带阻尼项的二阶非线性中立型Emden-Fowler微分方程的振动准则

摘要/Abstract

摘要：

该文研究一类带有更广泛而不失物理意义阻尼项的二阶非线性中立型Emden-Fowler时滞微分方程的振动性.利用指数函数变换、Riccati变换和不等式技巧，获得了该类方程几个新的振动定理，推广、改进和丰富了已有文献中的研究结果，并逐一给出例子说明了相应定理的实用效果.

关键词: 振动准则, Emden-Fowler方程, 中立型, 非线性微分方程, 阻尼项

Abstract:

In this paper, by using the methods of exponential function transformation, Riccati transformation and inequality techniques, we study the oscillation behavior for a class of second order nonlinear neutral delay Emden-Fowler differential equations with the most extensive and physically meaningful damping term. Several new oscillation theorems which extend and improve some known results in the literature recently are established and some examples are provided to illustrate the relevance of new theorems.

Key words: Oscillation criteria, Emden-Fowler equation, Neutral, Nonlinear differential equation, Damping term

中图分类号:

O175

仉志余,宋菲菲,李同兴,俞元洪. 带阻尼项的二阶非线性中立型Emden-Fowler微分方程的振动准则[J]. 数学物理学报, 2020, 40(4): 934-946.

Zhiyu Zhang,Feifei Song,Tongxing Li,Yuanhong Yu. Oscillation Criteria of Second Order Nonlinear Neutral Emden-Fowler Differential Equations with Damping[J]. Acta mathematica scientia,Series A, 2020, 40(4): 934-946.

参考文献 37

1	Wong J S W . On the generalized Emden-Fowler equation. SIAM Rev, 1975, 17, 339- 360 doi: 10.1137/1017036
2	Hale J K . Theory of Functional Differential Equations. New York: Springer, 1977
3	Elbert A . A half-linear second order differentail equation. Qualitative Theory of Differential Equations, 1979, 8, 153- 180
4	Elbert A . Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations. Ordinary and Partial Differential Equations, 1982, 9, 187- 212
5	仉志余, 周勇, 俞元洪. 非线性二阶泛函微分方程的振动准则. 系统科学与数学, 2000, 20 (1): 1- 10
	Zhang Z Y , Zhou Y , Yu Y H . Oscillation criteria for nonlinear functional differential equations of second order. J Sys Sci Math Scis, 2000, 20 (1): 1- 10
6	仉志余, 王晓霞, 林诗仲, 俞元洪. 非线性二阶中立型时滞微分方程的振动和非振动准则. 系统科学与数学, 2006, 26 (3): 325- 334 doi: 10.3969/j.issn.1000-0577.2006.03.008
	Zhang Z Y , Wang X X , Lin S Z , Yu Y H . Oscillation and nonoscillation criteria for second order nonlinear neutral delay differential equations. J Sys Sci Math Scis, 2006, 26 (3): 325- 334 doi: 10.3969/j.issn.1000-0577.2006.03.008
7	Liu H , Meng F , Liu P . Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation. Appl Math Comput, 2012, 219, 2739- 2748
8	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 . Oscillaition for Emden-Fowler delay differential equations of neutral type. Acta Math Sci, 2015, 35A (4): 803- 814 doi: 10.3969/j.issn.1003-3998.2015.04.016
9	Li T , Rogovchenko Y V . Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations. Monatsh Math, 2017, 184, 489- 500 doi: 10.1007/s00605-017-1039-9
10	Li T , Thandopani E , Graef J R , Tunc E . Oscillation of second-order Emden-Fowler neutral differential equations. Nonlinear Studies, 2013, 20 (1): 1- 8
11	Luo H Y , Liu J , Liu X , Zhu C Y . Oscillation behavior of a class of new generalized Emden-Fowler equations. Thermal Science, 2014, 18, 1567- 1572 doi: 10.2298/TSCI1405567L
12	吴英柱. 中立型Emden-Fowler泛函微分方程的振动准则. 数学的实践与认识, 2017, 47 (10): 277- 284
	Wu Y Z . Oscillation for Emden-Fowler functional differential equations of neutral type. Mathematics in Practice and theory, 2017, 47 (10): 277- 284
13	Agarwal P R , Bohner M , Li T , et al. Oscillation of second-order Emden-Fowler neutral delay differential equations. Anali di Matematica, 2014, 193, 1861- 1875 doi: 10.1007/s10231-013-0361-7
14	Grace S R , Džurina J , Tadlovskǎ I , et al. An improved approach for studying oscillation of second-order neutral differential equations. J Inequalities and Appl, 2018, 2018, 193- 205 doi: 10.1186/s13660-018-1767-y
15	Bohner M , Grace S R , Jadlovskǎ I . Oscillation criteria for second-order neutral delay differential equations. Electronic J Qualitative Thoery Diff Equ, 2017, 2017 (60): 1- 12
16	Wang R , Li Q . Oscillation and asymptotic properties of a class of second order Emden Fowler neutral differential equations. Springer Plus, 2016, 5 (1): 1956- 1970 doi: 10.1186/s40064-016-3622-2
17	Sun S , Li T , Han Z , Sun Y . Oscillation of second-order neutral functional differential equations with mixed nonlinearities. Abstr Appl Anal, 2011, 2011, 1- 15
18	Li T , Han Z , Zhang C , et al. On the oscilation of second-order Emden-Fowler neutral dfferential equations. J Appl Math Comput, 2011, 37, 601- 610 doi: 10.1007/s12190-010-0453-0
19	Zhang C , Li T , Sun B , Thandapani E . On the oscilation of higher-order half-linear delay differential equations. Appl Math Lett, 2011, 24, 1618- 1621 doi: 10.1016/j.aml.2011.04.015
20	Sun S , Li T , Han Z , et al. Oscilation theorems for second-order quasilinear neutral functional differential equations. Abstr Appl Anal, 2012, 2012, 1- 17
21	罗红英, 屈英. 二阶中立型广义Emden-Fowler方程的振动准则. 数学的实践与认识, 2016, 46 (1): 267- 274
	Luo H Y , Qu Y . Oscillation criteria for second-order generalized Emden-Fowler equations of the neutral type. Mathematics in Practice and Theory, 2016, 46 (1): 267- 274
22	吴英柱. Emden-Fowler型泛函微分方程的振动准则. 数学的实践与认识, 2016, 46 (5): 257- 264
	Wu Y Z . Oscillation criteria for Functional differential equations of Emden-Fowler type. Mathematics in Practice and theory, 2016, 46 (5): 257- 264
23	Erbe L , Hassan T S , Peterson A . Oscillation criteria for nonlinear damped dynamic equations on time scales. Appl Math Comput, 2008, 203, 343- 357
24	Zhang Q . Oscillation of second-order half-linear delay dynamic equations with damping on time scales. J Comput Appl Math, 2011, 235, 1180- 1188 doi: 10.1016/j.cam.2010.07.027
25	Saker S H , Agarwal R P , O'Regan D . Oscillation of second-order damped dynamic equations on time scales. J Math Anal Appl, 2007, 330, 1317- 133 doi: 10.1016/j.jmaa.2006.06.103
26	Rogovchenko Yu V , Tuncay F . Oscillation criteria for second-order nonlinear differential equations with damping. Nonlinear Anal, 2008, 69, 208- 221 doi: 10.1016/j.na.2007.05.012
27	张全信, 高丽. 时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则. 中国科学:数, 2010, 40 (7): 673- 682
	Zhang Q X , Gao L . Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales. Sci Sin Math, 2010, 40 (7): 673- 682
28	张全信, 高丽, 刘守华. 时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则(Ⅱ). 中国科学:数学, 2011, 41 (10): 885- 896
	Zhang Q X , Gao L , Liu S H . Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales(Ⅱ). Sci Sin Math, 2011, 41 (10): 885- 896
29	张全信, 高丽, 刘守华. 时间尺度上具阻尼项的二阶半线性时滞动力方程振动的新结果. 中国科学:数学, 2013, 43 (8): 793- 806
	Zhang Q X , Gao L , Liu S H . New oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales. Sci Sin Math, 2013, 43 (8): 793- 806
30	孙一冰, 韩振来, 孙书荣, 等. 时间尺度上一类二阶具阻尼项的半线性中立型时滞动力方程的振动性. 应用数学学报, 2013, 36A (3): 480- 494 doi: 10.3969/j.issn.0254-3079.2013.03.010
	Sun Y B , Han Z L , Sun S R , et al. Oscillation of a class of second order half-linear neutral delay dynamic equations with damping on time scales. Acta Mathematicae Applicatae Sinica, 2013, 36A (3): 480- 494 doi: 10.3969/j.issn.0254-3079.2013.03.010
31	杨甲山. 时间测度链上一类二阶Emden-Fowler型动态方程的振荡性. 应用数学学报, 2016, 39A (3): 334- 350
	Yang J S . Oscillation for a class of second-order Emden-Fowler dynamic equations on time scales. Acta Mathematicae Applicatae Sinica, 2016, 39A (3): 334- 350
32	杨甲山, 李同兴. 时间模上一类二阶阻尼Emden-Fowler型动态方程的振荡性. 数学物理学报, 2018, 38A (1): 134- 155 doi: 10.3969/j.issn.1003-3998.2018.01.013
	Yang J S , Li T X . Oscillation for a class of second-order damped Emden-Fowler dynamic equations on time scales. Acta Math Sci, 2018, 38A (1): 134- 155 doi: 10.3969/j.issn.1003-3998.2018.01.013
33	李文娟, 汤获, 俞元洪. 中立型Emden-Fowler微分方程的振动性. 数学物理学报, 2017, 37A (6): 1062- 1069 doi: 10.3969/j.issn.1003-3998.2017.06.006
	Li W J , Tang H , Yu Y H . Oscillation of the neutral Emden-Fowler differential equation. Acta Math Sci, 2017, 37A (6): 1062- 1069 doi: 10.3969/j.issn.1003-3998.2017.06.006
34	Tadie. Oscillation criteria for some semi-linear Emden-Fowler ODE//Constanda C, Kirsch A, et al. Integral Methods in Science and Engineering. Switzerland:Springer-Verlag, 2015:607-615
35	Ma Q , Liu A . Oscillation criteria of nonlinear fractional differential equations with damping term. Mathematica Applicata, 2016, 29, 291- 297
36	Greenberg M D . Advanced Engineering Mathematics. New Jersey: Prentice Hall, 1998
37	Leighton W . The dection of the oscillation of solitions of second order linear differential equation. Duke Math J, 1950, 17, 57- 62 doi: 10.1215/S0012-7094-50-01707-8

[1]	廖梦兰,郭斌. 具变指数源项和强阻尼项的波动方程解的渐近稳定性[J]. 数学物理学报, 2020, 40(1): 146-155.
[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]	李文娟,李书海,俞元洪. 非线性二阶中立型分布时滞微分方程的振动性[J]. 数学物理学报, 2019, 39(4): 812-822.
[4]	张根根,王晚生,肖爱国. 一类变延迟中立型微分方程梯形方法的渐近估计[J]. 数学物理学报, 2019, 39(3): 560-569.
[5]	谷海波,高彩霞. 基于Razumikhin-Type理论的中立性随机切换非线性系统的P阶矩稳定性与几乎必然稳定性[J]. 数学物理学报, 2018, 38(5): 970-983.
[6]	王慧灵, 高建芳. 一类带有多项延迟的非线性中立型延迟微分方程解析解的振动性分析[J]. 数学物理学报, 2018, 38(4): 740-749.
[7]	姚绍文, 程志波. 一类带阻尼项的奇性Duffing方程周期解的存在性[J]. 数学物理学报, 2018, 38(3): 543-548.
[8]	王晚生, 钟鹏, 赵新阳. 非线性中立型变延迟微分方程的长时间稳定性[J]. 数学物理学报, 2018, 38(1): 96-109.
[9]	杨甲山, 李同兴. 时间模上一类二阶阻尼Emden-Fowler型动态方程的振荡性[J]. 数学物理学报, 2018, 38(1): 134-155.
[10]	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.
[11]	曾云辉, 罗李平, 俞元洪. 广义中立型Emden-Fowler方程的振动准则[J]. 数学物理学报, 2016, 36(6): 1067-1081.
[12]	刘宏影, 吕学琴. 再生核结合配置法求解一类带有积分边值条件的四阶非线性微分方程[J]. 数学物理学报, 2016, 36(5): 928-936.
[13]	魏利, Agarwal Ravi P. 含有广义p-Laplacian算子的双曲型非线性微分方程的单调性方法[J]. 数学物理学报, 2015, 35(5): 867-877.
[14]	曾云辉, 罗李平, 俞元洪. 中立型Emden-Fowler时滞微分方程的振动性[J]. 数学物理学报, 2015, 35(4): 803-814.
[15]	狄华斐, 尚亚东. 一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性[J]. 数学物理学报, 2015, 35(3): 618-633.