A Krasnosel'skii theorem for nonclosed sets in Rd

Abstract

This work will be concerned with a Krasnosel'skii theorem for nonclosed bounded sets in Rd, and the following theorem will be obtained: For each d ⩾ 2, define f(d) = d2 − 2d+3 if d ≠ 3 and f(d)=2d+1 if d = 3. Let S be a nonempty bounded set in Rd, d ⩾ 2, and assume that cl S ∼ S is a finite union of convex components, each having closure a polytope. If every f(d) points of S see via S a common point, then there is a point p in cl S such that Bp ≡ s:s in S and (p,s]⊄ S is nowhere dense in S.