# Volume 69, № 9, 2017

### Estimation of the generalized Bessel – Struve transform in a certain space of generalized functions

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1155-1165

We investigate the so-called Bessel – Struve transform on certain class of generalized functions called Boehmians. By using different convolution products, we generate the Boehmian spaces, where the extended transform is well defined. We also show that the Bessel – Struve transform of a Boehmian is an isomorphism which is continuous with respect to a certain type of convergence.

### Exact solutions of the nonliear equation $u_{tt} = = a(t) uu_{xx} + b(t) u_x^2 + c(t) u $

Barannik A. F., Barannik A. F., Yuryk I. I.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1180-1186

Ans¨atzes that reduce the equation$u_{tt} = = a(t) uu_{xx} + b(t) u_x^2 + c(t) u $ to a system of two ordinary differential equations are defined. Also it is shown that the problem of constructing exact solutions of the form $u = \mu 1(t)x_2 + \mu 2(t)x\alpha , \alpha \in \bfR$, to this equation, reduces to integrating of a system of linear equations $\mu \prime \prime 1 = \Phi 1(t)\mu 1, \mu \prime \prime 2 = \Phi 2(t)\mu 2$, where $\Phi 1(t)$ and \Phi 2(t) are arbitrary predefined functions.

### Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1187-1197

We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz – Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, where $\alpha (x) \in L^{\infty} (\mathbb{R}^n) is log-Holder continuous both at the origin and at infinity, $\omega = (1+| x| ) \gamma (x)$ with some $\gamma (x) > 0$, and $1/q_1 (x) 1/q_2 (x) = \beta (x)/n$ when $q_1 (x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1 (x)$ satisfies the logarithmic continuity condition both locally and at infinity and $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$.

### Asymptotic representation of solutions of differential equations with rightly varying nonlinearities

Evtukhov V. M., Korepanova Е. S.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1198-1216

The conditions of existence of some types of power-mode solutions of a binomial nonautonomous ordinary differential equation with regularly varying nonlinearities are established.

### Reconstruction of the Sturm – Liouville operator with nonseparated boundary conditions and a spectral parameter in the boundary condition

Ibadzadeh Ch. G.,, Nabiev I.M.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1217-1223

We study the inverse problem for the Sturm – Liouville operator with nonseparated boundary conditions one of which contains a spectral parameter. The uniqueness theorem is presented and sufficient conditions for the solvability of the inverse problem are obtained.

### Points of upper and lower semicontinuity of multivalued functions ..................

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1224-1231

We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely. among other results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping $F : X \times Y \rightarrow \scr K (Z)$ is jointly upper semicontinuous on sets of the from $D \times \{ y_0\}$, where $D$ is a dense G\delta subset of $X$ and $y_0 \in Y$. A similar result is obtained for the joint lower semicontinuity of upper horizontally quasicontinuous mappings. These results improve some known results on the joint continuity of single-valued functions.

### Lie algebras associated with modules over polynomial rings

Petravchuk A. P., Sysak K. Ya.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1232-1241

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $K[x, y]$. The actions of $x$ and $ y$ determine linear operators P and Q on V as a vector space over $\mathbb{K}$. Define the Lie algebra $L_V = K\langle P,Q\rangle \rightthreetimes V$ as the semidirect product of two abelian Lie algebras with the natural action of $\mathbb{K}\langle P,Q\rangle$ on $V$. We show that if $\mathbb{K}[x, y]$-modules $V$ and $W$ are isomorphic or weakly isomorphic, then the corresponding associated Lie algebras $L_V$ and $L_W$ are isomorphic. The converse is not true: we construct two $\mathbb{K}[x, y]$-modules $V$ and $W$ of dimension 4 that are not weakly isomorphic but their associated Lie algebras are isomorphic. We characterize such pairs of $\mathbb{K}[x, y]$-modules of arbitrary dimension over K. We prove that indecomposable modules $V$ and $W$ with $\mathrm{d}\mathrm{i}\mathrm{m}\mathbb{K} V = \mathrm{d}\mathrm{i}\mathrm{m}KW \geq 7$ are weakly isomorphic if and only if their associated Lie algebras $L_V$ and $L_W$ are isomorphic.

### Differential equations with small stochastic summands under the Levy approximating conditions

Nikitin A. V., Samoilenko I. V.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1242-1249

The proposed methods enable us to study a model of stochastic evolution that includes Markov switchings and to identify the diffusion component and big jumps of perturbing process in the limiting equation. Big jumps of this type may describe rare catastrophic events in different applied problems. We consider the case where the perturbation of the system is determined by an impulse process in the nonclassical approximation scheme. Special attention is given to the asymptotic behavior of the generator of the analyzed evolutionary system.

### Total differential

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1250-1256

We present necessary and sufficient conditions for a continuous differential form to be the total differential.

### On q-congruences involving harmonic numbers

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1257-1265

We give some congruences involving $q$-harmonic numbers and alternating $q$-harmonic numbers of order $m$. Some of them are $q$-analogues of several known congruences.

### Mykhailo Pylypovych Kravchuk (27.09.1892 – 09.03.1942), famous ukrainian mathematician (on the 125th anniversary of his birth)

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1265-1269

### Well-posedness of the Dirichlet problem in a cylindrical domain for three-dimensional elliptic equations with degeneration of type and order

Aldashev S. A., Kitaibekov E. T.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1270-1274

The paper shows the unique solvability of the classical Dirichlet problem in cylindrical domain for three-dimensional elliptic equations with degeneration type and order.

### Finite groups with 2pqr elements of the maximal order

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1275-1279

Let $3 < p < q < r$ be odd prime numbers. In this paper, we prove that the finite groups with exactly $2pqr$ elements of maximal order are solvable.

### Descriptive complexity of the sizes of subsets of groups

Banakh T. O., Protasov I. V., Protasova K. D.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1280-1283

We study the Borel complexity of some basic families of subsets of a countable group (large, small, thin, rarefied, etc.) determined by the sizes of their elements. The obtained results are applied to the Czech – Stone compactification $\beta G$ of the group $G$. In particular, it is shown that the closure of the minimal ideal $\beta G$ has the $F_{\sigma \delta}$ type.

### Some holomorphic generalizations of loxodromic functions

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1284-1288

The functional equation of the form $f(qz) = p(z)f(z), z \in C\setminus \{ 0\} , q \in C\setminus \{ 0\} , | q| < 1$ is considered. For certain fixed elementary functions $p(z)$, holomorphic solutions of this equation are found. These solutions are some generalizations of loxodromic functions. Some of solutions are represented via the Schottky – Klein prime function.

### Karamata integral representations for functions generalizing regularly varying functions

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1289-1296

We consider the classes of functions that generalized regularly varying and receive Karamata’s type integral representations for this functions.