Oscillation of third order nonlinear delay dynamic equations on time scales

It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation (a(t){[r(t)xΔ(t)]Δ}γ)Δ+f(t,x(τ(t)))=0, on a time scale TT, where γ≥1γ≥1 is the quotient of odd positive integers, aa andr are positive rdrd-continuous functions on TT, and the so-called delay function τ:T→Tτ:T→T satisfies τ(t)≤tτ(t)≤t for t∈Tt∈T and limt→∞τ(t)=∞limt→∞τ(t)=∞ and f∈C(T×R,R)f∈C(T×R,R). Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when T=RT=R and T=NT=N involve and improve some oscillation results for third order delay differential and difference equations; when T=hNT=hN, T=qN0T=qN0 and T=N2T=N2 our oscillation results are essentially new. Some examples are given to illustrate the main results.