# Volume 68, № 9, 2016

### Boundary-value problem for nonlinear degenerated parabolic equations with variable delay

Bokalo M. M., Il’nyts’ka О. V.

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1155-1168

We study boundary-value problem with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. A priori estimates of this solution are obtained.

### Impulsive functional differential equations of fractional order with variable moments

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1169-1179

We establish some existence results for the solutions of initial-value problems for fractional-order impulsive functional differential equations with neutral-delay at variable moments.

### Orthogonal polynomials related to some Jacobi-type pencils

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1180-1190

We study a generalization of the class of orthonormal polynomials on the real axis. These polynomials satisfy the following relation: $(J_5 \lambda J_3)\vec{p}(\lambda) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $\vec{p}(\lambda) = (p_0(\lambda ), p_1(\lambda ), p_2(\lambda ),...)^T$, the superscript $T$ denotes the operation of transposition with the initial conditions $p_0(\lambda ) = 1,\; p_1(\lambda) = \alpha \lambda + \beta,\; \alpha > 0, \beta \in R$. Certain orthonormality conditions for the polynomials $\{ pn(\lambda )\}^{\infty}_n = 0$ are obtained. An explicit example of these polynomials is constructed.

### $T$-radical and strongly $T$-radical supplemented modules

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1191-1196

We define (strongly) t-radical supplemented modules and investigate some properties of these modules. These modules lie between strongly radical supplemented and strongly $\oplus$ -radical supplemented modules. We also study the relationship between these modules and present examples separating strongly $t$-radical supplemented modules, supplemented modules, and strongly $\oplus$-radical supplemented modules.

### Mutual winding angles of particles in Brownian stochastic flows with zero top Lyapunov exponent

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1197-1228

The investigation of the geometric properties of particles moving in stochastic flows leads to the study of their mutual winding angles. The same problem for independent Brownian motions was solved in [10]. We generalize these results to the case of isotropic Brownian stochastic flows with top Lyapunov exponent equal to zero.

### Classical solutions of parabolic initial-boundary value problems and Hormander spaces.

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1229-1239

For the second-order linear parabolic differential equations with complex-valued coefficients, we establish new sufficient conditions under which the generalized solutions of these problems are continuous. The conditions are formulated in the terms of belonging of the right-hand sides of these problems to certain anisotropic Ho¨rmander spaces.

### Estimates for the best bilinear approximations of the classes $B^r_{p,\theta}$ and singular numbers of integral operators

Romanyuk A. S., Romanyuk V. S.

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1240-1250

We obtain the exact-order estimates for the best bilinear approximations of the Nikol‘ski–Besov classes $B^r_{p,\theta}$ of periodic functions of several variables. We also find the orders for singular numbers of the integral operators with kernels from the classes $B^r_{p,\theta}$.

### $I_\lambda$-Double statistical convergence of order $α$ in topological groups

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1251-1258

We introduce а new notion, namely, $I_\lambda$-double statistical convergence of order \alpha in topological groups. Consequently, we investigate some inclusion relations between $I$ -double statistical and $I_\lambda$ -double statistical convergence of order $\alpha$ in topological groups. We also study many other related concepts.

### On the local behavior of open discrete mappings of the Orlicz – Sobolev classes

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1259-1272

The paper is devoted to the study of mappings with unbounded characteristic of quasiconformality and, in particular, of mappings with finite distortion extensively studied in recent years. We obtain theorems on equicontinuity of families of mappings that belong to the Orlicz–Sobolev class for $n \geq 3$, and have finite distortion. To do this, we also investigate some auxiliary classes of mappings, namely, we study the relationship between the so-called lower $Q$-mappings and some inequalities of the capacity type.

### Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1273-1285

Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two fixed and equal positive charges are found with the help of the Lyapunov center theorem.

### Conditions of existence of bounded and almost periodic solutions of nonlinear differential equation with perturbations of solutions

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1286-1296

We present the conditions of existence and uniqueness of bounded solutions of a nonlinear scalar differential equation $dx(t)/dt=f(x(t)+h(t)),\; t \in R$, in the case where a function $f$ is continuous on $R$ and a function $h$ is bounded and continuous. In addition, we study the case of an almost periodic function $h$.