2019
Том 71
№ 9

# Volume 71, № 6, 2019

Article (Russian)

### On the problem for the mixed-type degenerate equation with Caputo and Erdelyi – Kober operators of fractional order

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 723-738

UDC 517.9
We establish the existence and uniqueness of the solution of a local problem for the degenerate parabolic-hyperbolic-type equations with loaded terms containing the trace of solution in the Erdelyi – Kober integrals. Since, the trace of solution (i.e., $u(x, 0)$) appears in the Erdelyi – Kober integrals and the hyperbolic-type equation is degenerated in the line $y = 0$, we use some properties and estimates for hypergeometric functions and, in addition, some integral equalities in proving the existence and uniqueness of solution of the investigated problem.

Article (English)

### Classical Kantorovich operators revisited

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 739-747

UDC 517.5
The main object of this paper is to improve some known estimates for the classical Kantorovich operators. We obtain a quantitative Voronovskaya-type result in terms of the second moduli of continuity, which improves some previous results. In order to explain the nonmultiplicativity of the Kantorovich operators, we present a Chebyshev – Gr¨uss inequality. Two Gr¨uss –Voronovskaya theorems for Kantorovich operators are also considered.

Article (English)

### Fine spectra of tridiagonal Toeplitz matrices

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 748-760

UDC 514.74
The fine spectra of $n$-banded triangular Toeplitz matrices and $(2n+1)$-banded symmetric Toeplitz matrices were computed in ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) and ( M. Altun, Abstr. and Appl. Anal. – 2012. – Article ID 932785). As a continuation of these results, we compute the fine spectra of tridiagonal Toeplitz matrices. These matrices are, in general, not triangular and not symmetric.

Article (Ukrainian)

### Bounded solutions of the nonlinear Lyapunov equation and homoclinic chaos

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 761-773

UDC 517.9
The paper is devoted to the investigation of bounded solutions of a nonlinear Lyapunov-type problem in Banach and Hilbert spaces. Necessary and sufficient conditions for the existence of bounded solutions are obtained under the assumption that the homogeneous equation admits exponential dichotomy on the semiaxes. Conditions for the existence of homoclinic chaos in nonlinear evolution equations are presented.

Article (Russian)

### Lower bounds for the volume of the image of a ball

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 774-785

UDC 517.5
We consider ring $Q$-homeomorphisms with respect to $p$-modulus in the space $\Bbb R^{n}$ as $p>n$. We obtain a lower bound for the volume of the image of a ball under these mappings. We solve the extremal problems of minimization of functionals of the volume of the image of a ball and the area of the image of a sphere.

Article (Russian)

### Bojanov – Naidenov problem for the differentiable functions on the real line and the inequalities of various metrics

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 786-800

UDC 517.5
For given $r \in {\rm \bf N},$ $p,\lambda > 0$ and any fixed interval $[a, b]\subset {\rm \bf R}$ we solve the extremal problem $$\int\limits_{a}^{b} |x(t)|^q dt \to \sup, \quad q\ge p,$$ on a set of functions $x\in L^r_{\infty}$ such that $$\|x^{(r)}\|_{\infty} \le 1,\quad \|x\|_{p, \delta} \le \|\varphi_{\lambda, r}\|_{p, \delta}, \quad \delta \in (0, \pi/ \lambda],$$ where $$\|x\|_{p, \delta}:=\sup \{ \|x\|_{L_p[a,\, b]}\colon a, \,b \in {\rm \bf R}, \; 0< b-a \le \delta \}$$ and $\varphi_{\lambda, r}$ is the $(2\pi/\lambda)$-periodic Euler spline of order $r.$ In particular, we solve the same problem for the intermediate derivatives $x^{(k)},$ $k=1,\ldots,r-1,$ with $q \ge 1.$ In addition, we prove the inequalities of various metrics for the quantities $\|x\|_{p, \delta}.$

Article (Ukrainian)

### Discontinuity points of separately continuous mappings with at most countable set of values

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 801-807

UDC 517.51
We obtain a general result on the constancy of separately continuous mappings and their analogs, which implies the wellknown Sierpi´nski theorem. By using this result, we study the set of continuity points of separately continuous mappings with at most countably many values including, in particular, the mappings defined on the square of the Sorgenfrey line with values in the Bing plane.

Article (Ukrainian)

### Theory of multidimensional Delsarte – Lions transmutation operators. II

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 808-839

UDC 517.9
The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand – Levitan – Marchenko equations that describe these operators are studied by using sutable differential de Rham – Hodge – Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang – Mills equations. Soliton solutions of a certain class of dynamical systems are discussed.

Article (English)

### Finite speed of propagation for the thin-film equation in the spherical geometry

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 840-851

UDC 517.953
We show that a double degenerate thin-film equation obtained in modeling of a flow of viscous coating on the spherical surface has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the solution $u>0$ and $u=0.$ Using local entropy estimates, we also obtain the upper bound for the rate of the interface propagation.

Brief Communications (Russian)

### Finite simple groups with Hall $\{2, r\}$-subgroups, $r \in π(G) \backslash \{2, t\}$, $t \in π(G)$

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 852-857

UDC 512.542
We describe finite simple groups with Hall biprimary subgroups of even order that contain Sylow subgroups of odd order of the $G$-group, except one of Sylow's subgroups of odd order.

Brief Communications (English)

### Instability intervals for Hill’s equation with symmetric single-well potential

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 858-864

UDC 517.9
We deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of the instability intervals of Hill’s equation with symmetric single-well potentials by using an auxiliary eigenvalue problem. We also give bounds for the gaps of the Dirichlet and Neumann eigenvalues.