Oscillation criteria for second-order nonlinear dynamic equations

This paper concerns the oscillation of solutions to the second-order dynamic equation (r(t)x(Delta) (t))(Delta) + p(t)x(Delta) (t) + q(t)f(x(sigma) (t)) = 0, on a time scale T which is unbounded above. No sign conditions are imposed on r(t), p(t), and q(t). The function f is an element of C(R, R) is assumed to satisfy xf (x) > 0 and f' (x) > 0 for x not equal 0. In addition, there is no need to assume certain restrictive conditions and also the both cases integral(infinity)(t0) Delta t/r(t) = infinity and integral(infinity)(t0) Delta t/r(t) < infinity are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as T = Z, T = hZ, h > 0, or T = {t : t = q(k), k is an element of N-0, q > 1} and the space of harmonic numbers T = H-n. Some examples illustrating the importance of our results are also included.