Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 598191, 10 pages
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

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.


Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T:Ω2H a maximal monotone operator with T10. Let PΩ be the metric projection of H onto Ω. Suppose that, for any given xnH, βn>0, and enH, there exists x¯nΩ satisfying the following set-valued mapping equation: xn+enx¯n+βnT(x¯n) for all n0, where {βn}(0,+) with βn+ as n and {en} is regarded as an error sequence such that n=0en2<+. Let {αn}(0,1] be a real sequence such that αn0 as n and n=0αn=. For any fixed uΩ, define a sequence {xn} iteratively as xn+1=αnu+(1αn)PΩ(x¯nen) for all n0. Then {xn} converges strongly to a point zT10 as n, where z=limtJtu.