Comparison Theorems on Minimax Inequalities, Variational Inequalities, and Fixed Point Theorems
Kok-Keong Tan · Journal of the London Mathematical Society · 1983
Based on Ky Fan's lemma (which is an extension of the classical theorem of Knaster, Kuratowski and Mazurkiewicz to topological vector spaces), Fan's minimax inequality is slightly generalized simultaneously to non-compact convex sets and to a pair of functions. By applying the generalized inequality directly, Dugundji's and Granas's variational inequality in reflexive Banach spaces, which is an extension of Hartman's and Stampacchia's variational inequality, is generalized simultaneously to setvalued maps and to non-compact convex sets in topological vector spaces. This generalization in the single-valued case is in turn used to obtain a fixed point theorem for pseudo-contractive maps on a nonweakly compact convex subset of a Hilbert space, generalizing Browder's fixed point theorem for nonexpansive maps.