Existence results for globally efficient solutions of vector equilibrium problems via a generalized kkm principle

The aim of this article is to present new existence results for globally efficient solutions of a strong vector equilibrium problem given by a sum of two functions via a generalized KKM principle, and to establish the connectedness of the solutions set.     

Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces

In this paper, a generalized vector equilibrium problem with set-valued maps defined on a reflexive Banach space is considered. By using the recession method, we first give the conditions under which the solution set is non-empty, convex and weakly compact, and then extend it to the strong generalized vector equilibrium problem. This facilitates generalizing and modifying various existence theorems. Furthermore, the topological properties of the solution set are studied and it is shown that the solution set includes some boundary points.     

Existence of solutions for generalized vector quasi-equilibrium problems

This paper deals with three classes of generalized vector quasi-equilibrium problems with or without compact assumptions. Using the well-known Fan-KKM theorems, their existence theorems for them are established. Some examples are given to illustrate our results.     

Existence of solutions on weak vector equilibrium problems

Weak vector equilibrium problems with bi-variable mappings from product space of two bounded complete locally convex Hausdorff topological vector spaces to another topological vector space are studied. The existence theorems of solutions are proved by the FKKM fixed point theorem. Viscosity principle of vector equilibrium problems is dealt with. The relations between solutions of the vector equilibrium problem and those of its perturbation problem are presented.     

Existence and iterative algorithm of solutions for a class of bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

Existence results for proper efficient solutions of vector equilibrium problems and applications

In this paper, we present sufficient conditions for the existence of Henig efficient solutions, superefficient solutions and Henig globally efficient solutions of a vector equilibrium problem in topological vector spaces, using a well-known separation theorem in infinite dimensional spaces. As an application, using a scalarization technique, existence results for proper efficient solutions of generalized vector variational inequalities are given.     

The connectedness of the solutions set for generalized vector equilibrium problems

In this paper, we establish the connectedness of the sets of Henig efficient solutions, globally efficient solutions, weak efficient solutions, superefficient solutions and efficient solutions for a class of generalized vector equilibrium problems without the assumptions of monotonicity and compactness.     

Existence of solutions for systems of generalized quasi‐equilibrium problems

In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies. Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi‐equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature.     

Existence of solutions for a generalized vector quasivariational inequality

The paper deals with a generalization of a vector quasivariational inequality. An existence theorem for its solutions is established; it is based on a kind of minimax inequality, which is here established for continuous affine mappings and differs from previous results. Fan's section theorem for set-valued mappings is extended. An application for an equilibrium problem of a network with vector-valued cost functions is given.This research was supported by the National Nature Science Foundation of China.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

Recession methods for generalized vector equilibrium problems

By using the recession method, we give some necessary and/or sufficient condition of solutions of generalized vector equilibrium problems.     

Connectedness of the sets of weak efficient solutions for generalized vector equilibrium problems

In this paper, a generalized vector equilibrium problem is introduced and studied. A scalar characterization of weak efficient solutions for the generalized vector equilibrium problem is obtained. By using the scalarization result, the existence of the weak efficient solutions and the connectedness of the set of weak efficient solutions for the generalized vector equilibrium problem are proved in locally convex spaces.     

On the generalized vector equilibrium problems

In this paper, generalized vector equilibrium problems are studied and some existence theorems of solutions for these problems in the setting of topological vector spaces are proved. Sufficient conditions for the set of solutions to be compact and convex are given. Our results improve some recent results in this field.     

The Hölder continuity of solutions to generalized vector equilibrium problems

In this paper, by using a weaker assumption, we discuss the Hölder continuity of solution maps for two cases of parametric generalized vector equilibrium problems under the case that the solution map is a general set-valued one, but not a single-valued one. These results extend the recent ones in the literature. Several examples are given for the illustration of our results.     

Existence of solutions for generalized implicit vector variational-like inequalities

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompact assumptions on underlying convex sets.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

向量最优化弱有效解的特征和存在性

研究多目标凸向量优化问题在Gateaux可微条件下弱有效解的特性,并讨论一类非凸向量最优化问题弱有效解及与一变分不等式的等价性,给出了解的存在性。     

Existence results for strong vector equilibrium problems and their applications

New existence results for the strong vector equilibrium problem are presented, relying on a well-known separation theorem in infinite-dimensional spaces. The main results are applied to strong cone saddle-points and strong vector variational inequalities providing new existence results, and furthermore they allow recovery of an earlier result from the literature.     

On the existence and stability of approximate solutions of perturbed vector equilibrium problems

In this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of Painlevé-Kuratowski to the initial set. Then, we introduce the concept of approximate solution for equilibrium problem governed by set-valued maps and we study the stability of these solutions. The particular case of linear continuous operators is considered as well.