Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales
نویسندگان
چکیده
and Applied Analysis 3 where m ∈ N, α is the quotient of odd positive integers, t0 ∈ T, the time scale interval t0,∞ T {t ∈ T : t ≥ t0}, a ∈ Crd t0,∞ T, 0,∞ , p ∈ C t0,∞ T,R , τ, δ ∈ C T,T with limt→∞τ t limt→∞δ t ∞ and f ∈ C t0,∞ T×R,R satisfying the following conditions: i uf t, u > 0 for any t ∈ t0,∞ T and u/ 0. ii f t, u is nondecreasing in u for any t ∈ t0,∞ T. Since we are interested in the oscillatory behavior of solutions near infinity, we assume that supT ∞. By a solution of 1.7 we mean a nontrivial real-valued function x ∈ Crd Tx,∞ T,R , Tx ≥ t0, such that a t x t − p t x τ t Δ m α ∈ C1 rd Tx,∞ T,R and satisfies 1.7 on Tx,∞ . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of 1.7 is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. 2. Auxiliary Results We state the following conditions, which are needed in the sequel: H1 ∫∞ t0 a t −1/αΔt ∞; H2 there exist constants a1, b1 ∈ 0, 1 with a1 b1 < 1 such that −a1 ≤ p t ≤ b1 for all t ∈ t0,∞ T; H3 there exist constants a2, b2 ∈ 1,∞ such that −a2 ≤ p t ≤ −b2 for all t ∈ t0,∞ T; H4 there exist constants a3, b3 ∈ 1,∞ such that a3 ≤ p t ≤ b3 for all t ∈ t0,∞ T. Let k be a nonnegative integer and s, t ∈ T; we define two sequences of functions hk t, s and gk t, s as follows: hk t, s ⎧ ⎪ ⎨ ⎪ ⎩ 1 if k 0, ∫ t s hk−1 τ, s Δτ if k ≥ 1, gk t, s ⎧ ⎪ ⎨ ⎪ ⎩ 1 if k 0, ∫ t s gk−1 σ τ , s Δτ if k ≥ 1. 2.1 By Theorems 1.112 and 1.60 of 2 , we have hk t, s −1 gk s, t for all t, s ∈ T, ht k t, s ⎧ ⎨ ⎩ 0 if k 0, hk−1 t, s if k ≥ 1, gt k t, s ⎧ ⎨ ⎩ 0 if k 0, gk−1 σ t , s if k ≥ 1, 2.2 4 Abstract and Applied Analysis where gt k t, s and h Δt k t, s denote for each fixed s the derivatisve of gk t, s and hk t, s with respect to t, respectively. Lemma 2.1 see 23, 24 . Assume that s, t ∈ T and g ∈ Crd T × T,R , then ∫ t
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