# Volume 61, № 5, 2009

### Construction of the solutions of boundary-value problems for the laplace equation in domains of revolution with edged boundary

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 579-595

A method for the construction of high-precision approximate solutions of boundary-value problems for the Laplace equation in domains with corner points is proposed. We consider boundary-value problems for the three-dimensional Laplace equation in domains in the form of bodies of revolution whose meridional section has corner points. The solutions of the problems are constructed by using variational methods. For the numerical realization of these methods, we construct special solutions of the Laplace equation with singularities (or with singularities of their partial derivatives) on a certain ray originating at a corner point and directed outside the domain. To illustrate the proposed method, we construct the solutions of the Neumann problem and the problem of natural oscillations of ideal liquid in a spherical cavity.

### Inequalities for the inner radii of nonoverlapping domains and open sets

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 596-610

We generalize some classical results in the theory of extreme problems for nonoverlapping domains.

### Description of posets critical with respect to the nonnegativity of the quadratic Tits form

Bondarenko V. M., Stepochkina M. V.

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 611-624

We present the complete description of finite posets whose Tits form is not nonnegative but all proper subsets of which have nonnegative Tits forms. A similar result for positive forms was obtained by the authors earlier.

### On the instantaneous shrinking of the support of a solution to the Cauchy problem for an anisotropic parabolic equation

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 625-640

We study the phenomenon of instantaneous shrinking of the support of solution to the Cauchy problem for the parabolic equation with anisotropic degeneration, double nonlinearity, and strong absorption. In terms of the behavior of locally integrable initial data, we formulate necessary and sufficient conditions for the realization of instantaneous shrinking and establish the exact (in order) bilateral estimates for the size of the support of solution.

### On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems

Denisenko V. S., Martynyuk A. A., Slyn'ko V. I.

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 641-649

We propose a general principle of comparison for stability-preserving mappings and establish sufficient conditions of stability for the Takagi – Sugeno continuous fuzzy systems.

### Correct solvability of Solonnikov–Eidel’man parabolic initial-value problems

Ivasyshen S. D., Ivasyuk H. P.

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 650-671

We consider initial-value problems for a new class of systems of equations that combine the structures of Solonnikov parabolic systems and Eidel’man parabolic systems. We prove a theorem on the correct solvability of these problems in Hölder spaces of rapidly increasing functions and obtain an estimate for the norms of solutions via the corresponding norms of the right-hand sides of the problem. For the correctness of this estimate, the condition of the parabolicity of the system is not only sufficient but also necessary.

### Methods for the solution of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions

Luchka A. Y., Nesterenko O. B.

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 672-679

We establish conditions for the existence of solutions of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions. We also substantiate the applicability of iterative and projection-iterative methods for the solution of these problems.

### On one modulus inequality for mappings with finite length distortion

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 680-688

The Väisälä inequality, which is well known in the theory of quasilinear mappings, is extended to the class of mappings with finite length distortion.

### Kirkwood–Salsburg equation for a quantum lattice system of oscillators with many-particle interaction potentials

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 689-700

For a Gibbs system of one-dimensional quantum oscillators on a *d*-dimensional hypercubic lattice interacting via superstable pair and many-particle potentials of finite range, we prove the existence of a solution of the (lattice) Kirkwood–Salsburg equation for correlation functions depending on the Wiener paths. Some many-particle potentials may be nonpositive.

### Quintuplets of orthoprojectors associated by a linear relation

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 701-710

We consider the equation $α_1 P_1 + α_2 P_2 + … α_n P_n = I$ over orthoprojectors $P_1, … ,P_n$ in a Hilbert space. We show that the set of real parameters $(α_1, … α_n)$ for which there exists a solution of this equation in orthoprojectors contains an open set from $ℝ^5$.

### Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 711-715

We prove that $\max |p′(x)|$, where $p$ runs over the set of all algebraic polynomials of degree not higher than $n ≥ 3$ bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) for all $x ∈ (−1, 1)$ such that \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \).

### Asymptotic periodicity of trajectories of an interval

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 716-720

We consider dynamical systems generated by continuous mappings of an interval *I* into itself. We prove that the trajectory of an interval *J* ⊂ *I* is asymptotically periodic if and only if *J* contains an asymptotically periodic point.