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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 362526, 23 pages
Research Article

New Inequalities of Opial's Type on Time Scales and Some of Their Applications

College of Science Research Centre, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 31 December 2011; Accepted 14 March 2012

Academic Editor: Hassan A. El-Morshedy

Copyright © 2012 Samir H. Saker. 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 will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.