# Volume 50, № 2, 1998

### On the decomposition of a perturbed operator of weighted shift

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 155–161

We obtain results concerning the reducibility of an operator of weighted shift to a system of scalar operators of weighted shift.

### Nonlinear boundary-value problems for systems of ordinary differential equations

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 162–171

We consider nonlinear boundary-value problems (with Noetherian operator in the linear part) for systems of ordinary differential equations in the neighborhood of generating solutions. By using the Lyapunov — Schmidt method, we establish conditions for the existence of solutions of these boundary-value problems and propose iteration algorithms for their construction.

### Multidimensional integral-sum inequalities

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 172–177

We consider functional integral inequalities of the Bellman-Bihari type for discontinuous functions of many variables.

### Linear extensions of dynamical systems on a torus that possess Green-Samoilenko functions

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 178–188

By using Lyapunov functions with alternating signs, we study problems of regularity and weak regularity for some linear extensions of dynamical systems on a torus.

### Methods for the investigation of systems of differential equations with pulse influence

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 189–194

We establish consistency conditions for a system of differential equations with pulse influence and additional conditions. The applicability of approximate methods to problems of this type is justified.

### Stochastic dynamics and Boltzmann hierarchy. I

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 195–210

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.

### On an upper bound for the number of characteristic values of an operator function

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 211–224

We prove a theorem on an upper bound for the number of characteristic values of an operator-valued function that is holomorphic and bounded in a domain. This estimate is similar to the well-known inequality for zeros of a number function that is holomorphic and bounded in a domain. We derive several corollaries of the theorem proved, in particular, a statement on an estimate of the number of characteristic values of polynomial bundles of operators that lie in a given disk.

### The theory of the numerical-analytic method: Achievements and new trends of development. II

Ronto M. I., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 225–243

We analyze results concerning the application of the numerical-analytic method suggested by Samoilenko in 1965 to second-order differential equations.

### Invariant tori of linear countable systems of discrete equations given on an infinite-dimensional torus

Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 244–251

We study the problem of existence and uniqueness of invariant toroidal manifolds of countable systems of linear equations given on an infinite-dimensional torus.

### Hierarchy of the matrix Burgers equations and integrable reductions in the Davey-Stewartson system

Samoilenko V. G., Sidorenko Yu. M.

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 252–263

We investigate integrable reductions in the Davey-Stewartson model and introduce the hierarchy of the matrix Burgers equations. By using the method of nonlocal reductions in linear problems associated with the hierarchy of the Davey-Stewartson-II equations, we establish a nontrivial relation between these equations and a system of matrix Burgers equations. In an explicit form, we present reductions of the Davey-Stewartson-II model that admit linearization.

### Nonlinear differential equations with asymptotically stable solutions

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 264–273

We establish conditions of asymptotic stability of all solutions of the equation \(\frac{{dx}}{{dt}} = A(x)x\) , *t*≥0in a Banach space *E* in the case where σ(*A*(*x*)⊂{λ:Reλ<0}∀*x*∈*E*. We give an example of an equation with unstable solutions.

### Approximation of $\bar {\psi} - integrals$−integrals of periodic functions by Fourier sums (small smoothness). Iof periodic functions by Fourier sums (small smoothness). I

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 274-291

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.

### On the structure of the general solution of a degenerate linear system of second-order differential equations

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 292–298

We establish sufficient conditions for the existence of a general Cauchy-type solution and conditions for the solvability of the Cauchy problem for a system of second-order differential equations.

### Averaging method in multifrequency systems with delay

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 299–303

We justify the averaging method for systems with delay that pass through resonances in the process of evolution. We obtain an estimate of the error of the method that explicitly depends on a small parameter.

### On periodic solutions of nonlinear difference equations in the critical case

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 304–308

We establish conditions for the existence and uniqueness of a periodic solution of one nonlinear difference equation.

### Investigation of invariant sets with random perturbations with the use of the Lyapunov function

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 309–312

We consider invariant sets of the form V(*t, x*) *=* 0, where V(*t, x*) is the Lyapunov function of the corresponding deterministic system.