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V. Ya. Derr

Publications and source records attributed to V. Ya. Derr.

4 recordsLinked to original sources
arXiv2024

Regulated functions. *-integral

The present book gives a systematic overview of function theory and the theory of Stieltjes integral. In particular, we give a detailed account of the theory of functions of bounded variation and of the theory of regulated functions (= functions having finite one-sided limits at each point of their domain). We also present a detailed discussion of $σ$-continuous functions (= functions having at most countable set of points of discontinuity) and of the theory of Riemann-Stieltjes integral. We introduce the notion of *-integral, which allows us to integrate $σ$-continuous functions over functions of bounded variation. The limit theorems for *-integral are somewhat more convenient than their analogues for the general Lebesgue-Stieltjes integral. Some applications of *-integral are also discussed.

math.CA↗
arXiv2012

On uniform continuous dependence of solution of Cauchy problem on a parameter

Suppose that an $n$-dimensional Cauchy problem \frac{dx}{dt}=f(t,x,μ) (t \in I, μ\in M), x(t_0)=x^0 satisfies the conditions that guarantee existence, uniqueness and continuous dependence of solution x(t,t_0,μ) on parameter μin an open set M. We show that if one additionally requires that family \{f(t,x,\cdot)\}_{(t,x)} is equicontinuous, then the dependence of solution x(t,t_0,μ) on parameter μ\in M is uniformly continuous. An analogous result for a linear n \times n-dimensional Cauchy problem \frac{dX}{dt}=A(t,μ)X+Φ(t,μ) (t \in I, μ\in M), X(t_0,μ)=X^0(μ) is valid under the assumption that the integrals \int_I\|A(t,μ_1)-A(t,μ_2)\|dt and \int_I \|Φ(t,μ_1)-Φ(t,μ_2)\|dt can be made smaller than any given constant (uniformly with respect to μ_1, μ_2 \in M) provided that \|μ_1-μ_2\| is sufficiently small.

math.CA↗
arXiv2010

Disconjugacy of a second order linear differential equation and periodic solutions

The present paper is devoted to a new criterion for disconjugacy of a second order linear differential equation. Unlike most of the classical sufficient conditions for disconjugacy, our criterion does not involve assumptions on the smallness of the coefficients of the equation. We compare our criterion with several known criteria for disconjugacy, for which we provide detailed proofs, and discuss the applications of the property of disconjugacy to the problem of factorization of linear ordinary differential operators, and to the proof of the generalized Rolle's theorem. The paper is self-contained, and may serve as a brief introduction to theory of disconjugacy of a second order linear differential equation.

math.CA↗