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

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
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