Dynamical Behavior of Second-order Quasilinear Delay Damped Dynamic Equations on Time Scales
LI Jimeng1, YANG Jiashan2
1. School of Science, Shaoyang University, Shaoyang 422004, China; 2. School of Data Science and Software Engineering, Wuzhou University, Wuzhou 543002, China
Abstract:The dynamical behavior of certain second-order quasilinear variable delay damped dynamic equations are investigated on a time scale T, where the equations are noncanonical form (i.e.,∫t0∞ [a-1(s)e-b/a(s, t0)] 1/λ Δs<∞). By using the generalized Riccati transformation, and in combination with the time scales theory and the classical inequality, some new oscillation criteria for the equation are established. The results fully reflect the influential actions of delay functions and damping terms in system oscillation. Finally, some examples are given to show that our results extend, improve and enrich those reported in the literature, and that they have good effectiveness and practicability.
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:数学, 2010, 40(7):673-682(Zhang Q X, Gao Li. Oscillation criteria for second-order half-linear delay dynamic equations with damping on times cales. Sci. Sin. Math., 2010, 40(7):673-682)
[2]
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性的新结果. 中国科学:数学, 2013, 43(8):793-806(Zhang Q X, Gao Li, 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)
[3]
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 Li, 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)
[4]
Erbe L, Hassan T S, Peterson A. Oscillation criteria for nonlinear damped dynamie equations on time scales. Apple. Math. Comput, 2008, 203:343-357
[5]
Zhang Q X. Oscillation of second-order half-linear delay dynamic equations with damping on time scales. Journal of Computational and Applied Mathematics, 2011, 235:1180-1188
[6]
Bohner M, Peterson A. Dynamic equations on time scales,an introduction with applications. Boston:Birkhauser, 2001
[7]
Saker S H. Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J. Comput. Appl. Math., 2006, 187:123-141
[8]
Han Z L, Li T X, Sun S R, et al. Oscillation for second-order nonlinear delay dynamic equations on time scales. Advances in Difference Equations, 2009, 2009:1-13
[9]
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, 36(3):480-494(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, 36(3):480-494)
[10]
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)
[11]
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(Yang J S, Li T X. Oscillation for a class of second-order damped Emden-Fowler dynamic equations on time scales. Acta Mathematica Scientia, 2018, 38A(1):134-155)
[12]
Yang J S, Qin X W, Zhang X J. Oscillation criteria for certain second-order nonlinear neutral delay dynamic equations with damping on time scales. Mathematica Applicata, 2015, 28(2):439-448
[13]
Li T X, Saker S H. A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(12):4185-4188
[14]
Deng X H, Wang Q R, Zhou Z. Oscillation criteria for second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients on time scales. Sci. China Math., 2017, 60:113-132
[15]
杨甲山, 方彬. 时间测度链上一类二阶非线性时滞阻尼动力方程的振动性分析. 应用数学, 2017, 30(1):16-26(Yang J S, Fang B. Oscillation analysis of certain second-order nonlinear delay damped dynamic equations on time scales. Mathematica Applicata, 2017, 30(1):16-26)
[16]
杨甲山. 时间尺度上二阶Emden-Fowler型延迟动态方程的振动性. 振动与冲击, 2018, 37(16):154-161(Yang J S. Oscillation for a class of second-order Emden-Fowler-type delay dynamic equations on time scales. Journal of Vibration and Shock, 2018, 37(16):154-161)
[17]
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)
[18]
李继猛, 杨甲山. 时间尺度上二阶半线性时滞阻尼动力方程的振动性. 中山大学学报(自然科学版), 2019, 58(4):108-114(Li J M, Yang J S. Oscillation for second-order half-linear delay damped dynamic equations on time scales. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2019, 58(4):108-114)