On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil

TL;DR

This paper explores the relationships between various Stieltjes type integrals of Young, Dushnik, and Kurzweil, providing new convergence results and clarifying their connections in the context of integral equations.

Contribution

It systematically compares different Stieltjes type integrals and introduces new convergence results for Young and Dushnik integrals, using elementary methods.

Findings

Established relationships between Young, Dushnik, and Kurzweil integrals.

Presented new convergence results for Young and Dushnik integrals.

Clarified the role of these integrals in solving generalized integral equations.

Abstract

Integral equations of the form $$ x(t)=x(t_0)+\int_{t_0}^t d[A]\,x=f(t)-f(t_0)$$ are natural generalizations of systems of linear differential equations. Their main goal is that they admit solutions which need not be absolutely continuous. Up to now such equations have been considered by several authors starting with J. Kurzweil and T.H. Hildebrandt. These authors worked with several different concepts of the Stieltjes type integral like Young's (Hildebrandt), Kurzweil's (Kurzweil, Schwabik and Tvrd\'{y}), Dushnik's (H\"{o}nig) or Lebesgue's (Ashordia, Meng and Zhang). Thus an interesting question arises: what are the relationships between all these concepts? Our aim is to give an answer to this question. In addition, we present also convergence results that are new for the Young and Dushnik integrals. Let us emphasize that the proofs of all the assertions presented in this paper are based on rather elementary tools.