Fix-finite approximation property in F-spaces

TL;DR

This paper demonstrates that in F-spaces, continuous maps and multifunctions can be approximated arbitrarily closely by maps with finitely many fixed points, using topological approximation properties.

Contribution

It introduces a method to approximate continuous maps and multifunctions in F-spaces with finite fixed points leveraging the simplicial approximation property.

Findings

Existence of epsilon-approximations with finite fixed points for continuous maps.

Extension of approximation results to n-valued multifunctions.

Applicability to path and simply connected compact subsets.

Abstract

In this paper, with the aid of the simplicial approximation property, the Hopf's construction and Dugundji's homotopy extension Theorem, we first show that if C is a nonempty compact convex subset of an F-space (E; || ||); then for every positive real number epsilon and every subset D of E containing C and every continuous map f from D to C there exists a continuous map g from D to C which is epsilon-near to f and has only a finite number of fixed points. Secondly, by using this result and the simplicial approximation property, we establish that for any positive real number epsilon and every path and simply connected compact subset D of E containing C and for each continuous n-valued multifunction F from D to C there exists a continuous n-valued multifunction G from D to C which is epsilon-near to F and has only a finite number of fixed points.