The Banach fixed point principle viewed as a monotone convergence with respect to the Lorentz cone

TL;DR

This paper presents a novel perspective on the Banach fixed point principle by interpreting Picard iteration as a monotone process within an augmented space ordered by the Lorentz cone, revealing deep structural insights.

Contribution

It introduces a new approach that augments space and modifies iteration to demonstrate convergence via monotonicity related to the Lorentz cone, linking fixed point theory and order structures.

Findings

Picard iteration can be viewed as monotone increasing in an augmented space.

Convergence of the fixed point iteration is shown through Lorentz cone ordering.

The approach highlights the relationship between fixed point principles and ordered vector spaces.

Abstract

We augment the dimension of the Euclidean space by one and the Picard iteration of a contraction by a simple iteration on the real line such that the resulting iteration becomes monotone increasing and bounded with respect to the order defined by the Lorentz cone of the augmented space. This provides a different way of showing the convergence of the Picard iteration of a contraction, exhibiting the strong relationship between the Banach fixed point principle and the ordering structure of the Euclidean space ordered by the Lorentz cone.