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.


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