Journal of East China Normal University(Natural Science) >

2017 >Issue 3: 54 - 63

DOI: https://doi.org/10.3969/j.issn.1000-5641.2017.03.006

Mathematics

New results of oscillation for certain second-order nonlinear dynamic equations on time scales

  • YANG Jia-shan ,
  • ZHANG Xiao-jian
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  • 1. School of Information and Electronic Engineering, Wuzhou University, Wuzhou Guangxi 543002, China;
    2. Laboratory of Complex System Simulation and Intelligent Computing, Wuzhou University, Wuzhou Guangxi 543002, China;
    3. Department of Science and Information, Shaoyang University, Shaoyang Hunan 422004, China

Received date: 2016-08-28

  Online published: 2017-05-18

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Abstract

This paper is concerned with oscillatory behavior of the following second-order nonlinear neutral variable delay functional dynamic equations

[A（t）Φ（[x（t）+B（t）g（x（τ（t）））]Δ）]Δ+ƒ（t，x（δ（t）））=0

on a time scale T, where Φ（u）=|u|λ-1u (hereλ> 0 is an arbitrary constant). By using a couple of Riccati substitutions, the time scales theory and inequality technique, we establish two new oscillation criteria for the equations, these results deal with some cases not covered by existing results in the literature. Finally, two examples are presented to illustrate the importance of our theorems.

Key words： oscillation; time scales; functional dynamic equations; variable delay

Cite this article

YANG Jia-shan , ZHANG Xiao-jian . New results of oscillation for certain second-order nonlinear dynamic equations on time scales[J]. Journal of East China Normal University(Natural Science), 2017 , (3) : 54 -63 . DOI: 10.3969/j.issn.1000-5641.2017.03.006

References

[1] 张晓建, 杨甲山. 时间模上二阶非线性动态方程振荡性的新结果[J]. 浙江大学学报(理学版), 2014, 41(5): 499-505.
[2] 杨甲山, 黄劲. 时间模上一类二阶非线性动态方程振荡性的新准则[J]. 华东师范大学学报(自然科学版), 2015(3): 9-15.
[3] BOHNER M, PETERSON A. Dynamic Equations on Time Scales, an Introduction with Applications [M]. Boston: Birkhauser, 2001.
[4] SAKER S H. Oscillation of second-order nonlinear neutral delay dynamic equations on time scales [J]. J Comput Appl Math, 2006, 187: 123-141.
[5] 杨甲山, 谭伟明, 覃学文, 等. 时间模上二阶非线性阻尼动力方程的振动性分析[J]. 浙江大学学报(理学版), 2016, 43(1): 64-70.
[6] 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[J]. 应用数学学报, 2013, 36(3): 480-494.
[7] 杨甲山. 时间测度链上具非线性中立项的二阶阻尼动力方程的振动性[J]. 浙江大学学报(理学版), 2012, 39(3): 261-265.
[8] 杨甲山. 时间测度链上二阶动力方程的振动准则[J]. 华东师范大学学报(自然科学版), 2012(3): 17-23.
[9] 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[J]. 中国科学: 数学, 2010, 40(7): 673-682.
[10] 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)[J]. 中国科学: 数学, 2011, 41(10): 885-896.
[11] HAN Z L, LI T X, SUN S R, et al. Oscillation for second-order nonlinear delay dynamic equations on time scales [J]. Advances in Difference Equations, 2009, 2009: 1-13.
[12] 杨甲山, 方彬. 时间模上一类二阶非线性中立型泛函动态方程的振荡性[J]. 内蒙古师范大学学报(自然科学汉文版), 2016, 45(5): 603-609.
[13] 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[J]. 应用数学, 2017, 30(1): 16-26.
[14] 杨甲山, 谭伟明, 苏芳, 等. 时间模上二阶非线性中立型时滞泛函动态方程的振荡性[J]. 南开大学学报(自然科学版), 2015, 48(3): 24-31.
[15] 张晓建, 杨甲山. 时标上三阶时滞动力方程的振动性和渐近性[J]. 华东师范大学学报(自然科学版), 2014(3): 51-59.
[16] ERBE L, HASSAN T S, PETERSON A. Oscillation criteria for nonlinear damped dynamie equations on time scales [J]. Apple Math Comput, 2008, 203: 343-357.
[17] 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[J]. 中国科学: 数学, 2013, 43(8): 793-806.
[18] ZHANG Q X. Oscillation of second-order half-linear delay dynamic equations with damping on time scales [J]. Journal of Computational and Applied Mathematics, 2011, 235: 1180-1188.
[19] 杨甲山. 时间测度链上一类二阶Emden-Fowler型动态方程的振荡性[J]. 应用数学学报, 2016, 39(3): 334-350.

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