# Volume 60, № 1, 2008

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

### Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 8–27

We consider a class of fourth-order nonlinear difference equations of the form $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n\in {\mathbb N} $. We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence conditions of the sums $$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$

### Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 28–55

We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus homoclinic orbits.

### On one mathematical problem in the theory of nonlinear oscillations

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 56–62

We consider one mathematical problem that was discussed by the author and A. M. Samoilenko at the Third International Conference on the Theory of Nonlinear Oscillations (Transcarpathia, 1967).

### On one bifurcation in relaxation systems

Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 63–72

We establish conditions under which, in three-dimensional relaxation systems of the form $$\dot{x} = f(x, y, \mu),\quad, \varepsilon\dot{y} = g(x, y),\quad x= (x_1, x_2) \in {\mathbb R}^2,\quad y\in{\mathbb R },$$ where $0 < ε << 1, |μ| << 1, ƒ, g ∈ C_{∞}$ the so-called “blue-sky catastrophe” is observed, i.e., there appears a stable relaxation cycle whose period and length tend to infinity as μ tends to a certain critical value μ*(ε), μ*(0) 0 = 0.

### On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

Nenya O. I., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 73–80

Continuing our previous investigations, we give simple sufficient conditions for global stability
of the zero solution of the difference equation
*x*_{n+1} = *qx _{n }* +

*f*(

_{n }*x*,...,

_{n }*x*),

_{n-k }*n ∈ Z*, where nonlinear functions

*f*satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2

_{n }*k*+ 2) /3] disjoint subintervals, and, for

*q*from each subinterval, we present a global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition.

### Some modern aspects of the theory of impulsive differential equations

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 81–94

We give a brief survey of the main results obtained in recent years in the theory of impulsive differential equations.

### On extension of the Sturm-Liouville oscillation theory to problems with pulse parameters

Pokornyi Yu. V., Shabrov S. A., Zvereva M. B.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 95–99

Oscillation spectral properties (the number of zeros, their alternation for eigenfunctions, the simplicity of the spectrum, etc.) are described for the Sturm-Liouville problem with generalized coefficients.

### An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 100–106

A generalization of the classical Leray-Schauder fixed-point theorem based on the infinite-dimensional Borsuk-Ulam-type antipode construction is proposed. A new nonstandard proof of the classical Leray-Schauder fixed-point theorem and a study of the solution manifold of a nonlinear Hamilton-Jacobi-type equation are presented.

### Stability for retarded functional differential equations

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 107–126

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs.

### Periodic moving waves on 2D lattices with nearest-neighbor interactions

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 127–139

We study the existence of periodic moving waves on two-dimensional periodically forced lattices with linear coupling between nearest particles and with periodic nonlinear substrate potentials. Such discrete systems can model molecules adsorbed on a substrate crystal surface.

### Polynomial quasisolutions of linear second-order differential-difference equations

Cherepennikov V. B., Ermolaeva P. G.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 140–152

The second-order scalar linear difference-differential equation (LDDE) with delay $$\ddot{x}(t) + (p_0+p_1t)\dot{x}(t) = (a_0 +a_1t)x(t-1)+f(t)$$ is considered. This equation is investigated with the use of the method of polynomial quasisolutions based on the presentation of an unknown function in the form of polynomial $x(t)=\sum_{n=0}^{N}x_n t^n.$ After the substitution of this function into the initial equation, the residual $\Delta(t)=O(t^{N-1}),$ appears. The exact analytic representation of this residual is obtained. The close connection is demonstrated between the LDDE with varying coefficients and the model LDDE with constant coefficients whose solution structure is determined by roots of a characteristic quasipolynomial.