Theorem 6.2 Let X be a Banach space which has the FPP, let C be a closed convex subset of X, and suppose f : C → C is a nonexpansive mapping for which F δ ( f, C ) is nonempty and bounded for some δ > 0. Lemma 6.1 Under the assumptions of Theorem 6.1, α ( S K ) = 0.Ī Banach space X is said to have the FPP if each of its bounded closed convex subsets has the fixed point property for nonexpansive self-mappings. This theorem is an immediate consequence of the following lemma (Lemma 3.1 in ). If X is of hyperbolic type, in view of Lemma 3.1 some bounded convex subset H of K is mapped into itself by g. Suppose f : K → K is a nonexpansive mapping, and suppose.
Theorem 1.1 Suppose K is a nonempty closed and convex subset of a reflexive Banach space, and suppose K has a normal structure. Indeed, the following result is found in the original 1965 paper of Kirk. However, in this case the mapping has bounded orbits. Consider, for example, simple rotations in the plane. On the other hand, it is obvious that nontrivial nonexpansive mappings defined on unbounded sets may have fixed points.
UNBOUNDED COMPLETE METRIC SPACE FREE
It has been shown recently (see ) that the failure of the fixed point property for every unbounded convex closed set is not a characteristic of Hilbert spaces more precisely, for every unbounded closed convex set in c 0, there exists a fixed point free nonexpansive self-mapping of the set. The answer is ‘yes’ if X is a Hilbert space (see ). If a closed convex subset has the fixed point property for all nonexpansive self-mappings, then is it necessarily bounded? This has long been an open question in metric fixed point theory.
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