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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

p (t) x Δ (t) γ Δ +q(t)fx τ ( t )=0

on a time scale 𝕋, here γ1 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:
34K11Oscillation theory of functional-differential equations
34N05Dynamic equations on time scales or measure chains
39A10Additive difference equations
References:
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[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)
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[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
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[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