# Volume 65, № 1, 2013

### Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Berezansky Yu. M., Boichuk О. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

### Hybrid type generalized multivalued vector complementarity problems

Agarwal P., Ahmad M. K., Salahuddin

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 7-20

We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and obtain results on the equivalence between the generalized multivalued vector complementarity problems and the generalized multivalued vector variational inequality problems.

### Dynamics of periodic modes for the phenomenological equation of spin combustion

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 21-43

We consider a scalar parabolic equation in the circle of radius r. This problem is a gasless combustion phenomenological model in the surface of a cylinder of $r$ radius. We consider the problems of the existence, asymptotic form and stability of traveling waves and the nature of gaining, losing their stability.

### Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations

Blackmore D., Golenia J., Prykarpatsky A. K., Prykarpatsky Ya. A.

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 44-57

Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator. New two-dimensional Boole-type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.

### Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices

Diblik J., Fečkan M., Pospíšil M.

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 58-69

We represent a solution of a nonhomogeneous second-order differential equation with two delays using matrix functions under the assumption that the linear parts are given by permutable matrices.

### Limit theorems for one-dimensional boundary-value problems

Kodlyuk T. I., Mikhailets V. A., Reva N. V.

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 70-81

We study the limit with respect to a parameter in the uniform norm for solutions of general boundary-value problems for systems of linear ordinary differential equations of the first order. A generalization of the Kiguradze theorem (1987) to these problems is obtained. The conditions on the asymptotic behavior of the coefficients of the systems are weakened as much as possible. Sufficient conditions for the Green matrices to converge uniformly to the Green matrix of the limit boundary-value problem are found as well.

### New methods for the investigation of periodic solutions in ring systems of unidirectionally coupled oscillators

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 82-102

We consider special systems of ordinary differential equations, namely, ring systems of unidirectionally coupled oscillators. A new method is developed for the investigation of the problem of the existence and stability of periodic solutions for this class of systems. The specific feature of this approach is the use of certain auxiliary delay systems for the determination of cycles and for the analysis of their properties. The proposed method is illustrated by a specific example.

### Theorem on the existence of an invariant section over $\mathbb{R}^m$ for the indefinite monotone system in $\mathbb{R}^m \times \mathbb{R}^n$

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 103-118

We consider a nonlinear system on the direct product $\mathbb{R}^m \times \mathbb{R}^n$. For this system, under the conditions of indefinite coercivity and indefinite monotonicity, we establish the existence of a bounded Lipschitzian invariant section over $\mathbb{R}^m$.

### On the asymptotic properties of continuous solutions of the systems of nonlinear functional equations

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 119-125

For systems of nonlinear functional equations, we study asymptotic properties of their solutions continuously differentiable and bounded for $t \geq T > 0$ in a neighborhood of the singular point $t = +\infty$.

### Averaging of set-valued impulsive systems

Perestyuk N. A., Skripnik N. V.

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 126-142

We give a review of the development of ideas of the averaging method for some classes of set-valued impulsive systems (impulsive differential inclusions, impulsive differential equations and inclusions with Hukuhara derivative, and impulsive fuzzy differential equations and inclusions).

### On interaction of an elastic wall with a Poiseuille type flow

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 143-160

We study dynamics of a coupled system consisting of the $3D$ Navier-Stokes equations which is linearized near a certain Poiseuille type flow in an (unbounded) domain and a classical (possibly nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup $S_t$ on an appropriate phase space. Then under some conditions concerning the underlying (Poiseuille type) flow we prove the existence of a compact finite-dimensional global attractor for this semigroup and also show that $S_t$ is an exponentially stable $C_0$-semigroup of linear operators in the fully linear case. Since we do not assume any kind of mechanical damping in the plate component, this means that dissipation of the energy in the fluid flow due to viscosity is sufficient to stabilize the system.