Oscillation Theorems for Second‐Order Quasilinear Neutral Functional Differential Equations

Shurong Sun,Zhenlai Han,Tongxing Li,Hua Li

doi:10.1155/2012/819342

Abstract

New oscillation criteria are established for the second‐order nonlinear neutral functional differential equations of the form , t ≥ t0, where z(t) = x(t) + p(t)x(τ(t)), p ∈ C1([t0, ∞), [0, ∞)), and α ≥ 1. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results.

Highlights

This paper is concerned with the oscillation problem of the second-order nonlinear functional differential equation of the following form: rtztα−1z t f t, x σ t0, t ≥ t0, 1.1 where α ≥ 1 is a constant, ztxtptxτt
Dzurina 17 was concerned with the oscillation behavior of the solutions of the second-order neutral differential equations as follows atxtptxτtγqt xβ σ t 0, 1.11 where 0 ≤ p t ≤ p0 < ∞, γ is a the ratios of two positive odd integers, and obtained some new results under the following conditions
Letting t → ∞, by 1.5, we find z t → −∞, which is a contradiction. ii Suppose that z t > 0 for t ≥ t2 ≥ t1

Summary

Introduction

This paper is concerned with the oscillation problem of the second-order nonlinear functional differential equation of the following form: rtztα−1z t f t, x σ t. There is constant interest in obtaining new sufficient conditions for the oscillation or nonoscillation of the solutions of varietal types of the second-order equations, see, e.g., papers 2–17. Dzurina 17 was concerned with the oscillation behavior of the solutions of the second-order neutral differential equations as follows atxtptxτtγqt xβ σ t 0, 1.11 where 0 ≤ p t ≤ p0 < ∞, γ is a the ratios of two positive odd integers, and obtained some new results under the following conditions t0 a t. Similar to the proof of case i of Theorem 3.1, we obtain a contradiction. If there exists a function η ∈ C1 t0, ∞ , R , η t ≥ t, η t > 0, σ t ≤ η t for t ≥ t0 such that 4.1 holds, 1.1 is oscillatory

Examples

Conclusions