EZEvgeny Semenovich Zhukovskiy

Evgeny Semenovich Zhukovskiy

Tambov State University
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Evgeny Semenovich Zhukovskiy has a total of 56 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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    Stability of solutions of implicit nonautonomous difference equations in spaces with generalized distance

    by V. V. Anisimov, T. V. Zhukovskaya, E. S. Zhukovskiy, В. В. Анисимов, Т. В. Жуковская, Е. С. Жуковский

    Implicit nonautonomous first-order difference equations in spaces endowed with distance are considered. The distance is only required to be non-negative (the distance need not possess any of the standard properties of a metric). Analogs of classical definitions of stability with respect to initial data (Lyapunov, asymptotic, and exponential stability) are introduced, taking into account the possible non-uniqueness of the solution to the Cauchy problem. Under the assumption that the mapping of two arguments generating the equation under consideration is covering with respect to one argument and Lipschitz with respect to the other, statements are obtained on the stability with respect to the initial data of a particular solution to the Cauchy problem and of all solutions to the equation. As a corollary, the stability of a stationary solution to an implicit autonomous equation is investigated. Conditions for the existence of a stationary solution for such an autonomous equation are also obtained. It is demonstrated how the presented results imply well-known conditions for the exponential stability of equilibria of explicit difference equations in linear normed spaces. A scheme for applying the obtained results to the study of the stability of some functional equations with delay is proposed. An example of a study using this scheme of a specific functional equation in the space of Lebesgue measurable essentially bounded functions is given.

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    Некоторые порядковые свойства множеств решений краевых задач

    by Е. С. Жуковский, А. С. Патрина

    УДК 517.911.5, 517.927.4, 517.988.525 MSC: 34A36, 34A40, 34B15, 34C12 https://doi.org/10.21538/0134-4889-2026-32-2-100-111 Исследование выполнено за счет гранта Российского научного фонда (проект № 25-21-00819, https://rscf.ru/project/25-21-00819/). Исследуется двухточечная краевая задача для обыкновенного дифференциального уравнения с возрастающей (и необязательно

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    Existence and Properties of Solutions to the Generalized Nash Equilibrium Problem

    by A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy

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    Некоторые топологические свойства пространств с расстоянием

    by Evgeny Semenovich Zhukovskiy

    На множестве $X$ рассматривается функция расстояния $\rho\colon X^2\to\mathbb R_+$, удовлетворяющая аксиоме тождества: $\rho(x,y)=0$ равносильно $x=y$. По функции $\rho$ определяется топология, множество $U\subset X$ принадлежит этой топологии тогда и только тогда, когда для каждого $u\in U$ при некотором положительном $\delta$ выполнено $\{x\colon \rho(u,x)<\delta\}\subset U$. Для рассматриваемого топологического пространства исследуются свойства замкнутости и секвенциальной замкнутости, компактности, секвенциальной компактности и полной ограниченности. Библиография: 26 названий.

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    Topological Properties of Spaces with a Distance

    by E. S. Zhukovskiy

    A set \(X\) with a distance function \(\rho\colon X^2\to\mathbb R_+\) satisfying the identity axiom, that is, such that \(\rho(x,y)=0\) if and only if \(x=y\), is considered. The function \(\rho\) determines a topology on \(X\); a set \(U\subset X\) belongs to this topology if and only if, for each \(u\in U\), there exists a positive \(\delta\) such that \(\{x\colon \rho(u,x)<\delta\}\subset U\). Closedness, sequential closedness, compactness, sequential compactness, and total boundedness in topological spaces thus arising are studied.

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    Some topological properties of f-quasimetric spaces

    by Evgeny S. Zhukovskiy, Tatiana V. Zhukovskaya

    The properties of the f–quasimetric space (X,ρ) are studied. In a space as such, the distance ρ satisfies the identity axiom and the generalized triangle inequality: ρ(x,z) ≤ f(ρ(x,y),ρ(y; z)) for any x,y,z∈X. Here the function f is positive for positive arguments, continuous at the point (0,0), and f(0,0)=0. The symmetry of the distance is not assumed. The topology on X generated by the distance ρ is defined in the standard way. The properties of convergent sequences and sequentially compact sets are studied. Conditions are obtained under which the convergence in itself (mutual convergence) is necessary for the convergence of a sequence. The relationship between the rates of convergence of a fundamental sequence and its convergence in itself is considered. The concept of a sequentially precompact set is introduced. Conditions are obtained under which the closure of a sequentially precompact set is sequentially compact.

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    The Method of Comparison with a Model Equation in the Study of Inclusions in Vector Metric Spaces

    by E. S. Zhukovskiy, E. A. Panasenko

    For a given multivalued mapping \(F:X\rightrightarrows Y\) and a given element \(\tilde{y}\in Y\), the existence of a solution \(x\in X\) to the inclusion \(F(x)\ni\tilde{y}\) and its estimates are studied. The sets \(X\) and \(Y\) are endowed with vector-valued metrics \(\mathcal{P}_{X}^{E_{+}}\) and \(\mathcal{P}_{Y}^{M_{+}}\), whose values belong to cones \(E_{+}\) and \(M_{+}\) of a Banach space \(E\) and a linear topological space \(M\), respectively. The inclusion is compared with a “model” equation \(f(t)=0\), where \(f:E_{+}\to M\). It is assumed that \(f\) can be written as \(f(t)\equiv g(t,t)\), where the mapping \(g:{E}_{+}\times{E}_{+}\to M\) orderly covers the set \(\{0\}\subset M\) with respect to the first argument and is antitone with respect to the second argument and \(-g(0,0)\in M_{+}\). It is shown that, in this case, the equation \(f(t)=0\) has a solution \(t^{*}\in E_{+}\). Further, conditions on the connection between \(f(0)\) and \(F(x_{0})\) and between the increments of \(f(t)\) for \(t\in[0,t^{*}]\) and the increments of \(F(x)\) for all \(x\) in the ball of radius \(t^{*}\) centered at \(x_{0}\) for some \(x_{0}\) are formulated, and it is shown that the inclusion has a solution in the ball under these conditions. The results on the operator inclusion obtained in the paper are applied to studying an integral inclusion.

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    Метод сравнения с модельным уравнением в исследовании включений в векторных метрических пространствах

    by Е. С. Жуковский, Е. А. Панасенко

    УДК 517.988 + 517.968.4 MSC: 54E35, 47H04, 45G10 DOI: 10.21538/0134-4889-2024-30-2-68-85 Исследование выполнено за счет гранта Российского научного фонда № 24-21-00272, https://rscf.ru/project/24-21-00272/ .

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    Comparison method in the study of equations and inclusions

    by E.S. Zhukovskiy, I.D. Serova

    Монография посвящена исследованию разрешимости и нахождению оценок решений уравнений и включений. В главе 1 рассмотрены операторные уравнения и включения в частично упорядоченных пространствах и функциональные уравнения и включения с отклоняющимся аргументом, явно не разрешенные относительно искомой функции, в пространстве измеримых функций. В главах 2, 3 рассматриваются неявные, то есть не разрешенные относительно старшей производной обыкновенные дифференциальные уравнения и включения. Исследованы задача Коши, краевая задача и система управления.

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    Method of approximate solution of partial derivative equations

    by Tatiana V. Zhukovskaia, Evgeny S. Zhukovsky, Mikhail A. Rybakov, Anna S. Trofimova

    The article considers a partial differential equation of the form ∂u/∂t=f(t,x,y,u,∂u/∂x,∂u/∂y,(∂^2 u)/(∂x^2 ),(∂^2 u)/(∂y^2 ),(∂^2 u)/∂x∂y),(x,y)∈D⊂R^2,t≥0, with respect to an unknown function u, defined in a domain D of spatial variables x,y and for t≥0. A method for finding an approximate solution is proposed. The equation under consideration is replaced by an approximate one by introducing the shift operator S:D→D, which allows replacing at each step of the calculations the unknown values of the function u(x,y,t) on the right side with the values u(S(x,y),t), obtained at the previous step. The idea of the proposed method goes back to the idea of the Tonelli method, known for differential equations with respect to functions of one variable (with ordinary, not partial derivatives). The advantages of the proposed method are the simplicity of the obtained iteration relation and the possibility of application to a wide class of equations and boundary conditions. In the article, iteration formulas are obtained for solving a boundary value problem with the Dirichlet condition for spatial variables and with an initial or boundary condition for the variable t. Based on the proposed method, an approximate solution is obtained for a specific initial-boundary value problem for the heat conductivity equation in a square domain.

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