On the Oscillation of Even-Order Half-Linear Functional Difference Equations with Damping Term

Abstract

We investigate the oscillatory behavior of solutions of themth order half-linear functional difference equations with damping term of the formΔ[pnQ(Δm-1yn)]+qnQ(Δm-1yn)+rnQ(yτn)=0,n≥n0, wheremis even andQ(s)=sα-2s,α>1is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.

Oscillation criteria for higher-order neutral type difference equations

“…However, these similar results cannot be found under the conditions r(n) = 1 and τ n ≥ n . Moreover, for r(n) ̸ = 1 , our results are completely different from those in [10,11,15,17,18] by the condition τ n ≥ n .…”

“…For instance, they are frequently used for the study of distributed networks containing lossless transmission lines. Recently, much researches have been carried out on the oscillatory and asymptotic behaviour of solutions of higher-order neutral type delay and advanced difference equations ( [5,6,[8][9][10][11][12][14][15][16][17][18]). In these studies, the authors considered the case of n − τ ≤ n or τ n ≤ n for higher order neutral type linear delay difference equations of the form ∆ m (y n + p n y τn ) + q n y σn = 0, m ≥ 2, and they obtained many results on this assumption.…”

An improved approach for studying oscillation of second-order neutral delay differential equations

“…The interest in this subject has been reflected by extensive references in monographs [ 3 – 7 ]. We also refer the reader to the papers [ 8 – 10 ] and the references cited therein regarding similar discrete analogues of ( E ) and its particular cases and modifications.…”

Some Oscillation Results for Even Order Delay Difference Equations with a Sublinear Neutral Term

“…In the past few years, there is a great interest in studying the oscillatory and asymptotic behavior of solutions of higher order neutral type difference equations, since such type of equations naturally arises in the applications including problems in population dynamics or in cobweb models in economics and so on. The problem of finding sufficient conditions which ensure that all solutions of the neutral type difference equations are oscillatory has been investigated by many authors; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12] and the references cited therein. In all the results the neutral term is linear and few results are available when the neutral term is nonlinear; see [13][14][15][16][17][18][19][20][21].…”