Strong non-oscillation and Oscillation second order half-linear difference equation

Abstract

This article is devoted to the study of the signs of strong oscillation and non-oscillation of one
class of second-order quasilinear and linear difference equations. A lot of articles, monographs
and books are devoted to the question of the oscillatory properties of difference equations. More
strongly investigated are linear, quasilinear difference equations of the second order with various
methods. Among the various methods for studying the oscillation properties of differential and
difference equations, there are two main methods, one of which is called the "Riccati technique" ,
which proceeds from the theory of linear differential and difference equations, and the other is
the "variational principle" or simply the "variational method" . In most works devoted to the
oscillatory properties of differential and difference equations, the Riccati technique is used. This is
due to the fact that in the variational method the problem is reduced to studying the fulfillment
of some weighted inequality on the set of compactly supported sequences, which is an equally
difficult task. In this paper, using the results of the authors on the Hardy weight inequalities in
difference form and based on the variational principle, various necessary and sufficient conditions
for strong oscillation and non-oscillation for the two-term half-linear and linear difference equation
of the second order are obtained. As an application of the results obtained, criteria are given
for boundedness below and discreteness of the spectrum of a one-term difference operator of the
second order.