射的伪单调性，利用Ky Fan引理以及KKM映射对一般向量均衡问题建立了相应的存在性定理．     

The Essential Component of the Solution Set for Vector Equilibrium Problems

In the paper, by using Ky Fan’s section theorem, we obtain an existence theorem for vector equilibrium problems. Motivated by the ideas of Kinoshita and McLennan, we introduce the concept of the essential component of the solution set for vector equilibrium problems, and we prove that there exists at least one essential component of the solution set for every vector equilibrium problem satisfying some conditions. Research was partially supported by the Natural Science Foundation of Guangdong Province, P. R. China.     

Strong Vector Equilibrium Problems

In this paper, the existence of the solution for strong vector equilibrium problems is studied by using the separation theorem for convex sets. The arc-wise connectedness and the closedness of the strong solution set for vector equilibrium problems are discussed; and a necessary and sufficient condition for the strong solution is obtained.     

Semicontinuity of Solution Mappings to Parametric Generalized Vector Equilibrium Problems

In this article, stability results concerning the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mappings to parametric generalized vector equilibrium problems with neither the monotonicity of mappings nor any information of the solution mappings are established by using scalarization methods and a new density result.     

集值映射下的向量均衡问题解的存在性

利用广义的截口定理,本讨论集值映射和变序下的向量均衡问题解的存在性。     

On Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Equilibrium Constraints

In this article, necessary conditions of Fritz John type for weak efficient solutions of a nonsmooth vector equilibrium problem involving equilibrium constraints (VEPEC) in terms of the Clarke subdifferentials are established. Under constraint qualifications which are suitable for (VEPEC), necessary conditions of Kuhn-Tucker type for efficiency are derived. Under assumptions on generalized convexity of data, sufficient conditions for efficiency are developed. Some applications to vector variational inequalities and vector optimization problems with equilibrium constraints are also given.     

无限族demi压缩映射与平衡问题的收敛定理

在Hilbert空间中,为了找到无限个demi压缩映射公共不动点集和广义混合平衡问题解的公共元,本文介绍了一种迭代算法,得到关于公共元的强收敛定理,并给出例子说明结果.     

乘积G-凸空间内的非空交定理和广义矢量平衡问题组

利用作者在G-凸空间内对集值映象簇得到的一个极大元存在性定理,在非紧乘积G-空间内,对集值映象簇建立了某些新的非空交定理．作为应用, 在非紧乘积G-凸空间内, 对广义矢量平衡问题组证明了一些平衡存在性定理．这些定理统一、改进和推广了文献中一些重要的已知结果．     

平衡问题解的存在性

主要研究平衡问题解的存在性.通过对目标函数和可行集合的渐近分析,给出拟单调平衡问题解集非空的条件.进而用类似的方法研究了向量平衡问题解存在的条件,并将其应用到向量优化问题上.     

具有控制结构的向量均衡问题与解集的性质

提出一种具有控制结构的向量均衡问题与向量映射的新的伪单调性概念,得到具有控制结构的向量均衡问题解的存在性及其解集的紧凸性．作为应用,得到具有控制结构的向量变分不等式与互补问题的解．     

Existence Results for Set-Valued Vector Quasiequilibrium Problems

This paper deals with the set-valued vector quasiequilibrium problem of finding a point (z ₀,x ₀) of a set E×K such that (z ₀,x ₀)∈B(z ₀,x ₀)×A(z ₀,x ₀), and, for all η∈A(z ₀,x ₀),

Stampacchia向量均衡问题与向量相补问题

引进Stampacchia向量均衡问题与一种新的向量相补问题．用数值方法,得到它们的存在定理,并讨论Stampacchia广义向量变分不等式,向量隐相补问题与极小元问题的关系．     

Existence of a Solution and Variational Principles for Vector Equilibrium Problems

In this paper, we prove an existence result for a solution to the vector equilibrium problems. Then, we establish variational principles, that is, vector optimization formulations of set-valued maps for vector equilibrium problems. A perturbation function     

Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem. This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jinxing Province, China.     

Existence Results in a General Equilibrium Problem

In this article, we give sufficient conditions for the existence of solutions of a general model which includes as special cases many generalized vector quasi-equilibrium problems with set-valued maps. The obtained results generalize and improve several earlier results.     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.