On first and second order linear Stieltjes differential equations

Abstract

This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notion 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 the space of such functions. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillator.