*(English)*Zbl 0941.45001

Summary: We investigate systems of linear integral equations in the space ${\mathbb{G}}_{L}^{n}$ of $n$-vector valued functions which are regulated on the closed interval $[0,1]$ (i.e. such that can have only discontinuities of the first kind in $[0,1]$) and left-continuous in the corresponding open interval $(0,1)\xb7$ In particular, we are interested in systems of the form

where $f\in {\mathbb{G}}_{L}^{n}$, the columns of the $n\times n$-matrix valued function $A$ belong to ${\mathbb{G}}_{L}^{n}$, the entries of $B(t,.)$ have a bounded variation on $[0,1]$ for any $t\in [0,1]$ and the mapping $t\in [0,1]\to B(t,.)$ is regulated on $[0,1]$ and left-continuous on $(0,1)$ as the mapping with values in the space of $n\times n$-matrix valued functions of bounded variation on $[0,1]\xb7$ The integral stands for the Perron-Stieltjes one treated as the special case of the Kurzweil-Henstock integral.

In particular, we prove basic existence and uniqueness results for the given equation and obtain the explicit form of its adjoint equation. A special attention is paid to the Volterra (causal) type case. It is shown that in that case the given equation possesses a unique solution for any right-hand side from ${\mathbb{G}}_{L}^{n}$, and its representation by means of resolvent operators is given.