Copyright © 2009 Ravi P. Agarwal and A. Zafer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form (r(t)Φα(xΔ))Δ+f(t,xσ)=e(t), t∈[t0,∞)T with f(t,x)=q(t)Φα(x)+∑i=1nqi(t)Φβi(x), Φ∗(u)=|u|∗−1u, where [t0,∞)T is a time scale interval with t0∈T, the functions r,q,qi,e:[t0,∞)T→ℝ are right-dense continuous with r>0, σ is the forward jump operator, xσ(t):=x(σ(t)), and β1>⋯>βm>α>βm+1>⋯βn>0. All results obtained are new even for T=ℝ and T=ℤ. In the special case when T=ℝ and α=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.