Qiu, Yang-Cong; Wang, Qi-Ru Kamenev-type oscillation criteria of second-order nonlinear dynamic equations on time scales. (English) Zbl 1264.34013 Discrete Dyn. Nat. Soc. 2013, Article ID 315158, 12 p. (2013). Summary: Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form \((p(t)\psi(x(t))k \circ x^\Delta(t))^\Delta + f(t, x(\sigma(t))) = 0\). Two examples are included to show the significance of the results. Cited in 4 Documents MSC: 34A08 Fractional ordinary differential equations PDFBibTeX XMLCite \textit{Y.-C. Qiu} and \textit{Q.-R. Wang}, Discrete Dyn. Nat. Soc. 2013, Article ID 315158, 12 p. (2013; Zbl 1264.34013) Full Text: DOI References: [1] R. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1-26, 2002. · Zbl 1020.39008 [2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001 [3] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001 [4] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten [Ph.D. thesis], Universität Würzburg, 1988. · Zbl 0695.34001 [5] 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. · Zbl 0722.39001 [6] R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics, vol. 35, no. 1-2, pp. 3-22, 1999. · Zbl 0927.39003 [7] R. P. Agarwal, D. O’Regan, and S. H. Saker, “Philos-type oscillation criteria for second order half-linear dynamic equations on time scales,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 4, pp. 1085-1104, 2007. · Zbl 1139.34029 [8] A. Del Medico and Q. Kong, “Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain,” Journal of Mathematical Analysis and Applications, vol. 294, no. 2, pp. 621-643, 2004. · Zbl 1056.34050 [9] A. Del Medico and Q. Kong, “New Kamenev-type oscillation criteria for second-order differential equations on a measure chain,” Computers & Mathematics with Applications, vol. 50, no. 8-9, pp. 1211-1230, 2005. · Zbl 1085.39014 [10] O. Do and S. Hilger, “A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 147-158, 2002, Dynamic equations on time scales. · Zbl 1009.34033 [11] L. Erbe, T. S. Hassan, A. Peterson, and S. H. Saker, “Interval oscillation criteria for forced second-order nonlinear delay dynamic equations with oscillatory potential,” Dynamics of Continuous, Discrete & Impulsive Systems Series A, vol. 17, no. 4, pp. 533-542, 2010. · Zbl 1202.34162 [12] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear dynamic equations on time scales,” Journal of the London Mathematical Society, vol. 67, no. 3, pp. 701-714, 2003. · Zbl 1050.34042 [13] L. Erbe, A. Peterson, and S. H. Saker, “Kamenev-type oscillation criteria for second-order linear delay dynamic equations,” Dynamic Systems and Applications, vol. 15, no. 1, pp. 65-78, 2006. · Zbl 1104.34026 [14] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505-522, 2007. · Zbl 1125.34046 [15] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for a forced second-order nonlinear dynamic equation,” Journal of Difference Equations and Applications, vol. 14, no. 10-11, pp. 997-1009, 2008. · Zbl 1168.34025 [16] H. Huang and Q.-R. Wang, “Oscillation of second-order nonlinear dynamic equations on time scales,” Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 551-570, 2008. · Zbl 1202.34067 [17] R. M. Mathsen, Q.-R. Wang, and H.-W. Wu, “Oscillation for neutral dynamic functional equations on time scales,” Journal of Difference Equations and Applications, vol. 10, no. 7, pp. 651-659, 2004. · Zbl 1060.34038 [18] Y. C. Qiu and Q. R. Wang, “Interval oscillation criteria of second-order nonlinear dynamic equations on time scales,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 952932, 16 pages, 2012. · Zbl 1248.34042 [19] S. H. Saker, “Oscillation of nonlinear dynamic equation on time scales,” Applied Mathematics and Computation, vol. 148, pp. 81-91, 2004. · Zbl 1045.39012 [20] S. H. Saker, “Oscillation of second-order delay and neutral delay dynamic equations on time scales,” Dynamic Systems and Applications, vol. 16, no. 2, pp. 345-359, 2007. · Zbl 1147.34050 [21] S. H. Saker, R. P. Agarwal, and D. O’Regan, “Oscillation of second-order damped dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1317-1337, 2007. · Zbl 1128.34022 [22] S. H. Saker and D. O’Regan, “New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 423-434, 2011. · Zbl 1221.34245 [23] S. H. Saker, D. O’Regan, and R. P. Agarwal, “Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales,” Acta Mathematica Sinica, vol. 24, no. 9, pp. 1409-1432, 2008. · Zbl 1153.34040 [24] A. Tiryaki and A. Zafer, “Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 60, no. 1, pp. 49-63, 2005. · Zbl 1064.34021 [25] Q.-R. Wang, “Oscillation criteria for nonlinear second order damped differential equations,” Acta Mathematica Hungarica, vol. 102, no. 1-2, pp. 117-139, 2004. · Zbl 1052.34040 [26] Q.-R. Wang, “Interval criteria for oscillation of certain second order nonlinear differential equations,” Dynamics of Continuous, Discrete & Impulsive Systems, vol. 12, no. 6, pp. 769-781, 2005. · Zbl 1087.34015 [27] H.-W. Wu, Q.-R. Wang, and Y.-T. Xu, “Oscillation and asymptotics for nonlinear second-order differential equations,” Computers & Mathematics with Applications, vol. 48, no. 1-2, pp. 61-72, 2004. · Zbl 1073.34035 [28] Q. Yang, “Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 49-64, 2003. · Zbl 1030.34034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.