Solvability of abstract semilinear equations by a global diffeomorphism theorem

TL;DR

This paper presents a simplified proof of a global diffeomorphism theorem and applies it to establish unique solvability of certain abstract semilinear equations, including applications to second order Dirichlet problems involving the Laplace operator.

Contribution

Provides a new, simpler proof of the global diffeomorphism theorem and demonstrates its use in solving abstract semilinear equations with applications to PDE boundary value problems.

Findings

Simplified proof of the global diffeomorphism theorem

Establishment of unique solvability for specific semilinear equations

Application to second order Dirichlet problems with Laplace operator

Abstract

In this work we proivied a new simpler proof of the global diffeomorphism theorem from [9] which we further apply to consider unique solvability of some abstract semilinear equations. Applications to the second order Dirichlet problem driven by the Laplace operator are given.