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Boundedness and oscillation for nonlinear dynamic equations on a time scale. (English) Zbl 1055.39007

The authors consider the nonlinear dynamic equation \[ (p(t) x^\Delta)^\Delta+ q(t) (f\circ x^\sigma)= 0 \] on a time scale \(\mathbb{T}\subset\mathbb{R}\) with \(\sup\mathbb{T}=\infty\), where \(f\in C^1(\mathbb{R},\mathbb{R})\) satisfies that \(f'(x)\geq {f(x)\over x}> 0\) for \(x\neq 0\). Criteria are obtained for oscillation and asymptotic behavior of solutions based on the assumption that a corresponding linear equation is oscillatory. Explicit conditions for oscillation are derived as consequences, and several examples, including a nonlinear Emden-Fowler dynamic equation, are given for illustration.

MSC:

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