# Volume 69, № 8, 2017

### On the unique solvability of a nonlocal boundary-value problem for systems of loaded hyperbolic equations with impulsive actions

Assanova A. T., Bakirova E. A., Kadirbayeva Zh. M.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1011-1029

We consider a nonlocal boundary-value problem with impulsive actions for a system of loaded hyperbolic equations and establish the relationship between the unique solvability of this problem and the unique solvability of a family of two-point boundary-value problems with impulse actions for the system of the loaded ordinary differential equations by method of introduction of additional functions. Sufficient conditions are obtained for the existence of a unique solution to a family two-point boundary-value problems with impulsive effects for the system of loaded ordinary differential equations by using method of parametrization. The algorithms of finding the solutions are constructed. The conditions of unique solvability of the nonlocal boundary-value problem for a system of loaded hyperbolic equations with impulsive actions are established. The numerical realization of the algorithms of the method of parametrization is proposed for the solution of the family of two-point boundary-value problems with impulsive actions for the system of the loaded ordinary differential equations. The results are illustrated by specific examples.

### Surface measures on Banach manifolds with uniform structure

Bogdanskii Yu. V., Moravets’ka E. V.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1030-1048

We propose a method for the construction of associated measures on the surfaces of finite codimension embedded in a Banach manifold with uniform atlas.

### Globally robust stability analysis for stochastic Cohen – Grossberg neural networks with impulse control and time-varying delays

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1049-1060

By constructing suitable Lyapunov functionals, in combination with the matrix-inequality technique, a new simple sufficient linear matrix inequality condition is established for the globally robustly asymptotic stability of the stochastic Cohen – Grossberg neural networks with impulsive control and time-varying delays. This condition contains and improves some previous results from the earlier references.

### On an operator preserving inequalities between polynomials

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1061-1072

Let $P(z)$ be a polynomial of degree n. We consider an operator $D\alpha$ that maps $P(z)$ into $D\alpha P(z) := nP(z)+(\alpha z)P\prime (z)$ and establish some results concerning the estimates of $| D\alpha P(z)| $ on the disk $| z| = R \geq 1$, and thereby obtain extensions and generalizations of a number of well-known polynomial inequalities.

### Exact values of the best (α, β) -approximations of classes of convolutions with kernels that do not increase the number of sign changes

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1073-1083

We obtain the exact values of the best $(\alpha , \beta )$-approximations of the classes $K \ast F$ of periodic functions $K \ast f$ such that $f$ belongs to a given rearrangement-invariant set $F$ and $K$ is $2\pi$ -periodic kernel that do not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points $2k\pi /n$ and $2k\pi /n + h, n \in N, k \in Z, h \in (0, 2\pi /n)$. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

### Property of mixing of continuous classical systems with strong superstable interactions

Rebenko A. L., Tertychnyi M. V.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1084-1095

We consider an infinite system of point particles in $R^d$, interacting via a strong superstable two-body potential $\phi$ of finite range with radius $R$. In the language of correlation functions, we obtain a simple proof of decrease in correlations between two clusters (two groups of variables) the distance between which is larger than the radius of interaction. The established result is true for sufficiently small values of activity of the particles.

### On generalized Besov and Campanato spaces

Alieva L.R., Gahramanova Z. Sh., Rzaev R. M.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1096-1106

We study the generalized Besov spaces and the spaces defined by the conditions imposed on local oscillations of locally summable functions (in the work, these spaces are called generalized Campanato spaces).

### Sardaryan T. G. On the solvability of one system of nonlinear Hammerstein-type integral equations on the semiaxis

Khachatryan Kh. A., Sardaryan T. G., Terjyan Ts. E.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1107-1122

We study the problems of construction of positive summable and bounded solutions for the systems of nonlinear Hammerstein-type integral equations with difference kernels on the semiaxis. The indicated systems have direct applications to the kinetic theory of gases, the theory of radiation transfer in spectral lines, and the theory of nonlinear Ricker competition models for running waves.

### Jensen – Ostrowski inequalities and integration schemes via the Darboux expansion

Cerone P., Dragomir S. S., Kikianty E.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1123-1140

By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen – Ostrowski-type inequalities. The applications to quadrature rules and $f$ -divergence measures (specifically, for higher-order $\chi$ -divergence) are also given.

### Minimal nonsupersolvable and minimum nonnilpotent groups and their role in the study of the structure of finite groups

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1141-1144

We study the influence of minimal nonsupersoluble subgroups and minimal nonnilpotent subgroups (Schmidt subgroups) of a group on its structure.

### Expressing infinite matrices as sums of idempotents

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1145-1147

Let $\scr M_{Cf} (F)$ be the set of all column-finite $N \times N$ matrices over a field $F$. The following problem is studied: what elements of $\scr M_{Cf} (F)$ can be expressed as a sum of idempotents? The result states that every element of $\scr M_{Cf} (F)$ can be represented as the sum of 14 idempotents.

### Stability of fixed points for a class of quasilinear cascades in the space conv $R^n$

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1166-1179

The discrete dynamical systems (cascades) in semilinear metric space of nonempty convex compacts of finite-dimensional space are studied. Using the methods of convex geometry of H. Minkowski and A. D. Alexandrov the sufficient conditions of the stability of the fixed points were established. Under certain restrictions on the mappings generating the cascade, the problem of asymptotic stability of fixed point of the cascade was reduced to localization of the roots of a polynomial inside the unit circle in the complex plane. Examples of cascades in the plane were given.