Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 892691, 14 pages
doi:10.1155/2009/892691
Research Article

On Series-Like Iterative Equation with a General Boundary Restriction

Wei Song,1 Guo-qiu Yang,1 and Feng-chun Lei2

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

Received 26 August 2008; Revised 19 November 2008; Accepted 4 February 2009

Academic Editor: Tomas Dominguez Benavides

Copyright © 2009 Wei Song et al. 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

By means of Schauder fixed point theorem and Banach contraction principle, we investigate the existence and uniqueness of Lipschitz solutions of the equation 𝒫(f)∘f=F. Moreover, we get that the solution f depends continuously on F. As a corollary, we investigate the existence and uniqueness of Lipschitz solutions of the series-like iterative equation ∑n=1∞anfn(x)=F(x),  x∈𝔹 with a general boundary restriction, where F:𝔹→𝔸 is a given Lipschitz function, and 𝔹,𝔸 are compact convex subsets of ℝN with nonempty interior.