  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.
