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In this paper, we will consider the higher-order functional dynamic equations of the form on an above-unbounded time scale , where and , . The function is a rd-continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

In this paper we relate the oscillation problem of the nonlinear functional differential equation (a(t)x'(t))' + q(t) f (x(g(t)))= 0 and the nonlinear neutral functional differential equation (a(t) (x(t)+ p(t)x(g*(t)))'} + q(t)f (x(g(t))) = 0 to some linear second order ordinary differential equations. Recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for the nonlinear equations. Similar results for the forced nonlinear functional differential equation (a(t)x'(t))' + q(t) f (x(g(t)))= e(t) and the forced neutral functional differential equation (a(t) (x(t)+ mx(t — n))')' + q(t) f (x(g(t)))= e(t) are also established. The function f appeared in the above equations is not require to be monotone.

In this note we study the oscillatory behavior of solutions of the nth order nonlinear functional differential equation x(")(t) + q(t)f(x[g(t)]) = 0, n even, without assuming that the deviating argument is retarded or advanced. Sufficient conditions are established for all solutions of the equation to be oscillatory. The purpose of this note is to obtain some oscillation criteria for the n th order equation (1) X(n)(t) + q(t)f(x[g(t)]) = 0, n even, where the following conditions are assumed to hold: (2) q, g E C[[ to, x), R], q(t) is nonnegative and not identically zero on any interval of the form [ tI, x), t1 > to and lim ,0 g(t) = x; (3)fE C[R, R],fisnondecreasing0 0; (4) Jfo du/f(u) 0; (5) f(xy) 2 Kf(x)f(y) for x, y > 0 and K is a positive constant. In what follows, by a proper solution of (1), we mean a function x E Cn?fTX, x), R] which satisfies (1) for all sufficiently large t and sup{Ix(t)I: t 2 T} > 0 for any T 2 T,. A proper solution of (1) is called oscillatory if it has arbitrarily large zeros and it is called nonoscillatory otherwise. We will have an occasion to use the following two lemmas given in [1]. LEMMA 1. Let u be a positive and n-times differentiable function on an interval [to, xe). If u(n) is of constant sign and not identically zero on any interval of the form [t1, x), t1 2 to, then there exists a t 0u> to and an integer 1, 0 1 n with n + 1 even for u(n) 2 0 or n + 1 odd for U(n) 0 and such that 1 > 0 implies U(k)(t) > 0 for every t 2 tu (k = ,1,.. .,l1-1) and 1 O for every t 2 tu (kL=MM, 2 + t,...,nc1). LEMMA 2. If the function u is as in Lemma I and ,(n-l)(t,U(n)(t) ,: : 0 fr everyt 0 such that u[At] >MA"-1|U("()| for all large t. Moreover, if u is increasing, then u ( t) u[ t] 2m M/ 2tn|u ("")(t)|for all large t. REMARK. From the proof of Lemma 2 [1, Lemma 2], we can take 22-2 1 (n -1) We also need the following lemma. LEMMA 3. Let n = 1, and conditions (2)-(4) hold. If

In this paper, we present some new criteria for the oscillation of the differential equation with damping where p(t) and q(t) are allowed to change signs on [t0 ∞]. These criteria are obtained by using an integral averaging technique and can be applied in some special cases in which other classical oscillation results are not applicable.