# Volume 70, № 8, 2018

### Derivations of gamma (semi)hyperrings

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1011-1018

Differential $\Gamma$ -(semi)hyperrings are $\Gamma$ -(semi)hyperrings equipped with derivation, which is a linear unary function satisfying the Leibniz product rule.We introduce the notions of derivation and weak derivation on $\Gamma$ -hyperrings and $\Gamma$ -semihyperrings and obtain some important results relating to them in a specific way.

### The structure of fractional spaces generated by the two-dimensional difference operator on the half plane

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1019-1032

We consider a difference operator approximation $A^x_h$ of the differential operator $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$ defined in the region $R^{+} \times R$ with the boundary condition $u(0, x_2) = 0,\; x_2 \in R$. Here, the coefficients $a_{ii}(x), i = 1, 2$, are continuously differentiable, satisfy the uniform ellipticity condition $a^2_{11}(x) + a^2_{22}(x) \geq \delta > 0$. We investigate the structure of the fractional spaces generated by the analyzed difference operator. Theorems on well-posedness in a Holder space of difference elliptic problems are obtained as applications.

### Generalizations of Sherman’s inequality via Fink’s identity and Green’s function

Ivelic Bradanovic S., Latif M. A., Pečarić J. E.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1033-1043

New generalizations of Sherman’s inequality for $n$-convex functions are obtained by using Fink’s identity and Green’s function. By using inequalities for the Chebyshev functional, we establish some new Ostrowski- and Gruss-type inequalities related to these generalizations.

### Entire solutions of one linear implicit differential-difference equation in Banach spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1044-1057

We establish the existence and uniqueness conditions for the solution for the initial problem $Bu\prime (z) = Au(z + h) + f(z),\; z \in C, u(0) = u_0$ in the classes of entire functions of exponential type. Closed linear operators $A$ and $B$ act on Banach spaces and can be degenerate. We also present an example of application of abstract results to partial differential equations.

### Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1058-1071

Among all two-dimensional algebras of the second rank with unity $e$ over the field of complex numbers $C$, we find a semisimple algebra $B_0 = \{ c_1e + c_2\omega: c_k \in C, k = 1, 2\} , \omega^2 = e$, containing bases $(e_1, e_2)$, such that $e^4_1 + 2pe^2_1e^2_2 + e^4_2 = 0$ for every fixed $p > 1$. A domain $\{ (e1, e2)\}$ is described in the explicit form. We construct $B_0$ -valued “analytic” functions $\Phi$ such that their real-valued components satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl(\frac{\partial^4}{\partial x^4} + 2p \frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr) u(x, y) = 0,$$ where $x, y$ are real variables.

### Finite structurally uniform groups and commutative nilsemigroups

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1072-1084

Let $S$ be a finite semigroup. By $\mathrm{S}\mathrm{u}\mathrm{b}(S)$ we denote the lattice of all its subsemigroups. If $A \in \mathrm{S}\mathrm{u}\mathrm{b}(S)$, then by $h(A)$ we denote the height of the subsemigroup $A$ in the lattice $\mathrm{S}\mathrm{u}\mathrm{b}(S)$. A semigroup $S$ is called structurally uniform if, for any $A, B \in \mathrm{S}\mathrm{u}\mathrm{b}(S)$ the condition $h(A) = h(B) implies that A \sim = B$. We present a classification of finite structurally uniform groups and commutative nilsemigroups.

### Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1085-1096

We study an $n$-order system of ordinary differential equations with a nonlinearity of nonideal-relay-type with hysteresis and an external periodic perturbations. We consider the existence of solutions with periods equal or multiple to the period of the external perturbation and two points of switching within period. The problem is solved in the case where the collection of simple real eigenvalues of the matrix of the system contains an eigenvalue equal to zero. By a nonsingular transformation, the system is reduced to a canonical system of a special form that enables us to perform its analysis by analytic methods. We propose an approach to finding the points of switching for the representation point of the periodic solution and to a choice of the parameters of the nonlinearity and the feedback vector. We prove a theorem on necessary conditions for the existence of the periodic solutions of the system. Sufficient conditions for the existence of the required solutions are established. We also perform an analysis of stability of the solutions by using the point mapping and the fixed-point method. We present an example that confirms the accumulated results.

### Upper and lower Lebesgue classes of multivalued functions of two variables

Karlova O. O., Mykhailyuk V. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1097-1106

We introduce a functional Lebesgue classification of multivalued mappings and obtain results on upper and lower Lebesgue classifications of multivalued mappings $F : X \times Y \multimap Z$ for wide classes of spaces $X, Y$ and $Z$.

### On the fundamental solution of the Cauchy problem for Kolmogorov systems of the second order

Burtnyak I. V., Malyts’ka H. P.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1107-1117

We study the structure of the fundamental solution of the Cauchy problem for a class of ultraparabolic equations with finitely many groups of variables degenerating parabolicity.

### Sufficient conditions for bounded turning of analytic functions

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1118-1127

Let function $f$ be analytic in the open unit disk and be normalized such that $f(0) = f\prime (0) 1 = 0$. In this paper methods from the theory of first order differential subordinations are used for obtaining sufficient conditions for $f$ to bewith bounded turning, i.e., the read part of its first derivative to map the unit disk onto the right half plane. In addition, several open problems are posed.

### Transformation operators in controllability problems for the degenerate wave equation with variable coefficients

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1128-1142

We study the control system $w_{tt} = \cfrac1{\rho } (kw_x) x + \gamma w,\; w(0, t) = u(t),\; x \in (0, l), t \in (0, T)$, in special modified spaces of the Sobolev type. Here, $\rho , k,$ and \gamma are given functions on $[0, l)$; $u \in L^{\infty} (0, T)$ is a control, and $T > 0$ is a constant. The functions $\rho$ and $k$ are positive on $[0, l)$ and may tend to zero or to infinity as $x \rightarrow l$. The growth of distributions from these spaces is determined by the growth of $\rho$ and $k$ as $x \rightarrow l$. Applying the method of transformation operators, we establish necessary and sufficient conditions for the $L^{\infty}$ -controllability and approximate $L^{\infty}$ -controllability at a given time and at a free time.

### Approximation of periodic functions of many variables by functions of smaller number of variables in Orlicz metric spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1143-1148

For periodic functions of many variables, the method of their approximation is offered in the Orlicz spaces $L_{\varphi} (T^m)$. In this method, the functions are approximated by the sums of functions of smaller number of variables, each of which is piecewiswe-constant in one of variables for fixed values of the other variables. A Jackson-type inequality is investigated for these approximations in terms of the mixed module of continuity.

### On the asymptotic of associate sigma-functions and Jacobi theta-functions

Kharkevych Yu. I., Korenkov M. E.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1149-1152

For the associated sigma-functions, Jacobi theta-functions, and their logarithmic derivatives, we present asymptotic formulas valid outside an efficiently constructed exceptional sets of discs.