Global invertibility and implicit function theorems by mountain pass theorem

TL;DR

This paper develops new global invertibility and implicit function theorems using mountain pass techniques, extending previous results to operators with critical points and providing insights into solving nonlinear equations like x^3=y.

Contribution

It introduces a novel approach combining mountain pass theorem with operator invertibility, extending prior results to include operators with critical points.

Findings

Extended invertibility results to operators with critical points

Applied mountain pass theorem to implicit function problems

Provided methods for solving nonlinear equations like x^3=y

Abstract

We formulate some global invertibility and implicit function theorems. We extend the result of Idczak, Skowron and Walczak on the invertibility of the operators to the case of the operators with critical points. The proof relies on the Mountain Pass Theorem combined with the Palais-Smale condition guaranteeing the claim by the invertibility of the first or the third derivative. I. e. how to solve $x^3=y$?