# Volume 54, № 9, 2002

### On Some Problems of Polynomial Approximation of Entire Transcendental Functions

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1155-1162

For entire transcendental functions of finite generalized order, we obtain limit relations between the growth characteristic indicated above and sequences of their best polynomial approximations in certain Banach spaces (Hardy spaces, Bergman spaces, and spaces \(B\left( {p,q,{\lambda }} \right)\) ).

### On Abnormally Factorizable Finite Solvable Groups

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1163-1171

We study hereditary formations closed with respect to the operation of taking products of abnormal subgroups of finite solvable groups. We obtain a constructive description of solvable local hereditary formations of finite groups with the property indicated.

### Global λ-Stability of One Nonautonomous Quasilinear Second-Order Equation

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1172-1189

We establish sufficient conditions for the λ-stability of the trivial solution of one quasilinear differential equation of the second order.

### Topological Properties of Periodic Components of Structurally Stable Diffeomorphisms

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1190-1199

We consider periodic components of structurally stable diffeomorphisms on two-dimensional manifolds. We study properties of these components and give a topological description of their boundaries.

### Coconvex Pointwise Approximation

Dzyubenko H. A., Gilewicz J., Shevchuk I. A.

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1200-1212

Assume that a function *f* ∈ *C*[−1, 1] changes its convexity at a finite collection *Y* := {*y* _{1}, ... *y* _{s}} of *s* points *y* _{i} ∈ (−1, 1). For each *n* > *N*(*Y*), we construct an algebraic polynomial *P* _{n} of degree ≤ *n* that is coconvex with *f*, i.e., it changes its convexity at the same points *y* _{i} as *f* and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where *c* is an absolute constant, ω_{2}(*f*, *t*) is the second modulus of smoothness of *f*, and if *s* = 1, then *N*(*Y*) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.

### Approximation of Differentiable Periodic Functions by Their Biharmonic Poisson Integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1213-1219

We determine the exact values and asymptotic decompositions of upper bounds of approximations by biharmonic Poisson integrals on classes of periodic differentiable functions.

### Extremal Problems in Logarithmic Potential Theory

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1220-1236

We pose and solve an extremal problem of logarithmic potential theory that is dual to the main minimum problem in the theory of interior capacities of condensers but, in contrast to the latter, it is solvable even in the case of a nonclosed condenser. Its solution is a natural generalization of the classical notion of interior equilibrium measure of a set. A condenser is treated as a finite collection of signed sets such that the closures of sets with opposite signs are pairwise disjoint. We also prove several assertions on the continuity of extremals.

### Averaging of Boundary-Value Problems with Parameters for Multifrequency Impulsive Systems

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

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1237-1249

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillating systems with pulse influence at fixed times. We also obtain estimates for the deviation of solutions of the averaged problem from solutions of the original problem.

### On the Existence of a Generalized Solution of One Partial Differential System

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1250-1264

We propose a method for the construction of generalized solutions for some nondivergent partial differential systems using set-valued analogs of the generalized statement of the problem based on subdifferential calculus. We establish new sufficient conditions for the existence of solutions of a variational inequality with set-valued operator under weakened coerciveness conditions. We consider examples of a weighted *p*-Laplacian in the Sobolev spaces \(W_p^1 \left( \Omega \right)\) , *p* ≥ 2.

### Averaging of Systems with Slow Variables

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1265-1275

We justify the averaging method for systems with delay described by both “slow” and “fast” variables. The results obtained are applied to the analysis of one problem in control theory.

### On the Growth of Infinite-Order Subharmonic Functions in ℂ

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1276-1281

For infinite-order functions *u* subharmonic in \(\mathbb{C}\) with given restrictions on the Riesz masses of a disk of radius *r* ∈ (0, +∞), we find majorants for the functions \(B\left( {r,u} \right) = \max \left\{ {\left| {u\left( z \right)} \right|:\left| z \right| \leqslant r} \right\}\) and \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{B} \left( {r,u} \right) = \sup \left\{ {\left| {\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{u} \left( z \right)} \right|:\left| z \right| \leqslant r} \right\}\) , where \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{u}\) is a function conjugate to *u*.

### On the Sign of Solutions of Systems of Ordinary Differential Equations

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1282-1283

We show that Theorems 1 and 3 in A. G. Gritsai's paper “Monotonicity properties of solutions of systems of nonlinear differential equations” published in the collection of works *Approximate and Qualitative Methods in the Theory of Differential and Functional Differential-Equations* (Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, 1979) are not true in the formulation presented.

### BPS-States in F-Theory

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1284-1288

The spectra of BPS states in F-theory on elliptic fibered fourfolds are investigated.

### On the Possibility of Stabilization of Evolution Systems of Partial Differential Equations on $ℝ^n × [0, + ∞)$Using One-Dimensional Feedback Controls

Fardigola L. V., Sheveleva Yu. V.

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1289-1296

We establish conditions for the stabilizability of evolution systems of partial differential equations on $ℝ^n × [0, + ∞)$ by one-dimensional feedback controls. To prove these conditions, we use the Fourier-transform method. We obtain estimates for semialgebraic functions on semialgebraic sets by using the Tarski–Seidenberg theorem and its corollaries. We also give examples of stabilizable and nonstabilizable systems.