Oscillation Theorems for Second-Order Nonlinear Neutral Delay Differential Equations

Abstract

We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature.

Oscillation of nonlinear second-order neutral delay differential equations

“…The results obtained in this paper complement those reported in [6,8,11,13,21,26,28] since our criteria can be applied to the case where β = λ and do not require assumptions (1.3), (1.5), and (1.6). Furthermore, our criteria solve a problem posed in [7,Summary] because these results can be applied to equation ( Let β = λ = 1, a(t) = 1, p(t) = p 0 = 9/10, τ(t) = t/4, q(t) = α/t 2 , δ(t) = t/5, f(u) = u, L = L 0 = 1, and ϕ(t) = t. Then τ 0 = 1/4, Q(t) = min{q(t), q(τ(t))} = α/t 2 , and so lim sup …”

Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation

“…But, these conditions are very restrictive for applications; for example, letting τ (t) = t/2 and δ(t) = t − 1, we see that τ • δ = δ • τ . Therefore, our results also improve the results in [11,12,17] in the sense that we do not require that τ…”

“…have been studied by many authors utilizing different methods; see, for example, [2,6,[11][12][13][15][16][17] and the references therein. In reviewing the literature, it becomes apparent that results on the oscillatory and asymptotic behavior of third order neutral differential equations are relatively scarce and most such results are concerned with the cases where 0 ≤ p(t) ≤ p 0 < 1 and −1 ≤ p(t) < 0.…”

“…On the other hand, to the best of our knowledge, there are a few results regarding oscillatory and asymptotic behavior of third order neutral differential equations in case p(t) > 1; see [11,12,17]. In [11,17], the authors considered equations (1.8) and (1.6), respectively, under the assumption that τ • δ = δ • τ .…”

“…In [11,17], the authors considered equations (1.8) and (1.6), respectively, under the assumption that τ • δ = δ • τ . In [12], the authors studied equation (1.9) with τ • δ = δ • τ and τ • σ = σ • τ . But, these conditions are very restrictive for applications; for example, letting τ (t) = t/2 and δ(t) = t − 1, we see that τ • δ = δ • τ .…”