具有振动系数的二阶非线性中立型时滞动力方程的有界振动性

Abstract

Abstract:

In this paper, we investigate the oscillation of bounded solutions of second-order nonlinear neutral delay dynamic equation with oscillating coefficients of the form
(r(t)([y(t)+p(t)y(τ(t))]^Δ)^α)^Δ+f(t, y(δ(t)))=0
on an arbitrary time scale T, where p is an oscillating function defined on T and α>0 is a quotient of odd positive integers. We get some sufficient conditions for the oscillation of all bounded solutions of the equation by developing a Riccati transformation technique. The obtained results extend and complement some known results in which p(t)≥0 for t∈T is required. Several examples are presented to illustrate our main results.

Key words: Bounded oscillation, Second-order nonlinear neutral delay dynamic equation, Oscillating coefficient, Time scale, Riccati transformation

CLC Number:

39A10

CHEN Da-Xue. Bounded Oscillation for Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients[J].Acta mathematica scientia,Series A, 2013, 33(1): 98-113.

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References

[1] Chen D X. Oscillation of second-order Emden-Fowler neutral delay dynamic equations on time scales. Math Comput Modelling, 2010, 51: 1221--1229

[2] Chen D X. Oscillation and asymptotic behavior for nth-order nonlinear neutral delay dynamic equations on time scales. Acta Appl Math, 2010, 109: 703--719

[3] Chen D, Liu J. Oscillation theorems for second-order nonlinear neutral dynamic equations with distributed delay on time scales. J Systems Sci Math Sci, 2010, 30: 1--14

[4] Chen D X, Liu J C. Asymptotic behavior and oscillation of solutions of third-order nonlinear neutral delay dynamic equations on time scales. Can Appl Math Q, 2008, 16: 19--43

[5] Karpuz B. Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations. Appl Math Comput, 2009, 215: 2174--2183

[6] Hassan T S. Oscillation of third-order nonlinear delay dynamic equations on time scales. Math Comput Modelling, 2009, 49: 1573--1586

[7] Hilger S. Ein Maβkettenkalk\"{u}l mit Anwendung auf Zentrumsmannigfaltigkeiten [D]. W\"{u}rzburg: Universitä}t W\"{u}rzburg, 1988

[8] Spedding V. Taming Nature's numbers. New Scientist, 2003, 179: 28--31

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

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

[11] Hardy G H, Littlewood J E, P\'{o}lya G. Inequalities (second ed). Cambridge: Cambridge University Press, 1988

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

[13] Erbe L, Hassan T S, Peterson A. Oscillation criteria for nonlinear damped dynamic equations on time scales. Appl Math Comput,
2008, 203: 343--357

[14] Hassan T S. Oscillation criteria for half-linear dynamic equations on time scales. J Math Anal Appl, 2008, 345: 176--185

[15] Zafer A. On oscillation and nonoscillation of second-order dynamic equations. Appl Math Lett, 2009, 22: 136--141

[16] Ou L. Atkinson's super-linear oscillation theorem for matrix dynamic equations on a time scale. J Math Anal Appl, 2004, 299: 615--629

[17] Medico A D, Kong Q. Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain.
J Math Anal Appl, 2004, 294: 621--643

[18] Bohner M, Saker S H. Oscillation criteria for perturbed nonlinear dynamic equations. Math Comput Modelling, 2004, 40: 249--260

[19] Do\v{s}l\'{y} O, Hilger S. A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales. J Comput Appl Math, 2002, 141: 147--158

[20] Zhang B G, Shanliang Z. Oscillation of second-order nonlinear delay dynamic equations on time scales. Comput Math Appl, 2005, 49: 599--609

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

[22] Erbe L, Peterson A, Saker S H. Hille and Nehari type criteria for third-order dynamic equations. J Math Anal Appl, 2007, 329: 112--131

[23] Erbe L, Peterson A, Saker S H. Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J Comput Appl Math, 2005, 181: 92--102

[24] Agarwal R P, O'Regan D, Saker S H. Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J Math Anal Appl, 2004, 300: 203--217

[25] Saker S H. Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J Comput Appl Math, 2006, 187: 123--141

[26] Wu H W, Zhuang R K, Mathsen R M. Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl Math Comput, 2006, 178: 321--331

[27] Saker S H, Agarwal R P, O'Regan D. Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales.
Appl Anal, 2007, 86: 1--17

[28] Zhang S Y, Wang Q R. Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl Math Comput, 2010, 216: 2837--2848

[29] Saker S H, O'Regan D. New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution. Commun Nonlinear Sci Numer Simul, 2011, 16: 423--434

[30] Erbe L H, Kong Q, Zhang B G. Oscillation Theory for Functional Differential Equations. New York: Marcel Dekker, 1995

[31] Adivar M, Raffoul Y. A note on "tability and periodicity in dynamic delay equations" (Comput Math Appl, 2009, 58: 264--273. Comput Math Appl, 2010, 59: 3351--3354

[32] Karpuz B. Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients.
Electron J Qual Theory Differ Equ, 2009, 34: 1--14

[33] 李同兴, 韩振来, 张承慧, 孙一冰. 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, 32: 222--232

[34] Liu Ailian, Wu Hongwu, Zhu Siming, Mathsen R M. Oscillation for nonautonomous neutral dynamic delay equations on time scales. Acta Math Sci (Ser B Engl Ed), 2006, 26: 99--106

Bounded Oscillation for Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients

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