Oscillation of a class of nonlinear neutral difference equations of higher order

Oscillation of higher order neutral functional difference equations with positive and negative coefficients

“…Our results hold good for G(u) ≡ u, f n ≡ 0 and u n ≡ 0. The last but not the least, this paper corrects, generalizes and improves some of the results of [11,12,14,16,17,19,21].…”

“…Moreover, we observe that the existing papers in the literature do not have much to offer when p n satisfies (A4), (A6) or (A7). In this direction we find that, the authors in [12] have obtained sufficient conditions for the oscillation of solutions of the equation (i) Every solution of (1.10) oscillates for m even.…”

“…Unfortunately, the following example contradicts the above theorems of [12]. where m may be any odd or even integer.…”

“…The authors of the papers [12,19,21] have studied sub-linear equation, and their results do not hold for linear or super linear equations (i.e. (1.5) satisfying (H3) or (1.7) with α ≥ 1).…”

“…Thus, (3.16) and (1.11) are equivalent, when q n is monotonic. Now we quote a result from [12], which uses the condition (3.16). It is important to note that our Example 1.1 contradicts the above theorem, because, from the neutral equation (1.12), we find q n = 4 (n+1)/3 , which is monotonic.…”