Remarks on Extensions of the Himmelberg Fixed Point Theorem

Abstract

Recommended by Anthony To-Ming LauRecently, Jafari and Sehgal obtained an extension of the Himmelberg fixed point theorem based on the Kakutani fixed-point theorem. We give generalizations of the extension to almost convex sets instead of convex sets. We also give generalizations for a large class B of better admissible multimaps instead of the Kakutani maps. Our arguments are based on the KKM principle and some of previous results due to the second author.

Fixed Point Theorems for Discontinuous Maps on a Non‐convex Domain

“…(See Himmelberg (), Idzik () and Jafari and Sehgal () for an extension without local convexity, but with an additional condition. See also Komiya and Park () for a generalization to an almost convex S . ) Proposition Let X be a locally convex Hausdorff topological vector space, and a given subset S of X be convex.…”

“…Takao Fujimoto g(S) ≡ {g(x) | x ∈ S} is contained in a compact subset of S, then there exists at least one fixed point x in S. (See Himmelberg (1972), Idzik (1988) and Jafari and Sehgal (2007) for an extension without local convexity, but with an additional condition. See also Komiya and Park (2007) for a generalization to an almost convex S.)…”