Linear Stieltjes integral equations in Banach spaces. (English) Zbl 0937.34047

Schwabik, Štefan

Math. Bohem. 124, No. 4, 433-457 (1999).

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.

MSC:

34G10	Linear differential equations in abstract spaces
45N05	Abstract integral equations, integral equations in abstract spaces
45A05	Linear integral equations
26A39	Denjoy and Perron integrals, other special integrals
26A42	Integrals of Riemann, Stieltjes and Lebesgue type

Keywords:

linear Stieltjes integral equations; generalized linear differential equation; Banach space