Abstract and Applied Analysis
Volume 4 (1999), Issue 1, Pages 49-59

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

G. Li1 and J. K. Kim2

1Department of Mathematics, Yangzhou University, Yangzhou 225002, China
2Department of Mathematics, Kyungnam University, Kyungnam 631-701, Masan, Korea

Received 2 February 1999

Copyright © 1999 G. Li and J. K. Kim. 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 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.