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Oscillation of second-order nonlinear delay dynamic equations on time scales. (English) Zbl 1075.34061

Summary: We establish the equivalence of the oscillation of the nonlinear dynamic equations \[ x^{\Delta\Delta}(t)+p(t)f\bigl(x(t-\tau) \bigr)=0\quad\text{and }x^{\Delta\Delta}(t)+p(t)(f\circ x^\sigma)=0 \] on time scales, from which we obtain some oscillation criteria and comparison theorems for the first equation. Next, we obtain some new oscillation criteria for second-order linear dynamic equations on time scales.

MSC:

34K11 Oscillation theory of functional-differential equations
39A12 Discrete version of topics in analysis
34C41 Equivalence and asymptotic equivalence of ordinary differential equations
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References:

[1] Zhang, B. G.; Xinghua, D., Oscillation of delay differential equations on time scales, Mathl. Comput. Modelling, 36, 11-13, 1307-1318 (2002) · Zbl 1034.34080
[2] Erbe, L.; Peterson, A., Oscillation criteria for second order matrix dynamic equation on time scale, J. Comput. Appl. Math., 14, 169-186 (2002) · Zbl 1017.34030
[3] L. Erbe, A. Peterson and S.H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales (to appear).; L. Erbe, A. Peterson and S.H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales (to appear). · Zbl 1050.34042
[4] Erbe, L.; Peterson, A.; Rehak, P., Comparison theorems for linear dynamic equation on time scales, J. Math. Anal. Appl., 275, 418-438 (2002) · Zbl 1034.34042
[5] Bohner, M.; Peterson, A., Dynamic Equation on Time Scales: An Introduction with Application (2001), Birkhäuser: Birkhäuser Huntington, New York · Zbl 0978.39001
[6] Erbe, L. H.; Kong, Q.; Zhang, B. G., Oscillation Theory for Functional Differential Equations (1995), Marcel Dekker: Marcel Dekker Boston, MA
[7] Zhang, B. G.; Yang, B., Equivalence of oscillation of a class of neutral differential equations and ordinary differential equations, J. Anal. Appl., 16, 2, 451-462 (1997) · Zbl 0883.34070
[8] Yan, J., Oscillation theorems for second order linear differential equations, Acta Math. Appl. SINICA, 10, 167-174 (1987), (in Chinese) · Zbl 0632.34031
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