Existence of solutions for discontinuous systems of Stieltjes differential equations

Abstract

We introduce new transversality conditions in terms of derivatives with respect to a nondecreasing function $g: \mathbb R \longrightarrow \mathbb R$, and we use them to prove the existence of solutions for discontinuous systems of Stieltjes differential equations, i.e. differential equations where usual derivatives are replaced with derivatives with respect to the derivator $g$. The proof of our main existence result leans on some interesting new results on $g$-derivatives also proven in this paper.

Keywords

• discontinuous differential equations
• carathéodory solutions
• stieltjes differential equations