Oscillation of second-order nonlinear neutral delay dynamic equations on time scales

摘要

In this paper, some sufficient conditions for oscillation of the second-order nonlinear neutral delay dynamic equation (r(t)([y(t)+p(t)y(t-τ)]Δ)γ)Δ+f(t,y(t-δ))=0,on a time scale T are established; here γ⩾1 is an odd positive integer with r(t) and p(t) are rd-continuous functions defined on T. Our results as a special case when T=R and T=N, involve and improve some well-known oscillation results for second-order neutral delay differential and difference equations. When T=hN and T=qN={t:t=qk,k∈N,q>1}, i.e., for generalized neutral delay difference and q-neutral delay difference equations our results are essentially new and also can be applied on different types of time scales, e.g., T=N2={t2:t∈N} and T=Tn={tn:n∈N0} where {tn} is the set of harmonic numbers. Some examples illustrating our main results are given.