Integral Representation of Functions of Bounded Variation

Abstract

Functions of bounded variations form important transition between absolute continuous and singular functions. With Bainov’s introduction of impulsive differential equations having solutions of bounded variation, this class of functions had eventually entered into the theory of differential equations. However, the determination of existence of solutions is still problematic because the solutions of differential equations is usually at least absolute continuous which is disrupted by the solutions of bounded variations. As it is known, if f:[a,bλ]→Rn is of bounded variation then f is the sum of an absolute continuous function fa and a singular function fs where the total variation of fs generates a singular measure τ and fs is absolute continuous with respect to τ. In this paper we prove that a function of bounded variation f has two representations: one is f which was described with an absolute continuous part with respect to the Lebesgue measure λ, while in the other an integral with respect to τ forms the absolute continuous part and t(τ) defines the singular measure. Both representations are obtained as parameter transformation images of an absolute continuous function on total variation domain [a,bν].

Existence Theorem for Impulsive Differential Equations with Measurable Right Side for Handling Delay Problems

“…e extended impulsive differential equations mean changing some basic assumptions used in Bainov's model as described in equation (1) or in rewritten form in equations (3) and (10). Major changes include the time control of the impulses may have infinite discontinuities but has to be of bounded variation on every closed bounded interval, and the system dynamics is measurable as a function of the time and not necessarily continuous.…”

“…Putting these into equation (10) and changing 􏽥 g with g, we get the extended impulsive differential equation in t-scale as…”

“…is section summarises the concepts which will serve as the basis of most of our coming discussion and were developed in [10] and partly in [40]. We use the notations and concepts formulated in Section 1.1 and our starting point will be the integral equation (4).…”

“…From the interpretation of a pair of impulsive differential equation follows that the concept of initial value problem as presented in the Bainovian model, (1), (4), and (10) requires some clarifications. e Bainovian model discussed in Section 2.3.3 has form (4) or more likely (10) and the initial time t 0 ∈ [a, b λ ]\D λ cannot be a discontinuity point. From the analysis of equation (30) follows that any change between f ] ⟶ g ] or g ] ⟶ f ] takes place starting from the leftmost point of the domain/interval of the new dynamics.…”

“…It was proved in paper [40] that the Radon-Nikodym derivatives 􏽢 f ],λ and 􏽢 f ],τ fulfill the relations:…”