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Oscillation criteria for some new generalized Emden-Fowler dynamic equations on time scales. (English) Zbl 1277.34125

Summary: By means of novel analytical techniques, we establish several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale \(\mathbb T\), that is, \[ (r(t)|Z^\Delta(t)|^{\alpha - 1}Z^\Delta(t))^\Delta + f(t, x(\delta(t))) = 0 \] with respect to the case \(\int^\infty_{t_0} r^{-1/\alpha}(s)\Delta s = \infty\) and the case \(\int^\infty_{t_0} r^{-1/\alpha}(s)\Delta s < \infty\), where \(Z(t) = x(t) + p(t)x(\tau(t))\), \(\alpha\) is a constant, \(|f(t, u)| \geqslant q(t)|u^\beta|\), \(\beta\) is a constant satisfying \(\alpha \leqslant \beta > 0\), and \(r\), \(p\), and \(q\) are real valued right-dense continuous nonnegative functions defined on \(\mathbb T\).

MSC:

34N05 Dynamic equations on time scales or measure chains
34K11 Oscillation theory of functional-differential equations
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