Нелінійні коливання, 2001, № 4

On three solutions of the second order periodic boundary-value problem

Mon, 01 Jan 2001 00:00:00 GMT

On three solutions of the second order periodic boundary-value problem Draessler, J.; Rachůnková, I. We consider the periodic boundary-value problem x'' + a(t)x' + b(t)x = f(t, x, x'), x(') =x(2π), x'(0) = x' (2π), where a, b are Lebesgue integrable functions and f fulfils the Caratheodory conditions. We extend results about the Leray – Schauder topological degree and ´ present conditions implying nonzero values of the degree on sets defined by lower and upper functions. Using such results we prove the existence of at least three different solutions to the above problem.

To the problem of complementability of a periodic frame to a periodic basis

Mon, 01 Jan 2001 00:00:00 GMT

To the problem of complementability of a periodic frame to a periodic basis Burylko, O.A.; Davydenko, A.A. We obtain sufficient conditions (and necessary conditions in the simplest case) of complementability of a periodic frame to a periodic basis for the Euclidean space in terms of monodromy matrices of some linear system of differential equations built by using this periodic frame. We consider the problem of complementability for introducing local coordinates in a neighbourhood of a smooth m-dimensional invariant torus of a dynamic system in the Euclidean space R n if the dimensions satisfy the inequality m + 1 < n ≤ 2m.

Asymptotic stability for a thermoelectromagnetic material

Mon, 01 Jan 2001 00:00:00 GMT

Asymptotic stability for a thermoelectromagnetic material Amendola, G. In this work we consider a linear thermoelectromagnetic material, whose behaviour is characterized by two rate-type equations for the heat flux and the electric current density. We derive the restrictions imposed by the laws of thermodynamics on the constitutive equations and introduce the free energy which yields the existence of a domain of dependence. Uniqueness, existence and asymptotic stability theorems are then proved.

On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel

Mon, 01 Jan 2001 00:00:00 GMT

On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel Zolotenko, G.F. The problem of integrating the Laplace equation in a changing 3-dimensional region, with the initial and boundary conditions, is investigated. The paper is mainly devoted to the problem arising in dynamics of an inviscid incompressible fluid which partially fills a moving vessel and is in irrotational absolute motion. In this case the considered space region is bounded by the rigid vessel’s walls and the unknown free surface of fluid. The boundary conditions consist of the Neyman conditions on the rigid walls and the nonlinear kinematic and dynamic conditions on the free surface. Besides, the condition of a constancy of the region’s volume is imposed. The concept of a solution of this problem is analyzed. One distinguishes a certain class of solutions and proves their existence. An example of such a solution is given.