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

Let TT be a time scale (i.e., a closed nonempty subset of RR) with supT=+∞supT=+∞. Consider the second-order half-linear dynamic equation (r(t)(xΔ(t))α)Δ+p(t)xα(σ(t))=0,(r(t)(xΔ(t))α)Δ+p(t)xα(σ(t))=0, where r(t)>0,p(t)r(t)>0,p(t) are continuous, ∫t0∞(r(t))−1αΔt=∞, αα is a quotient of odd positive integers. In particular, no explicit sign assumptions are made with respect to the coefficient p(t)p(t). We give conditions under which every positive solution of the equations is strictly increasing. For α=1α=1, T=RT=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 equations which improve and extend earlier results. Some examples are given to illustrate our theorems.