## Abstract and Applied Analysis

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

#### Abstract

Let $G$ be a semitopological semigroup, $C$ a nonempty subset of a real Hilbert space $H$, and $\Im =\{T_{t}:t\in G\}$ a representation of $G$ as asymptotically nonexpansive type mappings of $C$ into itself. Let $L(x)=\{z\in H:\inf_{s\in G}\sup_{t\in G}\|T_{ts}x-z\|=\inf_{t\in G}\|T_{t}x-z\|\}$ for each $x\in C$ and $L(\Im)=\bigcap_{x\in C}L(x)$. In this paper, we prove that $\bigcap_{s\in G}\overline\mathrm{conv}\{T_{ts}x:t\in G\}\bigcap L(\Im)$ is nonempty for each $x\in C$ if and only if there exists a unique nonexpansive retraction $P$ of $C$ into $L(\Im)$ such that $PT_{s}=P$ for all $s\in G$ and $P(x)\in\overline\mathrm{conv}\{T_sx:s\in G\}$ for every $x\in C$. 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

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