{"title":"Nonoscillation Criteria for Nonlinear Delay Dynamic Systems on Time Scales","authors":"Xinli Zhang","volume":85,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":222,"pagesEnd":227,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9997821","abstract":"

In this paper, we consider the nonlinear delay dynamic system
\r\nxΔ(t) = p(t)f1(y(t)), yΔ(t) = −q(t)f2(x(t − τ )).
\r\nWe 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.<\/p>\r\n","references":" S. Hilger, \"Analysis on measure chains-a unified approach to continuous\r\nand discrete calculus,\u201d Results in Mathematics, vol. 18, no. 1-2, pp. 18-56,\r\n1990.\r\n R. Agarwal, M. Bohner, D. O\u2019Regan and A. Peterson, \"Dynamic equations\r\non time scales: a survey,\u201d J. Comput. Appl. Math., vol. 141, no. 1-2,\r\npp. 1-26, 2002.\r\n M. Bohner and A. Peterson, \"Dynamic Equations on Time Scales: An\r\nIntroduction with Applications,\u201d Birkh\u00a8auser, Boston, Mass, USA, 2001.\r\n S. Zhu, C. Sheng, \"Oscillation and Nonoscillation Criteria for Nonlinear\r\nDynamic Systems on Time Scales,\u201d Discrete Dynamics in Nature and\r\nSociety, vol. 2012, pp. 1-14, 2012.\r\n S. C. Fu, M. L. Lin, \"Oscillation and nonoscillation criteria for linear\r\ndynamic systems on time scales,\u201d Comput. Math. Appl., vol. 59, pp. 2552-\r\n2565, 2010.\r\n Y. J. Xu, Z. T. Xu, \"Oscillation criteria for two-dimensional dynamic\r\nsystems on time scales,\u201d J. Comput. Math. Appl., vol. 225, no. 1, pp.\r\n9-19, 2009.\r\n Lynn Erbe, Raziye Mert, \"Some new oscillation results for a nonlinear\r\ndynamic system on time scales,\u201d Appl. Maht. Comput., vol. 215, pp. 2405-\r\n2412, 2009.\r\n B.G.Zhang, S.Zhu, \"Oscillation of second-order nonlinear delay equations\r\non time scales,\u201d Comput.Math. Appl., vol. 49, pp. 599-609, 2005.\r\n L. Erbe, A. Peterson and S.H.Saker, \"Oscillation criteria for second-order\r\nnonlinear delay dynamic equations,\u201d J. Math. Anal. Appl., vol. 333, pp.\r\n505-522, 2007.\r\n Z. Han, S. Sun and B. Shi, \"Oscillation criteria for a class of second\r\norder Emden-Fowler delay dynamic equations on time scales,\u201d J. Math.\r\nAnal. Appl., vol. 334, no. 2, pp. 847-858, 2007.\r\n W. T. Li, \"Classification schemes for positive solutions of nonlinear\r\ndifferential systems,\u201d Math. and Comput. Modelling , vol. 36, pp. 411-\r\n418, 2002.\r\n W. T. Li, \"Classification schemes for nonoscillatory of two-dimensional\r\nnonlinear difference systems,\u201d Comput. Math. Appl., vol. 42, pp. 341-355,\r\n2001.\r\n Z. Q. Zhu and Q. R. Wang, \"Existence of nonoscillatory solutions to\r\nneutral dynamic equations on time scales,\u201d J. Math. Anal. Appl., vol. 335,\r\nno. 2, pp. 751-762, 2007.\r\n M. Bohner and A. Peterson, \"Advances in Dynamic Equations on Time\r\nScales,\u201d Birkh\u00a8auser, Boston, Mass, USA, 2003.\r\n I. Gyori and G. Ladas, \"Oscillation Theory of Delay Differential\r\nEquations with Applications,\u201d Clarendom Press, Oxford, 1991.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 85, 2014"}