New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales

∧ second-order half-linear dynamic equation (r(t)(x(t))) + p(t)x(σ (t)) = 0, where r(t) > 0, p(t) are continuous, ∫ ∞ t0 (r(t)) 1 α ∆t = ∞, α is a quotient of odd positive integers. In particular, no explicit sign assumptions aremadewith respect to the coefficient p(t). We give conditions under which every positive solution of the equations is strictly increasing. For α = 1, T = R, the result improves the original theorem [see: [Lynn Erbe, Oscillation theorems for ∧ second-order linear differential equation, Pacific J. Math. 35 (2) (1970) 337–343]]. As applications, we get two comparison theorems and an oscillation theorem for half-linear dynamic equationswhich improve and extend earlier results. Some examples are given to illustrate our theorems. © 2008 Published by Elsevier Ltd