Necessary and Sufficient Conditions for Fixed-Point Theorems, Minimax Inequalities and Related Theorems

Abstract

This paper provides necessary and sufficient conditions for fixed-point theorems, minimax inequalities and some related theorems defined on arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of equilibrium of minimax inequality without imposing any kind of convexity nor any restriction on topological space. The result then is employed to fully characterize fixed point theory, saddle point theory, and the FKKM theory. 2010 Mathematical subject classification: 49K35, 90C26, 55M20 and 91A10.

Cite this paper

@inproceedings{Tian2013NecessaryAS, title={Necessary and Sufficient Conditions for Fixed-Point Theorems, Minimax Inequalities and Related Theorems}, author={Guoqiang Tian}, year={2013} }