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.
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References
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Breen, M. A Krasnosel'skii theorem for nonclosed sets in Rd . J Geom 26, 105–114 (1986). https://doi.org/10.1007/BF01227836
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DOI: https://doi.org/10.1007/BF01227836