Inequalities for solutions to some nonlinear equations

TL;DR

This paper establishes conditions for the existence of solutions to certain nonlinear equations in Hilbert spaces and justifies a dynamical systems method to compute these solutions via a Cauchy problem.

Contribution

It provides new inequalities ensuring solutions exist and introduces a justified dynamical systems approach for solving nonlinear equations.

Findings

Derived sufficient conditions for solution existence.

Validated the dynamical systems method for solution computation.

Provided inequalities that bound solutions in Hilbert spaces.

Abstract

Let $F$ be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation $F(u)=0$ is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suitable assumptions.