# Volume 68, № 12, 2016

### Exponentially convergent method for an abstract nonlocal problem with integral nonlinearity

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1587-1597

We consider a problem for the first-order differential equation with unbounded operator coefficient in Banach space and a nonlinear integral nonlocal condition. An exponentially convergent method for the numerical solution of this problem is proposed and justified under assumption that the indicated operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein equation, discretization of this equation by the collocation method, and its subsequent solution by the fixed-point iteration method. Each iteration of the method involves the Sinc-based numerical evaluation of the exponential operator function represented by the Dunford – Cauchy integral over the hyperbola enveloping the spectrum of A. The integral part of the nonlocal condition is approximated by using the Clenshaw – Curtis quadrature formula.

### On generalized statistical and ideal convergence of metric-valued sequences

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1598-1606

We consider the notion of generalized density, namely, natural density of weight g recently introduced in [Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. –147, № 1. – P. 97 – 115] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results are also obtained in more general form using the notion of ideals. The entire investigation is performed in the setting of general metric spaces extending the recent results of Kucukaslan M., Deger U., Dovgoshey O. On statistical convergence of metric valued sequences, see Ukr. Math. J. – 2014. – 66, № 5. – P. 712 – 720.

### Numerical interpretation of the Gurov – Reshetnyak inequality on the real line

Didenko V. D., Korenovskii A. A., Tuah N. J.

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1607-1619

We find the “norm” of a power function in the Gurov – Reshetnyak class on the real line. Moreover, as a result of numerical experiments, we establish a lower bound for the norm of the operator of even extension from the semiaxis onto the entire real line in the Gurov – Reshetnyak class.

### On the uniqueness of representation by linear superpositions

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1620-1628

Let $Q$ be a set such that every function on $Q$ can be represented by linear superpositions. This representation is, in general, not unique. However, for some sets, it may be unique provided that the initial values of the representing functions are prescribed at some point of $Q$. We study the properties of these sets.

### On resolvent of the Levy process with matrix-exponential distribution of jumps

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1629-1640

We consider the representations of resolvent for a Levy process whose jumps have a matrix-exponential distribution.

### Realization of exact three-point difference schemes for nonlinear boundary-value problems on the semiaxis

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1641-1656

New algorithmic realization of exact three-point difference schemes via the three-point difference schemes of high order of accuracy is proposed for the numerical solution of boundary-value problems for systems of nonlinear ordinary differential equations on the semiaxis. We study the existence and uniqueness of the solution of three-point difference schemes and estimatie the rate of convergence. The results of numerical experiments are also presented.

### Some properties of the moduli of continuity of periodic functions in metric spaces

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1657-1664

Let $L_0(T)$) be the set of real-valued periodic measurable functions, let $\Psi : R^{+} \rightarrow R^{+}$ be the modulus of continuity, and let $$L_{\Psi} \equiv L_{\Psi} (T) = \left\{ f \in L_0(T) : \| f\| _{\Psi} := \frac1{2\pi} \int_T \Psi (| f(x)| )dx < \infty \right\}.$$ We study the properties of multiple modules of continuity for the functions from $L_{\Psi}$.

### Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1665-1682

We give a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary surfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of a harmonic function and its normal derivative on one of the two closed parts of the boundary according to the information about these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in the input data may completely destroy the procedure of finding the approximate solution. We describe and present a results for a procedure of regularization aimed at the stable determination of the required quantities based on the representation of the solution to the Cauchy problem in the form a single-layer potential. For the given data, this representation yields a system of boundary integral equations for two unknown densities. We establish the existence and uniqueness of these densities and propose a method for the numerical discretization in two- and three-dimensional domains. We also consider the cases of simply connected domains of solution and unbounded domains. Numerical examples are presented both for two- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracy with relatively small amount of computations.

### Perturbation and error analyses of partitioned LU factorization for block tridiagonal linear systems

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1683-1695

The perturbation and backward error analyses of the partitioned LU factorization for block tridiagonal matrices are presented. Moreover, we consider the perturbation bounds for the partitioned LU factorization for block tridiagonal linear systems. Finally, numerical examples are given to verify our results.

### Application of Faber polynomials to approximate solution of the Riemann problem

Pylak D., Sheshko M. A., Wójcik P.

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1696-1704

In the paper, Faber polynomials are used to derive an approximate solution of the Riemann problem on a Lyapunov curve. Moreover, an estimation of the error of the approximated solution is presented and proved.

### Order estimates for the approximative characteristics of functions from the classes $S_{p,θ}^{Ω} B(R^d)$ with a given majorant of generalized mixed modules of smoothness in the uniform metric

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1705-1714

We establish the exact-order estimates of approximation for the classes $S_{p,θ}^{Ω} B$ of functions of several variables defined on $R^d$ in the norm of $L_{\infty} (R^d)$ by entire functions of exponential type with supports of their Fourier transforms in the sets generated by the level surfaces of a function $\Omega$.

### Volodymyr Leonidovych Makarov (on his 75th birthday)

Korolyuk V. S., Lukovsky I. O., Nesterenko B. B., Nikitin A. G., Perestyuk N. A., Samoilenko A. M., Solodkii S. G., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1715-1717

### Some conditions for cyclic chief factors of finite groups

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1718-1722

A subgroup $H$ of a finite group $G$ is called $\scrM$ -supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H_1B$ is a proper subgroup of $G$ for every maximal subgroup $H_1$ of $H$. The main purpose of the paper is to study the influence of $\scrM$ -supplemented subgroups on the cyclic chief factors of finite groups.

### Index of volume 68 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1723-1728