# A Global Diffeomorphism Theorem for Fr\'{e}chet spaces

## TL;DR

This paper establishes conditions under which a local diffeomorphism between Fréchet spaces is global, extending Clarke's gradient theory and proving a mountain pass theorem in this infinite-dimensional setting.

## Contribution

It extends Clarke's generalized gradient theory to Fréchet spaces and proves a global diffeomorphism theorem using the Chang Palais-Smale condition.

## Key findings

- Extended Clarke's theory to Fréchet spaces
- Proved a mountain pass theorem for Lipschitz maps in Fréchet spaces
- Established conditions for global diffeomorphisms in infinite-dimensional spaces

## Abstract

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fr\'{e}chet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.