Attractors for relations in \(\sigma\)-compact spaces
\textit{C. Conley} [Isolated invariant sets and the Morse index, Regional Conference Series in Mathematics, No. 38, Providence, R.I.: American Mathematical Society (1978; Zbl 0397.34056)] introduced a topological definition of an attractor for a flow on a compact metric space and showed that each of these attractors has an associated Lyapunov function. Among the generalizations of Conley's results there is the following theorem from \textit{M. Hurley} [Topology Appl. 109, 201-210 (2001; Zbl 0968.37005)]: Let \(X\) be a locally compact and \(\sigma\)-compact metric space, \(f\) be a continuous map \(X\to X\) and \(A\subseteq X\) be closed, nonempty and \(f\)-invariant. Suppose that \(h: X\to [0, \infty)\) is a continuous function with the properties: (1) \(h^{-1}(0)=A\), (2) there is a neighbourhood \(W\) of \(A\) in \(X\) such that \(h^{-1}([0,1])\subseteq W\) and (3) \(h(f(x)) <f(x)\) whenever \(x\in W\setminus A\). Then \(A\) is a weak attractor for \(f\). The paper under review extends this Hurley result to the case of compact-valued continuous relations on \(X\).
scientific article; zbMATH DE number 2142290