Representation of bounded and compact linear operators in the Banach space of regulated functions is given in terms of Perron-Stieltjes integral.

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of...

In 1990, Hönig proved that the linear Volterra integral equation $$x\left(t\right)-\left(K\right){\int }_{\left[a,t\right]}\alpha \left(t,s\right)x\left(s\right)ds=f\left(t\right),t\in \left[a,b\right],$$ where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[a,b\right]$ of the real line $ℝ$, 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 $$x\left(t\right)-\left(K\right){\int }_{\left[a,t\right]}\alpha \left(t,s\right)x\left(s\right)dg\left(s\right)=f\left(t\right),t\in \left[a,b\right],$$ in a real-valued context.