Approximating fixed points of enriched contractions in Banach spaces

TL;DR

This paper introduces enriched contractions in Banach spaces, establishing their unique fixed points and providing iterative methods for approximation, thereby generalizing many classical fixed point results.

Contribution

It defines a broad class of mappings called enriched contractions, unifying and extending existing fixed point theorems in Banach spaces.

Findings

Unique fixed points for enriched contractions

Convergence of Kransnoselskij iterative scheme

Generalization of classical fixed point theorems

Abstract

We introduce a large class of mappings, called enriched contractions, which includes, amongst many other contractive type mappings, the Picard-Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Kransnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results in this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia type enriched contractions. Examples to illustrate the generality of our new concepts and the corresponding fixed point theorems are also given.