Existence of solution to an evolution equation and a justification of the DSM for equations with monotone operators

TL;DR

This paper proves the existence of solutions for an evolution equation related to the DSM method for monotone operators, requiring minimal smoothness assumptions and thereby justifying the method's applicability.

Contribution

It establishes the first proof of local and global existence of solutions under minimal smoothness conditions for the evolution equation in the DSM context.

Findings

Proved local and global existence of solutions.

Justified DSM for equations with monotone operators.

Reduced smoothness requirements for the operator.

Abstract

An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving ill-posed problems with monotone nonlinear operators $F$. Local and global existence of the unique solution to this evolution equation are proved, apparently for the firs time, under the only assumption that $F'(u)$ exists and is continuous with respect to $u$. The earlier published results required more smoothness of $F$. The Dynamical Systems method (DSM) for solving equations $F(u)=0$ with monotone Fr\'echet differentiable operator $F$ is justified under the above assumption apparently for the first time.