# Volume 70, № 5, 2018

### Second-order differential subordinations on a class of analytic functions defined by Rafid-operator

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 587-598

The purpose of the present paper is to introduce a new class of analytic functions by using the Rafid-integral operator and obtain some subordination results.

### Numerical solutions of fractional system, two-point BVPs using iterative reproducing kernel algorithm

Al-Smadi M., Altawallbeh Z., Ateiwi A. M., Komashinskaya I. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 599-610

We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product spaces are constructed in which the boundary conditions of the systems are satisfied. The reproducing kernel functions are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based on generating the orthonormal basis with an aim to formulate the solution throughout the evolution of the algorithm. The analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed components. In this connection, some numerical examples are presented to show the good performance and applicability of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution of fractional models arising in different fields of sciences and engineering.

### Divergence theorem in the $L_2$ -version. Application to the Dirichlet problem

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 611-624

We propose the $L_2$ -version of the divergence theorem. The Green and Poisson operators associated with the infinitedimensional version of the Dirichlet problem are investigated.

### On the approximation of the classes $W_{β}^rH^{α}$ by biharmonic Poisson integrals

Hrabova U. Z., Kalchuk I. V., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 625-634

We obtain asymptotic equalities for the least upper bounds of the deviations of biharmonic Poisson integrals from functions of the classes $W_{β}^rH^{α}$ in the case where $r > 2, 0 \leq \alpha < 1$.

### On Darboux vector in Lorentzian 5-space

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 635-641

We introduce the Darboux vector in the Lorentzian 5-space. We give some characterizations of this vector in the space. In addition, we consider some special cases in the space.

### Application of the method of averaging to the problems of optimal control for ordinary differential equations on the semiaxis

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 642-654

The method of averaging is applied to the nonlinear and linear (with respect to control) problems of optimal control on the semiaxis with small parameter and rapidly oscillating coefficients. It is shown that the solutions of the exact problem converge to the solutions of the averaged problem.

### Joint universality for $L$-functions from Selberg class and periodic Hurwitz zeta-functions

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 655-671

We obtain a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts consisting of $L$-functions from the Selberg class and periodic Hurwitz zeta-functions.

### Construction of intermediate differentiable functions

Maslyuchenko V. K., Mel'nik V. S.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 672-681

For given upper and lower semicontinuous real-valued functions $g$ and $h$, respectively, defined on a closed parallelepiped $X$ in $R^n$ and such that $g(x) < h(x)$ on $X$ and points $x_0 \in X$ and $y_0 \in (g(x_0), h(x_0))$, we construct a smooth function $f : X \rightarrow R$ such that $f(x_0) = y_0$ and $g(x) < f(x) < h(x)$ on $X$. We also present similar constructions for functions defined on separable Hilbert spaces and Asplund spaces.

### Continued-fractions representations of the functions $\mathrm{s}\mathrm{h} z, \mathrm{c}\mathrm{h} z, \mathrm{s}\mathrm{i}\mathrm{n} z, \mathrm{c}\mathrm{o}\mathrm{s} z$

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 682-698

We obtain the representations of the functions $\mathrm{s}\mathrm{h} z, \mathrm{c}\mathrm{h} z, \mathrm{s}\mathrm{i}\mathrm{n} z,$ and $\mathrm{c}\mathrm{o}\mathrm{s} z$ by quasireciprocal functional continued fractions of the Thiele type.

### Multiple modules of continuity and the best approximations of periodic functions in metric spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 699-707

It is proved that, under the condition $M_{\Psi} \Bigl( \frac 12\Bigr) < 1$, where $M_{\Psi}$ is a stretching function $\Psi$ in the space $L_{\Psi}$ , the Jackson inequalities $$\sup_n \sup_{f\in L_{\Psi}, f\not = \text{const}} \frac{E_{n-1}(f)_{\Psi} }{\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}} < \infty,$$ are true; here, $E_{n-1}(f)_{\Psi}$ is the best approximation of $f$ by trigonometric polynomials of degree at most $n - 1$ and $\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}$ is the modulus of continuity of $f$ of order $k$, $k \in N$. We study necessary and sufficient conditions for the function $f$ under which the following relation is true: $E_{n-1}(f)_{\Psi} \asymp \omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}.$

### Best approximations of the Cauchy – Szegö kernel in the mean on the unit circle

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 708-714

We compute the values of the best approximations of the Cauchy – Szeg¨o kernel in the mean on the unit circle by quasipolynomials with respect to the Takenaka – Malmquist system.

### Necessary and sufficient conditions for the absolute instability of solutions of linear differential-difference equations with self-adjoint operator coefficients

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 715-724

For linear differential-difference equations of retarded and neutral types with infinitely many deviations and self-adjoint operator coefficients, we present necessary and sufficient conditions for the absolute instability of the zero solutions.