具非线性中立项的二阶延迟微分方程的Philos型准则

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 169-186.

具非线性中立项的二阶延迟微分方程的Philos型准则

李继猛¹,杨甲山^2,*()

¹ 邵阳学院理学院湖南邵阳 422004
² 梧州学院大数据与软件工程学院广西梧州 543002

收稿日期:2019-03-08 出版日期:2020-02-26 发布日期:2020-04-08
通讯作者: 杨甲山 E-mail:syxyyjs@163.com
作者简介:李继猛, E-mail:syxyljm@163
基金资助:
国家自然科学基金项目(51765060);湖南省自然科学基金(12JJ3008);湖南省教育厅教学改革研究项目(2016JG671)

Philos-Type Criteria for Second-Order Delay Differential Equations with Nonlinear Neutral Term

Jimeng Li¹,Jiashan Yang^2,*()

¹ School of Science, Shaoyang University, Hunan Shaoyang 422004
² School of Data Science and Software Engineering, Wuzhou University, Guangxi Wuzhou 543002

Received:2019-03-08 Online:2020-02-26 Published:2020-04-08
Contact: Jiashan Yang E-mail:syxyyjs@163.com
Supported by:
the NSFC(51765060);the Natural Science Foundation of Hunan Province(12JJ3008);the Research Project on Teaching Reform of Hunan Education Department(2016JG671)

摘要/Abstract

摘要：

利用黎卡提变换技术，结合伯努利、杨氏不等式以及数学分析技巧，研究了具有非线性中立项的Emden-Fowler型微分方程

$\begin{eqnarray*}&&\{a(t)|[x(t)+p(t)x^\alpha(\tau(t))]'|^{\beta-1}[x(t)+p(t)x^\alpha(\tau(t))]'\}'\\&&+q(t)f(|x(\delta(t))|^{\gamma-1}x(\delta(t)))=0, t\geqslant t_0\end{eqnarray*}$

的振动性，获得了该方程的若干新Philos型振动定理，所举例子说明，这些准则不仅推广并改进了一些已有的结果，而且具有较好的实用性和可操作性.

关键词: 振动性, Emden-Fowler型微分方程, 非线性中立项, Riccati变换

Abstract:

We study the oscillatory behavior of Emden-Fowler-type differential equations with a nonlinear neutral term

$\{a(t)|[x(t)+p(t)x^\alpha(\tau(t))]'|^{\beta-1}[x(t)+p(t)x^\alpha(\tau(t))]'\}'+q(t)f(|x(\delta(t))|^{\gamma-1}x(\delta(t)))=0, t\geq t_0$

in this article. By using the generalized Riccati transformation, and Bernoulli inequality, Yang inequality and integral averaging technique, we establish some new oscillation criterions for the equations. The illustrative examples are provided to show that our results obtained extend and improve those reported in the literature, and have practicability and maneuverability.

Key words: Oscillation, Emden-Fowle differential equation, Nonlinear neutral, Riccati transformation

中图分类号:

O175.1

李继猛,杨甲山. 具非线性中立项的二阶延迟微分方程的Philos型准则[J]. 数学物理学报, 2020, 40(1): 169-186.

Jimeng Li,Jiashan Yang. Philos-Type Criteria for Second-Order Delay Differential Equations with Nonlinear Neutral Term[J]. Acta mathematica scientia,Series A, 2020, 40(1): 169-186.

参考文献 18

1	Agarwal R P, Bohner M, Li W T. Nonoscillation and Oscillation: Theory for Functional Differential Equations. New York: Marcel Dekker, 2004
2	Li T, Rogovchenko Y V. Oscillation theorems for second-order nonlinear neutral delay differential equations. Abstract and Applied Analysis, 2014, Article ID: 594190
3	Dzurina J , Stavrouakis I P . Oscillation criteria for second order delay differential equations. Appl Math Comput, 2003, 14 (3): 445- 453
4	Sun Y G , Meng F W . Note on the paper of Dzurina and Stavroulakis. Appl Math Comput, 2006, 27 (4): 1634- 1641
5	黄记洲, 符策红. 广义Emden-Fowler方程的振动性. 应用数学学报, 2015, 38 (6): 1126- 1135
	Huang Jizhou , Fu Cehong . Oscillation criteria of generalized Emden-Fowler equations. Acta Mathematicae Applicatae Sinica, 2015, 38 (6): 1126- 1135
6	Yang J S , Wang J J , Qin X W , et al. Oscillation of nonlinear second-order neutral delay differential equations. Journal of Nonlinear Sciences and Applications, 2017, 10 (5): 2727- 2734 doi: 10.22436/jnsa.010.05.39
7	杨甲山. 具非线性中立项的二阶变时滞微分方程的振荡性. 华东师范大学学报(自然科学版), 2016, (4): 30- 37 doi: 10.3969/j.issn.1000-5641.2016.04.004
	Yang Jiashan . Oscillation of second-order variable delay differential equations with nonlinear neutral term. Journal of East China Normal University (Natural Science), 2016, (4): 30- 37 doi: 10.3969/j.issn.1000-5641.2016.04.004
8	Sun S , Li T , Han Z , et al. Oscillation theorems for second-order quasilinear neutral functional differential equations. Abstract and Applied Analysis, 2012, 2012, 1- 17
9	Li T , Rogovchenko Yu V , et al. Oscillation of second-order neutral differential equations. Funkc Ekvac, 2013, 56, 111- 120 doi: 10.1619/fesi.56.111
10	Lin X , Tang X . Oscillation of solutions of neutral differential equations with a superlinear neutral term. Appl Math Lett, 2007, 20, 1016- 1022 doi: 10.1016/j.aml.2006.11.006
11	Zhang C , Agarwal R P , Bohner M , et al. New oscillation results for second-order neutral delay dynamic equations. Advances in Difference Equations, 2012, 2012, 227 doi: 10.1186/1687-1847-2012-227
12	Agarwal R P , Bohner M , Li T X , et al. Oscillation of second-order Emden-Fowler neutral delay differential equations. Annali di Matematica Pura ed Applicata, 2014, 193 (6): 1861- 1875 doi: 10.1007/s10231-013-0361-7
13	Agarwal R P , Bohner M , Li T , et al. Oscillation of second-order differential equations with a sublinear neutral term. Carpathian Journal of Mathematics, 2014, 30 (1): 1- 6
14	Han Z, Li T, Sun S, et al. Remarks on the paper[Appl Math Comput 207(2009)388-396]. Appl Math Comput, 2010, 215: 3998-4007
15	罗李平, 俞元洪, 罗振国. 三阶非线性中立型微分方程的振动分析. 系统科学与数学, 2016, 36 (4): 551- 559
	Luo L P , Yu Y H , Luo Z G . Oscillation analysis of third order nonlinear neutral differential equations. Journal of Systems Science and Complexity, 2016, 36 (4): 551- 559
16	曾云辉, 罗李平, 俞元洪. 中立型Emden-Fowler时滞微分方程的振动性. 数学物理学报, 2015, 35A (4): 803- 814 doi: 10.3969/j.issn.1003-3998.2015.04.016
	Zeng Y H , Luo L P , Yu Y H . Oscillation for Emden-Fowler delay differential equations of neutral type. Acta Mathematica Scientia, 2015, 35A (4): 803- 814 doi: 10.3969/j.issn.1003-3998.2015.04.016
17	Bohner M , Li T X . Kamenev-type criteria for nonlinear damped dynamic equations. Sci China Math, 2015, 58, 1445- 1452 doi: 10.1007/s11425-015-4974-8
18	杨甲山. 时间模上一类二阶非线性延迟动力系统的振动性分析. 应用数学学报, 2018, 41 (3): 388- 402
	Yang J S . Oscillation analysis of second-order nonlinear delay dynamic equations on time scales. Acta Mathematicae Applicatae Sinica, 2018, 41 (3): 388- 402

[1]	李继猛. 二阶广义Emden-Fowler型延迟微分方程的振荡性分析[J]. 数学物理学报, 2019, 39(5): 1041-1054.
[2]	仉志余,俞元洪,李淑萍,乔士柱. 二阶非线性中立型时滞微分方程的振动性[J]. 数学物理学报, 2019, 39(4): 797-811.
[3]	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.
[4]	罗李平, 罗振国, 邓义华. 脉冲扰动对非线性时滞双曲型分布参数系统振动的影响[J]. 数学物理学报, 2018, 38(2): 313-321.
[5]	李文娟, 汤获, 俞元洪. 中立型Emden-Fowler微分方程的振动性[J]. 数学物理学报, 2017, 37(6): 1062-1069.
[6]	王云竹, 高建芳. 一类非线性延迟微分方程θ-方法的数值解振动分析[J]. 数学物理学报, 2017, 37(2): 342-351.
[7]	杨甲山. 时间测度链上具正负系数的二阶阻尼动力方程的振动准则[J]. 数学物理学报, 2014, 34(2): 393-408.
[8]	陈大学. 具有振动系数的二阶非线性中立型时滞动力方程的有界振动性[J]. 数学物理学报, 2013, 33(1): 98-113.
[9]	林全文, 俞元洪. 二阶非线性振动的Philos型积分平均[J]. 数学物理学报, 2012, 32(4): 661-669.
[10]	李同兴, 韩振来, 张承慧, 孙一冰. 时间尺度上三阶Emden-Fowler动力方程的振动准则[J]. 数学物理学报, 2012, 32(1): 222-232.
[11]	冯春华. 一类简化的n个神经元时滞BAM神经网络模型的振动性[J]. 数学物理学报, 2011, 31(6): 1490-1501.
[12]	杨杰, 綦建刚, 景海斌. 奇异哈密顿微分系统的粘结引理和亏指数[J]. 数学物理学报, 2011, 31(3): 652-661.
[13]	汪红初, 胡适耕, 朱全新. 有界时滞非线性随机微分方程解的振动性和非振性[J]. 数学物理学报, 2010, 30(6): 1457-1464.
[14]	张超龙; 杨逢建; 杨建富 . 高阶非线性脉冲泛函微分方程的振动性[J]. 数学物理学报, 2008, 28(1): 188-200.
[15]	庄容坤;姚正安. 带强迫项的二阶非线性椭圆方程的区域振动准则[J]. 数学物理学报, 2007, 27(5): 855-861.