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Hille and Nehari type criteria for third-order dynamic equations. (English) Zbl 1128.39009

For the third order dynamic equation on an arbitrary time scale T with supT=,

x ΔΔΔ (t)+p(t)x(t)=0(1)

where p(t) is a positive real-valued rd-continuous function defined on T, the authors consider its oscillatory properties. Several sufficient conditions are obtained for oscillation of all solutions of (1).

The results given in this paper extend those established by E. Hille [Trans. Am. Math. Soc. 64, 234–252 (1948; Zbl 0031.35402)] and Z. Nehari [Trans. Am. Math. Soc. 85, 428–445 (1957; Zbl 0078.07602)] for second order differential equations. The oscillation criteria for (1) are new even for third order differential equations and the corresponding difference equations. Several examples illustrating the results are also given.

39A12Discrete version of topics in analysis
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A11Stability of difference equations (MSC2000)