# Volume 70, № 11, 2018

### Global existence results for neutral functional differential inclusions with state-dependent delay

Alaidarous E., Benchohra M., Medjadj I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1443-1456

We consider the existence of global solutions for a class of neutral functional differential inclusions with state-dependent delay. The proof of the main result is based on the semigroup theory and the Bohnenblust – Karlin fixed point theorem.

### Fredholm one-dimensional boundary-value problems with parameter in Sobolev spaces

Atlasiuk O. M., Mikhailets V. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1457-1465

For systems of linear differential equations on a compact interval, we investigate the dependence on a parameter $\varepsilon$ of the solutions to boundary-value problems in the Sobolev spaces $W^n_{\infty}$. We obtain a constructive criterion of the continuous dependence of the solutions of these problems on the parameter $\varepsilon$ for $\varepsilon = 0$. The degree of convergence of these solutions is established.

### Mapping properties for convolution involving hypergeometric series

Aouf M. K., Mostafa A. O., Zayed H. M.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1466-1475

We introduce sufficient conditions of (Gaussian) hypergeometric functions to be in a subclass of analytic functions. In addition, we investigate several mapping properties for convolution and integral convolution involving hypergeometric functions.

### Infinite-dimensional version of the Friedrichs inequality

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1476-1483

Two infinite-dimensional versions of the classical Friedrichs inequality are proposed.

### Free products of $n$-tuple semigroups

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1484-1498

We construct a free product of arbitrary n-tuple semigroups, introduce the notion of $n$-band of $n$-tuple semigroups and, in terms of this notion, describe the structure of the free product. We also construct a free commutative $n$-tuple semigroup of an arbitrary rank and characterize one-generated free commutative $n$-tuple semigroups. Moreover, we describe the least commutative congruence on a free $n$-tuple semigroup and establish that the semigroups of the constructed free commutative $n$-tuple semigroup are isomorphic and its automorphism group is isomorphic to the symmetric group.

### Subclass of $k$-uniformly starlike functions defined by symmetric $q$-derivative operator

Altinkaya S., Kanas S., Yal¸cin S.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1499-1510

The theory of $q$-analogs is frequently encountered in numerous areas, including fractals and dynamical systems. The $q$-derivatives and $q$-integrals play an important role in the study of $q$-deformed quantum-mechanical simple harmonic oscillators. We define a symmetric $q$-derivative operator and study a new family of univalent functions defined by using this operator. We establish some new relations between the functions satisfying analytic conditions related to conical sections.

### On the solvability of a finite group with $S$-seminormal Schmidt subgroups

Knyagina V. N., Monakhov V. S., Zubei E. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1511-1518

A finite nonnilpotent group is called a Schmidt group if all its proper subgroups are nilpotent. A subgroup $A$ is called $S$-seminormal (or $SS$-permutable) in a finite group $G$ if there is a subgroup B such that $G = AB$ and $A$ is permutable with every Sylow subgroup of B. We establish the criteria of solvability and $\pi$ -solvability of finite groups in which some Schmidt subgroups are $S$-seminormal. In particular, we prove the solvability of a finite group in which all supersoluble Schmidt subgroups of even order are $S$-seminormal.

### One multivalued discrete system and its properties

Komleva T. A., Plotnikov A. V., Plotnikova L. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1519-1524

We consider one multivalued discrete system and study its properties and the existence of its solution.

### Problem of shadow in the Lobachevski space

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1525-1532

We consider the problem of shadow in a hyperbolic space. This problem can be regarded as a problem of finding conditions guaranteeing that points belong to a generalized convex hull of the family of balls.

### A class of double crossed biproducts

Dong L. H., Li H. Y., Ma T. S.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1533-1540

Let $H$ be a bialgebra, let $A$ be an algebra and a left $H$-comodule coalgebra, let $B$ be an algebra and a right $H$-comodule coalgebra. Also let $f : H \otimes H \rightarrow A \otimes H, R : H \otimes A \rightarrow A \otimes H$, and $T : B \otimes H \rightarrow H \otimes B$ be linear maps. We present necessary and sufficient conditions for the one-sided Brzezi´nski’s crossed product algebra $A\#^f_RH_T\#B$ and the two-sided smash coproduct coalgebra $A \times H \times B$ to form a bialgebra, which generalizes the main results from [On Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct [Double-bosonization of braided groups and the construction of $U_q(g)$ // Math. Proc. Cambridge Phil. Soc. – 1999. – 125, № 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases.

### Evaluation of the weighted level of damping of bounded disturbances in descriptor systems

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1541-1552

We establish necessary and sufficient conditions for the validity of the upper bounds for the performance criteria of linear descriptor systems characterizing the weighted damping level of external and initial disturbances. The verification of these conditions is reduced to solving matrix equations and inequalities. The main statements are formulated with an aim of their subsequent application in the problems of robust stabilization and in the $H_{\infty}$ -optimization problems for descriptor control systems.

### On the lower estimate of the distortion of distance for one class of mappings

Markish A. A., Salimov R. R., Sevost'yanov E. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1553-1562

We study the behavior of one class of mappings with finite distortion in a neighborhood of the origin. Under certain conditions imposed on the characteristic of quasiconformality, we establish a lower estimate for the distortion of distance under mappings of the indicated kind.

### Procedure of stochastic approximation for the diffusion process with semi-Markov switchings

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1563-1570

We obtain sufficient conditions for the convergence of the procedure of stochastic approximation for the diffusion process in the case of a uniformly ergodic semi-Markov process of switchings of the regression function with the use of a small parameter in the scheme of series.

### A remark on John – Nirenberg theorem for martingales

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1571-1577

This paper is mainly devoted to establishing an extension of the John – Nirenberg theorem for martingales, more precisely, let $1 < p < \infty$ and $0 < q < \infty$. If the stochastic basis $(\scr {F_n})_n\geq 0$ is regular, then $BMO_{p,q} = BMO_1$ with the equivalent norms. Our method is to use a new atomic decomposition construction of the martingale Hardy space.

### On one theorem of G. Freud

Dadashova I. B., Mamedkhanov J. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1578-1584

We obtain a generalization and improvement of the G. Freud theorem on arbitrary Jordan curves in the complex plane.