an:01406926
Zbl 0937.34047
Schwabik, ??tefan
Linear Stieltjes integral equations in Banach spaces
EN
Math. Bohem. 124, No. 4, 433-457 (1999).
00063695
1999
j
34G10 45N05 45A05 26A39 26A42
linear Stieltjes integral equations; generalized linear differential equation; Banach space
Summary: The background of the theory is the Kurzweil approach to integration, based on Riemann-type integral sums. It is known that the Kurzweil theory leads to the (nonabsolutely convergent) Perron-Stieltjes integral in the finite-dimensional case. Here, basic results concerning equations of the form
\[
x(t) = x(a) + \int ^t_a \text{d} [A(s)] x(s) + f(t) - f(a)
\]
are presented on the basis of the Kurzweil-type Stieltjes integration. The author is looking for generally discontinuous solutions which belong to the space of Banach space-valued regulated functions in the case that \(A\) is a suitable operator-valued function and \(f\) is regulated.