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Oscillation of second-order sublinear dynamic equations with damping on isolated time scales. (English) Zbl 1210.34137

Summary: This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping
\[ x^{\Delta\Delta}(t)+q(t)x^{\Delta^\sigma}(t)+p(t)x^\alpha(\sigma(t))=0, \]
on an isolated time scale \(\mathbb T\) which is unbounded above, \(0<\alpha < 1\), and \(\alpha\) is the quotient of odd positive integers. As an application, we get that the difference equation
\[ \Delta^2x(n)+n^{-\gamma}\Delta x(n+1)+[(1/n(\ln n)^\beta)+b((-1)^n/(\ln n)^\beta)]x^\alpha (n+1)=0, \]
where \(\gamma >0\), \(\beta >0\), and \(b\) is any real number, is oscillatory.

MSC:

34N05 Dynamic equations on time scales or measure chains
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A21 Oscillation theory for difference equations
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References:

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