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Half-linear dynamic equations with mixed derivatives. (English) Zbl 1092.39004

The authors consider the second order half-linear dynamic equation in a time scale \[ (r(t)\Phi(x^\Delta))^\nabla + c(t)\Phi(x) = 0, \tag{1} \] where \(\Phi(x)=| x| ^{p-2}x\), \(p > 1\), \(r(t)\) is continuous, \(r(t) \neq 0\), \(c(t)\) is ld-continuous. The authors suppose that a time scale \(T\) is unbounded above. Equation (1) is said to be nonoscillatory if there exists \(\alpha \in T\) such that (1) is disconjugate on \([\alpha, \beta]\) for every \(\beta > \alpha\). In the opposite case, (1) is said to be oscillatory. The authors establish a theorem which relates disconjugacy of (1) to solvability of a Riccati-type equation and positivity of a \(p\)-degree functional. Using this result, oscillation (nonoscillation) statements are proved.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
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