Linear integral equations in the space of regulated functions

Tvrdý, Milan

Mathematica Bohemica, Tome 123 (1998) no. 2, pp. 177-212.

Résumé

n 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_0^1B(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).

Détail

DOI : 10.21136/MB.1998.126306

Classification : 26A39, 26A42, 45A05, 45B05, 45D05, 45F05, 47G10
Keywords: regulated function; Fredholm-Stieltjes integral equation; Volterra-Stieltjes integral equation; compact operator; Perron-Stieltjes integral; Kurzweil-Henstock integral; existence; uniqueness; resolvent operators; Kurzweil integral