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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 f1 ∈ F such that
for each x ∈ X the closure of the set {f1n(x) : n = 1,2,⋯} contains a point of M
(where f1n denotes the n-th iterate, under composition, of f1). 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 f-closure 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.
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