# Volume 57, № 12, 2005

### Construction of solutions for the problem of free oscillations of an ideal liquid in cavities of complex geometric form

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1587–1600

We consider the problem of free oscillations of an ideal incompressible liquid in cavities of complex geometric form. The domain filled with liquid is divided into subdomains of simpler geometric form. The original problem is reduced to the spectral problem for a part of the domain filled with liquid. To this end, we use solutions of auxiliary boundary-value problems in subdomains. We construct approximate solutions of the problem obtained using the variational method. We also consider the problem of the rational choice of a system of coordinate functions. Results of the numerical realization of the proposed method are presented.

### Best uniform approximation of a continuous compact-valued mapping by sets of continuous single-valued mappings

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1601–1618

We prove existence theorems and establish necessary and sufficient conditions and criteria for an extremal element for the problem of the best uniform approximation of a continuous compact-valued mapping by sets of continuous single-valued mappings.

### Rate of convergence for Szász-Bézier operators

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1619–1624

We estimate the rate of convergence for functions of bounded variation for the Bezier variant of the Szasz operators $S_{n, \alpha}(f, x)$. We study the rate of convergence of $S_{n, \alpha}(f, x)$ for the case $0 < \alpha < 1$.

### General time-dependent bounded perturbation of a strongly continuous semigroup

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1625–1632

We consider an evolution family whose generator is formed by a time-dependent bounded perturbation of a strongly continuous semigroup. We do not use the condition of the continuity of a perturbation. We prove a formula for a variation of a parameter and the corresponding generalization of the Dyson-Phillips theorem.

### Kolmogorov and linear widths of classes of s-monotone integrable functions

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1633–1652

Let $s \in \mathbb{N}$ and let $\Delta^s_+$ be the set of functions $x \mapsto \mathbb{R}$ on a finite interval $I$ such that the divided differences
$[x; t_0, ... , t_s ]$ of order $s$ of these functions are nonnegative for all collections of $s + 1$ distinct points $t_0,..., t_s \in I$.
For the classes $\Delta^s_+ B_p := \Delta^s_+ \bigcap B_p$ , where $B_p$ is the unit ball in $L_p$, we obtain orders of the Kolmogorov and linear widths in the spaces $L_q$ for $1 \leq q < p \leq \infty$.

### Cauchy problem with Riesz operator of fractional differentiation

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1653–1667

In the class of generalized functions of finite order, we establish the correct solvability of the Cauchy problem for a pseudodifferential equation whose symbols are homogeneous functions of order γ > 0. We prove a theorem on the localization property of a solution of this problem.

### On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1668–1676

We establish new properties of $C^1 (0, + ∞)$-solutions of the linear functional differential equation $\dot{x}(t) = ax(t) + bx(qt) + cx(qt)$ in the neighborhoods of the singular points $t = 0$$ and t = + ∞$.

### On one extremal problem for positive series

Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1677–1683

The approximation properties of the spaces $S^p_{\varphi}$ introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of $n$-term approximations of $q$-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set. Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way.

### Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1684–1697

We consider an abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations. This system describes, e.g., thermoelastic phenomena in various physical bodies. Several results on the existence of invariant exponentially attracting manifolds for similar problems were obtained earlier. In the present paper, we prove the existence of such an invariant manifold under less restrictive conditions for a broader class of problems.

### Spectral representation for generalized operator-valued Toeplitz kernels

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1698–1710

We prove an integral representation for operator-valued Toeplitz kernels. The proof is based on the spectral theory of the corresponding differential operator constructed from this kernel and acting in a Hilbert space.

### On a special critical case of stability of a nonautonomous essentially nonlinear system

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1711–1718

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous essentially nonlinear differential system in a special critical case.

### Destabilizing effect of random parametric perturbations of the white-noise type in some quasilinear continuous and discrete dynamical systems

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1719–1724

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.

### Index of volume 57 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1725-1729