# Volume 69, № 6, 2017

### Convergence of the spectral decomposition of a function from the class $W_{p,m}^1 (G),\; p > 1$, in the vector eigenfunctions of a differential operator of the third order

Abbasova Yu. G., Kurbanov V. M.

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 719-733

We consider, a third-order differential operator with matrix coefficients. The absolute and uniform convergence of the orthogonal expansion of a vector function from the class $W_{p,m}^1 (G),\; p > 1$, in the vector eigenfunctions of this operator is studied and the rate of uniform convergence of this expansion on $G = [0, 1]$ is estimated.

### On one boundary-value problem for elliptic differential-operator equations of the second order with quadratic spectral parameter

Aliev B. A., Kurbanova N. K., Yakubov Ya.

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 734-750

The problem of solvability of a boundary-value problem for a differential-operator equation of the second order on a finite interval is studied in a complex separable Hilbert space H in the case where the same spectral parameter appears in the equation in the form of a quadratic function and in the boundary conditions in the form of a linear function and, moreover, the boundary conditions are not separated. The asymptotic behavior of the eigenvalues of one homogeneous abstract boundary-value problem is also investigated. The asymptotic formulas for the eigenvalues are obtained and an application of the obtained results to partial differential equations is analyzed.

### Weakly perturbed operator equations in Banach spaces

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 751-764

We obtain the conditions of bifurcation of the solutions of weakly perturbed operator equations in Banach spaces from the point $\varepsilon = 0$ and propose a convergent iterative procedure for finding the solutions in the form of parts of the series in powers of $\varepsilon$ with pole at the point $\varepsilon = 0$.

### A generalized theorem of mean values of an analytic function and an algorism of the determination of mean values

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 765-787

We prove the mean-value theorem for functions analytic in starlike domains, propose an algorithm for finding the function of mean values, and study its analytic continuation. We present a differential equation for the function of mean values and the interpretation of the Lagrange formula for analytic functions in terms of the theory of differential equations. The set of points of the initial values of the function of mean values is analyzed and the upper of the radius of expansion of the function of mean values in Taylor’s series is established.

### Favard – Amerio theory for almost periodic functional-differential equations without the use of $H$-classes of these equations

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 788-802

The Favard – Amerio theory is constructed for almost periodic functional-differential equations in a Banach space without the use of $\scr H$ -classes of these equations. For linear equations, we present the first example of an almost periodic operator, which has no analogs in the classical Favard – Amerio theory.

### Level-crossing intensity for the density of the image of the Lebesgue measure under the action of a Brownian stochastic flow

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 803-822

We compute the level-crossing intensity for the density of the image of the Lebesgue measure under the action of a Brownian stochastic flow, which is a smooth approximation of the Arratia flow, and determine its asymptotic behavior as the height of the level tends to infinity.

### Substantiation of the collocation method for one class of systems of integral equations

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 823-835

We present the substantiation of the collocation method for a system surface integral equations of the boundary-value problem of conjugation for the Helmholtz equations. Moreover, we construct a sequence convergent to the exact solution of the boundary-value problem of conjugation and estimate its error.

### Jacobi operators and orthonormal matrix-valued polynomials. II

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 836-847

We use a system of operator-valued orthogonal polynomials to construct analogs of L. de Branges spaces and establish their relationship with the theory of nonself-adjoint operators.

### Tri-additive maps and local generalized $(α,β)$-derivations

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 848-853

Let $R$ be a prime ring with nontrivial idempotents. We characterize a tri-additive map $f : R^3 \rightarrow R$ such that $f(x, y, z) = 0$ for all $x, y, z \in R$ with $xy = yz = 0$. As an application, we show that, in a prime ring with nontrivial idempotents, any local generalized $(\alpha , \beta)$-derivation (or a generalized Jordan triple $(\alpha , \beta)$-derivation) is a generalized $(\alpha , \beta)$-derivation.

### Monads and tensor products

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 854-859

M. Zarichnyi defined an operation of tensor product for each functor that can be complemented to a monad. We investigate the existence of tensor product for functors which cannot be complemented to monads.

### On the absolute continuity of mappings distorting the moduli of cylinders

Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 860-864

We consider the mappings satisfying one modular inequality with respect to cylinders in the space. The distortion of modulus is majorized by an integral depending on a certain locally integrable function. We also prove a result on the absolute continuity of the analyzed mappings on lines.