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

• Main Authors: Haiyun Zhou, Shin Min Kang, Yeol Je Cho
• Format: Article
• Language: English
• Published: SpringerOpen 2008-03-01
• Series: Journal of Inequalities and Applications
• Online Access: http://dx.doi.org/10.1155/2008/598191

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 x̄n∈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.