# Volume 59, № 1, 2007

### A criterion for the existence of the unique invariant torus of a linear extension of dynamical systems

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 3–13

Under the assumption that a linear homogeneous system defined on the direct product of a torus and the Euclidean space is exponentially dichotomous on semiaxes, we obtain a necessary and sufficient condition for the existence of the unique invariant torus of the corresponding inhomogeneous linear system.

### FD-method for an eigenvalue problem with nonlinear potential

Gavrilyuk I. P., Klymenko A. V., Makarov V. L., Rossokhata N. O.

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 14–28

Using the functional discrete approach and Adomian polynomials, we propose a numerical algorithm for an eigenvalue problem with a potential that consists of a nonlinear autonomous part and a linear part depending on an independent variable. We prove that the rate of convergence of the algorithm is exponential and improves as the order number of an eigenvalue increases. We investigate the mutual influence of the piecewise-constant approximation of the linear part of the potential and the nonlinearity on the rate of convergence of the method. Theoretical results are confirmed by numerical data.

### Coconvex approximation of periodic functions

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 29–43

The Jackson inequality *E _{n } * (

*f*) ≤

*c*ω

_{3 }(

*f*, π /

*n*) connects the value of the best uniform approximation

*E*(

_{n }*f*) of a 2π-periodic function

*f*:

**R**→

**R**by trigonometric polynomials of order ≤

*n*— 1 with its third modulus of continuity ω

_{3 }(

*f, t*).

In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity on [—π, π) only at every point of a fixed finite set consisting of the even number of points are approximated by polynomials coconvex to them.

### Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 44–70

We consider the problem of nonlinear oscillations of an ideal incompressible liquid in a tank of a body-of-revolution shape. It is shown that the ordinary way of application of perturbation techniques results in the violation of solvability conditions of the problem. To avoid this contradiction we introduce some additional conditions and revise previously used approaches. We construct a discrete nonlinear model of the investigated problem on the basis of the Hamilton-Ostrogradskii variational formulation of the mechanical problem, preliminarily satisfying the kinematic boundary conditions and solvability conditions of the problem. Numerical examples testify to the efficiency of the constructed model.

### Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

Loveikin Yu. V., Parasyuk I. O.

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 71–98

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed.

### Investigation of the structure of the set of continuous solutions of systems of nonlinear difference equations with continuous argument

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 99–108

We study the structure of the set of continuous solutions for one class of systems of nonlinear difference equations with continuous argument in the neighborhoods of equilibrium states.

### Nonlocal Dirichlet problem for linear parabolic equations with degeneration

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 109–121

In the spaces of classical functions with power weight, we prove the correct solvability of the Dirichlet problem for parabolic equations with nonlocal integral condition with respect to the time variable and an arbitrary power order of degeneration of coefficients with respect to the time and space variables.

### Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 122–132

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients and prove a theorem on the estimation of its precision.

### Locally graded groups with normal nonmetacyclic subgroups

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 133–135

We establish the solvability of locally graded groups with normal nonmetacyclic subgroups and prove that the degree of solvability does not exceed 4.

### Representation of solutions of one integro-differential operator equation

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 136–139

We describe solutions of an integro-differential operator equation in the class of linear continuous operators acting in spaces of functions analytic in domains.

### On one class of topological *-algebras with standard identities

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 140–143

Let *A* be a unital semisimple topological nuclear *-algebra over *C* and let *Z* be its center.
Then *A* is topologically isomorphic to *M _{n }* (

*Z*) if and only if

*A*satisfies the standart identity and the maximality condition.