Fixed point theorems for families of nonexpansive mappings on unbounded sets (Q1080162)

Let X be a Banach space, C a closed and convex subset of X and S a semigroup of nonexpansive mappings T of C into itself. The semigroup S is said to be left- [right-] reversible if, for every \(T_ 1,T_ 2\in S\), there is a pair \(T_ 1',T_ 2'\in S\) such that \(T_ 1T_ 1'=T_ 2T_ 2'\) \([T_ 1'T_ 1=T_ 2'T_ 2]\). The author shows (Theorem 2) that if X is reflexive and has normal structure, then the semigroup S has a common fixed point in C if and only if there is an element \(x\in C\) such that the set \(\{\) Tx:T\(\in S\}\) is bounded. This theorem is then used to deduce a number of corollaries which generalize earlier results [such as those of the author, Proc. Am. Math. Soc. 81, 253-256 (1981; Zbl 0456.47054) and \textit{W. A. Kirk} and the reviewer, Stud. Math. 64, 127-138 (1979; Zbl 0412.47033)]. Related results deal with reformulations of the nonlinear mean ergodic theorem for a (somewhat more restricted) class of noncommutative semigroups. The proofs involve a technical reworking of the ideas developed for commuting families.