Improvement Results for Oscillatory Behavior of Second Order Neutral Differential Equations with Nonpositive Neutral Term

Abstract

In this paper we obtain new criteria for the oscillation of all solutions of second order neutral differential equations with nonpositive neutral term, which improve some of the results in [1].Examples are provided to illustrate the main results.

Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type

“…Example 3.1 Assume that T = R. Consider the second-order neutral delay differential equation Remark 3.1 Oscillation criteria established in this paper for equation (1.3) complement, on one hand, the results reported by Arul and Shobha [3] and Li et al [14] because we use assumption (1.5) rather than…”

“…However, for equations with nonpositive neutral coefficients, there are relatively fewer results in the literature; see [3,4,6,11,14,[16][17][18]. For instance, in the particular case of (1.1) when T = R, Li et al [14] studied the differential equation…”

“…Their results were improved by Arul and Shobha [3] who established new oscillation results for the solutions of (1.3). Seghar et al [16] discussed the difference equation…”

Oscillation Criteria for Second-Order Nonlinear Mixed Neutral Dynamic Equations With Non Positive Neutral Term on Time Scales H. A. Agwa, Ahmed M. M. Khodier and Heba M. Arafa

“…Qi Li et al [10] obtained oscillation criteria for the delay differential equation (r(t)((y(t) − p(t)y(τ (t))) ′ ) γ ) ′ + q(t)f (y(δ(t))) = 0. R. Arul and V. S. Shobha [4] improved the obtained results in [10] and E. Thandapani et al [14] obtained some results on oscillatory behavior of the second order neutral difference equation:…”

“…( 46). Remark 1 The results of [4], [9] and [10] can not be applied to (46) as p 2 (t) ̸ = 0 and f (t, x(τ 1 (t))) ̸ = 0 ̸ = g(t, x(τ 2 (t))), but according to Theorem 1 we obtain that every solution of ( 46) is almost oscillatory or converges to zero as t → ∞.…”