Oscillation criteria for half-linear dynamic equations on time scales. (English) Zbl 1156.34022

The author considers the second-order half-linear dynamic equation
\[ (r(t)(x^{\Delta}(t))^{\gamma})^{\Delta}+p(t)x^{\gamma}(t)=0\tag{1} \]
on an arbitrary time scale \(\mathbb{T}\) (\(\sup\mathbb T=\infty\)), where \(\gamma\) is the quotient of odd positive integers, \(r(t)\) and \(p(t)\) are positive rd-continuous functions on \(\mathbb{T}\). Main results of the paper are sufficient conditions for every solution of (1) to be oscillatory. As the author remarks, when \(\mathbb T=\mathbb R\), the obtained results improve several results known for differential equations and when \(\mathbb T=\mathbb N\), then they improve some results known for second order difference equations.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10 Additive difference equations
Full Text: DOI


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