Abstract
The present paper focuses on the oscillation of the third-order nonlinear neutral differential equations with damping and distributed delay. The oscillation of the third-order damped equations is often discussed by reducing the equations to the second-order ones. However, by applying the Riccati transformation and the integral averaging technique, we give an analytical method for the estimation of Riccati dynamic inequality to establish several oscillation criteria for the discussed equation, which show that any solution either oscillates or converges to zero. The results make significant improvement and extend the earlier works such as (Zhang et al. in Appl. Math. Lett. 25:1514–1519 2012). Finally, some examples are given to demonstrate the effectiveness of the obtained oscillation results.
Highlights
1 Introduction Differential equations arise in modeling situations to describe population growth, biology, economics, chemical reactions, neural networks, and so forth; see, e.g., [2–8]
We investigate the oscillatory behavior of a third-order neutral differential equation with damping and distributed delay
By Theorem 3.2 we know that any solution of (4.2) is oscillatory or converges to zero as t → ∞
Summary
Differential equations arise in modeling situations to describe population growth, biology, economics, chemical reactions, neural networks, and so forth; see, e.g., [2–8]. Letting b y(t) = x(t) + p(t, μ)x τ (t, μ) dμ, a a function x(t) is the solution of equation (1.1) if x(t) satisfies (1.1) on [Tx, ∞) for every t ≥ Tx ≥ t0 with x(t), α(t)y (t) and r(t)(α(t)y (t)) ∈ C1[Tx, ∞). Our focus is on the oscillation for third-order neutral differential equations with distributed delay and damping term, such as [46]. The main contribution in this paper is that we provide another method for the inequality estimation to discuss the oscillation of differential equations with damping and distributed delay on the basis of the Riccati transformation and the integral averaging technique. Proof We set that x(t) is the positive solution of (1.1) for [t0, ∞) It follows from (H4) and (H5) that x(τ (t, μ)) > 0 and x(g(t, ζ )) > 0 for t ≥ t1 with sufficiently large t1, respectively.
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