Boundary Value Problems
Volume 2008 (2008), Article ID 937138, 16 pages
doi:10.1155/2008/937138
Research Article

Existence of Solutions of Periodic Boundary Value Problems for Impulsive Functional Duffing Equations at Nonresonance Case

Xingyuan Liu1 and Yuji Liu2

1Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, China
2Department of Mathematics, Guangdong University of Business Studies, Guangzhou, Guangdong 510320, China

Received 19 June 2008; Accepted 27 August 2008

Academic Editor: Zhitao Zhang

Copyright © 2008 Xingyuan Liu and Yuji Liu. 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.

Abstract

This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: x′′(t)+αx′(t)+βx(t)=f(t,x(t),x(α1(t)),…,x(αn(t))), a.e.  t∈[0,T], Δx(tk)=Ik(x(tk),x′(tk)), k=1, …,m, Δx′(tk)=Jk(x(tk),x′(tk)), k=1,…,m, x(i)(0)=x(i)(T), i=0,1. Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer's fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results.