Nonoscillation Criteria for Nonlinear Delay Dynamic Systems on Time Scales
Authors: Xinli Zhang
In this paper, we consider the nonlinear delay dynamic system xΔ(t) = p(t)f1(y(t)), yΔ(t) = −q(t)f2(x(t − τ )). We obtain some necessary and sufficient conditions for the existence of nonoscillatory solutions with special asymptotic properties of the system. We generalize the known results in the literature. One example is given to illustrate the results.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1091590Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1146
 S. Hilger, "Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990.
 R. Agarwal, M. Bohner, D. O’Regan and A. Peterson, "Dynamic equations on time scales: a survey,” J. Comput. Appl. Math., vol. 141, no. 1-2, pp. 1-26, 2002.
 M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications,” Birkh¨auser, Boston, Mass, USA, 2001.
 S. Zhu, C. Sheng, "Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales,” Discrete Dynamics in Nature and Society, vol. 2012, pp. 1-14, 2012.
 S. C. Fu, M. L. Lin, "Oscillation and nonoscillation criteria for linear dynamic systems on time scales,” Comput. Math. Appl., vol. 59, pp. 2552- 2565, 2010.
 Y. J. Xu, Z. T. Xu, "Oscillation criteria for two-dimensional dynamic systems on time scales,” J. Comput. Math. Appl., vol. 225, no. 1, pp. 9-19, 2009.
 Lynn Erbe, Raziye Mert, "Some new oscillation results for a nonlinear dynamic system on time scales,” Appl. Maht. Comput., vol. 215, pp. 2405- 2412, 2009.
 B.G.Zhang, S.Zhu, "Oscillation of second-order nonlinear delay equations on time scales,” Comput.Math. Appl., vol. 49, pp. 599-609, 2005.
 L. Erbe, A. Peterson and S.H.Saker, "Oscillation criteria for second-order nonlinear delay dynamic equations,” J. Math. Anal. Appl., vol. 333, pp. 505-522, 2007.
 Z. Han, S. Sun and B. Shi, "Oscillation criteria for a class of second order Emden-Fowler delay dynamic equations on time scales,” J. Math. Anal. Appl., vol. 334, no. 2, pp. 847-858, 2007.
 W. T. Li, "Classification schemes for positive solutions of nonlinear differential systems,” Math. and Comput. Modelling , vol. 36, pp. 411- 418, 2002.
 W. T. Li, "Classification schemes for nonoscillatory of two-dimensional nonlinear difference systems,” Comput. Math. Appl., vol. 42, pp. 341-355, 2001.
 Z. Q. Zhu and Q. R. Wang, "Existence of nonoscillatory solutions to neutral dynamic equations on time scales,” J. Math. Anal. Appl., vol. 335, no. 2, pp. 751-762, 2007.
 M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,” Birkh¨auser, Boston, Mass, USA, 2003.
 I. Gyori and G. Ladas, "Oscillation Theory of Delay Differential Equations with Applications,” Clarendom Press, Oxford, 1991.