Oscillation theorems for second order neutral differential equations

Abstract

a b s t r a c tThe aim of this paper is to study the oscillation of the second order neutral differential equationsThe obtained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. The obtained comparison principles essentially simplify the examination of the studied equations.

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 …”

Oscillation of Second-Order Neutral Differential Equations

“…(24) are oscillatory provided that [4] and Hasanbulli and Rogovchenko [11] because we use assumption (2) rather than (3) and, on the other hand, those by Li et al [14] since our criteria can be applied to the case where 0 a pðtÞ a p 0 < y: Remark 9. Using methods di¤erent from those exploited in [10,11,14,15], we improve results of Han et al [10] and Li et al [15,Theorem 3.8] by removing assumptions sðtÞ a tðtÞ a t or t b sðtÞ b tðtÞ imposed in the cited papers and by providing su‰cient conditions which ensure that all solutions of (1) are oscillatory.…”

“…As mentioned by Baculíková and Džurina [4,Remark 4], assumptions ðH 2 Þ and ðH 3 Þ on functional arguments do not specify whether tðtÞ is a delayed or advanced argument. Furthermore, sðtÞ can be a delayed argument and sðtÞ À t can even oscillate.…”

“…Furthermore, sðtÞ can be a delayed argument and sðtÞ À t can even oscillate. However, to achieve such flexibility, we are forced to require, as in [4], monotonicity of t and that t s ¼ s t: The question regarding the study of oscillatory properties of equation (1) with other methods that do not require assumption ðH 3 Þ remains open at the moment.…”

“…A solution of (1) Li et al [14] studied oscillation of equation (1) under the assumptions that (2) holds, t and s are strictly increasing, pðtÞ > 1, and (4) is satisfied. Li et al [15] analyzed oscillatory properties of equation (1) in the case where ðH 1 Þ-ðH 3 Þ hold, tðtÞ b t, and sðtÞ b t: They derived a su‰cient condition which ensures that solutions of equation (1) We stress that theorems in [4,5,6,11] cannot be applied to (1) in the case where (2) holds. The purpose of this paper is to extend a new method for the analysis of the oscillation of equation (1) via comparison principles suggested by Baculíková and Džurina [4] for the study of (1) under the assumption (2).…”