为了进一步发展和完善时间模上动态方程的振荡性理论,研究了下面的时间模\,\textbf{T}\,上一类二阶中立型变时滞非线性的动态方程\,$[A(t)\phi([x(t)+B(t)g(x(\tau (t)))]^\Delta )]^\Delta +f(t,x(\delta(t)))=0$\,的振荡性, 其中\,$\phi (u)=\vert u\vert ^{\lambda-1}u(\lambda>0$\,为任意常数). 利用时间模上的微积分理论和不等式技巧,得到了该方程振荡的一些新准则. 最后, 举例说明了本文定理的应用.
In order to develop and improve the theory about oscillation of dynamic equations on time scales, this paper is concerned with the oscillatory behavior of the following second-order neutral variable delay nonlinear dynamic equation $[A(t)\phi ([x(t)+B(t)g(x(\tau (t)))]^\Delta)]^\Delta +f(t,x(\delta(t)))=0 on a time scale \textbf{T}, where phi (u)=\vert u\vert^{\lambda -1}u$, $\lambda $ is an arbitrary positive constant. By using the calculus theory on time scales and the inequality technique, we establish some new oscillation criteria for the equation. Finally, example is presented to illustrate the effects of our theorems.
[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]BOHNER M, PETERSON A. Dynamic Equations on Time Scales, AnIntroduction with Applications [M]. Boston: Birkhauser, 2001.
[3]AGARWAL R P, BOHNER M, GRACE S R, et al. Discrete Oscillation Theory[M]. New York: Hindawi Publishing Corporation, 2005.
[4]ZHANG Q X, GAO L. Oscillation criteria for second-order half-lineardelay dynamic equations with damping on time scales [J]. Sci SinMath, 2010, 40(7): 673-682.
[5]SAHINER Y. Oscillation of second order delay differential equationson time scales [J]. Nonlinear Analysis, TMA, 2005, 63: e1073-e1080.
[6]HAN Z L, SHI B, SUN S R. Oscillation of second-order delay dynamicequation on time scales [J]. Acta Scientiarum NaturaliumUniversitatis Sunyatseni, 2007, 46(6): 10-13.
[7]SUN S, HAN Z, ZHANG C. Oscillation of second order delay dynamicequations on time scales [J]. J Appl Math Comput, 2009, 30: 459-468.
[8]GRACE S R, AGARWAL R P, KAYMAKCALAN B, et al. Oscillation theoremsfor second order nonlinear dynamic equations [J]. J Appl MathComput, 2010, 32: 205-218.
[9]AGARWAL R P, O'REGAN D, SSKER S H. Oscillation criteria forsecond-order nonlinear neutral delay dynamic equations [J]. J MathAnal Appl, 2004, 300: 203-217.
[10]SAKER S H. Oscillation of second-order nonlinear neutral delaydynamic equations on time scales [J]. J Comput Appl Math, 2006, 187:123-141.
[11]SAHINER Y. Oscillation of second-order neutral delay and mixed-typedynamic equations on time scales [J]. Advances in DifferentEquations, 2006, 1-9.
[12]SAKER S H, AGARWAL R P, O'REGAN D. Oscillation results forsecond-order nonlinear neutral delay dynamic equations on time scales [J]. Applicable Analysis, 2007, 86: 1-17.
[13]WU H W, ZHUANG R K, MATHSEN R M. Oscillation criteria forsecond-order nonlinear neutral variable delay dynamic equations [J].Appl Math Comput, 2006, 178: 321-331.
[14]SAKER S H, O'REGAN D. New oscillation criteria for second-orderneutral function dynamic equation via the generalized Riccati substitution [J]. Commun Nonlinear Sci Numer Simulat, 2010, 16:423-434.
[15]SAKER S H. Oscillation of second-order neutral delay dynamic equations of Emden-Fowler type [J]. Dyn Sys Appl, 2006, 15: 629-644.
[16]韩振来, 孙书荣, 张承慧.时间尺度上二阶中立型时滞动力方程的振动性~[J]. 中山大学学报:自然科学版, 2010, 49(5): 21-24.
[17]杨甲山. 时标上一类具阻尼项的二阶动态方程的振荡性~[J].系统科学与数学, 2014, 34(6): 734-751.
[18]孙一冰, 韩振来, 李同兴. 二阶拟线性中立型动力方程振动准则~[J].济南大学学报: 自然科学版, 2010, 24(3): 308-311.
[19]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.
[20]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.
[21]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.
[22]杨甲山, 莫协强. 时间轴上一类二阶动态系统振荡的充分条件~[J].安徽大学学报: 自然科学版, 2014, 38(5): 1-6.
[23]李同兴, 韩振来. 时间尺度上二阶超线性动力方程振动性~[J].济南大学学报: 自然科学版, 2010, 24(2): 209-211.
[24]HAN Z L, LI T X, SUN S R, et al. Oscillation for second-ordernonlinear delay dynamic equations on time scales [J]. Advances in Difference Equations, 2009, 1-13.
[25]HAN Z L, LI T X, SUN S R, et al. On the oscillation of second-order neutral delay dynamic equations on time scales [J]. African Diaspora Journal of Mathematics, 2010, 9(1): 76-86.
[26]ZHANG C H, AGARWAL R P, BOHNER M, et al. New oscillation results for second-order neutral delay dynamic equations [J]. Advances in Difference Equations, 2012, 227-240.
[27]HAN Z L, LI T X, SUN S R, et al. Remarks on the paper [Appl. Math.Comput. 207 (2009) 388-396] [J]. Applied Mathematics and Computation, 2010, 215: 3998-4007.
[28]HAN Z L, LI T X, SUN S R, et al. On the oscillation of second-order neutral delay differential equations [J]. Advances in Difference Equations, 2010, 1-8.
[29]杨甲山, 苏芳. 时间测度链上一类二阶动力方程的振动性~[J].数学的实践与认识, 2014, 44(13): 265-270.
[30]杨甲山, 苏芳.具阻尼项的一类二阶非线性动态方程的振动准则\,(英文)~[J]. 应用数学,2014, 27(2): 392-404.
[31]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, 34A(2): 393-408.
[32]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.
中国综合性科技类核心期刊(北大核心)