Distributions, their primitives and integrals with applications to differential equations

TL;DR

This paper investigates the integrability of distributions with primitives as regulated functions, exploring their properties and applications to various distributional differential equations, including nonlinear and higher-order cases.

Contribution

It introduces new spaces of distributions and primitives, and applies primitive integrals to solve complex distributional differential equations.

Findings

Defined new spaces of distributions and primitives

Established properties of primitive integrals

Applied methods to nonlinear and higher-order distributional differential equations

Abstract

In this paper we will study integrability of distributions whose primitives are left regulated functions and locally or globally integrable in the Henstock--Kurzweil, Lebesgue or Riemann sense. Corresponding spaces of distributions and their primitives are defined and their properties are studied. Basic properties of primitive integrals are derived and applications to systems of first order nonlinear distributional differential equations and to an $m$th order distributional differential equation are presented. The domain of solutions can be unbounded, as shown by concrete examples.