# Volume 71, № 9, 2019

### Certain integrals involving ℵ-functions and Laguerre polynomials

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1159-1175

UDC 517.5

Our aim is to establish certain new integral formulas involving $\aleph$ -functions associated with Laguerre-type polynomials. We
also show how the main results presented in paper are general by demonstrating 18 integral formulas that involve simpler
known functions, e.g., the generalized hypergeometric function $_pF_q$ in a fairly systematic way.

### Numerical method for the solution of linear boundary-value problem for integrodifferential equations based on spline approximations

Assanova A. T., Bakirova E. A., Iskakova N. B.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1176-1191

UDC 517.642

We propose a numerical method for the solution of linear boundary-value problem for system of integrodifferential equations.
This method is based on the approximation of the integral term by a cubic spline and reduction of the original problem to
a linear boundary-value problem for a system of loaded differential equations. We also propose new algorithms for finding
the numerical solution and a method for the construction of approximate solution to the approximating boundary-value
problem.

### On spliced sequences and the density of points with respect to a matrix constructed by using a weight function

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1192-1207

UDC 517.5

Following the line of investigation in
[Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42],
for $y\in\mathbb{R}$ and a sequence $x=(x_n)\in\ell^\infty$ we define
а new notion of density $\delta_{g}$ with respect to a weight function $g$ of indices of the elements $x_n$ close to $y,$ where $ g\colon \mathbb{N}\to[
{0,\infty })$ is such that $ g(n) \to \infty $ and
$ n / g(n) \nrightarrow 0.$
We present the relationships between the densities
$\delta_{g}$ of indices of $(x_n)$ and the variation of the Ces\`aro-limit of $(x_n).$
Our main result states that if the set of limit points of $(x_n)$
is countable and $\delta_g(y)$ exists for any $y\in\mathbb{R},$ then $ \lim\nolimits_{n\to\infty}
\dfrac{1}{g(n)}\displaystyle\sum\nolimits_{i=1}^{n} x_i = \sum\nolimits_{y\in\mathbb{R}}\delta_g(y)\cdot y ,$ which is an extended and much more general form of the ``natural density version of the Osikiewicz theorem''.
Note that in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42],
the regularity of the matrix was used in the entire investigation, whereas in the present paper the investigation is actually performed with respect to a special type of matrix, which is not necessarily regular.

### Nonlocal multipoint (in time) problem for evolutionary pseudodifferential equations with analytic symbols in spaces of type $W$

Horodets’kyi V. V., Martynyuk O. V., Petryshyn R. I.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1208-1226

UDC 517.956

The correct solvability of a nonlocal multipoint (in time) problem for the evolution equations with differentiation operators of infinite order is established for an infinite time interval and an initial function, which is an element of the space of generalized functions of the type $ W'$.
The properties of the fundamental solution and the behavior of the solution as $ t \to + \infty $ are investigated.

### On meromorphic solutions of the systems of linear differential equations with meromorphic coefficients

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1227-1240

UDC 517.925.7

For systems of linear differential equations whose dimension can be decreased, we establish estimates for the growth of meromorphic vector solutions. As an essentially new feature, we can mention the fact that no additional restrictions are imposed on the order of growth of coefficients of the system.

### The radical formula for noncommutative rings

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1241-1248

UDC 512.5

We determine some classes of left modules satisfying the radical formula in a noncommutative ring.
We also show that, under a certain condition, a finitely generated module over an $HNP$-ring (the generalization of Dedekind domain) both satisfies the radical formula and can be decomposed into a direct sum of torsion modules and extending modules.

### Asyptotic bounds for the solutions of functional and differentialfunctional equations with constant and linear delays

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1249-1270

UDC 517.929

We establish asymptotic bounds for the solutions of functional and differential-functional equations with linearly transformed arguments and constant delays.

### A model of dynamical system for the attainment of consensus

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1271-1283

UDC 517.9 + 316.4

We propose a mathematical model for the diffusion of opinions, which eventually lead to the attainment of the state of consensus.
The theory of conflict dynamical systems with attractive interaction is used for the construction of the model.
The behavior of the model in the case of making binary decisions is described in detail and the behavior of trajectories in the decision-making model with many alternative positions is investigated.

### Method of local linear approximation for nonlinear discrete equations

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1284-1296

UDC 517.988.6

We obtain new conditions for the existence of bounded solutions of nonlinear discrete equations with application of the local linear approximation of these equations.