# Volume 55, № 4, 2003

### Asymptotic Solutions of Systems of Linear Degenerate Integro-Differential Equations

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 435-445

We construct asymptotic solutions of a singularly perturbed system of integro-differential equations in which the matrix coefficient of the derivative is degenerate at a point.

### Global Attractor of One Nonlinear Parabolic Equation

Kapustyan O. V., Shkundin D. V.

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 446-455

We apply the theory of multivalued semiflows to a nonlinear parabolic equation of the “reaction–diffusion” type in the case where it is impossible to prove the uniqueness of its solution. A multivalued semiflow is generated by solutions satisfying a certain estimate global in time. We establish the existence of a global compact attractor in the phase space for the multivalued semiflow generated by a nonlinear parabolic equation. We prove that this attractor is an upper-semicontinuous function of a parameter.

### On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 456-469

For 2π-periodic functions \(x \in L_\infty ^r \) and arbitrary *q* ∈ [1, ∞] and *p* ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality \(|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}, k, r \in N, k < r,\) which takes into account the number of changes in the sign of the derivatives ν(*x* ^{(k)}) over the period. Here, α = (*r* − *k* + 1/*q*)/(*r* + 1/*p*), ϕ_{ r } is the Euler perfect spline of degree *r*, \(\begin{gathered} \left\| {\left| x \right|} \right\|_p : = {\text{sup}}_{a,b \in {\text{R}}} \{ E_0 (x)_{L_p [a,b]} :x'(t) \ne 0{\text{ }}\forall t \in (a,b)\} , \\ {\text{ }} \\ {\text{ }}E_0 (x)_{L_p [a,b]} : = {\text{ inf}}_{c \in {\text{R}}} \left\| {x - c} \right\|_{L_p [a,b]} , \\ \\ \left\| x \right\|_{L_p [a,b]} : = \left\{ {\int\limits_a^b {\left| {x(t)} \right|^p dt} } \right\}^{1/p} {\text{ for }}0 < p < \infty , \\ \end{gathered} \) and \(\left\| x \right\|_{L_p [a,b]} : = {\text{ sup vrai}}_{t \in \left[ {a,b} \right]} \left| {x(t)} \right|\) . The inequality indicated turns into the equality for functions of the form *x*(*t*) = *a*ϕ_{ r }(*nt* + *b*), *a*, *b* ∈ **R**, *n* ∈ **N**. We also obtain an analog of this inequality in the case where *k* = 0 and *q* = ∞ and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.

### Groups with Hypercyclic Proper Quotient Groups

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 470-478

We continue the investigation of (solvable) groups all proper subgroups of which are hypercyclic. The monolithic case is studied completely; in the nonmonolithic case, however, one should impose certain additional conditions. We investigate groups all proper quotient groups of which possess supersolvable classes of conjugate elements.

### Interpolational Integral Continued Fractions

Khlobystov V. V., Makarov V. L., Mykhal'chuk B. R.

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 479-488

For nonlinear functionals defined on the space of piecewise-continuous functions, we construct an interpolational integral continued fraction on continual piecewise-continuous nodes and establish conditions for the existence and uniqueness of interpolants of this type.

### Harmonic Properties of Gauss Mappings in $H^3$

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 489-499

We consider some harmonic mappings related to hyperbolic Gauss mappings and Gauss mappings in the Obata sense.

### Best *n*-Term Approximations in Spaces with Nonsymmetric Metric

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 500-509

We determine exact values of *n*-term approximations of *q*-ellipsoids in the spaces \(S_\phi^{p, \mu}\) .

### Error Estimates for the Averaging Method for Pulse Oscillation Systems

Lakusta L. M., Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 510-524

We prove new theorems on the justification of the averaging method on a segment and semiaxis in multifrequency oscillation systems with pulse action at fixed times.

### On Measure-Valued Processes Generated by Differential Equations

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 525-532

We study the problem of representation of a homogeneous semigroup {Θ_{ t }}_{ t ≥ 0} of transformations of probability measures on \(\mathbb{R}^d \) in the form \(\Theta _t (\mu) = \mu \circ u_{\mu}^{-1} (\cdot ,t),\) where \(u_{\mu} :\mathbb{R}^d \times [0, T] \to \mathbb{R}^d\) satisfies a differential equation of a special form dependent on the measure μ. We give necessary and sufficient conditions for this representation.

### On One Generalization of the Hardy–Littlewood–Pólya Inequality

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 533-536

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.

### On a Jackson-Type Inequality in the Approximation of a Function by Linear Summation Methods in the Space $L_2$

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 537-545

We prove a statement on exact inequalities between the deviations of functions from their linear methods (in the metric of $L_2$) with multipliers defined by a continuous function and majorants determined as the scalar product of the squared modulus of continuity (of order r) in $L_2$ for the lth derivative of the function and a certain weight function θ. We obtain several corollaries of the general theorem.

### Whitney Interpolation Constants Bounded by 2 for *k* = 5, 6, 7

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 546-549

Let *f* ∈ *C*[0, 1], *k* = 5, 6, 7. We prove that if *f*(*i*/(*k* − 1)) = 0, *i* = 0, 1,..., *k* − 1, then \(\left| {f(x)} \right| \leqslant 2{\text{ }}\mathop {{\text{sup}}}\limits_{x,x + kh \in [0,1]} {\text{ }}\left| {\sum\limits_{j = 0}^k {( - 1)^j } \left( {\mathop {}\limits_j^k } \right)f(x + jh)} \right|.\)

### Reduction of Matrices over Bezout Rings of Stable Rank not Higher than 2

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 550-554

We prove that a commutative Bezout ring is an Hermitian ring if and only if it is a Bezout ring of stable rank 2. It is shown that a noncommutative Bezout ring of stable rank 1 is an Hermitian ring. This implies that a noncommutative semilocal Bezout ring is an Hermitian ring. We prove that the Bezout domain of stable rank 1 with two-element group of units is a ring of elementary divisors if and only if it is a duo-domain.

### Theorem on Conflict for a Pair of Stochastic Vectors

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 555-560

We investigate a mathematical model of conflict with a discrete collection of positions.

### On the Asymptotic Behavior of Solutions of Linear Differential Equations

Nguyen Minh Man, Nguyen The Hoan

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 561-569

We present sufficient conditions for the linear asymptotic equilibrium of linear differential equations in Hilbert and Banach spaces. The results obtained are applied to studying the asymptotic equivalence of two linear differential equations.

### A Note on the Recursive Sequence $x_{n + 1} = p_kx_n + p_{k − 1}x_{n − 1} +...+ p_1x_{n − k + 1}$

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 570-574

We present some comments on the behavior of solutions of the difference equation $x_{n + 1} = p_kx_n + p_{k − 1}x_{n − 1} +...+ p_1x_{n − k + 1}$, $n = −1, 0, 1,…,$ where $p_i ≥ 0, i = 1,..., k, k ∈ N$, and $x_{−k},..., x_{−1} ∈ R$.

### The third international conference on analytical theory of numbers and space mosaic devoted to Voronoi's memory (the first announcement)

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 575-576