Abstract and Applied Analysis

Nonlinear ergodic theorems for a semitopological semigroup of non-Lipschitzian mappings without convexity

G. Li and J. K. Kim

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Abstract

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H, and ={Tt:tG} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)={zH:infsGsuptGTtsxz=inftGTtxz} for each xC and L()=xCL(x). In this paper, we prove that sGconv¯{Ttsx:tG}L() is nonempty for each xC if and only if there exists a unique nonexpansive retraction P of C into L() such that PTs=P for all sG and P(x)conv¯{Tsx:sG} for every xC. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.

Article information

Source
Abstr. Appl. Anal., Volume 4, Number 1 (1999), 49-59.

Dates
First available in Project Euclid: 9 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1049907139

Digital Object Identifier
doi:10.1155/S1085337599000056

Mathematical Reviews number (MathSciNet)
MR1799460

Zentralblatt MATH identifier
0988.47039

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10
Secondary: 47H20

Citation

Li, G.; Kim, J. K. Nonlinear ergodic theorems for a semitopological semigroup of non-Lipschitzian mappings without convexity. Abstr. Appl. Anal. 4 (1999), no. 1, 49--59. doi:10.1155/S1085337599000056. https://projecteuclid.org/euclid.aaa/1049907139


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