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linear Stieltjes integral equations; generalized linear differential equation; Banach space; equation in Banach space
Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in \cite5. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite3). 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.
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