Convergence of trajectories of discrete dispersive dynamical systems (Q1014131)

This paper studies dynamical systems (called discrete dispersive dynamical systems), arising from set-valued maps \[ a: K\to 2^X\setminus\{\emptyset\}, \] where \(K\) is a closed subset of a compact metric space \(X\). Such maps were introduced by \textit{A. M. Rubinov} [Sib. Mat. Zh. 21, No. 4, 136--145 (1980; Zbl 0453.90024)], (for \(K= X\)), and this paper generalizes results of the author [Nonlinear Dyn. Syst. Theory 7, No. 3, 315--325 (2007; Zbl 1137.37007)]. A sequence \(\{x_t\}^\infty_{t=0}\subset K\) and \(x_{t+1}\in A(x_t)\) for all \(t\geq 0\). It is shown that (under certain conditions on the graph of \(a\)) that \(a\) has a trajectory. Necessary and sufficient conditions for uniform convergence of trajectories to the global attractor are given.