Asymptotically Unpredictable Solutions of Quasilinear Impulsive Systems

We propose the notion of piecewise continuous asymptotically unpredictable functions, and investigate the existence and uniqueness of such solutions in impulsive systems under sufficient conditions. This class of functions and the systems under discussion admit regular discontinuity moments. Some techniques for obtaining new piecewise continuous asymptotically unpredictable functions from a given one are provided. Even though piecewise continuous unpredictable functions are asymptotically unpredictable, it is demonstrated that the converse is not true. In other words, the set of discontinuous unpredictable functions is properly contained in the set of asymptotically unpredictable ones. Appropriate examples are given with regard to all theoretical results.