Nonoscillation of half-linear dynamic equations

The half-linear dynamic equation on time scales 􀀀 r.t/jy1j 􀀀1 sgn y1 1 C p.t/jy j 􀀀1 sgn y D 0; > 1; is here considered, under the condition R 1 r1=.1􀀀 /.s/1s < 1. 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 wellstudied 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.