An oscillation theorem for second order superlinear dynamic equations on time scales

In this paper, the oscillatory behavior of the second order superlinear dynamic equation equation(0.1)(r(t)xΔ(t))Δ+p(t)xα(σ(t))=0,α>1,is studied under the assumption∫∞Δtr(t)<∞,where r,p∈Crd(T,R),r(t)>0,Tr,p∈Crd(T,R),r(t)>0,T in our main theorem is assumed to be a regular time scale, αα is the quotient of odd positive integers. When the coefficient function p(t)p(t) is allowed to be negative for arbitrarily large values of t, we establish a sufficient condition for oscillation of all solutions of Eq. (0.1). As special cases, we get that the superlinear differential equation(r(t)x′(t))′+p(t)xα(t)=0,α>1,is oscillatory, if∫∞Rα(t)p(t)dt=∞,R(t)=∫t∞dsr(s),and the superlinear difference equationΔ(r(n)Δx(n))+p(n)xα(n+1)=0,α>1,is oscillatory, if∑∞Rα(n+1)p(n)=∞,R(n)=∑k=n∞1r(k).