Applications of Fixed Point Theorems to Generalized Saddle Points of Bifunctions on Chain-Complete Posets

Hongya Gao,Ying Liu,Jinlu Li

doi:10.1155/2014/580621

Abstract

We apply the extensions of the Abian-Brown fixed point theorem for set-valued mappings on chain-complete posets to examine the existence of generalized and extended saddle points of bifunctions defined on posets. We also study the generalized and extended equilibrium problems and the solvability of ordered variational inequalities on posets, which are equipped with a partial order relation and have neither an algebraic structure nor a topological structure.

Highlights

Let X be a topological vector space and let C be a subset of X
We prove the following theorem for the existence of generalized saddle point
We prove the following theorem for the existence of generalized equilibrium

Summary

Introduction

Let X be a topological vector space and let C be a subset of X. In economic theory and social sciences, there are some examples that both of the income and outcome spaces of mappings are posets, which are equipped with neither topological structure nor algebraic structure In these circumstances, the optimization problems will be orderoptimization problems, which are not normal optimization problems (with respect to real valued functions) and they cannot be solved by using the standard methods. In [10], Xie et al generalized the extensions of the Abian-Brown fixed point theorem provided in [8] from chain-complete posets to chain-complete preordered sets for set-valued mappings By using these generalizations and by applying the order-increasing upward property of set-valued mappings, they prove several existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chaincomplete preordered sets. We study the solvability of generalized and extended equilibrium problems of bifunctions and ordered variational inequalities on posets, which have neither an algebraic structure nor a topological structure

Several Extensions of the Abian-Brown Fixed Point Theorem on Posets

Generalized and Extended Saddle Points of Bifunctions on Posets

Generalized and Extended Equilibrium Problems of Bifunctions on Posets

Generalized and Extended Ordered Variational Inequalities on Posets

Concluding Remarks