On first and second order linear Stieltjes differential equations

TL;DR

This paper develops a comprehensive framework for solving first and second order Stieltjes differential equations, including definitions, existence and uniqueness results, Green's functions, and an application to the Stieltjes harmonic oscillator.

Contribution

It introduces the notion of Stieltjes derivatives on entire domains, defines n-times continuously Stieltjes-differentiable functions, and proves fundamental existence and uniqueness theorems.

Findings

Established existence and uniqueness of solutions for Stieltjes differential equations.

Derived Green's functions for various boundary value problems.

Applied the theory to analyze the Stieltjes harmonic oscillator.

Abstract

This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notions of Stieltjes derivative on the whole domain of the functions involved, provide a notion of n-times continuously Stieltjes-differentiable functions and prove existence and uniqueness results of Stieltjes differential equations in those spaces. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillator.