George Xian-Zhi Yuan

The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods

Abstract In this paper, we present two ways to study the existence of weight Nash-equilibria and Pareto equilibria for multiobjective games. One is the ‘fixed-point method’ which is well known; and the second is the application of ‘Ky Fan minimax inequality’, which is not often used in the study of optimization, game theory, and mathematical programming. As results, several existence theorems for weight Nash-equilibria and Pareto equilibria are established which improve and unify the corresponding results in the recently existing literatures.

The Knaster-Kuratowski and Mazurkiewicz theroy in hyperconvex metric spaces and some of its applications

The Knaster-Kuratowski and Mazurkiewicz principle is characterized in hyperconvex metric spaces, leading to a characterization theorem for a family of subsets with the finite intersection property in such setting. The theorem is illustrated by giving hyperconvex versions of Fans celebrated minimax principle and Fans best approximation theorem for set-valued mappings. These are applied to obtain formulations of the Browder-Fan fixed point theorem and the Schauder-Tychonoff fixed point theorem in hyperconvex metric spaces for set-valued mappings. In addition, existence theorems for saddle points, intersection theorems and Nash equilibria are obtained.

Fixed point theorems of upper semicontinuous multivalued mappings with applications in hyperconvex metric spaces

In the present paper, we establish two fixed point theorems for upper semicontinuous multivalued mappings in hyperconvex metric spaces and apply these to study coincidence point problems and minimax problems.

The essential components of coincident points for weakly inward and outward set-valued mappings

In this paper, by the concept of essential components of coincident points for set-valued mappings, we study the existence of essential components of both coincident and fixed points for non-self upper semicontinuous set-valued mappings in normed spaces. These results include corresponding results in the literature as special cases.

The existence of equilibria for noncompact generalized games

The purpose of this paper is to establish general existence of equilibria for noncompact generalized games (respectively, noncompact abstract economics) under general setting of noncompact conditions and in which the L-majorized preference mappings may not have lower semicontinuity, and constraint correspondences are only lower or upper semicontinuous. In our model, strategic (respectively, commodity) spaces are not compact, the set of players (respectively, agents) are countable or uncountable, and underlying spaces are either finite- or infinite-dimensional locally topological vector spaces. Our results might be regarded as a unified theory for the corresponding results in the existing literatures in the study of generalized games (respectively, abstract economics) theory.

Maximal elements and equilibria of generalized games for 𝒰-majorized and condensing correspondences

In this paper, we first give an existence theorem of maximal elements for a new type of preference correspondences which are 𝒰-majorized. Then some existence theorems for compact (resp., non-compact) qualitative games and generalized games in which the constraint or preference correspondences are 𝒰-majorized (resp., Ψ-condensing) are obtained in locally convex topological vector spaces.

SADDLE POINTS AND MINIMAX THEOREMS FOR VECTOR-VALUED MULTIFUNCTIONS ON H-SPACES

Abstract In this paper some existence theorems of loose saddle point, saddle point, and minimax problems for vector-valued multifunctions on H -spaces are proved. The results presented in this paper generalize some recent results in [1–4].

The Existence of Essentially Connected Components of Solutions for Variational Inequalities

The aim of this paper is to establish the existence of essentially connected components for Hartman-Stampacchia type variational inequalities for both set-valued and single-valued mappings in normed spaces. Our results show that each variational inequality problem has, at least, one connected component of its solutions which is stable though in general its solution set may not have a good behavior (i.e., not stable). Thus if a variational inequality problem has only one connected solution set, it must be stable. Here we don’t need to require the objective mapping to be either Lipschitz or differential.

On the existence of periodic solutions for nonlinear evolutions in Hilbert spaces

A periodic solutions for nonlinear evolution equations of the form du/dt ∈ -Au(t) + F(t,u(t)), t ∈ R, where F : R × H → 2 is a set-valued Caratheodory function are considered when both F and A are set-valued mappings and not necessarily linear.

A necessary and sufficient condition for upper hemicontinuous set-valued mappings without compact-values being upper demicontinuous

The purpose of this article is to give a characterization of an upper hemicontinuous mapping with non-empty convex values being upper demicontinuous, i.e., we show that an upper hemicontinuous set-valued mapping with non-empty convex values (not necessarily compact-valued) is upper demicontinuous if and only if the set-valued mapping has no interior asymptotic plane.