Some extensions of Krasnoselskii's fixed point result for real functions

TL;DR

This paper extends Krasnoselskii's fixed point theorem to non-self real functions, providing new proofs and broadening applicability to unbounded mappings and the real line, with implications for the Newton-Raphson method.

Contribution

It introduces novel extensions of Krasnoselskii's fixed point theorem for non-self functions and offers simplified proofs for existing results, including Hillam's and the Newton-Raphson convergence.

Findings

Extended Krasnoselskii's theorem to non-self functions

Provided a new proof for Hillam's fixed point result

Demonstrated global convergence of Newton-Raphson method

Abstract

We extend Krasnoselskii's fixed point result to non-self-real functions. We find a new and simple proof for Hillam's result. In our approach, we don't assume the image of the related mapping to be compact or bounded. In this way, we extend Hillam's result to self-mappings on $\mathbb R$. Finally, we present a new proof for the global convergence of the Newton-Raphson method.