In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space X. We extend results obtained by Štefan Schwabik and complete the theory so that it will be well applicable to prove results on the continuous dependence of solutions to generalized linear differential equations in a Banach space. By Schwabik, the integral ∫ b a d[F]g exists if F : [a, b] → L(X) has a bounded semi-variation on [a, b] and g : [a, b] → X is regulated on [a, b]. We prove that this integral has sense also if F is regulated on [a, b] and g has a bounded semi-variation on [a, b]. Furthermore, the integration by parts theorem is presented under the assumptions not covered by Schwabik (2001) and Naralenkov (2004), and the substitution formula is proved.
We study the singular periodic boundary value problem of the form (φ(u ' ))' + h(u)u ' = g(u) + e(t), u(0) = u(T), u ' (0) = u ' (T), where φ: R→R is an increasing and odd homeomorphism such that φ(R ) = R, h ∈ C[0, ∞), e ∈ L1[0, T] and g ∈ C(0, ∞) can have a space singularity at x = 0, i.e. lim sup x→0+ |g(x)| = ∞ may hold. We prove new existence results both for the case of an attractive singularity, when lim inf x→0+ g(x) = −∞, and for the case of a strong repulsive singularity, when lim x→0+ R 1 x g(ξ)dξ = ∞. In the latter case we assume that φ(y) = φp(y) = |y| p−2 y, p > 1, is the well-known p-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.
The paper deals with the boundary value problem u '' + k u = g(u) + e(t), u(0) = u(2π), u ' (0) = u ' (2π), where k ∈ R, g : (0, ∞) → R is continuous, e ∈ L[0, 2π] and lim x→0+ ∫1 x g(s) ds = ∞. In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.