2019
Том 71
№ 9

All Issues

Volume 51, № 2, 1999

Article (Russian)

On the connection between certain inequalities of the Kolmogorov type for periodic and nonperiodic functions

Babenko V. F., Selivanova S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 147–157

We obtain nonperiodic analogs of the known inequalities that estimateL p -norms of intermediate derivatives of a periodic function in terms of itsL -norms and higher derivative.

Article (Russian)

On equivariant extensions of a differential operator by the example of the Laplace operator in a circle

Burskii V. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 158–169

We propose a method for investigation of both correctness of the equivariant problem and the spectrum of the corresponding operator.

Article (Ukrainian)

Asymptotic integration of systems of integro-differential equations with degeneracies

Zavizion G. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 170–180

We construct asymptotic solutions of singularly perturbed homogeneous and heterogeneous systems of integro-differential Fredholm-type equations with degenerate matrix at the derivative.

Article (Russian)

On a problem of restoration of curves

Ligun A. A., Shumeiko A. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 181–189

We investigate the problem of asymptotically optimal placement of disks of the same radius under the condition of minimization of the Hausdorff distance between a given curve $Γ$ and the union of disks under study.

Article (Ukrainian)

Solution of boundary-value problems for elliptic equations in the space of distributions

Lopushanskaya G. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 190–203

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.

Article (Russian)

On the practical $μ$-stability of solutions of standard systems with delay

Martynyuk A. A., Sun' Chzhen-tsi

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 204–213

We study the problem of $μ$-stability of a dynamical system with delay. Conditions of the practical $μ$-stability are established for the general case and for a quasilinear system. The conditions suggested are illustrated by an example.

Article (Russian)

Stabilization for a finite time in problems with free boundary for some classes of nonlinear second-order equations

Berezansky Yu. M., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 214–223

We obtain estimates for the time of stabilization of solutions of problems with free boundary for one-dimensional quasilinear parabolic equations.

Article (Russian)

Continuous approximations of discontinuous nonlinearities of semilinear elliptic-type equations

Iskakov R. S., Pavlenko V. N.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 224–233

We obtain new variational principles of the existence of strong and semiregular solutions of principal boundary-value problems for elliptic-type second-order equations with discontinuous nonlinearity. We study a problem of proximity between the sets of solutions of an approximating problem with nonlinearity continuous in phase variable and solutions of the initial boundary-value problem with discontinuous nonlinearity.

Article (Ukrainian)

On analytic solutions of nonlinear differential functional equations with nonlinear deviations of arguments

Pelyukh G. P., Samoilenko A. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 234–240

We obtain the conditions for existence and uniqueness of an analytic solution of a nonlinear differential functional equation with nonlinear deviations of the argument

Article (English)

Uniqueness of solutions of impulsive hyperbolic differential-functional equations

Janiszewska-Walczak D.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 241–250

For impulsive partial differential-functional equations, we prove the theorems on existence and uniqueness of solutions and their continuous dependence on the right-hand sides of the equations.

Brief Communications (Ukrainian)

On the Levy-Baxter theorems for shot-noise fields. III

Buldygin V. V., Mel'nik V. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 251–254

We establish sufficient conditions for singularity of distributions of shot-noise fields with response functions of a certain form.

Brief Communications (Ukrainian)

On boundedness of square means for the logarithms of Blaschke products

Kondratyuk A. A., Kondratyuk Ya. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 255–259

We establish conditions for boundedness of square means of the logarithms of Blaschke products.

Brief Communications (Russian)

Imbedding of the images of operators and reflexivity of Banach spaces

Vershinin R. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 260–262

We establish a criterion of reflexivity for a separable Banach space in terms of the relation between the imbedding of the images, factorization, and majorization of operators acting in this space.

Article (Ukrainian)

On some integral equations with the generalized Legendre function

Sichkar' Yu, V., Virchenko N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 263–267

We solve and investigate an integral equation with the generalized associated Legendre functionP k m,n (z) by using the fractional integro-differential calculus.

Brief Communications (Ukrainian)

On the approximation of functions from Weyl-Nagy classes by Zygmund sums in the $L_q$ metric

Kostich M. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 268-270

In the integral metric, estimates exact by order are found for deviations of the Zygmund linear means from functions, which belong to Weyl-Nagy classes.

Brief Communications (Russian)

Representations of the additive group of a nuclear space in terms of self-adjoint operators

Lopotko O. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 271–274

We prove theorems on integral representations of the additive group of a real nuclear space in terms of self-adjoint operators. We assume that algebraic relations are realized in a dense invariant set of integral vectors.

Brief Communications (Russian)

On continuous dependence of solutions of linear differential equations on a parameter in a Banach space

Fam Ngok Boi, Nguen Tkhe Khoan

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 275–280

We prove the reduction theorem for linear differential equations in a Banach space in the case where the convergence is strong. This result is used to obtain the necessary and sufficient conditions for continuous dependence of solutions on a parameter.

Brief Communications (Ukrainian)

A linear periodic boundary-value problem for a second-order hyperbolic equation

Khoma L. G., Khoma N. H.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 281–284

We study the boundary-value problemu tt -u xx =g(x, t),u(0,t) =u (π,t) = 0,u(x, t +T) =u(x, t), 0 ≤x ≤ π,t ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of \(\frac{\pi }{q} - , \frac{{2\pi }}{{2s - 1}} - \) , and \(\frac{{4\pi }}{{2s - 1}}\) -periodic functions (q and s are natural numbers). We obtain the results only for sets of periods \(T_1 = (2p - 1)\frac{\pi }{q}, T_2 = (2p - 1)\frac{{2\pi }}{{2s - 1}}\) , and \(T_3 = (2p - 1)\frac{{4\pi }}{{2s - 1}}\) which characterize the classes of π-, 2π -, and 4π-periodic functions.

Brief Communications (Russian)

A resonance case of the existence of solutions of a quasilinear second-order differential system, which are represented by Fourier series with slowly varying parameters

Shchegolev S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 285–288

For a quasilinear second-order differential system, whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove, under certain conditions, the existence of a particular solution having a similar structure. This result is obtained in the case where the characteristic equation possesses purely imaginary roots, which satisfy a certain resonance relation.