Oscillation Criteria for Second-Order Superlinear Neutral Differential Equations

and Applied Analysis 3 where τ t ≤ t, σ t ≤ t, τ ′ t τ0 > 0, 0 ≤ p t ≤ p0 < ∞, and the authors obtained some oscillation criteria for 1.7 . However, there are few results regarding the oscillatory problem of 1.1 when τ t ≥ t and σ t ≥ t. Our aim in this paper is to establish some oscillation criteria for 1.1 under the case when τ t ≥ t and σ t ≥ t. The paper is organized as follows. In Section 2, we will establish an inequality to prove our results. In Section 3, some oscillation criteria are obtained for 1.1 . In Section 4, we give two examples to show the importance of the main results. All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough. 2. Lemma In this section, we give the following lemma, which we will use in the proofs of our main results. Lemma 2.1. Assume that α ≥ 1, a, b ∈ . If a ≥ 0, b ≥ 0, then a b ≥ 1 2α−1 a b α 2.1 holds. Proof. i Suppose that a 0 or b 0. Obviously, we have 2.1 . ii Suppose that a > 0, b > 0. Define the function g by g u u, u ∈ 0,∞ . Then g ′′ u α α − 1 uα−2 ≥ 0 for u > 0. Thus, g is a convex function. By the definition of convex function, for λ 1/2, a, b ∈ 0,∞ , we have g ( a b 2 ) ≤ g a g b 2 , 2.2