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Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. (English) Zbl 1075.39010
Authors’ abstract: We establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation \((c(t)(a(t)x^\Delta (t))^\Delta )^\Delta +q(t)f(x(t))=0,\) \(t\geq t_0,\) oscillates or converges to zero.

MSC:
39A12 Discrete version of topics in analysis
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[1] Agarwal, R.P.; Grace, S.R.; O’Regan, D., Oscillation theory for difference and functional differential equations, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0969.34062
[2] R.P. Agarwal, S.R. Grace, D. O’Regan, Oscillation Theory for Second order Dynamic Equations, Series in Mathematical Analysis and Applications, 5, Taylor & Francis, London, 2003.
[3] Bohner, E.A.; Hoffacker, J., Oscillation properties of an Emden-Fowler type equation on discrete time scales, J. differential equations appl., 9, 603-612, (2003) · Zbl 1038.39009
[4] Bohner, M.; Došlý, O.; Kratz, W., An oscillation theorem for discrete eigenvalue problems, Rocky mountain J. math., 33, 4, 1233-1260, (2003) · Zbl 1060.39003
[5] Bohner, M.; Peterson, A., Dynamic equations on time scalesan introduction with applications, (2001), Birkhäuser Boston
[6] Bohner, M.; Saker, S.H., Oscillation of second order nonlinear dynamic equations on time scales, Rocky mountain J. math., 34, 4, 1239-1254, (2004) · Zbl 1075.34028
[7] M. Bohner, S.H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, submitted for publication. · Zbl 1112.34019
[8] O. Došlý, S. Hilger, A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, in: R.P. Agarwal, M. Bohner, D. O’Regan (Eds.), special issue on Dynamic Equations on Time Scales, J. Comput. Appl. Math. 141 (2002) 147-158. · Zbl 1009.34033
[9] Erbe, L., Oscillation criteria for second order linear equations on a time scale, Canad. appl. math. quart., 9, 1-31, (2001) · Zbl 1050.39024
[10] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. modelling, boundary value problems and related topics, 32, 571-585, (2000) · Zbl 0963.34020
[11] Erbe, L.; Peterson, A., Riccati equations on a measure chain, (), 193-199 · Zbl 1008.34006
[12] Erbe, L.; Peterson, A., Boundedness and oscillation for nonlinear dynamic equations on a time scale, Proc. amer. math. soc., 132, 735-744, (2004) · Zbl 1055.39007
[13] Erbe, L.; Peterson, A.; Rehak, P., Comparison theorems for linear dynamic equations on time scales, J. math. anal. appl., 275, 418-438, (2002) · Zbl 1034.34042
[14] Erbe, L.; Peterson, A.; Saker, S.H., Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London math. soc., 67, 701-714, (2003) · Zbl 1050.34042
[15] G.Sh. Guseinov, B. Kaymakçalan, On a disconjugacy criterion for second order dynamic equations on time scales, in: R.P. Agarwal, M. Bohner, D. O’Regan (Eds.), special issue on Dynamic Equations on Time Scales, J. Comput. Appl. Math. 141 (2002) 187-196. · Zbl 1014.34023
[16] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Resultate math., 18, 18-56, (1990) · Zbl 0722.39001
[17] S. Huff, G. Olumolode, N. Pennington, A. Peterson, Oscillation of an Euler-Cauchy dynamic equation, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, Discrete and Continuous Dynamical Systems, 2002, pp. 24-27. · Zbl 1052.39007
[18] Kamenev, I.V., An integral criterion for oscillation of linear differential equations of second order, Mat. zametki, 23, 249-251, (1978) · Zbl 0386.34032
[19] Saker, S.H., Oscillation of nonlinear dynamic equations on time scales, Appl. math. comput., 148, 81-91, (2004) · Zbl 1045.39012
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