# Volume 50, № 3, 1998

### Central manifolds of quasilinear parabolic equations

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 315–328

We investigate central manifolds of quasilinear parabolic equations of arbitrary order in an unbounded domain. We suggest an algorithm for the construction of an approximate central manifold in the form of asymptotically convergent power series. We describe the application of the results obtained in the theory of stability.

### New classes of exact solutions for a problem of many bodies that attract one another according to an arbitrary law depending on the distances between bodies

Gadomskii L. Ya., Grebenikov E. A., Gurskaya A. R., Zemtsova N. I.

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 329–337

The existence of a 5-parameter family of exact solutions is proved for differential equations describing the motion of many bodies that attract one another according to an arbitrary law depending on the distances between the bodies.

### On the stability of a trivial invariant torus of one class of impulsive systems

Dudzyanyi S. I., Perestyuk N. A.

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 338–349

We consider the problem of asymptotic stability of the trivial invariant torus of one class of impulsive systems. Sufficient criteria of asymptotic stability are obtained by the method of freezing in one case, and by the direct Lyapunov method for the investigation of stability of solutions of impulsive systems in another case.

### Periodic solutions of systems of two linear first-order ordinary differential equations with degenerate asymmetric matrix with derivatives

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 350–356

We establish sufficient conditions for the existence of a periodic solution of a system of two linear firstorder ordinary differential equations with degenerate asymmetric matrix with derivatives in the case of an arbitrary periodic inhomogeneity.

### Stochastic dynamics and Boltzmann hierarchy. II

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 372–387

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.

### Approximation of $\bar {\psi} - \text{Integrals}$ of periodic functions by Fourier sums (small smoothness). IIof periodic functions by Fourier sums (small smoothness). II

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 388-400

We investigate the rate of convergence of Fourier series on the classes $L^{\bar {\psi}} - \text{N}$ in the uniform and integral metrics. The results obtained are extended to the case where the classes $L^{\bar {\psi}} - \text{N}$ are the classes of convolutions of functions from $\text{N}$ with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets $L^{\bar {\psi}} - \text{N}$, which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.

### Conditional stability of quasitoroidal manifolds of systems of partial differential equations

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 401–408

We investigate the problem of conditional stability of quasitoroidal manifolds of a system of partial differential equations.

### On linear homogeneous almost periodic systems that satisfy the Favard condition

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 409–413

We prove the existence of a linear homogeneous almost periodic system of differential equations that has nontrivial bounded solutions and is such that all systems from a certain neighborhood of it have no nontrivial almost periodic solutions.

### Linear and nonlinear representations of Galilei groups in two-dimensional space-time

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 414-423

We study Galilei groups represented as groups of Lie transformations in the space of two independent variables and one dependent variable. We classify the representations of the groups *A G* _{1}(1,1), *A G* _{2}(1,1), *A G* _{3}(1,1), *A ~G* _{1} (1,1), *A ~G* _{2} (1,1), and A ~G_{3}(1,1) in the class of Lie vector fields.

### On necessary conditions for the weak regularity of a linear extension of a dynamical system on a torus

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 424–425

We give a necessary condition for the weak regularity of a linear extension of a dynamical system on a torus in the form of Fourier series with respect to a part of the variables.

### An iteration method for the problem of averaging in the standard form

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 426–429

An iteration method for the enhancement of the precision of an approximate solution of the problem of averaging in the standard form is considered.

### On the application of approximation of the central manifold of a stationary point to one critical case

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 430–432

We establish conditions under which a central manifold can be replaced by its approximation in the reduction principle for ordinary differential equations in a critical case of one zero root.

### Investigation of a linear evolution system in the Banach space with random times of perturbations

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 433–436

For a linear evolution system given in the Banach space and characterized by pulse perturbations at random times, we establish conditions for the existence of a unique solution of the Cauchy problem and investigate the stability of the zero solution.

### On the limit polynomial for a solution of an elliptic equation of the fourth order with constant coefficients

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 437–444

We show that a solution of the Dirichlet problem for an elliptic equation of the fourth order with constant coefficients, whose right-hand side is periodic in all variables except one and exponentially decreases, converges at infinity to a certain polynomial of the first degree in the nonperiodic variable. Coefficients of this polynomial are determined.

### A method of asymptotic expansion of an *m*-parameter family of solutions for a quasilinear system of differential equations

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 445–450

By using the method of asymptotic expansion of an *m*-parameter family of solutions, we obtain the asymptotic expansion of solutions of a quasilinear system of differential equations.

### Conditions for exponential stability and dichotomy of pulse linear extensions of dynamical systems on a torus

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 451–453

Conditions for exponential stability and dichotomy of pulse linear extensions of dynamical systems on a torus are investigated.

### All-Ukrainian scientific conference “New approaches to the solution of differential equations”

Ptashnik B. I., Samoilenko A. M.

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 454–455