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

Zhang, B. G.; Shanliang, Zhu

Comput. Math. Appl. 49, No. 4, 599-609 (2005).

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

Keywords:

Oscillation; Time scales; Riccati technique; Delay dynamic equation

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
[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
[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