On Some Boundedness and Convergence Properties of a Class of Switching Maps in Probabilistic Metric Spaces with Applications to Switched Dynamic Systems

Abstract

This paper investigates some boundedness and convergence properties of sequences which are generated iteratively through switched mappings defined on probabilistic metric spaces as well as conditions of existence and uniqueness of fixed points. Such switching mappings are built from a set of primary self-mappings selected through switching laws. The switching laws govern the switching process in between primary self-mappings when constructing the switching map. The primary self-mappings are not necessarily contractive but if at least one of them is contractive then there always exist switching maps which exhibit convergence properties and have a unique fixed point. If at least one of the self-mappings is nonexpansive or an appropriate combination given by the switching law is nonexpansive, then sequences are bounded although not convergent, in general. Some illustrative examples are also given.