Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation x Delta(t)+p(t)x(tau(t))=0fort is an element of[t0,infinity)T,where p is an element of Crd([t0,infinity)T,R0+) and tau is an element of Crd([t0,infinity)T,T) satisfy tau(t) infinity tau(t)=infinity. As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: integral tau(t)sigma(t)p(eta)Delta eta <= 1e for all large t. Also, with some examples, we show that newly presented results are sharp