Common fixed point theorems for semigroups on metric spaces

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 ( O f ( x , y ) ) ) for f ∈ S and x , y in M , where δ ( O f ( 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 { f n ( 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 ) .