A Krasnosel′skiĭ-type theorem involving $p$-arcs
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- by Jean B. Chan PDF
- Proc. Amer. Math. Soc. 102 (1988), 667-676 Request permission
Abstract:
Let $p$ be a point in ${E_2}$. A convex arc joining a pair of distinct points $x$ and $y$ in ${E_2}$ is called a $p$-arc if it is contained in the simplex with vertices $x,y$, and $p$. In this paper, we prove the following Krasnosel’skii-type theorem: Let $S$ be a compact simply connected set in ${E_2}$ and let $p$ be a point not in $S$. If for each three points ${x_{1,}}{x_2}$, and ${x_3}$ of $S$ there exists at least one point $y \in S$ such that $y$ and ${x_i}\left ( {i = 1,2,3} \right )$ can be joined by $p$-arcs in $S$, then there exists a point $k \in S$ such that every point $x \in S$ can be joined to $k$ by some $p$-arc in $S$.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 667-676
- MSC: Primary 52A35; Secondary 52A30
- DOI: https://doi.org/10.1090/S0002-9939-1988-0929000-4
- MathSciNet review: 929000