Cosmic Divergence, Weak Cosmic Convergence, and Fixed Points at Infinity

TL;DR

This paper investigates the asymptotic behavior of fixed-point iterations for non-expansive operators without fixed points, disproving a conjecture about universal cosmic convergence and exploring the nature of fixed points at infinity.

Contribution

It provides counterexamples to the conjecture that cosmic convergence always occurs and offers new insights into fixed points at infinity in infinite-dimensional spaces.

Findings

Counterexample disproving universal cosmic convergence

Demonstration of weak cosmic convergence in infinite dimensions

Interpretation of cosmic accumulation points as fixed points at infinity

Abstract

To characterize the asymptotic behavior of fixed-point iterations of non-expansive operators with no fixed points, Bauschke et al. [Fixed Point Theory Appl. (2016)] recently studied cosmic convergence and conjectured that cosmic convergence always holds. This paper presents a cosmically divergent counter example, which disproves this conjecture. This paper also demonstrates, with a counter example, that cosmic convergence can be weak in infinite dimensions. Finally, this paper shows positive results relating to cosmic convergence that provide an interpretation of cosmic accumulation points as fixed points at infinity.