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Abstract and Applied Analysis
Volume 2010, Article ID 978137, 14 pages
Research Article

Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations

1College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 30 March 2009; Revised 15 December 2009; Accepted 11 January 2010

Academic Editor: John Mallet-Paret

Copyright © 2010 Rong Cheng and Jianhua Hu. 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.


The existence of multiple periodic solutions of the following differential delay equation ๐‘ฅ ๎…ž ( ๐‘ก ) = โˆ’ ๐‘“ ( ๐‘ฅ ( ๐‘ก โˆ’ ๐‘Ÿ ) ) is established by applying variational approaches directly, where ๐‘ฅ โˆˆ โ„ , ๐‘“ โˆˆ ๐ถ ( โ„ , โ„ ) and ๐‘Ÿ > 0 is a given constant. This means that we do not need to use Kaplan and Yorke's reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974) is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.