Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces. (4th December 2007)

• Record Type: Journal Article
• Authors: Cho, Yeol Je; Kang, Shin Min; Zhou, Haiyun
• Other Names: Thompson H. Bevan Academic Editor.
• Is Part Of: Journal of inequalities and applications. Volume 2008(2008)
• Publication Date: 2007-12-04
• Subjects: Inequalities (Mathematics) -- Periodicals; 512.97
• DOI: 10.1155/2008/598191
• Languages: English
• ISSNs: 1029-242X

Abstract

Abstract : LetH be a real Hilbert space, Ω a nonempty closed convex subset ofH, andT : Ω → 2 H a maximal monotone operator withT − 1 0 ≠ ∅ . LetP Ω be the metric projection ofH ontoΩ . Suppose that, for any givenx n ∈ H, β n > 0, ande n ∈ H, there existsx ̄ n ∈ Ω satisfying the following set-valued mapping equation:x n + e n ∈ x ¯ n + β n T ( x ¯ n ) for alln ≥ 0, where{ β n } ⊂ ( 0, + ∞ ) withβ n → + ∞ asn → ∞ and{ e n } is regarded as an error sequence such that∑ n = 0 ∞ ∥ e n ∥ 2 < + ∞ . Let{ α n } ⊂ ( 0, 1 ] be a real sequence such thatα n → 0 asn → ∞ and∑ n = 0 ∞ α n = ∞ . For any fixedu ∈ Ω, define a sequence{ x n } iteratively asx n + 1 = α n u + ( 1 − α n ) P Ω ( x ¯ n − e n ) for alln ≥ 0 . Then{ x n } converges strongly to a pointz ∈ T − 1 0 asn → ∞, wherez = lim t → ∞ J t u .