On Oscillation of a Certain Class of Third-Order Nonlinear Functional Dynamic Equations on Time Scales

In this paper, we establish some new sufficient conditions for oscillation of the third order nonlinear functional dynamic equation [p(t)[(r(t)x(Delta)(t))(Delta)]](Delta) + q(t) f (x(tau(t))) = 0, for t is an element of [t(0), infinity)tau, on a time scale T, where gamma > 0 is the quotient of odd positive integers, p, q, r and T are positive rd-continuous functions defined on T and f is an element of C(R,R), uf (u) > 0 and f(u)/u(gamma) >= K > 0, for u not equal 0. The results provided substantial improvement over those obtained by Yu and Wang [J. Comp. App!. Math. 225 (2009), 531-540) and Hassan [Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comp. Modelling 49 (2009), 1573-1586], in the sense that, our results can be applied when 0 < gamma < 1, (tau o sigma) (t) not equal (sigma o tau) (t), and do not require that integral(infinity)(t0) q(t)Delta t = infinity. Some examples illustrating the main results are given.