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
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
Academic Editor: H. Bevan Thompson
Copyright © 2008 Yeol Je Cho 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
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.