Limits of eventual families of sets with application to algorithms for the common fixed point problem

TL;DR

This paper introduces an abstract framework for analyzing convergence of iterative algorithms using the concept of eventual families of sets, with applications to fixed point problems.

Contribution

It develops a novel abstract approach based on eventual families and multiset extensions for convergence analysis in fixed point algorithms.

Findings

Defined eventual families and their accumulation points.

Extended the framework to multisets and multifamilies.

Applied the framework to analyze convergence of fixed point algorithms.

Abstract

We present an abstract framework for asymptotic analysis of convergence based on the notions of eventual families of sets that we define. A family of subsets of a given set is called here an "eventual family" if it is upper hereditary with respect to inclusion. We define accumulation points of eventual families in a Hausdorff Topological space and define the "image family" of an eventual family. Focusing on eventual families in the set of the integers enables us to talk about sequences of points. We expand our work to the notion of a "multiset" which is a modification of the concept of a set that allows for multiple instances of its elements and enable the development of "multifamilies" which are either "increasing" or "decreasing". The abstract structure created here is motivated by, and feeds back to, our look at the convergence analysis of an iterative process for asymptotically finding a common fixed point of a family of operators.