Fixed-point theorems for families of contraction mappings.

Let X be a nonempty, bounded, closed and convex subset of a Banach space B. A mapping fιX-*X is called a contraction mapping if 11 fix) — f(y) \\ ̂ 11 x — y |1 for all x, y e X. Let % be a nonempty commutative family of contraction mappings of X into itself. The following results are obtained. (i) Suppose there is a compact subset M of X and a mapping /i6% such that for each xe X the closure of the set {fi(%)' w = 1,2, •} contains a point of ikf (where / * denotes the n iterate, under composition, of /i). Then there is a point xeM such that fix) = x for each / e g . (ii) If X is weakly compact and the norm of B strictly convex, and if for each / e g the /-closure of X is nonempty, then there is a point xeX which is fixed under each / e g . A third theorem, for finite families, is given where the hypotheses are in terms of weak compactness and a concept of Brodskii and Milman called normal structure.