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

Tvrdý, Milan

Math. Bohem. 123, No. 2, 177-212 (1998).

Summary: We investigate systems of linear integral equations in the space \({{\mathbb G}^n_L}\) 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 \(f\in {{\mathbb G}^n_L}\), the columns of the \(n\times n\)-matrix valued function \(A\) belong to \({{\mathbb G}^n_L}\), 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.
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 \({{\mathbb G}^n_L}\), and its representation by means of resolvent operators is given.

MSC:

45B05	Fredholm integral equations
45D05	Volterra integral equations

Keywords:

regulated function; Fredholm-Stieltjes integral equation; Volterra-Stieltjes integral equation; compact operator; Perron-Stieltjes integral; Kurzweil-Henstock integral; existence; uniqueness; resolvent operators