On Singularly Perturbed Retarded Functional Differential Equations

✍ Scribed by Zvi Artstein; Marshall Slemrod

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 160 KB
• Volume: 171
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.2000.3840

✦ Synopsis

Interest in singularly perturbed delay and functional differential equations stems from both the analytical mathematics that emerges and from realistic applications where both delays and perturbations play a role. The discussion in the next section examines possible appearances of singular perturbations in the presence of delays. In particular, we point out the distinction between the case where the delay in the fast dynamics is in the slow scale and the case where the delay occurs in the fast scale. The paper concentrates on the latter case. In fact, in such a case the semi-group approach (see, e.g., Hale [15]) is very effective, and a theory analogous to the one available for ordinary differential equations can be developed. We do that in the rest of the paper, exploiting, however, the special structure of delayed and retarded functional differential equations. Here the oscillatory phenomena and the existence of periodic trajectories generated by the delay (see, e.g., Diekmann et al. [10]) make it appealing to develop