# Volume 70, № 2, 2018

### Application of the Faber polynomials in proving combinatorial identities

Abdullayev F. G., Imash kyzy M., Savchuk V. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 151-164

We study the possibility of application of the Faber polynomials in proving some combinatorial identities. It is shown that the coefficients of Faber polynomials of mutually inverse conformal mappings generate a pair of mutually invertible relations. We prove two identities relating the coefficients of Faber polynomials and the coefficients of Laurent expansions of the corresponding conformal mappings. Some examples are presented.

### On the rigidity of rank gradient in a group of intermediate growth

Grigorchuk R. I., Kravchenko R.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 165-176

We introduce and investigate a rigidity property of rank gradient for an example of a group $\scr G$ of intermediate growth constructed by the first author in [Grigorcuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal. i Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. It is shown that $\scr G$ is normally $(f, g)$-RG rigid, where$ f(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n)$ and $g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{l}\mathrm{o}\mathrm{g}(n))$.

### Almost periodic solutions of Lotka – Volterra systems with diffusion and impulsive action

Dvornyk A. V., Struk O. O., Tkachenko V. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 177-192

We establish sufficient conditions for the existence and asymptotic stability of positive piecewise continuous almost periodic solutions for the Lotka –Volterra systems of differential equations with diffusion and impulsive action.

### Linear and nonlinear heat equations on a $p$ -adic ball

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 193-205

We study the Vladimirov fractional differentiation operator $D^{\alpha}_N,\; \alpha > 0,\; N \in Z$, on a $p$-adic ball B$B_N = \{ x \in Q_p : | x|_p \leq p^N\}$. To its known interpretations via the restriction of a similar operator to $Q_p$ and via a certain stochastic process on $B_N$, we add an interpretation as a pseudodifferential operator in terms of the Pontryagin duality on the additive group of $B_N$. We investigate the Green function of $D^{\alpha}_N$ and a nonlinear equation on $B_N$, an analog of the classical equation of porous medium.

### Application of the method of averaging to the problems of optimal control over functional-differential equations

Koval’chuk T. V., Kravets V. I., Mohyl'ova V. V., Stanzhitskii A. N.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 206-215

We study the application of the method of averaging to the problems of optimal control over functional-differential equations. The procedure of averaging allows us to replace the original problem with the problem of optimal control by a system of ordinary differential equations. It is proved that the optimal control over the averaged problem is almost optimal for the exact problem. The optimal control problems are investigated on finite and infinite horizons.

### Limit theorems for the solutions of boundary-value problems

Mikhailets V. A., Pelekhata O. B., Reva N. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 216-223

We study the uniform limit with respect to a parameter for the solutions of a sequence of general boundary-value problems for systems of linear ordinary differential equations of any order on a finite interval. An essential generalization of the Kiguradze theorem (1987) for these problems is obtained.

### Kolmogorov widths and bilinear approximations of the classes of periodic functions of one and many variables

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 224-235

We obtain the exact order estimates for the Kolmogorov widths of the classes $W^g_p$ of periodic functions of one variable generated by the integral operators with kernels $g(x, y)$ from the Nikol’skii – Besov classes $B^r_{p,\theta}$. We also study the behavior of bilinear approximations to the classes $W^r_{p,\alpha}$ of periodic multivariate functions with bounded mixed derivative in the spaces $L_{q_1,q_2}$ for some relations between the parameters $r_1, p, q_1, q_2$.

### Asymptotic $Σ$-solutions to singularly perturbed Benjamin – Bona – Mahony equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 236-254

We study the problem of construction of asymptotic $\Sigma$ -solutions to the singularly perturbed Benjamin – Bona – Mahony equation with variable coefficients. An algorithm for the construction of solutions is described. We determine main and first terms of the asymptotic solution. The theorems on the accuracy with which the indicated asymptotic solution satisfies the considered equation are also proved.

### On equations with generalized periodic right-hand side

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 255-279

Periodic solutions are studied for second-order differential equations with generalized forcing. Analytical bifurcation results are derived with application to forced harmonic and Duffing oscillators.

### Least-squares method in the theory of matrix differential-algebraic boundary-value problems

Chuiko S. M., Dzyuba M. V., Nesmelova O. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 280-292

We use the scheme of the classical least-squares method for the construction of approximate pseudosolutions of a linear matrix boundary-value problem for a system of differential-algebraic equations.

### Differential-geometric structure and the Lax – Sato integrability of a class of dispersionless heavenly type equations

Hentosh О. Ye., Pritula N. N., Prykarpatsky Ya. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 293-297

This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.