Oscillation of Second Order Difference Equation with a Sub-linear Neutral Term

Abstract

This paper deals with the oscillation of a certain class of second order difference equations with a sub-linear neutral term. Using some inequalities and Riccati type transformation, four new oscillation criteria are obtained. Examples are included to illustrate the main results.

Oscillation of second order difference equation with several super-linear neutral terms

“…In this paper, by using the Riccati transformation, the summation averaging technique and a new comparison theorem, we prove some suffcient conditions which are new, extend and complement some of the results established in [2,5,6,9,10,[12][13][14][15][16][17][18] either for the case m = 1 or for the case α i = 1 for i = 1, 2, 3, . .…”

“…In view of the above discussion, in this paper we choose to examine the oscillatory behavior of such type of difference equations because similar properties for difference equations with linear neutral term received great attention of the researchers. The oscillatory behavior of second order difference equations with sub-linear neutral term, that is, x α (nk) with 0 < α < 1, was studied in [2,6,14,17,18], and therefore in this paper we investigate the oscillatory properties of solutions of Eq. (1.1) with super-linear neutral terms, that is,…”

“…Example 3.2 Consider the second order neutral difference equation 2 x(n) + 1 2 n x 7/3 (n -1) + 1 3 n x 3 (n -2) + 3 n x 1/3 (n -1) = 0, n ≥ 1.…”

Oscillation results for nonlinear second order difference equations with mixed neutral terms

“…In the qualitative analysis of such systems, indeed, the oscillatory behavior of solutions of equations, where the rate of the growth depends not only on the current and the past states but also on rate of change in the past, play an important role [22][23][24]. In the light of this motivation and justification, different results have been reported regarding the asymptotic behavior of second order difference equations with neutral terms [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. For relevant results on the application of oscillation theory, the reader can consult [18,41,42].…”

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

“…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].…”