# Volume 71, № 10, 2019

### A problem of extreme decomposition of the complex plane with free poles

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1299-1320

UDC 517.54

We study a problem of nonoverlapping domains
with free poles on radial systems.
Our main results strengthen and generalize several known
results obtained in the investigation of this problem.

### A criterion of solvability of resonant equations and construction of their solutions

Boichuk О. A., Feruk V. A., Makarov V. L.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1321-1330

UDC 517.983

We establish conditions for the existence and determine the general structure of solutions of resonant and iterative equations in a Banach
space and their algorithmic realization.

### Generalized moment representations and multivariate multipoint Padé-type approximants

Chernetska L. O., Holub A. P., Pozharskiy O. A.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1331-1346

UDC 517.53

Dzyadyk's method of generalized moment representations is used to construct and study bivariate two-point Pad\'e-type approximants.

### On a Frankl-type boundary-value problem for a mixed-type degenerating equation

Islomov B. I., Ochilova N. K., Sadarangani K. S.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1347-1359

UDC 517.9

We investigate the existence and uniqueness of solutions for an analog of the Frankl-type boundary-value problem for a parabolic-hyperbolic-type equation.
The uniqueness of solution is proved by using the extreme principle and the existence is proved by the method of integral equations.

### Hyperbolic systems in Gelfand and Shilov spaces

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1360-1373

UDC 517.956.32, 517.955.2

For systems hyperbolic in Shilov's sense with time-dependent coefficients, the properties of the Green function are studied in the $S$-type spaces.
For systems of this kind in the indicated spaces, we establish the correct solvability of the Cauchy problem.
It is shown that, for each $\beta>1,$ the space ${S_0^\beta}'$ of Gelfand and Shilov distributions is the class of well-posedness of this problem.

### Robust stabilization and weighted suppression of bounded disturbances in descriptor control systems

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1374-1388

UDC 517.925.51; 681.5.03

We establish necessary and sufficient conditions for the existence of dynamic regulators guaranteeing the prescribed estimation of the weighted damping level of bounded disturbances and the asymptotic stability of linear descriptor systems.
An algorithm of construction of these regulators in the problems of robust stabilization and generalized $H_\infty$-optimization is proposed for the descriptor systems with controlled and observed outputs.
The main computational procedures of the algorithm are reduced to the solution of linear matrix inequalities with additional rank restrictions.
The efficiency of the algorithm is demonstrated with the help of an illustrative example of descriptor stabilization system with bounded disturbances.

### A class of meromorphic Bazilevič-type functions defined by a differential operator

Ahmad Q. Z., Arif M., Khan N., Noor K. I., Orhan H.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1389-1404

UDC 517.54

We define a new subclass of meromorphic Bazileviˇc-type functions by using a differential operator. We study some
interesting properties, such as the arc length, the growth of coefficients, and the integral representation of functions from
this class.

### Approximative characteristics of the Nikol’skii – Besov classes of functions $S_{1, θ}^r B(\mathbb{R}^d)$

Radchenko O. Ya., Yanchenko S. Ya.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1405-1421

UDC 517.51

We establish the exact-order estimates for the approximation of the classes $S^{\boldsymbol{r}}_{1,\theta}B \left(\mathbb{R}^d\right)$ by entire functions of exponential type with supports of their Fourier transforms lying in a step hyperbolic cross.
The error of approximation is estimated in the metric of the Lebesgue space $L_q\left(\mathbb{R}^d\right),\; 1 < q \leq \infty.$

### Second order parallel tensors on $S$ -manifolds and semi-parallel hypersurfaces of $S$ -space forms

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1422-1429

UDC 515.12

We study a second order parallel symmetric tensor in an $\mathcal{S}$-manifold and we deduce that there is no semi-parallel hypersurface in $\mathcal{S}$-space forms $\widetilde{M}^{2n+s}(c)$
with $c\neq s.$

### Ruin probabilities for risk models with constant interest

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1430-1434

UDC 519.21

We consider continuous-time risk models with $m$-dependent claim sizes and constant interest rate. Under some special conditions, we obtain the upper bound for the infinite-time ruin probability. Our approach is based on the martingale methods.

### Local nearrings with multiplicative Shmidt group

Raievska I. Yu., Raievska M. Yu.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1435-1440

UDC 517.6

We propose a classification of finite local nearrings with multiplicative Shmidt group.
Moreover, it is shown that there are no nearrings with identity
on the Shmidt groups.