It is the purpose of this paper to give oscillation criteria for the third order nonlinear functional dynamic equationon a time scale T, where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on T, and the function g : T → T satisfies lim t→∞ g(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 equations. Some examples are given to illustrate the main results.
This paper is concerned with oscillation of the second-order half-linear dynamic equationon a time scale T where γ is the quotient of odd positive integers, r(t) and p(t) are positive rd-continuous functions on T. Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085-1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375-387] and our results in the special cases when T = R and T = Z involve and improve some oscillation results for second-order differential and difference equations; and when T = hZ, T = q N 0 and T = N 2 0 , etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.
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