Vol. 18, No. 2, 1966

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Fixed-point theorems for families of contraction mappings

Lawrence Peter Belluce and William A. Kirk

Vol. 18 (1966), No. 2, 213–217

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)xyfor 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.

Mathematical Subject Classification
Primary: 47.85
Received: 5 April 1965
Published: 1 August 1966
Lawrence Peter Belluce
William A. Kirk