# Volume 57, № 1, 2005

### On Some Euler Sequence Spaces of Nonabsolute Type

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 3–17

In the present paper, the Euler sequence spaces $e_0^r$ and $e^r_c$ of nonabsolute type which are the $BK$-spaces including the spaces $c_0$ and $c$ have been introduced and proved that the spaces $e_0^r$ and $e^r_c$ are linearly i somorphic to the spaces $c_0$ and $c$, respectively. Furthemore, some inclusion theorems have been given. Additionally, the $\alpha-, \beta-, \gamma-$ and continuous duals of the spaces $e_0^r$ and $e^r_c$ have been computed and their basis have been constructed. Finally, the necessary and sufficient conditions on an infinite matrix belonging to the classes $(e^r_c :\; {l}_p)$ and $(e^r_c :\; c)$ have been determined and the characterizations of some other classes of infinite matrices have also been derived by means of a given basic lemma, where $1 \leq p \leq \infty$.

### Twist Functors and *D*-Branes

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 18–31

We discuss a categorical approach to the investigation of topological *D*-branes. Twist functors and their induced action on the cohomology ring of a manifold are studied. A nontrivial spherical object of the derived category of coherent sheaves of a reduced plane singular curve of degree 3 is constructed.

### Problem of Conjugation of Solutions of the Lame Wave Equation in Domains with Piecewise-Smooth Boundaries

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 32–46

We study the problem of conjugation of solutions of the Lame wave equation in domains containing singular lines (sets of angular points) and conic points. We show that solutions of the Lame wave equation have power-type singularities near nonsmoothnesses of boundary surfaces and determine their asymptotics. Taking these asymptotics into account and using the introduced simple-layer, double-layer, and volume elastic retarded potentials, we reduce the problem to a system of functional equations and formulate conditions for its solvability.

### Pointwise Estimates for the Coconvex Approximation of Differentiable Functions

Dzyubenko H. A., Zalizko V. D.

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 47–59

We obtain pointwise estimates for the coconvex approximation of functions of the class $W^r,\; r > 3$.

### Necessary and Sufficient Conditions for the Solvability of the Gauss Variational Problem

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 60–83

We investigate the well-known Gauss variational problem considered over classes of Radon measures associated with a system of sets in a locally compact space. Under fairly general assumptions, we obtain necessary and sufficient conditions for its solvability. As an auxiliary result, we describe potentials of vague and (or) strong limit points of minimizing sequences of measures. The results obtained are also specified for the Newton kernel in $\mathbb{R}^n$.

### Integral Form of Bounded Solutions of Some Systems of Differential Equations

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 84–93

We investigate the well-known Gauss variational problem considered over classes of Radon measures associated with a system of sets in a locally compact space. Under fairly general assumptions, we obtain necessary and sufficient conditions for its solvability. As an auxiliary result, we describe potentials of vague and (or) strong limit points of minimizing sequences of measures. The results obtained are also specified for the Newton kernel in $\mathbb{R}^n$.

### One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 94–101

We investigate the existence of a separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with a one-point set of points of discontinuity in the case where the topological spaces $X$ and $Y$ satisfy conditions of compactness type. In particular, for the compact spaces $X$ and $Y$ and the nonizolated points $x_0 \in X$ and $y_0 \in Y$, we show that the separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with the set of points of discontinuity $\{(x_0, y_0)\}$ exists if and only if sequences of nonempty functionally open set exist in $X$ and $Y$ and converge to $x_0$ and $y_0$, respectively.

### Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 102–110

We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated.

### Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 111–124

We construct asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter.

### Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 125–130

We construct an algorithm for the two-sided approximation of a solution of a multipoint boundary-value problem for a quasilinear differential equation under assumptions that are two-sided analogs of the Pokornyi B-monotonicity of the right-hand side of the equation. We establish conditions for the monotonicity of successive approximations and their uniform convergence to a solution of the problem.

### On the Exponential Stability of Some Nonlinear Systems

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 131–136

By using Lyapunov functions, we obtain, for the first time, necessary and sufficient conditions for the exponential stability of some nonlinear systems of differential and difference equations.

### On the Asymptotic Behavior of Solutions of Differential Systems

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 137–142

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let $ϕ (t)$ be a scalar continuous monotonically increasing positive function tending to ∞ as $t → ∞$. It is established that if all solutions of a differential system satisfy the inequality $$\left\| {x(t;t_0 ,\;x_0 )} \right\| \leqslant M\frac{{\varphi (t_0 )}}{{\varphi (t)}}\quad \operatorname{for} \;all\quad t \geqslant t_0 ,\quad x_0 \in \left\{ {x:\left\| x \right\| \leqslant \alpha } \right\},$$ then the solution $x(t; t_0, x_0)$ of this differential system tends to 0 faster than $M\frac{{\varphi (t_0 )}}{{\varphi (t)}}$.

### The Skorobogat'ko international mathematical conference

Ptashnik B. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 143-144