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.
In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem u '' + k u = f(t, u), u(0) = u(2 π), u 0 (0) = u 0 (2π), k ∈ R, k ≠ 0. These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.