Advances in Difference Equations
Volume 2009 (2009), Article ID 938706, 20 pages
Research Article

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

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
2Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Received 21 January 2009; Accepted 19 April 2009

Academic Editor: Mariella Cecchi

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.


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 t0T, the functions r,q,qi,e:[t0,)T 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= and T=. In the special case when T= 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.