Attractors for relations in σ-compact spaces

We develop a theory of discrete multi-valued dynamical systems for attractors and Lyapunov functions. In [Topology Appl. 109 (2001) 201–210], Hurley proved that if X is a locally compact and σ-compact space and the zero set Z of a Lyapunov function is closed, then Z is a weak attractor. Here we extend the above result to the case of compact-valued continuous relation on X.