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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 Δ (t)] α ) Δ +p(t)x α (σ(t))=f(t)

on a time scale 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)] α ) ' +csintx α (t)=cost,

where α=p/q, p,q are odd positive integers.

34N05Dynamic equations on time scales or measure chains
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34K11Oscillation theory of functional-differential equations