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

数学物理学报 ›› 2014, Vol. 34 ›› Issue (2): 393-408.

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

杨甲山

梧州学院数理系广西梧州 543002

收稿日期:2012-05-28 修回日期:2013-07-26 出版日期:2014-04-25 发布日期:2014-04-25
基金资助:
湖南省自然科学基金项目(12JJ6006)、湖南省科技厅基金项目(2012FJ3107)、广西教育厅科研项目(2013YB223)和国家自然科学基金项目(11071222)资助.

Oscillation Criteria for Second-Order Dynamic Equations with Positive and Negative Coefficients and Damping on Time Scales

YANG Jia-Shan

Department of Mathematics and Physics, Wuzhou University, Guangxi Wuzhou 543002

Received:2012-05-28 Revised:2013-07-26 Online:2014-04-25 Published:2014-04-25
Supported by:
湖南省自然科学基金项目(12JJ6006)、湖南省科技厅基金项目(2012FJ3107)、广西教育厅科研项目(2013YB223)和国家自然科学基金项目(11071222)资助.

摘要/Abstract

摘要：

研究了时间测度链上一类具正负系数和阻尼项及非线性中立项的二阶变时滞非线性动力方程的振荡性. 利用时间测度链上的有关理论及广义Riccati变换,结合大量不等式技巧, 建立了该方程若干新的振动准则,这些准则不仅推广和改进了一些已知的结果,
而且在时间测度链上统一了二阶非线性时滞阻尼微分方程和二阶非线性时滞阻尼差分方程的振动性质.

关键词: 振动性, 动力方程, 时间测度链, 正负系数, 阻尼项, 非线性中立项

Abstract:

The oscillation for certain second-order nonlinear variable delay dynamic equations with positive and negative coefficients and damping and nonlinear neutral on time scales is discussed. By using the time scales theory and the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the inequations. Our results extend and improve some known results, but also unify the oscillation of second-order nonlinear delay differential equations with damping and second-order nonlinear delay difference equations with damping.

Key words: Oscillation, Dynamic equations, Time scales, Positive and negative coefficients, Damping, Nonlinear neutral

中图分类号:

34k11

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.

YANG Jia-Shan. Oscillation Criteria for Second-Order Dynamic Equations with Positive and Negative Coefficients and Damping on Time Scales[J]. Acta mathematica scientia,Series A, 2014, 34(2): 393-408.

参考文献

[1] Hilger S. Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math, 1990, 18: 18--56

[2] Agarwal R P, Grace S R, O'Regan D. Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and
Sublinear Dynamic Equations. Dordrecht: Kluwer Academic, 2002

[3] Agarwal R P, Bohner M, Grace S R, et al. Discrete Oscillation Theory. New York: Hindawi Publishing Corporation, 2005

[4] Bohner M, Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkhauser, 2001

[5] Bohner M, Peterson A. Advances in Dynamic Equations on Time Scales. Boston: Birkhauser,2003

[6] Agarwal R P, Bohner M, O'Regan D, et al. Peterson, Dynamic equations on time scales: a survey. J Comput Appl Math, 2002, 141: 1--26

[7] Bohner M, Saker S H. Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mountain J Math, 2004, 34: 1239--1254

[8] Erbe L. Oscillation criteria for second order linear equations on a time scale. Can Appl Math Q, 2001, 9: 345--375

[9] Erbe L, Peterson A, Rehák P. Comparison theorems for linear dynamic equations on time scales. J Math Anal Appl, 2002, 275: 418--438

[10] Sun S, Han Z, Zhang C, Oscillation of second order delay dynamic equations on time scales. J Appl Math Comput, 2009, 30: 459--468

[11] Grace S R, Agarwal R P, Kaymakcalan B, et al. Oscillation theorems for second order nonlinear dynamic equations.
J Appl Math Comput, 2010, 32: 205--218

[12] Agarwal R P, Bohner M, Saker S H. Oscillation of second order delay dynamic equations. Can Appl Math Q, 2005, 13: 1--18

[13] Sahiner Y. Oscillation of second order delay differential equations on time scales. Nonlinear Anal TMA, 2005, 63: 1073--1080

[14] Saker S H. Oscillation criteria of second-order half-linear dynamic equations on time scales. J Comput Appl Math, 2005, {\bf 177}: 375--387

[15] Erbe L, Peterson A, Saker S H. Oscillation criteria for second order nonlinear delay dynamic equations. J Math Anal Appl, 2007, 333: 505--522

[16] Erbe L, Hassan T S, Peterson A. Oscillation criteria for nonlinear functional neutral dynamic equations on time scales. J Difference Equ Appl, 2009, 15(11/12): 1097--1116

[17] Grace S R, Bohner M, Agarwal R P. On the oscillation of second-order half-linear dynamic equations. J Difference Equ Appl, 2009, 15(5): 451--460

[18] Agarwal R P, Bohner M, Li W T. Nonoscillation and Oscillation: Theory for Functional Differential Equations. New York:Marcel Dekker, 2004.

[19] Tuna A, Kutukcu S. Some integral inequalities on time scales. Applied Mathematics and Mechanics, 2008, 29(1): 23--29

[20] Bohner M. Some oscillation criteria for first order delay dynamic equations. Far East J Appl Math, 2005, 18(3): 289--304

[21] Zhang Q X. Oscillation of second-order half-linear delay dynamic equations with damping on time scales. Journal of Computational and Applied, 2011, 235: 1180--1188

[22] 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

[23] 杨甲山. 时间测度链上一类具阻尼项的二阶动力方程的振动准则. 上海交通大学学报, 2012, 46(9): 1529--1533

[24] 欧柳曼, 朱思铭.时标动力方程的稳定性分析. 数学物理学报,2008, 28A(2): 308--319

[25] 杨甲山. 时间测度链上具非线性中立项的二阶动力方程的振动性. 中国科学院研究生院学报, 2012, 29(6): 731--737

[26] 李同兴, 韩振来, 张承慧, 等.时间尺度上三阶Emden-Fowler动力方程的振动准则. 数学物理学报,2012, 32A(1): 222--232