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and Applied Analysis 3 Since Δzn > 0 and zn > 0 eventually, there exists a positive constant M such that zn−l ≥ M for all n ≥ n2. Using this and the positivity of anΔm−1zn in (18) and letting n → ∞, we obtain ∞ ∑ n=n1 Qn < ∞ (19) which is a contradiction to (12).This completes the proof. Remark 6. In the above theorem, we did not impose any condition on β and hence our result is more general than some of the existing results in the literature. In the following, we present other oscillation criteria using Lemma 3. Theorem 7. Let condition (2) hold. Assume that there is a positive decreasing real sequence {ρn} tending to zero such that P(n) is positive for all n ≥ N ∈ N(n0). If (i)

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