Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral. (English) Zbl 1090.45001

Summary: In 1990, C. S. Hönig proved that the linear Volterra integral equation \[ x(t) - (K) \int _{[a, t]} \alpha (t,s)x(s)\,\text ds = f (t), \quad t \in [a,b] \] where the functions are Banach space-valued and \(f\) is a Kurzweil integrable function defined on a compact interval \([a, b]\) of the real line \(\mathbb R\), admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation \[ x(t) - (K) \int _{[a, t]} \alpha (t,s)x(s)\,\text dg(s) =f (t), \quad t \in [a, b] \] in a real-valued context.


45A05 Linear integral equations
26A39 Denjoy and Perron integrals, other special integrals
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