# Volume 71, № 1, 2019

### On 2-dimensional surfaces in 3-dimensional and 4-dimensional Euclidean spaces. Results and unsolved problems

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 3-36

We present a survey of the results obtained for 2-dimensional surfaces in $E^3$ and $E^4$ and connected with the Gaussian curvature and Gaussian torsion. In this connection, we consider the Monge –Amp´ere equations, obtain the generalizations of Bernstein’s integral formula, and establish some lower estimates for the exterior diameter of surfaces in $E^3$.

### Existence of positive solutions for a coupled system of nonlinear fractional differential equations

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 37-46

We study the following nonlinear boundary-value problems for fractional differential equations $$D^{\alpha} u(t) = f(t, v(t),D^{\beta - 1}v(t)), t > 0,\\ D^{\beta} v(t) = g(t, u(t),D^{\alpha - 1}u(t)), t > 0,\\ u > 0,\; v > 0 \in (0,\infty), \lim_{t\rightarrow 0+} u(t) = \lim_{t\rightarrow 0+} v(t) = 0,$$ where $1 < \alpha \leq 2$ and $1 < \beta \leq 2$. Under certain conditions on $f$ and $g$, the existence of positive solutions is obtained by applying the Schauder fixed-point theorem.

### Correctness of one nonlocal boundary-value problem with constant coefficient for the nonlinear mixed-type equation of the second kind of the second order in the space

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 47-58

Under certain conditions imposed on the coefficients of the nonlinear mixed-type equation of the second kind of the second order in the space, we prove the correctness of the solution of a nonlocal boundary-value problem.

### Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 59-72

We first classify space-like surfaces in the Minkowski space $E^4_1$, de Sitter space $S^3_1$, and hyperbolic space $H^3$ with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also present some explicit examples.

### On the asymptotic of solutions of second-order differential equations with rapidly varying nonlinearities

Chernikova A. G., Evtukhov V. M.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 73-91

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

### Antinormal composition operators on $L^2$ -space of an atomic measure space

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 92-98

Let $L^2(\mu)$ denotes the Hilbert space associated with a $\sigma$ -finite atomic measure $\mu$. We propose a characterization of antinormal composition operators on $L^2(\mu)$.

### Green’s functional for higher-order ordinary differential equations with general nonlocal conditions and variable principal coefficient

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 99-116

The method of Green’s functional is a little-known technique for the construction of fundamental solutions to linear ordinary differential equations (ODE) with nonlocal conditions. We apply this technique to a higher order linear ODE involving general nonlocal conditions. A novel kind of adjoint problem and Green’s functional are constructed for the completely inhomogeneous problem. Several illustrative applications of the theoretical results are provided.

### On the post-Darwin approximation of the Maxwell – Lorentz equations of motion of point charges in the absence of neutrality

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 117-128

The existence of holomorphic (in time) solutions of the nonrelativistic equations of motion of nonneutral systems of point charges that do not contain inverse powers of the velocity of light greater than three is proved by using the Cauchy theorem. The indicated equations contain time derivatives of the accelerations of charges.

### Asymptotic estimates for the solutions of a differential-functional equation with linear delay

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 129-138

We establish new properties of the solutions of a differential-functional equation with linearly transformed argument.

### On the solvability of the main inverse problem of stochastic differential systems

Ibraeva G. T., Tleubergenov M. I.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 139-145

Using the quasi-inversion method we obtain necessary and sufficient conditions for the solvability of the main (according to Galiullin’s classification) inverse problem in the class of first-order Itˆo stochastic differential systems with random perturbations from the class of processes with independent increments, with diffusion degenerate in a part of variables and with given properties, depending on a part of variables.