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Interval criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments. (English) Zbl 1205.34126

The authors consider the dynamic equation
\[ (r(t)\Phi_\alpha(x^\Delta(t)))^\Delta+p(t)\Phi_\beta(x(\tau(t)))+ q(t)\Phi_\gamma(x(\theta(t)))=e(t) \]
on a time scale interval \([t_0,\infty)\), where \(\Phi_\ast(u)=|u|^{\ast-1}u\) with \(\beta,\gamma\geq\alpha>0\), \(r,p,q,e:\mathbb{T}\to\mathbb{R}\) are rd-continuous with \(r>0\) nondecreasing, \(\theta,\tau:\mathbb{T}\to\mathbb{T}\) are nondecreasing rd-continuous with \(\theta(t)\geq t\), \(\tau(t)\leq t\) and \(\tau(t)\to\infty\) as \(t\to\infty\). Sufficient conditions (which are of interval type) guaranteeing oscillation of all solutions to this equation are established. A Riccati like transformation plays an important role in the proof. The cases with nabla derivatives or mixed derivatives are also discussed. Four examples (for the case \(\mathbb{T}=\mathbb{Z}\)) are given.
Reviewer: Pavel Rehak (Brno)

MSC:

34N05 Dynamic equations on time scales or measure chains
34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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