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

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

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.