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We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ(t) = 0, on a time scale T ,w here γ> 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscilla- tion of Hille-Kneser type. Our results in the special case when T = R involve the well- knownHille-Kneser-typecriteriaofsecond-orderlineardifferentialequationsestablished by Hille. For the case of the second-order half-linear differential equation, our results ex- tend and improve some earlier results of Li and Yeh and are related to some work of Dosl´ y and ˇ Reh´ ak and some results of ˇ

Hille-Kneser-type criteria for second-order dynamic equations on time scales