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

M De La Sen,A Ibeas

doi:10.1155/2015/836283

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.

Highlights

The background literature on fixed point theory and applications and associated convergence properties in metric spaces, Banach spaces, probabilistic metric spaces, Menger spaces, and some fuzzy-type versions is very abundant
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
The theory focused on probabilistic metric spaces, including their specialization to Menger spaces, is abundant

Summary

Introduction

The background literature on fixed point theory and applications and associated convergence properties in metric spaces, Banach spaces, probabilistic metric spaces, Menger spaces, and some fuzzy-type versions is very abundant. 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.

Results

Conclusion