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Oscillation of third order nonlinear functional dynamic equations on time scales. (English) Zbl 1213.34116

The purpose of the paper is to give oscillation criteria for the third order nonlinear functional dynamic equation

${\left(a\left(t\right){\left[{\left(r\left(t\right){x}^{{\Delta }}\left(t\right)\right)}^{{\Delta }}\right]}^{\gamma }\right)}^{{\Delta }}+f\left(t,x\left(g\left(t\right)\right)\right)=0$

on a time scale $𝕋$, where $\gamma$ is a quotient of odd positive integers, $a$ and $r$ are positive rd-continuous functions on $𝕋$, and the function $g:𝕋\to 𝕋$ satisfies ${lim}_{t\to \infty }g\left(t\right)=\infty$ and $f\in C\left(𝕋×ℝ,ℝ\right)$. Some examples are given to illustrate the main results.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34K11 Oscillation theory of functional-differential equations
##### Keywords:
oscillation theory; dynamic equations on time scales
##### References:
 [1] Agarwal R., Bohner M. and Saker S. H., Oscillation of second order delay dynamic equations, Canad. Appl. Math. Quart., 13, 1–17, (2005) [2] Bohner M. and Peterson A., Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, (2001) [3] Bohner M. and Peterson A., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, (2003) [4] Bohner M. and Saker S. H., Oscillation of second order half-linear dynamic equations on discrete time scales, Internat. J. Difference Equs., 1, 208–218, (2006) [5] Gera M., Graef J. R. and Gregus M., On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations, Nonlinear Anal., 32, 417–425, (1998) · Zbl 0945.34021 · doi:10.1016/S0362-546X(97)00483-5 [6] Došlý O. and Hilger E., A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Special Issue on Dynamic Equations on Time Scales (Agarwal P. P., Bohner M. and O’Regan D., eds.), J. Comp. Appl. Math., 141(1–2), 571–585, (2002) [7] Elabbasy E. M. and Hassan T. S., Oscillation of third order nonlinear functional differential equations, Diff. Eq. Appl., submitted [8] Erbe L., Hassan T. S. and Peterson A., Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comp., 203, 343–357, (2008) · Zbl 1162.39005 · doi:10.1016/j.amc.2008.04.038 [9] Erbe L., Hassan T. S. and Peterson A., Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, J. Diff. Eq. Appl., 15, 1097–1115, (2009) · Zbl 1193.34135 · doi:10.1080/10236190902785199 [10] Erbe L., Hassan T. S., Peterson A. and Saker S. H., Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dynam. Sys. Th., 9, 51–68, (2009) [11] Erbe L., Hassan T. S., Peterson A. and Saker S. H., Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Diff. Equ., 3, 227–245, (2008) [12] Erbe L., Peterson A. and Saker S. H., Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. Comp. Appl. Math., 181, 92–102, (2005) · Zbl 1075.39010 · doi:10.1016/j.cam.2004.11.021 [13] Erbe L., Peterson A. and Saker S. H., Hille and Nehari type criteria for third order dynamic equations, J. Math. Anal. Appl., 329, 112–131, (2007) · Zbl 1128.39009 · doi:10.1016/j.jmaa.2006.06.033 [14] Erbe L., Peterson A. and Saker S. H., Oscillation and asymptotic behavior a third-order nonlinear dynamic equation, Canad. Quart. Appl. Math., 14, 2, (2006) [15] Hardy G. H., Littlewood J. E. and Polya G., Inequalities, second ed., Cambridge University Press, Cambridge, (1988) [16] Hassan T. S., Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345, 176–185, (2008) · Zbl 1156.34022 · doi:10.1016/j.jmaa.2008.04.019 [17] Hilger S., Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math., 18, 18–56, (1990) [18] Kac V. and Cheung P., Quantum Calculus, Universitext, (2002) [19] Zhang B. G. and Deng X., Oscillation of delay differential equations on time scales, Math. Comp. Mod., 36, 1307–1318, (2002) · Zbl 1034.34080 · doi:10.1016/S0895-7177(02)00278-9 [20] Şahiner Y. and Stavroulakis I. S., Oscillation of first order delay dynamic equations, Dynam. Systems Appl., 15, 645–655, (2006) [21] Wu H., Zhuang R. and Mathsen R. M., Oscillation criteria of second-order nonlinear neutral variable delay dynamic equations, Appl. Math. Comp., 178, 321–331, (2006)