New algorithms for the approximation of fixed points and fractal functions

This article is devoted to explore the abilities of an iterative scheme for the approximation of fixed points of self-maps, called the N-algorithm, defined in a previous paper. In a first part of the article, the algorithm is modified in order to consider operators with asymptotic properties, namely nearly uniform contractions and nearly asymptotically nonexpansive mappings. Sufficient conditions on the (normed or quasi-normed) underlying space and the operator are given in order to ensure weak or strong convergence of the new algorithm to a fixed point. Afterwards, the definition of fractal functions as fixed points of the Read-Bajraktarević operator is considered, giving very general conditions for their existence when the operator is nonexpansive. This is done in the framework of the Hilbert space L2(J), where J is a real compact interval. The capacity of the N-algorithm for the approximation of these fractal functions is proved as well. In the last part of the paper, the fractal convolution of operators on L2(J) is studied. The text explores the properties of the convolution when its components satisfy determined conditions, closely related to the existence of fixed points. •Introduction of a new iterative algorithm for the approximation of fixed points.•Study of the convergence of this numerical procedure.•Definition of integrable fractal functions as fixed points of nonexpansive operators.•Approximation of these maps by means of the N-iteration.•Fractal convolution of operators acting on the space of square integrable functions.