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Forced oscillation of second-order half-linear dynamic equations on time scales. (English) Zbl 1205.34134

Summary: We establish a new interval oscillation criterion for the second-order half-linear dynamic equation
\[ (r(t)[x^\Delta(t)]^\alpha)^\Delta+p(t)x^\alpha(\sigma(t))=f(t) \]
on a time scale \(\mathbb T\) which is unbounded. This criterion is an extension of the oscillation result for second order linear dynamic equation established by L. Erbe, A. Peterson and S. H. Saker [J. Difference Equ. Appl. 14, No. 10–11, 997–1009 (2008; Zbl 1168.34025)]. As an application, we obtain a sufficient condition for oscillation of the second-order half-linear differential equation
\[ ([x'(t)]^\alpha)'+c\sin tx^\alpha(t)=\cos t, \]
where \(\alpha=p/q\), \(p, q\) are odd positive integers.

MSC:

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

Citations:

Zbl 1168.34025
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References:

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