Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients

Abstract

A criterion is established on the bounded solutions of type higher-order nonlinear neutral differential equations of type oscillatory or tending to zero at infinitywhere t ≥ t0 , n ≥ 2, a, p are positive, r, q, φ are allowed to alternate in sign infinitely many times, F, G are continuous functions, and κ, τ, σ are strictly increasing unbounded continuous delay functions.

Necessary and sufficient conditions on the asymptotic behavior of second‐order neutral delay dynamic equations with positive and negative coefficients

“…The following lemma is a particular case of ,Lemma 2.3 and ,Lemma 2. Lemma Let , and be fixed.…”

“…In , the author studies oscillation of unbounded solutions of where the arguments of the equation are as stated previously. Thus, Theorem extends the main result of (for second‐order equations) to . One may see that the method introduced here is not applicable to get information about the unbounded solutions of when n ≥ 3 in some cases.…”

“…Therefore, z x and z x are both positive on OEt 2 , 1/ T and so are y x and y x by (10) and (11). Moreover, we have`y D 1, where`y is defined by (16). Considering the fact that lim t!1 z x .t/ is finite (recall that z x is eventually positive and decreasing), we get from (12) that…”

Oscillation criteria for higher‐order nonlinear dynamic equations with Laplacians and a deviating argument on time scales

“…Recently, using Riccati substitution, Hassan and Kong obtained asymptotics and oscillation criteria for the n th‐order half‐linear dynamic equation with deviating argument where α [1, n − 1]: = α 1 ⋯ α n − 1 ; and Grace and Hassan further studied the asymptotics and oscillation for the higher‐order nonlinear dynamic equation with Laplacians and deviating argument However, the establishment of the results in requires the restriction on the time scale that g ∗ ∘ σ = σ ∘ g ∗ with , which is hardly satisfied, see conclusion 1 in for such a special case. For more results on dynamic equations, we refer the reader to the papers .…”

Oscillation Criteria for Solutions to Nonlinear Dynamic Equations of Higher Order

“…In the literature many papers discuss the behavior of solutions for certain classes of dynamic equations; we refer the reader to [1,3,5,9,11,12,13,15,18,19,20,23,24,25,26,29,21,27] and the references cited therein. In particular these papers present oscillatory criteria and asymptotic behavior for first, second and third order dynamic equations on time scales and some interesting results were obtained for special cases of (1.1); see [10,14,16,17,28].…”