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Asymptotic properties of solutions of certain third-order dynamic equations. (English) Zbl 1239.34112

Summary: The well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r 2 (t)((r 1 (t)x Δ (t)) Δ ) γ ) Δ +q(t)f(x(t))=0

on time scale 𝕋, where γ1 is a ratio of odd positive integers. Our results are essentially new even for third-order differential and difference equations, i.e., when 𝕋= and 𝕋=. Two examples of dynamic equations on different time scales are given to show the applications of our main results.

34N05Dynamic equations on time scales or measure chains
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
[1]Bohner, M.; Peterson, A.: Dynamic equations on time scales, an introduction with applications, (2001)
[2]Bohner, M.; Peterson, A.: Advances in dynamic equations on time scales, (2003)
[3]Fite, W. B.: Concerning the zeros of the solutions of certain differential equations, Trans. amer. Math. soc. 19, 341-352 (1918)
[4]Hille, E.: Non-oscillation theorems, Trans. amer. Math. soc. 64, 234-252 (1948) · Zbl 0031.35402 · doi:10.2307/1990500
[5]Nehari, Z.: Oscillation criteria for second order linear differential equations, Trans. amer. Math. soc. 85, 428-445 (1957) · Zbl 0078.07602 · doi:10.2307/1992939
[6]Erbe, L.; Hassan, T. S.; Peterson, A.; Saker, S. H.: Oscillation criteria for half-linear delay equations on time scales, Nonlinear dyn. Syst. theory 9, 51-68 (2009) · Zbl 1173.34037
[7]Anderson, D. R.: Oscillation and nonoscillation criteria for two-dimensional time-scale systems of first-order nonlinear dynamic equations, Electron. J. Differential equations 2009, 1-13 (2009) · Zbl 1185.34140 · doi:emis:journals/EJDE/Volumes/2009/24/abstr.html
[8]Fu, S. -C.; Li, M. -L.: Oscillation and nonoscillation criteria for linear dynamic systems on time scales, Comput. math. Appl. 59, 2552-2562 (2010) · Zbl 1193.34181 · doi:10.1016/j.camwa.2010.01.014
[9]Erbe, L.; Peterson, A.; Saker, S. H.: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. comput. Appl. math. 181, 92-102 (2005) · Zbl 1075.39010 · doi:10.1016/j.cam.2004.11.021
[10]Erbe, L.; Peterson, A.; Saker, S. H.: Oscillation and asymptotic behavior of a third-order dynamic equation, Can. appl. Math. Q. 14, No. 2, 129-147 (2006) · Zbl 1145.34329
[11]Erbe, L.; Peterson, A.; Saker, S. H.: Hille and Nehari type criteria for third order dynamic equation, J. math. Anal. appl. 329, 112-131 (2007) · Zbl 1128.39009 · doi:10.1016/j.jmaa.2006.06.033
[12]Hassan, T. S.: Oscillation of third order nonlinear delay dynamic equations on time scales, Math. comput. Modelling 49, 1573-1586 (2009) · Zbl 1175.34086 · doi:10.1016/j.mcm.2008.12.011
[13]Yu, Z. H.; Wang, Q. R.: Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales, J. comput. Appl. math. 225, 531-540 (2009) · Zbl 1165.34029 · doi:10.1016/j.cam.2008.08.017