# Volume 49, № 2, 1997

### On extremal problems for symmetric disjoint domains

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 179–185

We study two extremal problems for the product of powers of conformal radii of symmetric disjoint domains.

### Problem with nonlocal conditions for weakly nonlinear hyperbolic equations

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 186–195

For weakly nonlinear hyperbolic equations of order *n, n*≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem.

### Phase transition in an exactly solvable model of interacting bosons

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 196–205

In the formalism of the grand canonical ensemble, we study a model system of a lattice Bose gas with repulsive hard-core interaction on a perfect graph. We show that the corresponding ideal system may undergo a phase transition (Bose-Einstein condensation). For a system of interacting particles, we obtain an explicit expression for pressure in the thermodynamic limit. The analysis of this expression demonstrates that the phase transition does not take place in the indicated system.

### Mean oscillations and the convergence of Poisson integrals

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 206–222

We establish conditions for mean oscillations of a periodic summable function under which the summability of its Fourier series (conjugate series) by the Abel-Poisson method at a given point implies the convergence of Steklov means (the existence of the conjugate function) at the indicated point. Similar results are also obtained for the Poisson integral in ℝ_{+}^{n+1}.

### Periodic solutions of systems of differential equations with random right-hand sides

Danilov V. Ya., Martynyuk D. I., Stanzhitskii A. N.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 223–227

We prove a theorem on the existence of periodic solutions of a system of differential equations with random right-hand sides and small parameter of the form *dx/dt=εX(t, x, ξ(t))* in a neighborhood of the equilibrium state of the averaged deterministic system *dx/dt*=ε*X* _{0}(*t*).

### Potential fields with axial symmetry and algebras of monogenic functions of vector variables. III

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 228–243

We obtain new representations of the potential and flow function of three-dimensional potential solenoidal fields with axial symmetry, study principal algebraic analytic properties of monogenic functions of vector variables with values in an infinite-dimensional Banach algebra of even Fourier series, and establish the relationship between these functions and the axially symmetric potential or the Stokes flow function. The developed approach to the description of the indicated fields is an analog of the method of analytic functions in the complex plane used for the description of two-dimensional potential fields.

### Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 244–254

We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an *a priori* estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small.

### On direct decompositions in modules over group rings

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 255–261

In the theory of infinite groups, one of the most important useful generalizations of the classical Maschke theorem is the Kovačs-Newman theorem, which establishes sufficient conditions for the existence of *G*-invariant complements in modules over a periodic group *G* finite over the center. We genralize the Kovačs-Newman theorem to the case of modules over a group ring *KG*, where *K* is a Dedekind domain.

### On a limit theorem for an additive functional on a nonrecurrent Markovian chain

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 262–271

We establish conditions under which the distribution of an additive functional on a nonrecurrent Markovian chain is asymptotically normal.

### Weakly nonlinear boundary-value problems for operator equations with pulse influence

Boichuk О. A., Samoilenko A. M., Zhuravlev V. F.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 272–288

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle.

### On finite-dimensional approximation of solutions of ill-posed problems

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 289–295

We show that the modified method for finite-dimensional approximation of solutions of Fredholm integral equations of the first kind presented in this paper is more economical than traditional methods for finite-dimensional approximation.

### The solvability of a boundary-value periodic problem

Khoma G. P., Petrovskii Ya. B.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 302–308

In the space of functions *B* _{a} ^{3+} ={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T_{3}/2)=g(x, −t)}, we establish that if the condition *aT* _{ 3 } *(2s−1)=4πk, (4πk, a (2s−1))=1*, *k* ∈ ℤ, *s* ∈ ℕ, is satisfied, then the linear problem *u* _{ u } *−a* ^{ 2 } *u* _{ xx } *=g(x, t), u(0, t)=u(π, t)=0, u(x, t+T* _{ 3 })*=u(x, t)*, ℝ^{2}, is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator.

### Existence of the Vejvoda-Shtedry spaces

Botyuk A. O., Khoma G. P., Khoma L. G.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 302–308

We investigate the linear periodic problem *u* _{ tt } *−u* _{ xx } *=F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t)*, ∈ ℝ^{2}, and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry spaces.

### On the problem on periodic solutions of one class of systems of difference equations

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 309–314

The scheme of the Samoilenko numerical-analytic method for finding periodic solutions in the form of a uniformly convergent sequence of periodic functions is applied to one class of difference equations.

### A projective method for the construction of solutions of the problem of normal symmetric oscillations of viscous liquid

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 315–320

We propose a variational formulation of the spectral problem of normal symmetric oscillations of viscous liquid. On the basis of this formulation, we construct a projective method for the determination of real eigenvalues of the problem. We present the numerical realization of this method in the case of a spherical cavity.