2018
Том 70
№ 2

Volume 54, № 12, 2002

Article (Russian)

Vladimir Nikolaevich Koshlyakov (On His 80th Birthday)

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1587-1588

Anniversaries (Ukrainian)

Mykola Ivanovych Shkil' (On His 70th Birthday)

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1589-1591

Article (Ukrainian)

Dirichlet-Type Problems for Systems of Partial Differential Equations Unresolved with Respect to the Highest Time Derivative

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1592-1602

We establish conditions for the correct solvability of problems for systems of partial differential equations unresolved with respect to the highest time derivative with Dirichlet-type conditions with respect to time and periodic conditions with respect to space variables. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of these problems.

Article (Russian)

On the Best Polynomial Approximations of $2π$-Periodic Functions and Exact Values of $n$-Widths of Functional Classes in the Space $L_2$

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1603-1615

To solve extremal problems of approximation theory in the space $L_2$, we use τ-moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of $n$-widths of functional classes determined by these moduli.

Article (Ukrainian)

Compound Poisson Processes with Two-Sided Reflection

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1616-1625

We consider a compound oscillating Poisson process with two-sided reflection. This process is defined by an upper-semicontinuous compound Poisson process ξ(t) and its functionals, namely the first-exit time of ξ(t) from an interval and the first-exit time of ξ(t) across the upper and lower levels. We study the main characteristics of this oscillating process in terms of the potential and resolvent of the process ξ(t) introduced by Korolyuk. For this purpose, we refine the Pecherskii identities and some other results for upper-semicontinuous Poisson processes.

Article (Russian)

On the Existence of Periodic Solutions of Nonlinear Difference Equations

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1626-1633

We obtain new sufficient conditions for the existence and uniqueness of an N-periodic solution (N is a positive integer) of a nonlinear difference equation with continuous argument of the form x(t + 1) = f(x(t)) and investigate the properties of this solution.

Article (Russian)

On the Solution of the Exterior Dirichlet Problem for an Axisymmetric Potential

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1634-1641

For an unbounded domain of the meridian plane with bounded smooth boundary that satisfies certain additional conditions, we develop a method for the reduction of the Dirichlet problem for an axisymmetric potential to Fredholm integral equations. In the case where the boundary of the domain is a unit circle, we obtain a solution of the exterior Dirichlet problem in explicit form.

Article (Russian)

New Integral Transformations and Their Applications to Some Boundary-Value Problems of Mathematical Physics

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1642-1652

We construct new integral transformations and present their applications to the construction of exact solutions of some boundary-value problems of mathematical physics. We solve the problem of diffraction of acoustic waves in a circular cone truncated by two spherical surfaces. We also solve the initial boundary-value problem of the theory of heat conduction for the same truncated cone under nonzero initial conditions.

Article (Ukrainian)

Approximation of Analytic Periodic Functions by de la Vallée-Poussin Sums

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1653-1669

We investigate the approximation properties of the de la Vallée-Poussin sums on the classes $C_{\beta }^q H_{\omega }$ . We obtain asymptotic equalities that, in certain cases, guarantee the solvability of the Kolmogorov–Nikol'skii problem for the de la Vallée-Poussin sums on the classes $C_{\beta }^q H_{\omega }$ .

Article (Ukrainian)

Norms of Multipliers and Best Approximations of Holomorphic Functions of Many Variables

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1669-1680

We show that the Lebesgue–Landau constants of linear methods for summation of Taylor series of functions holomorphic in a polydisk and in the unit ball from $\mathbb{C}^m$ over triangular domains do not depend on the number m. On the basis of this fact, we find a relation between the complete and partial best approximations of holomorphic functions in a polydisk and in the unit ball from $\mathbb{C}^m$ .

Article (Russian)

Some Local Contour-Solid Theorems for Finely Holomorphic Functions

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1680-1687

We prove some local contour-solid theorems for finely holomorphic functions defined on sets of the complex plane that are finely open with nonpolar complements.

Article (Ukrainian)

Linear Singularly Perturbed Systems

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1688-1693

We investigate the solvability of the Cauchy problem for a linear singularly perturbed homogeneous system in the case of a singular pencil of matrices.

Brief Communications (Russian)

On Kolmogorov-Type Inequalities with Integrable Highest Derivative

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1694-1697

We obtain the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}$$ for 2π-periodic functions $x \in L_1^r$ and any k, rN, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type.

Brief Communications (Ukrainian)

Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1698-1703

We prove the existence of continuously differentiable solutions $x:(0,{\rho ]} \to \mathbb{R}^n$ such that $$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)/t} \right),{ }t \to + 0$$ or $$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t \to + 0,$$ where $${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$ $${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },{ \rho is sufficiently small}{.}$$

Brief Communications (Ukrainian)

Approximate Synthesis of Optimal Bounded Control for a Parabolic Boundary-Value Problem

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1704-1709

We consider the approximate optimal control based on the principle of feedback relation (synthesis) for a parabolic boundary-value problem. We represent the feedback operator as Fourier series in the eigenfunctions of the Laplace operator, which does not enable us to use these results in practice. In view of this fact, we justify the convergence of approximate controls, switching points, and values of the quality criterion to the exact values of the corresponding variables.

Article (Russian)

Klee Theorem for Linearly Convex Sets

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1710-1713

We prove a complex analog of the classical Klee theorem for strongly linearly convex closed sets.

Brief Communications (English)

On the Stability of Semilinear Nonautonomous Evolution Equations in Banach Spaces and Its Application to Strongly Parabolic Equations

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1714-1719

The paper is concerned with the exponential stability of the zero solution of strongly nonautonomous parabolic equations. Conditions are found on time-dependent coefficients of a parabolic equation under which its solutions converge exponentially to 0 as t → ∞.

Chronicles (Ukrainian)

The international scientific conference devoted to the 90-th birthday of В. V. Gnedenko

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1720-1722

Index (Ukrainian)

Index of volume 54 of Ukrainian Mathematical Journal

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1723-1728