Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 892691, 14 pages
doi:10.1155/2009/892691
On series-like iterative equation with a general boundary restriction
Wei Song1
, Guo-Qiu Yang1
and Feng-Chun Lei3
1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
3Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
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.