We establish criteria of Hille-Nehari type for the half-linear second order dynamic equation (r(t)Φ(y~Δ))~Δ +p(t)Φ(y~σ) = 0, Φ(u) = |u|~(α_1) sgn u, α > 1, on time scales, under the condition ∫ r~(1/(1-α))(s) Δs < ∞. As a particular important case we get that there is a (non-improvable) critical oscillation constant which may be different from the one known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. Along with the results of the previous paper by the author, which dealt with the condition ∫ r~(1/(1-α))(s) Δs < ∞, a quite complete discussion on generalized Hille-Nehari type criteria involving the best possible constants is provided. To prove these criteria, appropriate modifications of the approaches known from the linear case (α = 2) or the continuous case (T = R) cannot be used in a general case, and thus we apply a new method. As applications of the main results we state criteria for strong (non)oscillation, examine a generalized Euler type equation, and establish criteria of Kneser type. Examples from q-calculus and h-calculus, and a Hardy type inequality are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.
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