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Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales. (English) Zbl 1125.34047

The authors establish several new oscillation criteria for second order dynamic equations \[ x^{\Delta\Delta}(t)+p(t)\,x^\gamma(\tau(t))=0 \] of Emden-Fowler type on an unbounded time scale \({\mathbb T}\) (which is by definition any nonempty closed subset of \({\mathbb R}\)). The exponent \(\gamma\) is a quotient of odd positive integers, \(p(\cdot)\) is positive and rd-continuous, and \(\tau:{\mathbb T}\to{\mathbb T}\) is rd-continuous, sublinear, i.e., \(\tau(t)\leq t\), and \(\tau(t)\to\infty\) as \(t\to\infty\). The main tool in deriving these oscillation criteria is a Riccati technique and, in some cases, the Keller–Pötzsche time scale chain rule.

MSC:

34K11 Oscillation theory of functional-differential equations
39A10 Additive difference equations
39A13 Difference equations, scaling (\(q\)-differences)
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