# Volume 69, № 11, 2017

### Two-weighted inequalities for Riesz potential in $p$-convex weighted modular Banach function spaces

Bandaliyev R. A., Guliyev V. S., Hasanov S. G.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1443-1454

The main goal of this paper is to prove a two-weight boundedness for Riesz potential from one weighted Banach function space to another weighted Banach function space. In particular, we obtain a two-weight boundedness for Riesz potential and find sufficient conditions on the weights for boundedness of Riesz potential in weighted Musielak – Orlicz spaces.

### Multi-objective nonlinear sum of fractional optimization problems with non-convex constraints using duality based branch and bound algorithm

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1455-1471

The present paper investigates the solution of multiobjective nonlinear sum of fractional optimization problems. A duality based branch and bound cut method is developed to obtain efficient solution and the methodology is validate by proving the theoretical assertions for the solution. The present method is the extension of the work P. P. Shen, Y. P. Duan and Y. G. Pei[19] which developed for single objective sum of ratios nonlinear optimization problem. The proposed method is coded in matlab (version 2014b). Two numerical problems are considered for solving by using the proposed method and global optimal solution is obtained.

### Sharp Remez-type inequalities of various metrics in the classes of functions with а given comparison function

Gaydabura A. E., Kofanov V. A.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1472-1485

For any $p \in [1,\infty ],\; \omega > 0, \;\beta \in (0, 2\omega )$, and any measurable set $B \subset I_d := [0, d], \mu B \leq \beta$, we obtain the following sharp Remez-type inequality of various metrics $$E_0(x)\infty \leq \frac{\| \varphi \|_{\infty} }{E_0 (\varphi )L_p(I_{2\omega} \setminus B_1)}\| x\|_{ L_p(I_d\setminus B)}$$ on the classes $S_{\varphi} (\omega )$ of $d$-periodic $(d \geq 2\omega)$ functions $x$ with a given sine-shaped $2\omega$ -periodic comparison function $\varphi$, where $B_1 := [(\omega \beta )/2, (\omega + \beta )/2], E_0(f)L_p(G)$ is the best approximation of the function $f$ by constants in the metric of the space $L_p(G)$. In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and splines.

### Elliptic problems with boundary conditions of higher orders in Hörmander spaces

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1486-1504

In a class of inner product H¨ormander spaces, we study a general elliptic problem for which the maximum order of the boundary conditions is not smaller than the order of the elliptic equation. The role of the order of regularity of these spaces is played by an arbitrary radial positive function $R_O$-varying at infinity in the sense of Avakumovi´c. We prove that the operator of the problem under investigation is bounded and Fredholm on the appropriate pairs of the indicated H¨ormander spaces. A theorem on isomorphism generated by this operator is proved. For the generalized solutions of this problem, we establish a local a priori estimate and prove the theorem on the local regularity of these solutions in H¨ormander spaces. As an application, we establish new sufficient conditions of continuity for the given generalized derivatives of the solutions.

### On rationally loxodromic holomorphic functions

Khrystiyanyn A. Ya., Lukivska Dz. V.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1505-1514

We consider a functional equation of the form $f(qz) = R(z)f(z)$, where $R(z)$ is a rational function, $z \in C\setminus \{ 0\},\;q \in C\setminus \{ 0\},\; | q| < 1$. Holomorphic solutions of this equation are obtained. These solutions can be regarded as generalizations of p-loxodromic functions.

### On a Banach algebra generated by the Bergman operator, constant coefficients, and finitely generated groups of shifts

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1515-1523

We study a Banach algebra generated by the Bergman operator, constant coefficients, and shifts formed by finitely generated commutative groups of hyperbolic transformations of the unit disk acting in the Lebesgue space $L_p, p > 1$, and obtain an effective criterion for the operators from the considered Banach algebra to be Fredholm operators.

### On matrix operators on the series space $|\bar{N}_p^θ|_k$

Mohapatra R. N., Sarigol M. A.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1524-1533

Recently, the space $|\bar{N}_p^θ|_k$ has been generated from the set of $k$-absolutely convergent series $\ell_k$ as the set of series summable by the absolute weighted method. In the paper, we investigate some properties of this space, such as $\beta$ -duality and the relationship with \ell k and then show that each element in the classes $\Bigl(|\bar{N}_p|,\;|\bar{N}_p^θ|_k\Bigr)$ and $\Bigl(|\bar{N}_p^θ|_k,\;|\bar{N}_q|\Bigr)$ of infinite matrices corresponds to a continuous linear operator and also characterizes these classes. Hence, in the special case, we deduce some well-known results of Sarıg¨ol, Bosanquet, Orhan, and Sunouchi.

### Common fixed-point theorems for hybrid generalized $(F, ϕ)$ -contractions under the common limit range property with applications ......................

Ahmadullah M., Imdad M., Nashine H. K.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1534-1550

We consider a relatively new hybrid generalized $F$ -contraction involving a pair of mappings and use this contraction to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying generalized $(F, \varphi)$-contraction condition with the common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying the rational-type Hardy – Rogers $(F, \varphi)$-contractive condition is also proved. We generalize and improve several results available from the existing literature. As applications of our results, we prove two theorems for the existence of solutions of certain system of functional equations encountered in dynamic programming and the Volterra integral inclusion. Moreover, we provide an illustrative example.

### Type of some nuclear subalgebras of the Toeplitz algebra generated by the inverse subsemigroups of a bicyclic semigroup

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1551-1563

We construct a $Z$-graded structure of the Toeplitz algebra and consider nuclear $C\ast$ -subalgebras of the Toeplitz algebra generated by the inverse subsemigroups of the bicyclic semigroup. The types of these algebras with respect to the Toeplitz algebra are determined. In addition, it is shown, that considered algebras are equipped with the structrure of Hilbert $C\ast$ -modules.

### On equicontinuity of homeomorphisms of the Orlicz and Orlicz – Sobolev classes in the closure of a domain

Petrov E. A., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1564-1576

We investigate the behavior of homeomorphisms of the Orlicz – Sobolev classes in the closure of a domain. The theorems on equicontinuity of the indicated classes are obtained in terms of the prime ends of regular domains. In particular, it is shown that indicated classes are equicontinuous in domains with certain restrictions imposed on their boundaries provided that the corresponding inner dilatation of order p has a majorant of finite mean oscillation at every point. We also prove theorems on the (pointwise) equicontinuity of the analyzed classes in the case of locally connected boundaries.

### Approximation of the Bergman kernels by rational functions with fixed poles

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1577-1584

We solve the problem of best rational approximations of the Bergman kernels on the unit circle of the complex plane in the quadratic and uniform metrics.