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Oscillation of second-order delay dynamic equations on time scales. (English) Zbl 1180.34069

Summary: By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations

${\left(p\left(t\right){\left({x}^{{\Delta }}\left(t\right)\right)}^{\gamma }\right)}^{{\Delta }}+q\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right)=0$

on a time scale $𝕋$, here $\gamma \ge 1$ is a quotient of odd positive integers with $p$ and $q$ real-valued positive rd-continuous functions defined on $𝕋$. Our results improve and extend some results established by S. H. Saker [J. Comput. Appl. Math. 177, No. 2, 375–387 (2005; Zbl 1082.34032)] but also unify the oscillation of the second order nonlinear delay differential equation and the second order nonlinear delay difference equation.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34N05 Dynamic equations on time scales or measure chains 39A10 Additive difference equations
##### Keywords:
oscillation; second order; delay dynamic equations; time scale
##### References:
 [1] Hilger, S.: Analysis on measure chains–a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990) [2] Agarwal, R.P., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141, 1–26 (2002) · Zbl 1020.39008 · doi:10.1016/S0377-0427(01)00432-0 [3] Bohner, M., Peterson, A.: Dynamic Equations on Time Scales, an Introduction with Applications. Birkhauser, Boston (2001) [4] Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003) [5] Bohner, M., Saker, S.H.: Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mt. J. Math. 34, 1239–1254 (2004) · Zbl 1075.34028 · doi:10.1216/rmjm/1181069797 [6] Erbe, L.H.: Oscillation results for second order linear equations on a time scale. J. Differ. Equ. Appl. 8, 1061–1071 (2002) · Zbl 1021.34012 · doi:10.1080/10236190290015317 [7] Saker, S.H.: Oscillation criteria of second-order half-linear dynamic equations on time scales. J. Comput. Appl. Math. 177, 375–387 (2005) · Zbl 1082.34032 · doi:10.1016/j.cam.2004.09.028 [8] Agarwal, R.P., Bohner, M., Saker, S.H.: Oscillation of second order delay dynamic equations. Can. Appl. Math. Q. 13, 1–18 (2005) [9] Erbe, L., Peterson, A., Saker, S.H.: Oscillation criteria for second-order nonlinear delay dynamic equations. J. Math. Anal. Appl. 333, 505–522 (2007) · Zbl 1125.34046 · doi:10.1016/j.jmaa.2006.10.055 [10] Han, Z., Sun, S., Shi, B.: Oscillation criteria for a class of second order Emden-Fowler delay dynamic equations on time scales. J. Math. Anal. Appl. 334, 847–858 (2007) · Zbl 1125.34047 · doi:10.1016/j.jmaa.2007.01.004 [11] Sahiner, Y.: Oscillation of second-order delay differential equations on time scales. Nonlinear Anal. TMA 63, 1073–1080 (2005) · Zbl 1224.34294 · doi:10.1016/j.na.2005.01.062 [12] Zhang, B.G., Shanliang, Z.: Oscillation of second order nonlinear delay dynamic equations on time scales. Comput. Math. Appl. 49, 599–609 (2005) · Zbl 1075.34061 · doi:10.1016/j.camwa.2004.04.038