Volume 1, Issue 3

We discuss the dynamics of the positive solutions of the delay difference equation in the title for some special values of the parameters A , p and k and we pose a conjecture and two open problems.

It is proved that for any sequence of real numbers satisfying R k ≥ k ( k ≥ 1) and R k = o ( k log 2 k ), k → ∞, there exists a orthonormal system , x ∈ (0; 1), such that none of its subsystems with n k ≤ R k ( k ≥ 1) is a convergence subsystem.

The paper gives sufficient conditions for the existence and nonuniqueness of monotone solutions of a nonlinear ordinary differential equation of the second order subject to two nonlinear boundary conditions one of which is two-point and the other is integral. The proof is based on an existence result for a problem with functional boundary conditions obtained by the author in [Brykalov, Differential Equations 27: 1415-1420, 1991].

Two-dimensional boundary value problems of flow of a viscous micropolar fluid are investigated in the case of linearization by Ozeen's method.

Sturm's type theorems on separation of zeros of solutions are proved for the second order linear differential equations with delayed argument.

It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.

We have obtained the estimate in the terms of partial and mixed moduli of continuity of deviation of Cesáro ( C, ) means of the sequence of rectangular partial sums of n -multiple ( n > 1) conjugate trigonometric series from n -multiple truncated conjugate function. This estimate implies the result on the -convergence (⋋ ≥ 1) of ( C, ) means , provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown, that the -convergence cannot be replaced by ordinary convergence.

Sufficient conditions for the solvability of two-point boundary value problems for the system are given, where f 1 and f 2 : [ a 1 , a 2 ] × R 2 → R are continuous functions.

A classification of classes of equivalent linear differential equations with respect to ω -limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

The problem of a stressed state in a nonhomogeneous infinite plane consisting of two different anisotropic half-planes and having slits of the finite number on the interface line is investigated. It is assumed that a difference between the displacement and stress vector values is given on interface line segments; on the edges of slits we have the following data: boundary values of the stress vector (problem of stress) or displacement vector values on the one side of slits, and stress vector values on the other side (mixed problem). Solutions are constructed in quadratures.