# Volume 69, № 10, 2017

### Transitivity of the surface measures on Banach manifolds with uniform structure

Bogdanskii Yu. V., Moravets’ka E. V.

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1299-1309

We perform the analysis of transitivity of associated measures on the surfaces with finite codimension imbedded in a Banach manifold with uniform atlas.

### Order estimates of the $L_q$-norms of generalized derivatives of the Dirichlet-type kernels with an arbitrary choice of harmonics

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1299-1309

We obtain the exact order estimates of the norms of generalized derivatives of the Dirichlet-type kernels with an arbitrary choice of harmonics in the space $L_q,\; 2 < q < \infty$.

### Lie-algebraic structure of the Lax-integrable (2| 1+ 1) -dimensional supersymmetric matrix dynamical systems

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1310-1323

By using a specially constructed Backlund transformation, we obtain the Hamiltonian representation for the hierarchy of Laxtype flows on the dual space to the Lie algebra of matrix superintegral-differential operators with one anticommutative variable, coupled with suitable evolutions of eigenfunctions and adjoint eigenfunctions of the associated spectral problems. We also propose the Hamiltonian description of the corresponding set of the hierarchies of additional homogeneous symmetries (squared eigenfunction symmetries). The connection between these hierarchies and the Lax-integrable (2| 1+1)-dimensional supersymmetric matrix nonlinear dynamical systems and their triple Lax-type linearizations is analyzed.

### Asymptotic behavior of the solutions of second-order differential equations with rapidly varying nonlinearities

Chernikova A. G., Evtukhov V. M.

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1345-1363

We establish conditions for the existence of one class of solutions of two-term nonautonomous differential equations of the second-order with rapidly varying nonlinearities and the asymptotic representations for these solutions and their first-order derivatives as и $t \uparrow \omega (\omega \leq +\infty )$.

### Cauchy problem for matrix factorizations of the Helmholtz equation

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1364-1371

We study the Cauchy problem for a system of elliptic equations of the first order with constant coefficients factorizing the Helmholtz operator in a two-dimensional bounded domain. An approximate solution of this problem based on the method of Carleman matrix is constructed.

### On one method for the solution of an analog of the Cauchy problem for a polycaloric equation with singular Bessel operator

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1372-1384

We study an analog of the Cauchy problem for an inhomogeneous singular polycaloric (polyparabolic) equation with Bessel operator. By using the Erd´elyi–Kober operator of the fractional order, we deduce an explicit formula for the solution of the formulated problem.

### $b$-coercive convolution equations in weighted function spaces and applications

Musaev H. K., Shakhmurov V. B.

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1385-1405

We study the $b$-separability properties of elliptic convolution operators in weighted Besov spaces and establish sharp estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in addition, play the role of negative generators of analytic semigroups. Moreover, the maximal $b$-regularity properties of the Cauchy problem for a parabolic convolution equation are established. Finally, these results are applied to obtain the maximal regularity properties for anisotropic integro-differential equations and the system of infinitely many convolution equations.

### Symmetric α-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudodifferential equation

Osipchuk M. M., Portenko N. I.

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1406-1421

We consider a pseudodifferential equation of parabolic type with operator of fractional differentiation with respect to a space variable generating a symmetric $\alpha$ -stable process in a multidimensional Euclidean space with an initial condition and a boundary condition imposed on the values of an unknown function at the points of the boundary of a given domain. The last condition is quite similar to the condition of the so-called third (mixed) boundary-value problem in the theory of differential equations with the difference that a traditional (co)normal derivative is replaced in our problem with a pseudodifferential operator. Another specific feature of the analyzed problem is the two-sided character of the boundary condition, i.e., a consequence of the fact that, in the case of \alpha with values between 1 and 2, the corresponding process reaches the boundary making infinitely many visits to both the interior and exterior regions with respect to the boundary.

### A generalization of WUU rings

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1422-1428

We define the class of UNI rings and present the results of their comprehensive investigations in connection with clean rings, namely, our main result describes commutative UNI clean rings up to an isomorphism. This new concept is a common generalization of the so-called UU rings examined by Danchev – Lam in (Publ. Math. Debrecen, 2016) and of the so-called WUU rings studied by the author in (Tsukuba J. Math., 2016).

### Generalized higher derivations on algebras

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1429-1436

We study the structure of generalized higher derivations on an algebra ${\scr A}$ and show that there exists a one-to-one correspondence between the set of all generalized higher derivations $\{ G_k\}^n_{k =0}$ on ${\scr A}$ with $G_0 = I$ and the set of all sequences $\{ g_k\}^n_{k = 0}$ of generalized derivations on ${\scr A}$ with $g_0 = 0$.

### On the construction of solutions of linear differential equations according to given sequences

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1437-1440

We consider the problem under what conditions the equation $f\prime \prime + Af = 0$ possesses an entire (meromorphic) solution with given sequences of zeros (poles) and critical points. The results are extended to equations of higher orders.