Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

FC-空间中含伪单调集值映象的广义向量平衡问题

在FC-空间内引入和研究了几类含伪单调集值映象的广义向量平衡问题．利用第二作者的KKM定理,在FC空间内对这些广义向量平衡问题证明了平衡点的存在性定理,改进与推广了近期文献中的相应结果．     

Generalized Equilibrium Problems for Quasimonotone and Pseudomonotone Bifunctions

By using quasimonotone and pseudomonotone bifunctions, we derive sufficient conditions which include weak coercivity conditions for existence of equilibrium points. As a consequence, we improve some recent results on the existence of such solutions.     

From Scalar to Vector Equilibrium Problems in the Quasimonotone Case

In a unified approach, existence results for quasimonotone vector equilibrium problems and quasimonotone (multivalued) vector variational inequality problems are derived from an existence result for a scalar equilibrium problem involving two (rather than one) quasimonotone bifunctions. The results in the vector case are not only obtained in a new way, but they are also stronger versions of earlier existence results.     

A Note on Equilibrium Problems with Properly Quasimonotone Bifunctions

In this paper, we consider some well-known equilibrium problems and their duals in a topological Hausdorff vector space X for a bifunction F defined on K x K,where K is a convex subset of X. Some necessary conditions are investigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is defined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sharp in order to solve the dual equilibrium problem on every convex set.     

Equilibrium Problems in the Quasimonotone Case

Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works.     

Generalized Vector Equilibrium Problems for Pseudomonotone Multivalued Bifunctions<Superscript><Emphasis Type="Bold">1</Emphasis></Superscript>

In this paper, we apply a new version of the Brézis, Nirenberg, and Stampacchia theorem; we use pseudomonotonicity and some coercivity conditions to establish some existence result for a solution of generalized vector equilibrium problems for multivalued bifunctions. The proper quasiconvexity of multivalued bifunctions is introduced and existence theorems for generalized vector equilibrium problems related to multivalued mappings with the KKM property are obtained. The new results extend and modify various existence theorems for similar problems.     

Geometric Properties and Coincidence Theorems with Applications to Generalized Vector Equilibrium Problems

The present paper is divided into two parts. In the first part, we derive a Fan-KKM type theorem and establish some Fan type geometric properties of convex spaces. By applying our results, we also obtain some coincidence theorems and fixed-point theorems in the setting of convex spaces. The second part deals with the applications of our coincidence theorem to establish some existence results for a solution to the generalized vector equilibrium problems.     

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.     

Generalized Vector Equilibrium Problems with Trifunctions

In this paper, we study the existence of strong and weak solutions of the generalized vector equilibrium problems for trifunctions. Two special classes of vector-valued trifunctions are introdcued, called the classes of (SPM) and (GPM), respectively. Some existence results for strong solutions associated to functions of these classes are given.     

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.     

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

Generalized Vector Equilibrium Problems in Generalized Convex Spaces

In this paper, we introduce and study a class of abstract generalized vector equilibrium problems (AGVEP) in generalized convex spaces which includes most vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems, and generalized vector variational inequality problems as special cases. By using the generalized GKKM and generalized SKKM type theorems due to the first author, some new existence results of equilibrium points for the AGVEP are established in noncompact generalized convex spaces. As consequences, some recent results in the literature are obtained under much weaker assumptions.     

具强制条件的向量均衡问题

研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性．     

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.     

Convergence Results for Weak Efficiency in Vector Optimization Problems with Equilibrium Constraints

In this paper, we consider weak efficiency in vector optimization problems with equilibrium constraints. We obtain results on the convergence of the marginal map, the value, and the solution sets. This research was supported by Hong Kong Polytechnic University under Grant G-YY33 and by National Science Foundation of China under Grant 10401043. The authors express their gratitude to Professor Guangya Chen for careful reading of an earlier version of this paper. The authors are grateful to the anonymous referees for helpful comments and constructive suggestions.     

扰动广义向量平衡问题解集的下半连续性

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.     

Combined Relaxation Method for Mixed Equilibrium Problems

We consider a general class of equilibrium problems which involve a single-valued mapping and a nonsmooth bifunction. Such mixed equilibrium problems are solved with a combined relaxation method using an auxiliary iteration of a splitting-type method for constructing a separating hyperplane. We prove the convergence of the method under the assumption that the dual of the mixed equilibrium problem is solvable. Convergence rates are also derived.S. Schaible - This author gratefully acknowledges partial support from the National Science Council of Taiwan.J.C. Yao - His research was partially supported by Grant NSC 93-2115-M-110-011     

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.     

Sensitivity Analysis for Vector Equilibrium Problems under Functional Perturbations

This article is devoted to sensitivity analysis for vector equilibrium problems under functional perturbations. We show that the solution mapping is upper semicontinuous. Sufficient conditions for lower semicontinuity and Hölder continuity of the solution mapping are established. Finally, we derive some corollaries for special cases of vector equilibrium problems as examples.