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Solution properties on discrete time scales. (English) Zbl 1032.39006

The authors study oscillatory and asymptotic properties of the nonlinear dynamic equation \[ u^{\Delta^2}(t)+p(t)u^\gamma(\sigma(t))=0 \tag{*} \] on an unbounded time scale \(\mathbb {T}\) consisting of only isolated points, where \(p(t)\) is either nonnegative or nonpositive function, and \(\gamma\) is a quotient of odd positive integers. If \(p(t)\leq 0\) (and eventually nontrivial), then equation (\(\ast\)) is shown to be nonoscillatory, and solution comparisons and asymptotic properties are presented.
The main result in the case where \(p(t)\geq 0\) (and eventually nontrivial) is the following oscillation criterion: Let \(a\in\mathbb {T}\), \(a\geq 0\), and \(\gamma>1\). If \(\int_a^\infty\sigma(t)p(t) \Delta t=\infty\), then equation (\(\ast\)) is oscillatory.
Reviewer: Pavel Rehak (Brno)

MSC:

39A12 Discrete version of topics in analysis
93C70 Time-scale analysis and singular perturbations in control/observation systems
39A11 Stability of difference equations (MSC2000)
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