# Volume 51, № 2, 1999

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

Babenko V. F., Selivanova S. A.

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

We obtain nonperiodic analogs of the known inequalities that estimate*L* _{ p }-norms of intermediate derivatives of a periodic function in terms of its*L* _{∞}-norms and higher derivative.

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

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.

### Asymptotic integration of systems of integro-differential equations with degeneracies

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.

### On a problem of restoration of curves

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.

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

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 order*2m*. 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.

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

Martynyuk A. A., Sun' Chzhen-tsi

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.

### 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.

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.

### Continuous approximations of discontinuous nonlinearities of semilinear elliptic-type equations

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.

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

Pelyukh G. P., Samoilenko A. M.

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

### Uniqueness of solutions of impulsive hyperbolic differential-functional equations

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.

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

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.

### On boundedness of square means for the logarithms of Blaschke products

Kondratyuk A. A., Kondratyuk Ya. V.

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

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

### Imbedding of the images of operators and reflexivity of Banach spaces

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.

### On some integral equations with the generalized Legendre function

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

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

We solve and investigate an integral equation with the generalized associated Legendre function*P* _{ k } ^{ m,n } (z) by using the fractional integro-differential calculus.

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

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.

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

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.

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

Fam Ngok Boi, Nguen Tkhe Khoan

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.

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

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

We study the boundary-value problem*u* _{ 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.

### 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

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.