Nonoscillation of half-linear dynamic equations

Abstract

The half-linear dynamic equation on time scales (r(t)|yΔ|α−1sgnyΔ)Δ+p(t)|yσ|α−1sgnyσ=0,α>1, is here considered, under the condition ∫∞r1/(1−α)(s)Δs<∞. Two new characterizations of nonoscillation to this equation are provided, namely in terms of solvability to a related weighted integral Riccati type inequality and in terms of the convergence of a certain function sequence involving the coefficients r and p. These relations are then applied to obtain new oscillation criteria and a comparison theorem, which extend the results known from the continuous or the linear case. The results are new even in the well-studied difference equation setting. A basic classification of nonoscillatory solutions to the equation is also presented. The paper is concluded by indicating some directions for a future research.