Abstract:The oscillation of certain second-order generalized Emden-Fowler dynamic equations with damping on time scales is discussed. By using the calculus theory on time scales and the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the equations. Our results extend and improve some known results, an example is given to illustrate the main results of this article.
1 HILGERS.Analysis on measure chains-A unified approach to continuous and discrete calculus[J]. Results in Mathematics, 1990, 18(1/2):18-56. DOI: 10.1007/BF03323153 2 BOHNERM, PETERSONA.Dynamic Equations on Time Scales:An Introduction with Applications[M]. Boston: Birkhauser, 2001. 3 AGARWALR P, BOHNERM, LIW T.Nonoscillation and Oscillation: Theory for Functional Differential Equations[M]. New York: Marcel Dekker, 2004. DOI: 10.1201/9780203025741 4 ZHANGQ X, GAOL. Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales[J]. Scientia Sinica Mathematica, 2010, 40(7): 673-682. 5 SAKERS H.Oscillation criteria of second-order half-linear dynamic equations on time scales[J]. Journal of Computational & Applied Mathematics, 2005, 177(2): 375-387. DOI: 10.1016/j.cam.2004.09.028 6 HANZ L,SHIB,SUNS R.Oscillation of second-order delay dynamic equations on time scales[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2007,46(6):10-13. DOI: 10.3321/j.issn:0529-6579.2007.06.003 7 ERBEL, HASSANT S, PETERSONA. Oscillation criteria for nonlinear damped dynamic equations on time scales[J]. Applied Mathematics Computation, 2008, 203(1): 343-357. DOI: 10.1016/j.amc.2008.04.038 8 CHENW S, HANZ L, SUNS R, et al. Oscillation behavior of a class of second-order dynamic equations with damping on time scales[J]. Discrete Dynamics in Nature and Society, 2010(3): 907130. DOI: 10.1155/2010/907130 9 ZHANGQ X.Oscillation of second-order half-linear delay dynamic equations with damping on time scales[J]. Journal of Computational and Applied Mathematics, 2011, 235(5): 1180-1188. DOI: 10.1016/j.cam.2010.07.027 10 ZHANGQ X, GAOL, LIUS H.Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales[J]. Scientia Sinica Mathematica, 2011, 41(10): 885-896.doi:10.3969/j.issn.1672-7010.2013.01.002 11 ZHANGQ X, GAOL, LIUS H.Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales[J].Scientia Sinica Mathematica,2013,43(8):793-806.doi:10.1360/012012-392 12 YANGJ S.Oscillation for a class of second-order emden-fowler dynamic equations on time scales[J]. Acta Mathematicae Applicatae Sinica, 2016, 39(3): 334-350. 13 SUNY B, HANZ L,SUNS R, et al. Oscillation of a class of second order half-linear neutral delay dynamic equations with damping on time scales[J]. Acta Mathematicae Applicatae Sinica, 2013, 36(3): 480-494. DOI:10.3969/j.issn.0254-3079.2013.03.010 14 YANGJ S, LIT X. Oscillation for a class of second-order damped Emden-Fowler dynamic equations on time scales[J]. Acta Mathematica Scientia, 2018, 38A(1): 134-155. DOI:10.3969/j.issn.1003-3998.2018.01.013 15 DENGX H, WANGQ R, ZHOUZ. Oscillation criteria for second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients on time scales[J]. Science China Mathematics, 2017,60(1):113-132. DOI:10.1007/s11425-016-0070-y 16 YANGJ S, ZHANGX J.New results of oscillation for certain second-order nonlinear dynamic equations on time scales[J]. Journal of East China Normal University (Natural Science), 2017(3): 54-63. DOI:10.3969/j.issn.1000-5641.2017.03.006 17 YANGJ S.Oscillation of solutions for a class of second-order nonlinear variable delay difference equation[J].Journal of Yantai University (Natural Science and Engineering Edition), 2012,25(2):90-94. DOI:10.3969/j.issn.1004-8820.2012.02.004 18 YANGJ S,SUNW B.Oscillation of second order difference equations with positive and negative coefficients[J].Journal of Shangdong University(Natural Science),2011,46(8): 59-63 19 YANGJ S.Oscillation for a class of second-order dynamic equations with damping on time scales[J]. Journal of Systems Science and Mathematical Sciences, 2014, 34(6): 734-751. 20 ZHANGX J.Oscillatory criteria for certain second-order nonlinear dynamic equations on time scales[J]. Journal of Zhejiang University (Science Edition), 2018, 45(2): 136-142. DOI:10.3785/j.issn.1008-9497.2018.02.002