Abstract
We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations on a time scale , where , . Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main results.
Highlights
We are concerned with oscillation behavior of the second order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t xα τ1 t q2 t xβ τ2 t
0, 1.1 on an arbitrary time scale T, where ztxtptx τ0 t, γ, α, and β are quotient of odd positive integers such that 0 < α < γ < β, r, p, q1, and q2 are rd-continuous functions on T, and r, q1, and q2 are positive, −1 < −p0 ≤ p t < 1, p0 > 0; the so-called delay functions τi : T → T satisfy that τi t ≤ t for t ∈ T and τi t → ∞ as t → ∞, for i 0, 1, 2, and there exists a function τ : T → T which satisfies that τ t ≤ τ1 t, τ t ≤ τ2 t, and τ t → ∞ as t → ∞
Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that sup T ∞ and define the time scale interval t0, ∞ T by t0, ∞ T : t0, ∞ ∩ T
Summary
We are concerned with oscillation behavior of the second order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t xα τ1 t q2 t xβ τ2 t1.1 on an arbitrary time scale T, where ztxtptx τ0 t , γ, α, and β are quotient of odd positive integers such that 0 < α < γ < β, r, p, q1, and q2 are rd-continuous functions on T, and r, q1, and q2 are positive, −1 < −p0 ≤ p t < 1, p0 > 0; the so-called delay functions τi : T → T satisfy that τi t ≤ t for t ∈ T and τi t → ∞ as t → ∞, for i 0, 1, 2, and there exists a function τ : T → T which satisfies that τ t ≤ τ1 t , τ t ≤ τ2 t , and τ t → ∞ as t → ∞.Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that sup T ∞ and define the time scale interval t0, ∞ T by t0, ∞ T : t0, ∞ ∩ T.Advances in Difference EquationsWe will consider the two cases ∞ Δt t0 r1/γ t ∞,∞ Δt < ∞. t0 r1/γ tRecently, there has been a large number of papers devoted to the delay dynamic equations on time scales, and we refer the reader to the papers in 1–17 .Agarwal et al 1 , Sahiner 10 , Saker , Saker et al , and Wu et al 15 studied the second-order nonlinear neutral delay dynamic equations on time scales rtytptyτtΔγΔft, y δ t0, t ∈ T, 1.4 where 0 ≤ p t < 1, and 1.2 holds. We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t xα τ1 t q2 t xβ τ2 t 0 on a time scale T, where ztxtptx τ0 t , 0 < α < γ < β. By developing a Riccati transformation technique some sufficient conditions for oscillation of all solutions of 1.1 on time scales are established.
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