Weakly convergent fixed point iterations for weakly sequentially non-expansive mappings

TL;DR

This paper introduces a new framework for the weak convergence of fixed point iterations for a broader class of mappings, using asymptotic bounds and weak sequential non-expansiveness in Opial spaces.

Contribution

It develops a general convergence framework based on asymptotic bounds and weak sequential non-expansiveness, extending classical results to more general mappings.

Findings

Established weak convergence under weaker assumptions than Lipschitz continuity.

Extended fixed point iteration convergence results to non-expansive mappings in reflexive Opial spaces.

Removed the need for geometric assumptions like uniform convexity.

Abstract

Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that a suitable contraction property can be verified. However, such conditions are typically too restrictive for more complex nonlinear equations that lack key structural features such as monotonicity or convexity. In this paper, we develop a general framework for the weak convergence of fixed point iterations based on asymptotic bounds. In particular, we introduce and exploit a weak sequential non-expansiveness property, which is significantly weaker than the global Lipschitz assumptions commonly employed in this context. This approach permits to extend classical convergence results to a broader class of mappings in general (reflexive) Opial spaces, without relying on additional geometric assumptions such as uniform convexity.