Let X be a nonempty, bounded,
closed and convex subset of a Banach space B. A mapping f : X → X is called a
contraction mapping if ∥f(x) −f(y)∥≦∥x−y∥ for all x,y ∈ X. Let F 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 f_{1} ∈ F such that
for each x ∈ X the closure of the set {f_{1}^{n}(x) : n = 1,2,⋯} contains a point of M
(where f_{1}^{n} denotes the nth iterate, under composition, of f_{1}). Then there is a point
x ∈ M such that f(x) = x for each f ∈ F.
(ii) If X is weakly compact and the norm of B strictly convex, and if for each
f ∈ F the fclosure of X is nonempty, then there is a point x ∈ X which is fixed
under each f ∈ F. 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.
