The Approximation of Fixed Points of Robust Mappings

Abstract

The study of the existence results of fixed points of discontinuous mappings was started in 1976 by Caristi [2]. However, to the best of our knowledge, serious effort has yet been devoted to the numerical computations of fixed points of discontinuous mappings. In this work we propose a method to find fixed points of a discontinuous robust mapping using the integral global minimization approach. In Section 2, we explain the reason for introducing the concept of robustness for global minimization. In Section 3, we define robust mappings and upper robust functions and study basic properties of robust mappings. In Section 4, we consider the existence results of fixed points of robust mappings. The computation of the fixed points of an robust mapping is deduced to finding the set of global minima of an robust function. Thus, we can use the integral global optimization method to find them. Numerical examples are also presented in this section to demonstrate that this approach is competitive with other algorithms.