New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

and Applied Analysis 3 By using weaker conditions than in 4, 10 , Karpuz et al. 1 have established oscillation criteria for differential equation [ x t ∑ i∈R ri t x αi t ]′′ ∑ i∈P pi t x ( βi t ) − ∑ i∈Q qi t x ( γi t ) f t . 1.6 In this paper, we shall continue in the direction to study the oscillatory properties of 1.1 and 1.2 . We establish new oscillation criteria for 1.1 and 1.2 , which extend and improve the corresponding results in 1, 4, 10 . We also give two examples to illustrate our main results. 2. Main Results The following properties of the set L1 t0,∞ in 1 are needed for our subsequent discussion. Property 1. If f ∈ L1 t0,∞ and f ∈ C t0,∞ ,R , then lim inft→∞f t 0. Corollary 2.1. Suppose that f ∈ L1 t0,∞ and limt→∞f t exists; then limt→∞f t 0. Property 2. If f ∈ C t0,∞ ,R and f ∈ L1 t1,∞ , where t1 ≥ t0, then we have f ∈ L1 t0,∞ . Property 3. Let t1 be such that g t1 ≥ t0. Suppose g ∈ D t1,∞ with lim supt→∞g ′ t < ∞ and f ∈ C t0,∞ ,R . If f ◦ g ∈ L1 t1,∞ holds, then f ∈ L1 t0,∞ . Property 4. Let t1 be such that g t1 ≥ t0. Suppose g ∈ D t1,∞ with lim inft→∞g ′ t > 0, f ◦g ∈ C t1,∞ ,R , and f ∈ C t0,∞ ,R . If f ∈ L1 t0,∞ holds, then f ◦ g ∈ L1 t1,∞ holds. For simplicity, we denote the set of bounded functions by B t0,∞ : { f ∈ C t0,∞ ,R : ‖f‖ < ∞ } , 2.1 where ‖f‖ : sup∣f t ∣, t ≥ t0 } . 2.2 For an arbitrary function ψ : Q → P, which satisfies A1 – A3 , we denote the function φ : t0,∞ → R by