# Volume 57, № 7, 2005

### Supplement to the Mergelyan Theorem on the Denseness of Algebraic Polynomials in the Space $C_w^0$

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 867–878

We give a supplement to the theorem on the denseness of polynomials in the space $C_w^0$ established by Mergelyan in 1956 for the case where algebraic polynomials are dense in $C_w^0$. In the case indicated, we give a complete description of all functions that can be approximated by algebraic polynomials in seminorm.

### A Stochastic Analog of Bogolyubov's Second Theorem

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 879–894

We establish an estimate for the rate at which a solution of an ordinary differential equation subject to the action of an ergodic random process converges to a stationary solution of a deterministic averaged system on time intervals of order $e^{1/ερ}$ for some $0 < ρ < 1$.

### On a Weak Solution of an Equation for an Evolution Flow with Interaction

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 895–903

We prove that a stochastic differential equation for an evolution flow with interaction whose coefficients do not satisfy the global Lipschitz condition has a weak solution.

### A Note on the Asymptotic Stability of Fuzzy Differential Equations

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 904–911

We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and the comparison principle for Lyapunov-like functions, we give sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations.

### Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

Khoma-Mohyl's'ka S. H., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 912–921

We consider the periodic boundary-value problem $u_{tt} − u_{xx} = g(x, t),\; u(0, t) = u(π, t) = 0,\; u(x, t + ω) = u(x, t)$. By representing a solution of this problem in the form $u(x, t) = u^0(x, t) + ũ(x, t)$, where $u^0(x, t)$ is a solution of the corresponding homogeneous problem and $ũ(x, t)$ is the exact solution of the inhomogeneous equation such that $ũ(x, t + ω) u_x = ũ(x, t)$, we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier.

### Conditions for Synchronization of One Oscillation System

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 922–945

Using methods of perturbation theory, we investigate the global behavior of trajectories on a toroidal attractor and in its neighborhood for a system of differential equations that arises in the study of synchronization of oscillations in the mathematical model of an optical laser.

### Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 946–971

We consider classes of $2\pi$-periodic functions that are representable in terms of convolutions with fixed kernels $\Psi_{\overline{\beta}}$ whose Fourier coefficients tend to zero with the exponential rate. We compute exact values of the best approximations of these classes of functions in a uniform and an integral metrics. In some cases, the results obtained enable us to determine exact values of the Kolmogorov, Bernstein, and linear widths for the classes considered in the metrics of spaces $C$ and $L$.

### Removability of an Isolated Singularity of Solutions of Nonlinear Elliptic Equations with Absorption

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 972–988

We prove *a priori* estimates for singular solutions of nonlinear elliptic equations with absorption. Using these estimates, we establish precise conditions for the behavior of the absorption term of the equation under which solutions with point singularities do not exist.

### On the Relationship between Properties of Solutions of Difference Equations and the Corresponding Differential Equations

Stanzhitskii A. N., Tkachuk A. M.

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 989–996

We establish conditions under which the existence of a periodic solution of a differential equation is preserved if a solution of the corresponding difference equation possesses the same property. We prove the convergence of periodic solutions of a system of difference equations to a periodic solution of a system of differential equations. Analogous problems are considered for bounded solutions.

### Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry

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

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 997–1001

By using the method of a priori estimates, we establish differential inequalities for energetic norms in $W^l_{2,r}$ of solutions of problems with a free bound in media with the fractal geometry for one-dimensional evolutionary equation. On the basis of these inequalities, we obtain estimates for the stabilization time $T$.

### On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight

Radzievskaya E. I., Radzievskii G. V.

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 1002–1006

Let $α=\{α_j\}_{j∈N}$ be a nondecreasing sequence of positive numbers and let $l_{1,α}$ be the space of real sequences $ξ=\{ξ_j\}_{j∈N}$ for which $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. We associate every sequence $ξ$ from $l_{1,α}$ with a sequence $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, where $ϕ(·)$ is a permutation of the natural series such that $|ξ_{φ(j)}| ⩾ |ξ_{φ(j+1)}|,\; j ∈ ℕ$. If $p$ is a bounded seminorm on $l_{1,α}$ and $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, then $$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$ Using this equality, we obtain several known statements.

### Igor Skrypnyk

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 1007-1008