# Volume 70, № 10, 2018

### A study of modules over rings and their extensions

Al - Hashmi S. A., Nauman S. K.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1299-1312

We study the transfer of properties of some types of modules under identity preserving ring homomorphisms. These studies seem to be overlooked in literature. We have picked a few types of modules and provided proofs for these properties that are transferable and suitable counterexamples for the properties that are not transferable.

### To the theory of nonlocal problems with integral conditions for systems of equations of hyperbolic type

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1313-1323

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations of the second order. By method of introduction of functional parameters, the investigated problem is reduced to an equivalent problem formed by the Goursat problem for a system of hyperbolic equations with parameters and integral relations. Algorithms for finding approximate solutions of this problem are constructed and their convergence to the exact solution is demonstrated. Sufficient conditions for the unique solvability of the equivalent problem are obtained in terms of the initial data. Moreover, the conditions of unique solvability of the nonlocal problem with integral conditions for system of hyperbolic equations are established in terms of the coefficients of the system and kernels in the integral conditions.

### Fredholm one-dimensional boundary-value problems in Sobolev spaces

Atlasiuk O. M., Mikhailets V. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1324-1333

For systems of ordinary differential equations on a compact interval, we investigate the character of solvability of the most general linear boundary-value problems in Sobolev spaces. We find the indexes of these problems and obtain a criterion of their well-posedness.

### Boundedness of $L$-index for the composition of entire functions of several variables

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1334-1344

We consider the following compositions of entire functions $F(z) = f \bigl( \Phi (z)\bigr) $ and $H(z,w) = G(\Phi 1(z),\Phi 2(w))$, where f$f : C \rightarrow C, \Phi : C^n \rightarrow C,\; \Phi_1 : C^n \rightarrow C, \Phi_2 : C^m \rightarrow C$, and establish conditions guaranteeing the equivalence of boundedness of the $l$-index of the function $f$ to the boundedness of the $L$-index of the function $F$ in joint variables, where $l$ : $C \rightarrow R_{+}$ is a continuous function and $$L(z) = \Bigl( l\bigl( \Phi (z)\bigr) \bigm| \frac{\partial \Phi (z)}{\partial z_1}\bigm| ,..., l \bigl( \Phi (z) \bigr) \bigm|\frac{\partial \Phi (z)}{\partial z_n} \bigm| \Bigr).$$ Under certain additional restrictions imposed on the function $H$, we construct a function $\widetilde{L} $ such that $H$ has a bounded $\widetilde{ L}$ -index in joint variables provided that the function $G$ has a bounded $L$-index in joint variables. This solves a problem posed by Sheremeta.

### Generalized characteristics of smoothness and some extreme problems of the approximation theory of functions in the space $L_2 (R)$. II

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1345-1373

In the second part of the paper, we establish the exact Jackson-type inequalities for the characteristic of smoothness $\Lambda^w$ on the classes of functions $L^{\alpha}_2 (R)$ defined by the fractional derivatives of order $\alpha \in (0,\infty )$ in the space $L_2(R)$. The exact values of the mean $\nu$ -widths for the classes of functions, defined by the generalized characteristics of smoothness $\omega w$ and $\Lambda w$ are also computed in $L_2(R)$.

### Nonlocal boundary-value problem for a second-order partial differential equation in an unbounded strip

Il'kiv V. S., Symotyuk M. M., Volyanska I. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1374-1381

The conditions of well-posedness of a nonlocal boundary-value problem are established for a second-order linear partial differential equation in an unbounded strip in the case where the real parts of the roots of its characteristic equation are different and nonzero.

### Commutative сomplex algebras of the second rank with unity and some cases of the plane orthotropy. II

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1382-1389

For an algebra $B_0 = \{ c_1e + c_2\omega : c_k \in C, k = 1, 2\} , e_2 = \omega 2 = e, e\omega = \omega e = \omega$, over the field of complex numbers $C$, we сonsider arbitrary bases $(e, e_2)$, such that$e + 2pe^2_2 + e^4_2 = 0$ for any fixed $p > 1$. We study $B_0$ -valued “analytic” functions $\Phi (xe+ye_2) = U_1(x, y)e + U_2(x, y)ie + U_3(x, y)e_2 + U_4(x, y)ie_2$ such that their real-valued components $U_k, k = 1, 4$, satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl( frac{\partial^4}{\partial x^4} + 2p\frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr)u(x, y) = 0,$$ here, $x$ and $y$ are real variables. All functions $\Phi$ for which $U_1 \equiv u$ are described in the case of a simply connected domain. Particular solutions of the equilibrium system of equations in displacements are found in the form of linear combinations of the components $U_k , k = 1, 4$, of the function $\Phi$ for some plane orthotropic media.

### On the inverse scattering problem for the one-dimensional Schrödinger equation with growing potential

Guseinov I. M., Khanmamedov A. Kh.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1390-1402

We consider a one-dimensional Schrödinger equation on the entire axis whose potential rapidly decreases at the left end and infinitely increases at the right end. By the method of transformation operators, we study the inverse scattering problem. We establish conditions for the scattering data under which the inverse problem is solvable. The basic Marchenko-type integral equations are investigated and their unique solvability is established.

### Superfractality of the set of incomplete sums of one positive series

Markitan V. P., Pratsiovytyi M. V., Savchenko I. O.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1403-1416

We consider a family of convergent positive normed series with real terms defined by the conditions $$\sum ^{\infty}_{n=1} d_n = \underbrace{c_1 + ...+c_1}_{a_1} + \underbrace{c_2 + ...+c_2}_{a_2} + ... + \underbrace{c_n + ...+c_n}_{a_n} + \widetilde{ r_n} = 1,$$ where $(a_n)$ is a nondecreasing sequence of real numbers. The structural properties of these series are investigated. For a partial case, namely, $(a_n) = 2^{n - 1}, c_n = (n + 1)\widetilde {r_n}, n \in N$, we study the geometry of the series (i.e., the properties of cylindrical sets, metric relations generated by them, and topological and metric properties of the set of all incomplete sums of the series). For the infinite Bernoulli convolution determined we describe its Lebesgue structure (discrete, absolutely continuous, and singular components) and spectral properties, as well as the behavior of the absolute value of the characteristic function at infinity. We also study the finite autoconvolutions of distributions of this kind.

### Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1417-1428

In the present paper, we apply Hopf’s lemma to the contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. This inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iran. Math. Soc. – 2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. – P. 17 – 26, and Sibel Sular, Cihan O¨ zgu¨r, Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Moreover, we also compute another inequality for the squared norm of the second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities acquired in [I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 and K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. The inequalities proved in the paper either generalize or improve all inequalities available in the literature and related to the squared norm of the second fundamental form for contact CR-warped product of submanifolds of any almost contact metric manifold.

### Caccioppoli-type estimates for a class of nonlinear differential operators

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1429-1438

We establish Caccioppoli-type estimates for a class of nonlinear differential equations with the aid of a differential identity that generalizes the well-known multidimensional Picone’s formula. In special cases, these estimates give the Finsler $p$-Laplacian, the $p$-Laplacian and the pseudo-$p$-Laplacian.

### International conference on the mathematical analysis, differential equations, and their applications (MADEA-8) devoted to the 80th birthday of Academician A. M. Samoilenko

Abdullayev F. G., Samoilenko A. M., Savchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1439-1440