On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

and Applied Analysis 3 where c0 0, ci n n − 1 · · · n − i 1 , and cn n!. Since τ > 0, ∫ t t0 x s ds ∫ t τ t0 τ x s − τ ds, 2.5 we have ∫ t τ t0 τ x s − τ ds ≥ ∫ t t0 τ x s − τ ds. 2.6 This together with 2.4 and 2.5 yield ∫ t t0 t − s e s ds ≥ − n−1 ∑ i 0 ci t − t0 n−ix n−i−1 t0 − cn ∫ t0 τ t0 x s − τ ds ∫ t t0 [ cnx s − τ − t − s q s x s − τ ] ds. 2.7 For given t and s t > s , set F x cnx − t − s q s x, x > 0, 0 < λ < 1. 2.8 It is not difficult to see that F x obtains its minimum at x cn/ t − s n q s 1/ 1−λ and Fmin λ − 1 λ 1−λ n! λ/ λ−1 [ t − s q s ]1/ 1−λ Q t, s . 2.9 It implies that cnx s − τ − t − s q s x s − τ ≥ Q t, s . 2.10 Therefore, for any τ > 0, multiplying 2.7 by t−t0 −n, using 2.10 , and taking lim inf on both sides of 2.7 , we get a contradiction with 2.1 . This completes the proof of Theorem 2.1. Theorem 2.2. Assume that 0 < λ < 1 and τ < 0. If 2.1 and 2.2 hold, then all solutions of 1.1 satisfying x t O t are oscillatory for any τ < 0. Proof. Let x t be a nonoscillatory solution of 1.1 satisfying x t O t . Without loss of generality, we may assume that x t > 0 for t ≥ t0, and there exists a positive constant M > 0 such that x t ≤ Mt. Similar to the corresponding computation in Theorem 2.1 and noting that τ < 0, we have ∫ t t0 t − s e s ds ≥ − n−1 ∑ i 0 ci t − t0 n−ix n−i−1 t0 cn ∫ t0 t0 τ x s − τ ds − cn ∫ t t τ x s − τ ds ∫ t t0 [ cnx s − τ − t − s q s x s − τ ] ds. 2.11 4 Abstract and Applied Analysis Since x t ≤ Mt, we get ∫ t t τ x s − τ ds ≤ M n 1 [ t − τ n 1 − t 1 ] . 2.12 Then, for any τ < 0, multiplying 2.11 by t − t0 −n, using 2.10 and 2.12 , and taking lim inf on both sides of 2.11 , we get a contradiction with 2.1 . This completes the proof of Theorem 2.2. The main results in this paper can also be extended to the case of time-varying delay. That is, we can consider the following equation: x n t q t |x σ t |λ−1x σ t e t , 2.13 where σ t is continuously differentiable on t0,∞ , limt→∞σ t ∞, and σ ′ t > 0 for t sufficiently large. Without loss of generality, say σ ′ t > 0 for t ≥ t0. Similar to the analysis as before, we have the following delay-independent and derivative-dependent oscillation criteria for 2.13 . Theorem 2.3. Assume that 0 < λ < 1 and σ t ≤ t. If lim sup t→∞ 1 t − t0 n ∫ t t0 [ t − s e s Q̃ t, s ] ds ∞, lim inf t→∞ 1 t − t0 n ∫ t t0 [ t − s e s − Q̃ t, s ] ds −∞, 2.14 where Q̃ t, s λ − 1 λ 1−λ n!σ ′ s )λ/ λ−1 [ t − s q s ]1/ 1−λ , 2.15 q s max{−q s , 0}, then all solutions of 2.13 are oscillatory. Theorem 2.4. Assume that 0 < λ < 1 and σ t ≥ t. If 2.1 and 2.2 hold, and there exists a continuous function φ t ≥ 0 on t0,∞ such that ∫ t σ−1 t φ s ds O t n , where σ−1 is the inverse of σ t , then all solutions of 2.13 satisfying x t O φ t are oscillatory. 3. Examples In this section, we work out two examples to illustrate the main results. Example 3.1. Consider the following equation: x n t t sin t|x t − τ |λ−1x t − τ t cos t, t ≥ 0, 3.1 Abstract and Applied Analysis 5 where τ / 0, α ≥ 0, β > 0, and 0 < λ < 1 are constants. Note that Q t, s ≥ Λ t − s n/ 1−λ s 1−λ , 3.2and Applied Analysis 5 where τ / 0, α ≥ 0, β > 0, and 0 < λ < 1 are constants. Note that Q t, s ≥ Λ t − s n/ 1−λ s 1−λ , 3.2 where Λ λ − 1 λ 1−λ n! λ/ λ−1 < 0. We have ∫ t