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

TL;DR

The paper extends Stieltjes differential equations to systems with multiple derivators $\boldsymbol g=(g_1,\dots,g_n)$ and analyzes the IVP $x'_{\boldsymbol g}(t)=f(t,x(t))$, $x(t_0)=x_0$. It develops Lebesgue–Stieltjes measure theory for finite sums of derivators, clarifies continuity notions, and classifies derivator topologies, enabling toolkits for existence and uniqueness proofs. It proves a sharp Osgood-type uniqueness result and establishes local existence and uniqueness under Montel–Tonelli-type growth and Carathéodory conditions, using fixed-point arguments in the $\hat g$-integral framework. Collectively, these results correct and extend prior work, providing a rigorous framework for multi-derivative impulsive/time-scale models through $x'_{g_i}$ and $f_i$.

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.