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

Hassan, Taher S.; Erbe, Lynn; Peterson, Allan

J. Difference Equ. Appl. 17, No. 4, 505-523 (2011).

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.

Reviewer: Roman Šimon Hilscher (Brno)

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