具非线性中立项的广义Emden-Fowler微分方程的振动性

数学物理学报 ›› 2018, Vol. 38 ›› Issue (4): 728-739.

具非线性中立项的广义Emden-Fowler微分方程的振动性

张晓建

邵阳学院理学院湖南邵阳 422004

收稿日期:2017-04-06 修回日期:2017-09-19 出版日期:2018-08-26 发布日期:2018-08-26
作者简介:张晓建,E-mail:zxj2006a@163.com
基金资助:
湖南省教育厅科研项目（10C1189）和湖南省教育厅教学改革研究项目（2016jg671）

Oscillation of Generalized Emden-Fowler Differential Equations with Nonlinear Neutral Term

Zhang Xiaojian

Faculty of Science, Shaoyang University, Hunan Shaoyang 422004

Received:2017-04-06 Revised:2017-09-19 Online:2018-08-26 Published:2018-08-26
Supported by:
Supported by the Scientific Research Fund of Education Department of Hunan Province (10Cl189) and the Teaching Reform Project of Education Department of Hunan Province (2016jg671)

摘要/Abstract

摘要： 研究了一类具有一个非线性中立项的二阶非线性变时滞广义Emden-Fowler型泛函微分方程
{a（t）|[x（t）+p（t）x^α（τ（t））]'|^β-1[x（t）+p（t）x^α（τ（t））]'}'+q（t）f（|x（δ（t））|^γ-1x（δ（t）））=0（t ≥ t₀）的振动性.利用广义的Riccati变换、Bernoulli不等式和Yang不等式，在两种情形
∫_t₀^+∞ a^-1/β（t）dt=+∞和∫_t₀^+∞ a^-1/β（t）dt <+∞
下建立了该类方程振动的若干新的判别准则，这些准则推广且改进了现有文献中的一些结果，所举例子说明这些定理的条件是比较宽松的.

关键词: 振动性, 泛函微分方程, 非线性中立项, Riccati变换

Abstract: We study the oscillatory behavior of a certain class of second-order nonlinear variable delay generalized Emden-Fowler functional differential equations with a nonlinear neutral term of the form
{a(t)|[x(t)+p(t)x^α(τ(t))]'|^β-1[x(t)+p(t)x^α(τ(t))]'}'+q(t)f(|x(δ(t))|^γ-1x(δ(t)))=0(t ≥ t₀). By using the generalized Riccati transformation and Bernoulli's inequality and Yang's inequality, we establish some new oscillation criteria for the equations under the cases when
∫_t₀^+∞ a^-1/β(t)dt=+∞ and ∫_t₀^+∞ a^-1/β(t)dt < +∞,
the results obtained extend and improve some related results reported in the literature. Examples are provided to illustrate assumptions in our theorems are less restrictive.

Key words: Oscillation, Functional differential equation, Nonlinear neutral, Riccati transformation

中图分类号:

O175.1

张晓建. 具非线性中立项的广义Emden-Fowler微分方程的振动性[J]. 数学物理学报, 2018, 38(4): 728-739.

Zhang Xiaojian. Oscillation of Generalized Emden-Fowler Differential Equations with Nonlinear Neutral Term[J]. Acta mathematica scientia,Series A, 2018, 38(4): 728-739.

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