On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

We investigate the oscillation of the following higher-order functional differential equation: where q t and e t are continuous functions on t 0 , ∞ , 1 > λ > 0 and τ / 0 are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the case τ > 0 and the case τ < 0 under the assumption that the potentials q t and e t change signs on t 0 , ∞

Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities

We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form (r(t)Phi(alpha)(x(Delta)))(Delta) + f(t,x(sigma)) = e(t), t is an element of [t(0), infinity)(T) with f (t, x) = q(t) Phi(alpha)(x) + Sigma(n)(i=1)q(i)(t)Phi(beta i)(x), Phi(*)(u) = vertical bar u vertical bar*(-1) u, where [t(0), infinity)(T) is a time scale interval with t(0) is an element of T, the functions r, q, q(i), e : [t(0), infinity)(T) -> R are right-dense continuous with r > 0, sigma is the forward jump operator, x(sigma) (t) := x(sigma(t)), and beta(1) > ... > beta(m) > alpha > beta(m+1) > ... beta(n) > 0. All results obtained are new even for T = R and T = Z. In the special case when T = R and alpha = 1 our theorems reduce to (Y. G. Sun and J. S. W. Wong, Journal of Mathematical Analysis and Applications. 337 (2007), 549-560). Therefore, our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale. Copyright (C) 2009 R. P. Agarwal and A. Zafer.Publisher's Versio