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 ) x Δ ( t ) ] Δ } γ ) Δ + f ( t , x ( τ ( t ) ) ) = 0 , on a time scale T , where γ ≥ 1 is the quotient of odd positive integers, a and r are positive r d -continuous functions on T , and the so-called delay function τ : T → T satisfies τ ( t ) ≤ t for t ∈ T and lim t → ∞ τ ( t ) = ∞ and 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 = R and T = N involve and improve some oscillation results for third order delay differential and difference equations; when T = h N , T = q N 0  and T = N 2 our oscillation results are essentially new. Some examples are given to illustrate the main results.