Journal of Inequalities and Applications 
Volume 2008 (2008), Article ID 598191, 10 pages
doi:10.1155/2008/598191
Research Article

Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces

Yeol Je Cho,1 Shin Min Kang,2 and Haiyun Zhou3

 1Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea
2Department of Mathematics and the RINS, Gyeongsang National University, Chinju 660-701, South Korea
3Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
Received 1 March 2007; Accepted 27 November 2007

Recommended by H. Bevan Thompson

Abstract

Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T:Ω→2H a maximal monotone operator with T−10 ≠ ∅. Let PΩ be the metric projection of H onto Ω. Suppose that, for any given xn∈H, βn>0, and en∈H, there exists x¯n∈Ω satisfying the following set-valued mapping equation: xn+en∈x¯n+βnT(x¯n) for all n≥0, where {βn}⊂(0,+∞) with βn→+∞ as n→∞ and {en} is regarded as an error sequence such that ∑n=0∞‖en‖2<+∞. Let {αn}⊂(0,1] be a real sequence such that αn→0 as n→∞ and ∑n=0∞αn=∞. For any fixed u∈Ω, define a sequence {xn} iteratively as xn+1=αnu+(1−αn)PΩ(x¯n−en) for all n≥0. Then {xn} converges strongly to a point z∈T−10 as n→∞, where z=limt→∞Jtu.