# Volume 59, № 4, 2007

### Existence of a solution and averaging of nonlinear multifrequency problems with delay

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 435–446

The method of averaging over the fast variables is applied to the investigation of multifrequency systems with linearly transformed argument. We prove the existence of solutions of the initial-and boundary-value problems in a small neighborhood of the solution of the averaged problem and estimate the error of the method of averaging for slow variables.

### Evaluation of the probability of bankruptcy for a model of insurance company

Bondarev B. V., Zhmykhova T. V.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 447–457

A problem of calculating the probability of ruin of an insurance company in infinite number of steps is considered in the case where this company is able to invest its capital to a bank deposit at every time. As a distribution describing claim amounts to the insurance company, the gamma distribution with parameters $n$ and $\alpha$ is chosen.

### Conditions of oscillatory or nonoscillatory nature of solutions for a class of second-order semilinear differential equations

Evtukhov V. M., Vasileva N. S.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 458–466

For a class of second-order semilinear differential equations, we prove the theorems on oscillatory or nonoscillatory nature of all proper solutions. These theorems are analogs of the well-known Kneser theorems for linear differential equations.

### Method of averaging in the problem of stability of motion of a body suspended from a string

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 467–475

On the basis of the Bogolyubov-Mitropol’skii method of averaging, we study the problem of stability of the vertical rotation of a body suspended from a string.

### Application of asymptotic methods to the investigation of one-frequency nonlinear oscillations of cylindrical shells interacting with moving fluid

Koval’chuk P. S., Kruk L. A., Kubenko V. D.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 476–487

We demonstrate the applicability of the Bogolyubov-Mitropol’skii asymptotic method to the construction of one-frequency solutions of a system of nonlinear equations used to describe the multimode free, forced, and parametrically excited vibrations of cylindrical shells interacting with moving fluid.

### Alternating Lyapunov functions in the theory of linear extensions of dynamical systems on a torus

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 488–500

We consider a series of problems connected with the application of quadratic-form Lyapunov functions to the investigation of the properties of regularity of linear extensions of dynamic systems on a torus.

### On solutions of linear functional differential equations with linearly transformed argument on a semiaxis

Denysenko N. L., Samoilenko A. M.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 501–513

We establish conditions under which solutions of a system of linear functional differential equations on a semiaxis are determined as solutions of a certain system of ordinary differential equations.

### Asymptotic equivalence of impulsive systems

Misyats O. O., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 514–521

We present sufficient conditions for the asymptotic equivalence of a nonlinear impulsive system and a nonlinear system without pulses. We also consider the case of the asymptotic equivalence of a weakly nonlinear impulsive system and a linear system with pulses.

*L*_{2 }-invariants and Morse - Smale flows on manifolds

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 522-533

We study the homotopy invariants of free cochain and Hilbert complexes.
These *L*_{2 } -invariants are applied to the calculations of exact values of minimal numbers of closed orbits of some indexes of nonsingular Morse - Smale flows on manifolds of large dimensions.

### Deterministic chaos of a spherical pendulum under limited excitation

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 534–548

We investigate the appearance, development, and vanishing of deterministic chaos in a “spherical pendulum-electric motor of limited power” dynamical system. Chaotic attractors discovered in the system are described in detail.

### Evgen Yakovich Khruslov (on his 75 th birthday)

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Pastur L. A., Samoilenko A. M., Sharko V. V.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 549-550

### On equivalent bases in Banach spaces

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 551–554

We present several generalizations of the classical Bari theorem on the Riesz basis property of close systems in Hilbert spaces to Banach spaces. We introduce the corresponding definitions and formulate theorems on the basis property of close systems in Banach spaces.

### Two-limit problems for almost semicontinuous processes defined on a Markov chain

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 555–565

We consider almost upper-semicontinuous processes defined on a finite Markov chain. The distributions of functionals associated with the exit of these processes from a finite interval are studied. We also consider some modifications of these processes.

### Middle-sectioned simplices in a four-dimensional affine space

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 566–570

We present the general geometric description and the Euler – Poincare characteristics of middle-sectioned simplexes in the four-dimensional affine space. We demonstrate the relation between such geometrical objects and four-dimensional analogs of the triangular Serpinski napkin.

### Generalization of the Kneser theorem on zeros of solutions of the equation $y" + p (t) y = 0$

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 571–576

We obtain conditions of the oscillation of solutions of the equation $y" + p(t)Ay = 0$ in the Banach space, where $A$ is a bounded linear operator and $p : R_+ \rightarrow R_+$ is a continuous function.