Classical finite dimensional fixed point methods for generalized functions

TL;DR

This paper extends classical fixed point theorems to the framework of generalized smooth functions, enabling the analysis of nonlinear singular problems involving distributions while preserving key properties of smooth functions.

Contribution

It introduces fixed point theorems within the generalized smooth functions framework, broadening the scope of nonlinear analysis to include singular and distributional problems.

Findings

Proved Banach, Newton-Raphson, and Brouwer fixed point theorems for generalized smooth functions.

Demonstrated applicability to equations involving distributions and singularities.

Extended classical fixed point results to a minimal generalized function setting.

Abstract

We prove Banach, Newton-Raphson and Brouwer fixed point theorems in the framework of generalized smooth functions, a minimal extension of Colombeau's theory (and hence of classical distribution theory) which makes it possible to model nonlinear singular problems, while at the same time sharing a number of fundamental properties with ordinary smooth functions, such as the closure with respect to composition and several non trivial classical theorems of the calculus. The proved results allows one to deal with equations of the form F(x)=0, where F is a generalized smooth function, in particular, a Sobolev-Schwartz distribution. We consider examples with singularities that are not included in the classical version of these theorems.