A DSM proof of surjectivity of monotone nonlinear mappings

TL;DR

This paper provides a simple, short proof using the Dynamical Systems Method to establish the surjectivity of twice Frechet differentiable, coercive monotone nonlinear mappings, a fundamental result in monotone operator theory.

Contribution

It introduces a novel proof technique for surjectivity of monotone nonlinear mappings using the DSM, simplifying existing proofs in the field.

Findings

Proves surjectivity under Frechet differentiability and coercivity.

Utilizes the Dynamical Systems Method for the proof.

Simplifies the understanding of monotone operator surjectivity.

Abstract

We prove that if $F$ is twice Frechet differentiable and coercivity conditions hold, then $F$ is surjective, i.e., the equation $F(u)=h$ is solvable for every $h\in H$. This is a basic result in the theory of monotone operators. Our aim is to give a simple and short proof of this result based on the Dynamical Systems Method (DSM), developed in the monograph A.G. Ramm, Dynamical systems method, Elsevier, Amsterdam, 2007.