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The generalized linear differential equation $dx=d\left[a\right(t\left)\right]x+df$ where $A,f\in B{V}_n^{loc}\left(J\right)$ and the matrices $I-{\Delta }^{-}A\left(t\right),I+{\Delta }^{+}A\left(t\right)$ are regular, can be transformed $\frac{dy}{ds}=B\left(s\right)y+g\left(s\right)$ using the notion of a logarithimc prolongation along an increasing function. This method enables to derive various results about generalized LDE from the well-known properties of ordinary LDE. As an example, the variational stability of the generalized LDE is investigated.

The first section consists of auxiliary results about nondecreasing real functions. In the second section a new characterization of relatively compact sets of regulated functions in the sup-norm topology is brought, and the third section includes, among others, an analogue of Helly's Choice Theorem in the space of regulated functions.

Assume that for any $t$ from an interval $[a,b]$ a real number $u\left(t\right)$ is given. Summarizing all these numbers $u\left(t\right)$ is no problem in case of an absolutely convergent series ${\sum }_{t\in [a,b]}u\left(t\right)$. The paper gives a rule how to summarize a series of this type which is not absolutely convergent, using a theory of generalized Perron (or Kurzweil) integral.

This paper deals with regulated functions having values in a Banach space. In particular, families of equiregulated functions are considered and criteria for relative compactness in the space of regulated functions are given.