# Volume 69, № 1, 2017

### Regularity of the mild solution of a parabolic equation with stochastic measure

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 3-16

We study a stochastic parabolic differential equation driven by a general stochastic measure. The existence, uniqueness, and Holder regularity of the mild solution are established.

### Haar’s condition and joint polynomiality of separate polynomial functions

Kosovan V. M., Maslyuchenko V. K., Voloshyn H. A.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 17-27

For systems of functions $F = \{ f_n \in K^X : n \in N\}$ and $G = \{ g_n \in K^Y : n \in N\}$ we consider an $F$ -polynomial $f = \sum^n_{k=1}\lambda_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1} \lambda_{k,j} f_k \otimes g_j$, and an $F \otimes G$-polynomial $(f_k\otimes g_j)(x, y) = = f_k(x)g_j(y)$, where $(f_k\otimes g_j)(x, y) = f_k(x)g_j(y)$. By using the well-known Haar’s condition from the approximation theory we study the following question: under what assumptions every function $h : X \times Y \rightarrow K$, such that all $x$-sections $h^x = h(x, \cdot )$ are $G$-polynomials and all $y$-sections $h_y = h(\cdot , y)$ are $F$ -polynomials, is an $F \otimes G$-polynomialy. A similar problem is investigated for functions of $n$ variables.

### Asymptotic properties of $M$-estimates of parameters in a nonlinear regression model with discrete time and singular spectrum

Ivanov O. V., Orlovs’kyi I. V.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 28-51

We study a nonlinear regression model with discrete time and observations errors whose spectrum is singular. Sufficient conditions are obtained for the consistency, asymptotic uniqueness and asymptotic normality of the $M$-estimates of the unknown parameters.

### Darboux problem for the generalized Euler – Poisson – Darboux equation

Ismoilov A. I., Mamanazarov A. O., Urinov A. K.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 52-70

In the article Euler – Poisson – Darboux equation was considered in the characteristic triangle and Darboux problem was investigated. The solution of the problem was found by Riemann’s method. Theorems on the existence and uniqueness of the solution were proved.

### The properties on differential-difference polynomials

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 71-82

The main aim of this paper is to improve some classical results on the distribution of zeros for differential polynomials and differential-difference polynomials. We present some results on the distribution of zeros of $[f(z)^nf(z + c)]^{(k)} \alpha (z)$ and $[f(z)^n(f(z + c) f(z))]^{(k)} \alpha (z)$ and give some examples to show that the results are best possible in a certain sense.

### Continuity of the solutions of one-dimensional boundary-value problems in Hölder spaces with respect to the parameter

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 83-91

We introduce the most general class of linear boundary-value problems for systems of ordinary differential equations of order $r \geq 2$ whose solutions belong to the complex Hölder space $C^{n+r,\alpha} ([a, b])$, where $n \in Z_{+},\; 0 < \alpha \leq 1$ и $[a, b] \subset R$, and $[a, b] \subset R$. We establish sufficient conditions under which the solutions of these problems continuously depend on the parameter in the H¨older space $C^{n+r,\alpha} ([a, b])$.

### The Nehari manifold approach for a $p(x)$ -Laplacian problem with nonlinear boundary conditions

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 92-103

We consider a class of $p(x)$-Laplacian equations that involve nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method of Drabek and Pohozaev, together with the recent idea from Brown and Wu.

### A generalization of semiperfect modules

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 104-112

A module $M$ is called radical semiperfect, if $\frac MN$ has a projective cover whenever $\mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M$. We study various properties of these modules. It is proved that every left $R$-module is radical semiperfect if and only if $R$ is left perfect. Moreover, radical lifting modules are defined as a generalization of lifting modules.

### Bezout rings of stable ranк 1.5 and the decomposition of a complete linear group into its multiple subgroups

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 113-120

A ring $R$ is called a ring of stable rank 1.5 if, for any triple $a, b, c \in R, c \not = 0$, such that $aR + bR + cR = R$, there exists $r \in R$ such that $(a + br)R + cR = R$. It is proved that a commutative Bezout domain has a stable rank 1.5 if and only if every invertible matrix $A$ can be represented in the form $A = HLU$, where $L, U$ are elements of the groups of lower and upper unitriangular matrices (triangular matrices with 1 on the diagonal) and the matrix $H$ belongs to the group $$\bf{G} \Phi = \{ H \in \mathrm{G}\mathrm{L}n(R) | \exists H_1 \in \mathrm{G}\mathrm{L}_n(R) : H\Phi = \Phi H_1\},$$ where $\Phi = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g} (\varphi 1, \varphi 2,..., \varphi n), \varphi 1| \varphi 2| ... | \varphi n, \varphi n \not = 0$.

### Nonlocal problem with integral conditions for a high-order hyperbolic equation

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 121-131

We study the solvability of a nonlocal problem with integral conditions for a high-order hyperbolic equation with predominant mixed derivative. The posed problem is reduced to the integral equation and the existence of its solution is proved by the help of a priori estimates.

### On the 100th birthday of outstanding mathematician and mechanic Yurii Oleksiiovych Mytropol’s’kyi (03.01.1917 – 14.06.2008)

Berezansky Yu. M., Boichuk A. A., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Parasyuk I. O., Perestyuk N. A., Samoilenko A. M., Sharkovsky O. M.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 132-144