This paper is a -continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) + ∫ ţa d[A(s)]x(s) + f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem Ф(t) = I+ ∫td d[A(s)]Ф(s) as well as the variation-of-constants formula for the former equation.
The McShane integral of functions f : I → ! defined on an m-dimensional interval I is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.
The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions.