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Oscillation criteria for second-order nonlinear dynamic equations

Research paper by Taher S Hassan

Indexed on: 02 Oct '12Published on: 02 Oct '12Published in: Advances in difference equations



Abstract

This paper concerns the oscillation of solutions to the second-order dynamic equationon a time scale TOpen image in new window which is unbounded above. No sign conditions are imposed on r(t)Open image in new window, p(t)Open image in new window, and q(t)Open image in new window. The function f∈C(R,R)Open image in new window is assumed to satisfy xf(x)>0Open image in new window and f′(x)>0Open image in new window for x≠0Open image in new window. In addition, there is no need to assume certain restrictive conditions and also the both casesare 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=ZOpen image in new window, T=hZOpen image in new window, h>0Open image in new window, or T={t:t=qk,k∈N0,q>1}Open image in new window and the space of harmonic numbers T=HnOpen image in new window. Some examples illustrating the importance of our results are also included.MSC:34K11, 39A10, 39A99.