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

Rong Cheng,Jianhua Hu

doi:10.1155/2010/978137

Abstract

The existence of multiple periodic solutions of the following differential delay equation is established by applying variational approaches directly, where , and 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.

Highlights

We are concerned in this paper with the search for 4r-periodic solutions r > 0 of a class of differential delay equations with the following form x t −f x t − r, 1.1 where x ∈ R, f ∈ C R, R, and r > 0 is a given constant
Guo and Yu 19 do not use Kaplan and Yorke’s reduction technique and apply variational methods directly to study the existence of multiple periodic solutions of 1.1 with x and f being vectors in Rn
H is the set of those functions satisfying

Summary

Introduction

We are concerned in this paper with the search for 4r-periodic solutions r > 0 of a class of differential delay equations with the following form x t −f x t − r , 1.1 where x ∈ R, f ∈ C R, R , and r > 0 is a given constant. Following the reduction idea of Kaplan and Yorke, Li and He 17, 18 were able to translate 1.1 with more than one delay to a coupled Hamiltonian system They used variational approaches to study the coupled Hamiltonian system and obtained some existence results of multiple periodic solutions of the equations. Guo and Yu 19 do not use Kaplan and Yorke’s reduction technique and apply variational methods directly to study the existence of multiple periodic solutions of 1.1 with x and f being vectors in Rn. That is to say they do not reduce the existence problem of 1.1 to an existence problem of a related coupled Hamiltonian system. We will use 31, Theorem 2.4 to prove the main result

Variational Functional on Hilbert Space

Proof of the Main Result