On oscillatory nonlinear fourth-order difference equations with delays

Abstract

In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form ∆ 2 (r(n)∆ 2 (y(n) + p(n)y(n − m))) + q(n)G(y(n − k)) = 0 is studied under the assumption ∞ n=0 n r(n) < ∞. New oscillation criteria have been established which generalize some of the existing results in the literature.

Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

“…From Lemma 1, one may have eight possible cases for n ≥ n 1 . From (19) and (H 1 ), we see that ∞ n�n 0 B(n, n 0 ) cannot be bounded, and by Lemma 2, case (i) − (iv) are impossible. Now, let us assume that one of the cases (v) − (viii) holds.…”

“…But the term on the right side of the above inequality tends to ∞ as n ⟶ ∞ due to (19), which contradicts the fact that {u(n)} is decreasing. is completes the proof.…”

“…In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the last several years, there has been a constant interest in getting sufficient conditions for oscillatory behavior of different classes of fourth-order difference equations with or without deviating arguments, (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], and the references cited therein). In particular, the authors in [5,8,13,15,18,19] established oscillation results for (1) under the following assumptions:…”

Oscillation of Nonlinear Neutral Delay Difference Equations of Fourth Order

“…In [18], Tripathy established new oscillation criteria for all solution oscillatory for the class of fourth-order neutral functional differential equations of the form…”