Ekeland's inverse function theorem in graded Fr\'echet spaces revisited for multifunctions

TL;DR

This paper revisits Ekeland's inverse function theorem for set-valued mappings in graded Fréchet spaces, providing implicit function theorems and applying them to differential equations with non-smooth data.

Contribution

It introduces new implicit function theorems for multifunctions in Fréchet spaces using variational principles and Lebesgue's theorem.

Findings

Established implicit function theorems for set-valued maps in Fréchet spaces.

Applied the theorems to prove existence of solutions for differential equations with non-smooth data.

Utilized Lebesgue's Dominated Convergence Theorem and Ekeland's variational principle in proofs.

Abstract

In this paper, we present some implicit function theorems for set-valued mappings between Fr\'echet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fr\'echet spaces with non-smooth data is given.