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Classical finite dimensional fixed point methods for generalized functions

Kevin Islami, George Apaaboah, Paolo Giordano

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.

Classical finite dimensional fixed point methods for generalized functions

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.
Paper Structure (10 sections, 12 theorems, 41 equations)

This paper contains 10 sections, 12 theorems, 41 equations.

Table of Contents

  1. Introduction: solving nonlinear equations with generalized functions
  2. Preliminary notions
  3. The ring of scalars
  4. Topologies on $\widetilde{\mathbb{R}}^{n}$
  5. Banach fixed point theorem in $n$-dimensions
  6. Banach fixed point theorem in $d$-dimensions
  7. Newton-Raphson method in $d$-dimensions
  8. Quadratic convergence of Newton method
  9. Examples
  10. Conclusions

Key Result

Lemma 3

Let $x\in\widetilde{\mathbb{R}}$. Then the following are equivalent:

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Definition 4
  • Theorem 5
  • Theorem 6: Fermat-Reyes theorem for GSF
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Lemma 10
  • ...and 18 more