International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 2, Pages 377-386
doi:10.1155/S0161171299223770
Common fixed point theorems for semigroups on metric spaces
Department of Mathematics, National Cheng Kung University, Tainan 70101, Taiwan
Received 21 December 1995; Revised 16 February 1998
Copyright © 1999 Young-Ye Huang and Chung-Chien Hong. 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.
Abstract
This paper consists of two main results. The first one shows that
if S is a left reversible semigroup of selfmaps on a complete metric space (M,d) such that there is a gauge function φ for which d(f(x),f(y))≤φ(δ(Of (x,y))) for f∈S and x,y in M, where δ(Of (x,y)) denotes the diameter of the orbit of x,y under f, then S has a unique common fixed point ξ in M and, moreover, for any f in S and x in M, the sequence of iterates {fn(x)} converges to ξ. The second result is a common fixed point theorem for a left reversible uniformly Lipschitzian semigroup of selfmaps on a bounded hyperconvex metric space (M,d).