Periodic points of multi-valued $\varepsilon$-contractive maps

Abstract

Let $(X,d)$ be a nonempty metric space, and let $(2^{X},H_{d})$ be the hyperspace of all nonempty compact subsets of $X$ with the Hausdorff metric. Let $F\colon X\rightarrow 2^{X}$ be an $\varepsilon$-contractive map. A general condition is given that guarantees the existence of a periodic point of $F$ (the theorem extends a result of Edelstein to multi-valued maps). The condition holds when $X$ is compact; hence, $F$ has a periodic point when $X$ is compact. It is shown that $F$ has a fixed point (a point $p\in F(p)$) if $X$ is a continuum. Applications to single-valued $\varepsilon$-expansive maps are given.