Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 75672, 11 pages
Trench's Perturbation Theorem for Dynamic Equations
1Department of Mathematics and Statistics, Missouri University of Science and Technology, 65401, MO, USA
2Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, Beograd 11000, Serbia
Received 10 June 2007; Accepted 31 October 2007
Copyright © 2007 Martin Bohner and Stevo Stević. 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 consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.