# Volume 68, № 6, 2016

### Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths of the classes of $(ψ,β)$-differential functions in $L_2$. I

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 723-745

For the generalized moduli of continuity, including the ordinary moduli of continuity and various their modifications, we establish the exact constants for Jackson-type inequalities in the classes of $2\pi$ -periodic functions in the space $L_2$ with $(\psi , \beta)$-derivatives, introduced by Stepanets. These results take into account the classification of $(\psi , \beta)$-derivatives and enable us to consider the major part of Jackson-type inequalities obtained earlier in the classes of differentiable functions $L_2^r,\; r \in N$, from the common point of view.

### Conitnuity of the solutions of one-dimensional boundary-value problems with respect to the parameter in slobodetsky spaces

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 746-756

For the system of linear ordinary differential equations of the first order, we study the broadest class of inhomogeneous boundary-value problems whose solutions belong to the Slobodetsky space $W^{s+1}_p ((a, b),C^m)$ with $m \in N,\; s > 0$, and $p \in (1,\infty )$. We prove a theorem on the Fredholm property of these problems. We also establish conditions under which the problems are uniquely solvable in the Slobodetsky space and their solutions are continuous in this space with respect to the parameter.

### Generalizations of the shadow problem

Stefanchuk M. V., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 757-763

We solve the shadow problem in the n-dimensional Euclidean space $N^n$ for a family of sets obtained from any convex domain with nonempty interior with the help of parallel translations and homotheties. We determine the number of balls with centers on the sphere, sufficient for giving a shadow in the $n$-dimensional complex (hypercomplex) space.

### On partial solutions of one equation with multiple characteristics and some properties of the fundamental solution

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 763-787

We construct partial solutions of odd-order equations with multiple characteristics and study some of their properties.

### Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 786-799

In Hilbert Hörmander spaces, we study the initial-boundary-value problems for arbitrary parabolic differential equations of the second order with Dirichlet boundary conditions or general boundary conditions of the first order in the case where the solutions of these problems belong to the space $H^{2,1,\varphi}$. It is shown that the operators corresponding to these problems are isomorphisms between suitable Hörmander spaces. The regularity of the functions that form these spaces is characterized by a couple of numerical parameters and a functional parameter $\varphi$ slowly varying at infinity in Karamata’s sense. Due to the presence of the parameter $\varphi$, the Hörmander spaces describe the regularity of the functions more precisely than the anisotropic Sobolev spaces.

### On the problem of exact controllability of neutral systems with time delay

Barkhayev P. Yu., Rabah R., Sklyar G. M.

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 800-815

We study the problem of exact controllability for a broad class of neutral and mixed systems with time delay. We consider an equivalent operator model in a Hilbert space and formulate steering conditions for controllable states in the form of a vector moment problem. The existence of a basis of eigenvectors of the operator of system with delay significantly simplifies the form of the moment problem. A change of function in the control by a feedback law modi es the system structure in order to guarantee the existence of a basis of eigenvectors of the corresponding operator. We prove a criterion for the exact controllability and determine the exact critical time of control.

### Multiple Haar basis and $m$-term approximations of functions from the Besov classes. II

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 816-825

We establish the exact-order estimates for the best $m$-term approximations in the multiple Haar basis $\mathrm{H}^d$ of functions from the Besov classes in the Lebesgue spaces $L_q(I^d)$. We also present a practical algorithm of the construction of the extreme nonlinear m-term aggregates (in a sense of the exact-order estimates for approximations).

### Fixed-point theorems for integral-type contractions on partial metric spaces

Altun I., Masiha H. P., Sabetghadam F.

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 826-834

We present some fixed-point results for single-valued mappings on partial metric spaces satisfying a contractive condition of integral type.

### Almost Menger property in bitopological spaces

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 835-841

We introduce the notion of almost Menger property in bitopological spaces. We give some characterizations in terms of $(i, j)$-regular open sets and almost continuous surjection. We also investigate the notion of almost $\gamma$ -set in the bitopological context.

### Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 842-852

We characterize the boundedness and compactness of the differences of weighted differentiation composition operators $D^n_{\varphi_1, u_1} - D^n_{\varphi_2, u_2}$, where $n \in N_0, u_1, u_2 \in H(D)$, and $\varphi_1, \varphi_2 \in S(D)$, from mixed-norm spaces $H(p, q, \phi)$, where $0 < p,\; q < \infty$ and \phi is normal, to weighted-type spaces $H^{\infty}_v$.

### Congruences on regular semigroups with $Q$-inverse transversals

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 853-859

We give congruences on a regular semigroup with a $Q$-inverse transversal $S^o$ by the congruence pair (abstractly), which consists of congruences on the structural component parts $R$ and $\Lambda$. We prove that the set of all congruences for this kind of semigroups is a complete lattice.

### Representations for the generalized inverses of a modified operator

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 860-864

Some explicit representations for the generalized inverses of a modified operator $A + YGZ$ are derived under some conditions, where $A, Y, Z$, and $G$ are operators between Banach spaces. These results generalize the recent works about the Drazin inverse and the Moore – Penrose inverse of complex matrices and Hilbert-space operators.

### Determination of the lower coefficient in a parabolic equation with strong power degeneration

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 922-932

We establish conditions for the existence and uniqueness of the classical solution to the inverse problem of identification of the time-dependent coefficient at the first derivative in a one-dimensional degenerate parabolic equation. The Dirichlet boundary conditions and the integral condition of overdetermination are imposed. We study the case of strong power degeneration.