Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients

Asymptotic behaviour of solutions of the difference Schrödinger equation

“…The proof of Theorem 2.1 is given in Appendix A. Theorem 2.1 is a difference analogue of the similar results for differential equations, obtained in [10].…”

Parametric Resonance in Adiabatic Oscillators

“…The latter follows by induction if we use the assumption of the proposition and the fact that matrices A i1... i l also belong to Ξ. To see this we note that each of these matrices is actually a mean value of the sum of matrix products A i1... ip (t)Y i1... is (t) [12]. We have already said that the sum and the product of two matrices from the class Ξ is a matrix that also belong to Ξ.…”

“…(1.3). In paper [12], the averaging changes of variable were represented in the most general form. We will apply these results to the following system with oscillatory decreasing coefficients:…”

“…(1.1), has a discrete type: for every 0 < ρ ≤ 1 there exists a finite set of values of parameter λ that produce unbounded solutions. In paper [12], the adiabatic oscillator of the form (1.2) with function In this work, we will be interested in the existence of unbounded solutions (instability of solutions) of equation…”

Method of averaging for systems with main part vanishing at infinity

“…The averaging change of variable having the form (2.19) was introduced for the first time in the paper [9]. Now suppose that the averaged system…”