Generalized recurrence in impulsive semidynamical systems

Abstract

We aim to introduce the generalized recurrence into the theory of impulsive semidynamical systems. Similarly to Auslander's construction in [J. Auslander, Generalized recurrence in dynamical systems, Contrib. Differential Equations 3 (1964), 65-74], we present two different characterizations, respectively, by Lyapunov functions and higher prolongations. In fact, we show that if the phase space is a locally compact separable metric space, then the generalized recurrent set is the same as the quasi prolongational recurrent set. Also, we see that many new phenomena appear for the impulse effects in the semidynamical system.