**
MATHEMATICA BOHEMICA, Vol. 123, No. 2, pp. 177-212 (1998)
**

#
Linear integral equations in the space

of regulated functions

##
Milan Tvrdy

* Milan Tvrdy,* Matematicky ustav AV CR, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail: ` tvrdy@math.cas.cz`

**Abstract:** In this paper we investigate systems of linear integral equations in the space ${{\Bbb G}^n_L}$ 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 $(0,1).$ In particular, we are interested in systems of the form

x(t) - A(t)x(0) - \int_{0}^{1}B(t,s)[\text{d} x(s)] = f(t),

where $f\in{{\Bbb G}^n_L}$, the columns of the $n\times n$-matrix valued function $A$ belong to ${{\Bbb G}^n_L}$, the entries of $B(t,\ldotp)$ have a bounded variation on ${[0,1]}$ for any $t\in{[0,1]}$ and the mapping $t\in{[0,1]} \to B(t,\ldotp)$ 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 ${[0,1]}.$ The integral stands for the Perron-Stieltjes one treated as the special case of the Kurzweil-Henstock integral. \endgraf 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 ${{\Bbb G}^n_L}$, and its representation by means of resolvent operators is given. \endgraf The results presented cover e.g. the results known for systems of linear generalized differential equations

x(t) - x(0) - \int_0^t [\text{d} A(s)] x(s) = f(t) - f(0)

as well as systems of Stieltjes integral equations

x(t) - \int_0^1 [\text{d}_s K(t,s)] x(s) = g(t) \quad\text{or}\quad x(t) - \int_0^t [\text{d}_s K(t,s)] x(s) = g(t).

**Keywords:** regulated function, Fredholm-Stieltjes integral equation, Volterra-Stieltjes integral equation, compact operator, Perron-Stieltjes integral, Kurzweil integral

**Classification (MSC2000):** 45A05, 47G10, 26A39, 26A42

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