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Oscillation of third order nonlinear delay dynamic equations on time scales. (English) Zbl 1175.34086

Summary: This paper gives oscillation criteria for the third order nonlinear delay dynamic equation

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

on a time scale $𝕋$ where $\gamma \ge 1$ is the quotient of odd positive integers, $a$ and $r$ are positive $rd$-continuous functions on $𝕋$, and the so-called delay function $\tau :𝕋\to 𝕋$ satisfies $\tau \left(t\right)\le t$ for $t\in 𝕋$ and ${lim}_{t\to \infty }\tau \left(t\right)=\infty$ and $f\in C\left(𝕋×ℝ,ℝ\right)$. Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when $𝕋=ℝ$ and $𝕋=ℕ$ involve and improve some oscillation results for third order delay differential and difference equations; when $𝕋=hℕ$, $𝕋={q}^{{ℕ}_{0}}$ and $𝕋={ℕ}^{2}$ our oscillation results are essentially new. Some examples are given to illustrate the main results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 39A10 Additive difference equations
##### References:
 [1] Hilger, S.: Analysis on measure chains – a unified approach to continuous and discrete calculus, Results math. 18, 18-56 (1990) · Zbl 0722.39001 [2] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with applications, (2001) [3] Kac, V.; Chueng, P.: Quantum calculus, universitext, (2002) [4] Bohner, M.; Peterson, A.: Advances in dynamic equations on time scales, (2003) [5] Došlý, O.; Hilger, E.: A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, J. comput. Appl. math. 141, No. 1–2, 571-585 (2002) · Zbl 1009.34033 · doi:10.1016/S0377-0427(01)00442-3 [6] Erbe, L.; Hassan, T. S.; Peterson, A.: Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. math. Comput. 203, 343-357 (2008) · Zbl 1162.39005 · doi:10.1016/j.amc.2008.04.038 [7] L. Erbe, T.S. Hassan, A. Peterson, Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, J. Difference. Equ. Appl. (in press) · Zbl 1193.34135 · doi:10.1080/10236190902785199 [8] Erbe, L.; Hassan, T. S.; Peterson, A.; Saker, S. H.: Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear dyn. Syst. theory 9, No. 1, 51-68 (2009) · Zbl 1173.34037 [9] L. Erbe, T.S. Hassan, A. Peterson, S.H. Saker, Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Diff. Equ. (in press) · Zbl 1173.34037 [10] 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 [11] Erbe, L.; Peterson, A.; Saker, S. H.: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. compu. Appl. math. 181, 92-102 (2005) · Zbl 1075.39010 · doi:10.1016/j.cam.2004.11.021 [12] Erbe, L.; Peterson, A.; 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 [13] Erbe, L.; Peterson, A.; Saker, S. H.: Oscillation and asymptotic behavior a third-order nonlinear dynamic equation, Can. appl. Math. Q. 14, No. 2, 129-147 (2006) · Zbl 1145.34329 [14] Gera, M.; Graef, J. R.; 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 [15] Hardy, G. H.; Littlewood, J. E.; Polya, G.: Inequalities, (1988) [16] Baculíková, B.; Elabbasy, E. M.; Saker, S. H.; Urina, J. Dž: Oscillation criteria for third order nonlinear differential equations, Math. slovaka 58, 201-220 (2008) · Zbl 1174.34052 · doi:10.2478/s12175-008-0068-1 [17] Zhang, B. G.; Deng, X.: Oscillation of delay differential equations on time scales, Math. comput. Modelling 36, 1307-1318 (2002)