**
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: ` schwabik@math.cas.cz`

**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

**Full text of the article:**

[Previous Article] [Contents of this Number]

*
© 2004—2005 ELibM and
FIZ Karlsruhe / Zentralblatt MATH
for the EMIS Electronic Edition
*