Continuous dependence of fixed points on parameters, remetrization theorems, and an application to the initial value problem

Abstract

We show that the recent result of Bełdzi ński et al. on continuous dependence of fixed points on parameters, which extends a classical parametrized version of the Banach fixed point theorem, may be treated as a particular case of the contraction principle for a Nemytskii operator on a space of continuous functions with values in a metric space. Next, we establish a generalization of the result of Bełdzi ński et al. involving an equicontinuous family of mappings having contractive pth iterates for some p ∈ N. As an application, we obtain a theorem on continuous dependence of solutions of Cauchy's problem on both the initial values and the parameters. Finally, we establish two remetrization theorems and we discuss a possibility of deriving our main result from the Banach contraction principle by using a remetrization of the space.