Oscillation Criteria for Half-Linear Delay Dynamic Equations on Time Scales

This paper is concerned with oscillation of the second-order halflinear delay dynamic equation (r(t)(x)) + p(t)x(τ(t)) = 0, on a time scale T, where γ ≥ 1 is the quotient of odd positive integers, p(t), and τ : T → T are positive rd-continuous functions on T, r(t) is positive and (delta) differentiable, τ(t) ≤ t, and limt→∞ τ(t) = ∞. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results in the special cases when T = R and T = N involve and improve some oscillation results for second-order differential and difference equations; and when T = hN, T = q0 and T = N our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.