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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.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A12Discrete version of topics in analysis
Full Text: DOI
[1] R.P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: a survey, in: R.P. Agarwal, M. Bohner, D. O’Regan (Eds.), Dynamic Equations on Time Scales (Preprint in Ulmer Seminare 5), J. Comput. Appl. Math. 141 (2002) 1 -- 26 (special issue). · Zbl 1020.39008
[2] Akın, E.; Erbe, L.; Kaymakçalan, B.; Peterson, A.: Oscillation results for a dynamic equation on a time scale. J. differential equations appl. 7, 793-810 (2001) · Zbl 1002.39024
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