## Oscillation criteria for second order sublinear dynamic equations with damping term.(English)Zbl 1228.34146

The authors establish new oscillation criteria for the second order dynamic equation of the form
$(r(t)\,x^\Delta(t))^\Delta+p(t)\,x^\Delta+q(t)\,f(x^\sigma(t))=0,\quad t\in{\mathbb T},$
where $$r$$, $$p$$, $$q$$ are real and rd-continuous functions on $${\mathbb T}$$. Here, $${\mathbb T}$$ is a time scale (i.e., a nonempty closed subset of $${\mathbb R}$$) which is unbounded above and which satisfies a certain restriction – called Condition (C) – saying that $${\mathbb T}$$ does not contain a converging sequence of right-scattered points. The novelty of the results is that no sign restriction is imposed on the coefficients.
The authors define an oscillatory solution as a solution which is neither eventually positive nor eventually negative. Note that a different concept of oscillation is also used in the literature, which is characteristic for second order dynamic equations with $$r(t)\neq0$$, (see, e.g., [M. Bohner and C. C. Tisdell, Pac. J. Math. 230, No. 1, 59–71 (2007; Zbl 1160.34029)] or [O. Došlý and S. Hilger, J. Comput. Appl. Math. 141, No. 1–2, 147–158 (2002Zbl 1009.34033]). This alternative concept is based on the existence of infinitely many generalized zeros in $${\mathbb T}$$, i.e., the existence of points $$t$$ or intervals $$(t,\sigma(t)]_{\mathbb T}$$ satisfying $$r(t)\,x(t)\,x^\sigma(t)\leq0$$. It would be interesting to compare the results of the present paper with the above definition of oscillation.

### MSC:

 34N05 Dynamic equations on time scales or measure chains 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Citations:

Zbl 1160.34029; Zbl 1009.34033
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