This article is cited in 1 paper

MATHEMATICS

On some properties of *-integral

V. Ya. Derr

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: This work continues the author's research on the theory of regulated functions and *-integral. The possibility to express a regulated function as a sum of right-continuous and left-continuous functions (called $rl$-representation) is studied. A limit theorem for the *-integral is proved. It allows approximating discontinuous integrands and integrators by sequences of absolutely continuous functions. A new result on $\delta$-correctness of the solution of an ordinary linear differential equation with generalized functions in coefficients is proved. This solution is defined via a quasi-differential equation. A formula for the total variation of an indefinite *-integral of a $\sigma$-continuous function with respect to a function of bounded variation is given. It generalizes the well-known formula for computing the total variation of an absolutely continuous function. The formula is also interesting in the case of an indefinite $RS$-integral.

Keywords: regulated functions, $\sigma$-continuous functions, $rl$-representation, *-integral, quasi-differential equation, generalized functions, $\delta$-correctness.

UDC: 517.518.126, 517.911

MSC: 26B30, 26A42, 34K10

Received: 20.12.2022
Accepted: 25.01.2023

DOI: 10.35634/vm230105

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