Fixed Point Theorems for Hypersequences and the Foundation of Generalized Differential Geometry I: The Simplified Algebra

TL;DR

This paper develops a fixed point theorem for hypersequences within Colombeau Generalized Functions, enabling the analysis of differential equations with distributional data and establishing a foundation for generalized differential geometry and manifolds.

Contribution

It introduces a fixed point theorem for hypersequences in Colombeau algebras, extending classical differential geometry to include generalized manifolds and differential calculus.

Findings

Fixed point theorem for hypersequences proved in Colombeau context

Classical manifolds discretely embedded in generalized manifolds

Distributions discretely embedded in Colombeau algebra

Abstract

Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to develop new environments and tools capable of handling such situations. The foundations of a Generalized Differential Geometry is set having Classical Differential Geometry as a discontinuous subcase, a fixed point theorem for hypersequences is proved in the context of Colombeau Generalized Functions and it is shown how it can be used to obtain existence and uniqueness of differential equations whose data involve products of distributions. Thus also setting the foundations of a Generalized Analysis. The strain is also picked up setting the foundations of generalized manifolds and shown that each classical manifold can be discretely embedded in a generalized manifold in such a way that the differential structure of the latter is a natural extension of the differential structure of the former. It is inferred that $\ {\cal{D}}^{\prime}(\Omega)$ is discretely embedded in $\ {\cal{G}}(\Omega)$, that the elements of $\ C^{\infty}(\Omega)$ form a grid of equidistant points in $\ {\cal{G}}(\Omega)$ and that association in $\ {\cal{G}}(\Omega)$ is a topological and not an algebraic notion. Ergo, classical solutions to differential equations are scarce. These achievements reckoned upon the Generalized Differential Calculus invented by the first author and his collaborators. Hopefully, Generalized Differential Calculus and the developments presented in this paper, may be of interest to those working in Analysis, Applied Mathematics, Geometry and Physics.