A Krasnosel′skiĭ-type theorem involving $p$-arcs

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 $S$.