Periodic points of multi-valued \(\varepsilon\)-contractive maps
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 Hausdoff metric. A multivalued map \(F:X\to 2^X\) is said to be \(\varepsilon\)-contractive (\(\varepsilon>0\)) provided that for all \(x,y\in X\) with \(x\not= y\) and \(d(x,y)<\varepsilon\), \(H_d(F(x),F(y))<d(x,y)\). A point \(x\in X\) is said to be periodic for \(F:X\to 2^X\) if there are finitely many points \(x_0=x,x_1,\dots,x_n\) in \(X\) such that \(x_i\in F(x_{i-1})\) for each \(i=1,\dots,n\) and \(x\in F(x_n)\). The author establishes a general condition that guarantees the existence of a periodic point for an \(\varepsilon\)-contractive multivalued map \(F:X\to 2^X\). This condition holds when \(X\) is compact. Further the author proves that when \(X\) is a continuum (i.e., a nonempty compact connected metric space), a multivalued \(\varepsilon\)-contractive map \(F:X\to 2^X\) has a fixed point. As applications the author obtains theorems about the existence of periodic and fixed points for an \(\varepsilon\)-expansive open map \(f:Y\subset X\to X\).
scientific article; zbMATH DE number 2062974