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Linear integral equations in the space of regulated functions. (English) Zbl 0941.45001

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

$x\left(t\right)-A\left(t\right)x\left(0\right)-{\int }_{0}^{1}B\left(t,s\right)\left[\text{d}x\left(s\right)\right]=f\left(t\right),$

where $f\in {𝔾}_{L}^{n}$, the columns of the $n×n$-matrix valued function $A$ belong to ${𝔾}_{L}^{n}$, the entries of $B\left(t,.\right)$ have a bounded variation on $\left[0,1\right]$ for any $t\in \left[0,1\right]$ and the mapping $t\in \left[0,1\right]\to B\left(t,.\right)$ is regulated on $\left[0,1\right]$ and left-continuous on $\left(0,1\right)$ as the mapping with values in the space of $n×n$-matrix valued functions of bounded variation on $\left[0,1\right]·$ 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 ${𝔾}_{L}^{n}$, and its representation by means of resolvent operators is given.

##### MSC:
 45B05 Fredholm integral equations 45D05 Volterra integral equations