# Volume 64, № 10, 2012

### Banach Manifolds with Bounded Structure and the Gauss?Ostrogradskii Formula

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1299-1313

We propose a version of the Gauss - Ostrogradskii formula for a Banach manifold with uniform atlas.

### Vibrating Systems with Rigid Light-Weight Inclusions: Asymptotics of the Spectrum and Eigenspaces

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1314-1329

We study the asymptotic behavior of the eigenvalues and eigenfunctions of a singularly perturbed boundary-value problem for a second-order elliptic operator. The problem describes the eigenmodes of an elastic system with finite number of stiff light-weight inclusions of arbitrary shape. The leading terms of the asymptotic representation of eigenelements are constructed with regard for their multiplicity. The justification of the asymptotic formulas is based on the uniform resolvent convergence of a certain family of unbounded self-adjoint operators.

*D*-homothetic deformation of normal almost contact metric manifolds

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1330-1329

The object of the present paper is to study a transformation called the $D$-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a $(2n + 1)$-dimensional normal almost contact metric manifold, the Ricci operator $Q$ commutes with the structure tensor $\phi$ under certain conditions, and the operator $Q\phi - \phi Q$ is invariant under a $D$-homothetic deformation. We also discuss the invariance of $\eta$-Einstein manifolds, $\phi$-sectional curvature, and the local $\phi$-Ricci symmetry under a $D$-homothetic deformation. Finally, we prove the existence of such manifolds by a concrete example.

### Asymptotics of Solutions of Nonautonomous Second-Order Ordinary Differential Equations Asymptotically Close to Linear Equations

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1346-1364

Asymptotic representations are obtained for a broad class of monotone solutions of nonautonomous binary differential equations of the second order that are close in a certain sense to linear equations.

### A new method of generating of traveling wave solutions for coupled nonlinear equations

Ding Shanyu, Jiuli Yin, Lixin Tian, Xinghua Fan

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1365-1372

A new algebraic transformation method is constructed for finding traveling-wave solutions of complicated nonlinear wave equations on the basis of simpler ones. The generalized Dullin - Gottwald - Holm (DGH) equation and mKdV equations are chosen to illustrate our method. The solutions of the DGH equation can be obtained directly from solutions of the mKdV equation. Conditions under which different solutions appear are also given. Abundant traveling-wave solutions of the generalized DGH equation are obtained, including periodic solutions, smooth solutions with decay, solitary solutions, and kink solutions.

### Generalization of One Sufficient Condition for Fourier Multipliers

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1373-1380

We obtain new sufficient conditions for Fourier multipliers in the Hardy spaces $H_p(\mathbb{R}^n),\; 0 < p < 2$. These conditions are given in terms of the simultaneous behavior of a function and its derivatives. The results of this paper generalize the corresponding theorems of A. Miachi.

### Solution of a Linear Second-Order Differential Equation with Coefficients Analytic in the Vicinity of a Fuchsian Zero Point

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1381-1393

We obtain a solution of a second-order differential equation with coefficients analytic near a Fuchsian zero point. This solution is expressed via the hypergeometric functions and the fractional-order hypergeometric functions introduced in this paper.

### FD-method for solving the nonlinear Klein - Gordon equation

Dragunov D. V., Makarov V. L., Sember D. A.

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1394-1415

We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein-Gordon equation. Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are illustrated by a numerical example.

### Kolmogorov widths of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables in the space $L_q$

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1416-1425

We obtain exact-order estimates for the Kolmogorov widths of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables in the space $L_q$ for $1 ≤ p, q ≤ ∞$.

### Uniformly distributed ridge approximation of some classes of harmonic functions

Babenko V. F., Levchenko D. A.

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1426-1431

We determine the exact values of the uniformly distributed ridge approximation of some classes of harmonic functions of two variables.

### On split metacyclic groups

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1432-1437

Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.

### International conference "Theory of approximation of functions and its applications" dedicated to the 70 th birthday of the corresponding member of NASU Professor O. I. Stepanets (1942 - 2007)

Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sokolenko I. V.

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1438-1440