Hille and Nehari type criteria for third-order dynamic equations (Q868770)

For the third order dynamic equation on an arbitrary time scale \(T\) with \(\sup T=\infty\), \[ x^{\Delta\Delta\Delta}(t)+p(t)x(t)=0 \eqno{(1)} \] 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 \textit{E. Hille} [Trans. Am. Math. Soc. 64, 234--252 (1948; Zbl 0031.35402)] and \textit{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.