2019
Том 71
№ 10

# Volume 71, № 4, 2019

Article (Russian)

### On one homogeneity test based on the kernel-type estimators of the distribution density

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 443-454

We construct a homogeneity test based on the kernel-type estimators of the distribution density and investigate its consistency.

Article (English)

### Resonant equations with classical orthogonal polynomials. II

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 455-470

UDC 517.9
We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm of finding their particular and general solutions in the closed form. The algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra, as well as of the square operator equations $A^2u = f$ , e.g., of the biharmonic equation.

Article (Russian)

### Conditions of solvability and representation of the solutions of equations with operator matrices

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 471-485

We propose new methods for the construction of generalized inverse operator matrices for the operator matrices in Banach spaces. The solvability criteria and the formulas for representations of the general solutions of operator equations with operator matrices are obtained. As an application, we consider the relationship between the obtained formulas and the well-known Frobenius formula for the construction of the matrix inverse to a nondegenerate block matrix.

Article (Russian)

### Consistent criteria for hypotheses testing

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 486-501

We investigate statistical structures that admit consistent criteria for hypotheses testing and establish necessary and sufficient conditions for the existence of consistent criteria for hypotheses testing.

Article (English)

### Stochastic differential equations for eigenvalues and eigenvectors of a $G$-Wishart process with drift

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 502-515

We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of the $G$-Wishart process defined according to a $G$-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence between the entries of the $G$-Brownian motion matrix, we assume in our model that their quadratic covariations are zero. An intermediate result, which states that the eigenvalues never collide is also obtained. This extends Bru’s results obtained for the classical Wishart process (1989).

Article (Ukrainian)

### Approximation of bounded holomorphic and harmonic functions by Fejér means

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 516-542

We compute the exact values of the exact upper bounds on the classes of bounded holomorphic and harmonic functions in a unit disk for the remainders in a Voronovskaya-type formula in the case of approximation by Fej´er means. We also present some consequences that are of independent interest.

Article (Ukrainian)

### Singular integral equation equivalent in the space of smooth functions to an ordinary differential equation, method of successive approximations for the construction of its smooth solutions and its nonsmooth solutions

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 543-563

We propose a singular integral equation whose definition is extended to a singular point by additional conditions. In the space of smooth functions, this equation becomes equivalent, by the indicated extended definition, to an ordinary differential equation, whereas in the space of piecewise discontinuous functions, it becomes equivalent to an impulsive differential equation. For smooth solutions of the singular equation, we substantiate the method of successive approximations. For the ordinary differential equation, this method turns into a new algorithm for the construction of successive approximations. For the investigated equation, we specify a solution of new type, which is equivalent, for the impulsive differential equation, to a solution with discontinuity of the second kind (a “solution with needle”). We propose an algorithmic formula for the general solution of the initial-value problem for the impulsive differential equation.

Article (Ukrainian)

### Order and canonical product of the Weierstrass $R$-integral

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 564-570

The order of the $R$-integral is specified and its representation in the form of the canonical Weierstrass product is found.

Brief Communications (English)

### Existence results for a class of Kirchhoff-type systems with combined nonlinear effects

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 571-580

UDC 517.9
We study the existence of positive solutions for a nonlinear system $$M_1 \bigl(\int_{\Omega} |\nabla u|^p dx\bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x|^{ap}| \nabla u|^{p-2}\nabla u) = \lambda | x| (a+1)p+c_1f(u, \upsilon ),\; x \in \Omega ,$$ $$M2 \bigl( \int_{ \Omega }| \nabla v| q dx \bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x| bq|\nabla \upsilon | q 2\nabla \upsilon ) = \lambda | x| (b+1)q+c_2g(u, \upsilon ),\; x \in \Omega ,$$ $$u = \upsilon = 0, x \in \partial \Omega ,$$ where $\Omega$ is a bounded smooth domain in $R^N$ with $0 \in \Omega,\; 1 < p, q < N, 0 \leq a < \cfrac{N-p}{p}, 0 \leq b < \cfrac{N-q}{p},$ а $c_1, c_2, \lambda$ are positive parameters. Here, $M_1,M_2, f$, and g satisfy certain conditions. We use the method of sub- and supersolutions to establish our results.