MATHEMATICA BOHEMICA, Vol. 124, No. 4, pp. 433-457 (1999)

Linear Stieltjes integral equations in Banach spaces

Stefan Schwabik

Stefan Schwabik, Matematicky ustav AV CR, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail:

Abstract: Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in \cite{5}. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite{3}). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case.
Here basic results concerning equations of the form $$ x(t) = x(a) +\int_a^t \dd[A(s)]x(s) +f(t) - f(a) $$ are presented on the basis of the Kurzweil type Stieltjes integration. We are 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.

Keywords: linear Stieltjes integral equations, generalized linear differential equation, equation in Banach space

Classification (MSC2000): 34G10, 45N05

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