Existence and uniqueness of solution for Stieltjes differential equations with several derivators

TL;DR

This paper investigates the existence and uniqueness of solutions for systems of differential equations involving multiple Stieltjes derivatives, establishing conditions under which solutions are unique or exist, and exploring related properties of Lebesgue-Stieltjes integrals.

Contribution

It introduces new results on existence and uniqueness for systems with different Stieltjes derivatives, extending classical conditions like Osgood and Montel-Tonelli.

Findings

Unique solutions under Osgood and Montel-Tonelli conditions

Existence results for solutions under certain conditions

Properties of Lebesgue-Stieltjes integrals for nondecreasing maps

Abstract

In this paper, we study some existence and uniqueness results for systems of differential equations in which each of equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have one solution under the Osgood condition, or even, the Montel-Tonelli condition. We also explore some results guaranteeing the existence of solution under these conditions. Along the way, we obtain some interesting properties for the Lebesgue-Stieltjes integral associated to a finite sum of nondecreasing and left-continuous maps, as well as a characterization of the pseudometric topologies defined by this type of maps.