Hard Implicit Function Theorem via the DSM

TL;DR

This paper introduces a new implicit function theorem using the Dynamical Systems Method (DSM), enabling solutions to nonlinear operator equations with unbounded derivatives, expanding the scope of implicit function results.

Contribution

It provides sufficient conditions for a hard implicit function theorem based on DSM, addressing cases with smoothing operators and unbounded derivatives.

Findings

Established a new implicit function theorem using DSM.

Applicable to nonlinear operators with smoothing derivatives.

Extended implicit function theory to unbounded inverse cases.

Abstract

Sufficient conditions are given for a hard implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when the Fr\'echet derivative of the nonlinear operator is a smoothing operator, so that its inverse is an unbounded operator.