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Oscillation criteria of second-order half-linear dynamic equations on time scales. (English) Zbl 1082.34032

The subject of the paper is oscillation of solutions of second-order delta differential equations of the type \[ (p(t)(x^\Delta(t))^\gamma)^\Delta+q(t)x^\gamma(t)=0\tag{1} \] on quite general time scales. Here, \(\gamma>1\) is an odd positive integer and \(p,\;q\) are positive right-dense continuous functions. The function \((1/p)^{1/ \gamma}\) may be integrable (in the sense of time scale analysis) at \(+\infty\) or not. Sufficient conditions for the oscillatory character of all non-trivial solutions of (1) are given in both cases. The main theorems are applicable, in particular, to half-linear difference equations and half-linear ordinary differential equations, unifying and extending previous results. At the end of the paper, some examples are discussed to illustrate the range of applicability of the main results.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A12 Discrete version of topics in analysis
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