Fixed Points, Maximal Elements and Equilibria of Generalized Games in Abstract Convex Spaces

Yan-Mei Du

doi:10.1155/2012/842515

Abstract

We firstly prove some new fixed point theorems for set‐valued mappings in noncompact abstract convex space. Next, two existence theorems of maximal elements for class of 𝒜C,θ mapping and 𝒜C,θ‐majorized mapping are obtained. As in applications, we establish new equilibria existence theorems for qualitative games and generalized games. Our theorems improve and generalize the most known results in recent literature.

Highlights

Since Borglin and Keiding 1 proved a new existence theorem for a compact generalized games abstract economy with KF-majorized preference correspondences
An abstract convex space E, D; Γ consists of a topological space E, a nonempty set D, and a mapping Γ : D → 2E with nonempty values ΓA Γ A for each A ∈ D
Let X be a topological space, and Y be a nonempty subset of an abstract convex space E; Γ

Summary

Introduction

Since Borglin and Keiding 1 proved a new existence theorem for a compact generalized games abstract economy with KF-majorized preference correspondences. Following their ideas, many authors studied the existence of equilibria for generalized games, for example; see Ding 2 , Shen 3 , Chowdhury et al 4 , Briec and Horvath 5 , Ding and Wang 6 , Kim et al 7 , Kim and Tan 8 , Lin et al 9 , Lin and Liu , Du and Deng , and so forth. Many authors established existence theorems of maximal elements and equilibria of generalized games with majorized correspondences in H-space, G-convex space, and FCspace, respectively. Our results generalized and improve the corresponding results due to Ding and Feng 20 , Ding and Wang 6 , Park 22, 24 , Yuan , Chowdhury et al 4 , Tan and Yuan , Borglin-Keiding 1 , Yannelis , and so forth

Preliminaries

Fixed Point Theorems

Existence of Maximal Elements

Existence of Equilibrium of Points