# Volume 71, № 7, 2019

### The classification of naturally graded Zinbiel algebras with characteristic sequence equal to $(n - p,\, p) $

Adashev J. K., Ladra M., Omirov B. A.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 867-883

UDC 512.5

This work is a continuation of the description of some classes of nilpotent Zinbiel algebras.
We focus on the study of Zinbiel algebras with restrictions imposed on gradation and characteristic sequence.

Namely, we obtain the classification of naturally graded Zinbiel algebras with characteristic sequence equal to $ (n-p, p)$.

### New general solutions of ordinary differential equations and the methods of solving the boundary-value problems

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 884-905

UDC 517.624

New general solutions of ordinary differential equations are introduced and their properties are established. We develop
new methods of solving the boundary-value problems based on the construction and solving of the systems of algebraic
equations for arbitrary vectors of the general solutions. An approach to finding the initial approximation to the solution of
a nonlinear boundary-value problem is proposed.

### Uniqueness of difference-differential polynomials of meromorphic functions

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 906-914

UDC 517.9

We investigate the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions
sharing a small function ignoring multiplicity and obtain some results that extend the results of K. Liu, X. L. Liu, and
T. B. Cao.

### On the approximation of functions from the Hölder class given on a segment by their biharmonic Poisson operators

Zhyhallo K. M., Zhyhallo T. V.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 915-921

UDC 517.5

We obtain the exact equality for the upper bounds of deviations of biharmonic Poisson operators on the Hölder classes of functions continuous on the segment $[-1;1]$.

### On the problem of Frattini duality in the theory of Fitting classes

Nanying Yang, Shuya Zhao, Vorob’ev N. T.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 922-929

UDC 512.542

We determine the application of the Frattini duality to the description of multiple local Fitting classes.
In particular, we establish a necessary and sufficient
condition for the local Fitting class to be a formation.

### Limit theorems for the solutions of linear boundary-value problems for systems of differential equations

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 930-937

UDC 517.927

We establish sufficient conditions for the sequence of solutions
of general boundary-value problems for systems of linear ordinary differential equations of any order on a finite interval to be convergent in the uniform norm.

### On the equicontinuity of families of mappings in the case of variable domains

Sevost'yanov E. A., Skvortsov S. A.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 938-951

UDC 517.5

We study the problem of local behavior of maps in the closure of a domain in the Euclidean space. The equicontinuity of
families of these mappings is established in the case where the mapped domain is not fixed. We separately consider the
domains with bad and good boundaries, as well as the homeomorphisms and maps with branching.

### On the Favard theory without $H$ -classes for differential-functional equations in Banach spaces

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 952-967

UDC 517.937

We obtain necessary and sufficient conditions for the existence and uniqueness of solutions bounded on the real axis with
precompact sets of values of linear almost periodic differential-functional equations in Banach spaces.

### Notes on the lightlike hypersurfaces along spacelike submanifolds

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 968-975

UDC 514.7

In the light of the method of construction of lightlike hypersurfaces along spacelike submanifolds, we give a relation between
the second fundamental form of a spacelike submanifold and the screen second fundamental form of the corresponding
lightlike hypersurface. In addition, we investigate the conditions for a lightlike hypersurface of this kind to be screen
conformal.

### Characterization of weakly Berwald fourth root metrics

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 976-995

UDC 514.7

In recent studies, it is shown that the theory of fourth root metrics plays a very important role in physics, theory of space-time structures, gravitation, and general relativity.
The class of weakly Berwald metrics contains the class of Berwald metrics as a special case.
We establish the necessary and sufficient condition under which the fourth root Finsler space with an $(\alpha, \beta)$-metric is a weakly Berwald space.

### Inequalities for the inner radii of nonorevlapping domains

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 996-1002

UDC 517.54

We consider the problem of maximum of the functional
$$
r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),
$$
where $B_{0},\ldots,B_{n}$, $n\geq 2$, are pairwise disjoint
domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$,
$k=\overline{1,n},$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner
radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to
$a$).
Show that it attains its maximum at a configuration of domains $B_{k}$ and points $a_{k}$ possessing rotational $n$-symmetry.
This problem was solved by Dubinin for $\gamma=1$ and by Kuz’mina for
$0<\gamma< 1$.
Later, Kovalev solved this problem for $n\geqslant5$ under an
additional assumption that the angles between neighboring linear
segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$.
We generalize this problem to the case of arbitrary
locations of the systems of points in the complex plane and obtain some
estimates for the functional for all $n$ and $\gamma\in (1, n]$.

### The order of coexistence of homoclinic trajectories for interval maps

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 1003-1008

UDC 517.9

А nonperiodic trajectory of a discrete dynamical system is called $n$-homoclinic if its $\alpha$- and $\omega$-limit sets coincide and form the same cycle of period $n.$
We prove the statement formulated in that the ordering $1 \triangleright 3 \triangleright 5 \triangleright 7 \triangleright \ldots \triangleright 2 \cdot 1 \triangleright 2 \cdot 3\triangleright 2 \cdot 5 \triangleright \ldots \triangleright 2^2 \cdot 1 \triangleright 2^2 \cdot 3 \triangleright 2^2 \cdot 5 \triangleright \ldots $ determines the coexistence of homoclinic trajectories of one-dimensional systems:
If a one-dimensional dynamical system possesses an $n$-homoclinic trajectory, then it also has an $m$-homoclinic trajectory for each $m$ such that $ n \triangleright m .$
It is also proved that every one-dimensional dynamical system with a cycle of period $ n \neq 2^i $ also possesses an $n$-homoclinic trajectory.