Nonlinear ergodic theorems for a semitopological semigroup of non-Lipschitzian mappings without convexity. (1999)

• Record Type: Journal Article
• Authors: Li, G.; Kim, J. K.
• Is Part Of: Abstract and applied analysis. Volume 4:Number 1(1999)
• Page Start: 49
• Page End: 59
• Publication Date: 1999
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/S1085337599000056
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : LetG be a semitopological semigroup, C a nonempty subset of a real Hilbert spaceH, andℑ = { T t : t ∈ G } a representation ofG as asymptotically nonexpansive type mappings ofC into itself. LetL ( x ) = { z ∈ H : inf s ∈ G sup t ∈ G ‖ T t s x − z ‖ = inf t ∈ G ‖ T t x − z ‖ } for eachx ∈ C andL ( ℑ ) = ∩ x ∈ C L ( x ) . In this paper, we prove that∩ s ∈ G conv ¯ { T t s x : t ∈ G } ∩ L ( ℑ ) is nonempty for eachx ∈ C if and only if there exists a unique nonexpansive retractionP ofC intoL ( ℑ ) such thatP T s = P for alls ∈ G andP ( x ) ∈ conv ¯ { T s x : s ∈ G } for everyx ∈ C . Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.