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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Fixed points in nonconvex domains

Authors: Eric Chandler and Gary Faulkner
Journal: Proc. Amer. Math. Soc. 80 (1980), 635-638
MSC: Primary 47H10; Secondary 47H09
MathSciNet review: 587942
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Abstract: A lemma of Janos is used to prove that nonexpansive self-maps which ``shrink'' a compact set X away from its boundary in $ \overline {{\text{co}}} \;X$ have fixed points in X. The lemma is further employed to derive a version, for nonexpansive maps on star-shaped sets, of Janos and Solomon's fixed-point theorem for continuous maps on spaces having an attractor for compact sets.

References [Enhancements On Off] (What's this?)

  • [1] W. G. Dotson, Jr., Fixed-point theorems for non-expansive mappings on star-shaped subsets of Banach spaces, J. London Math. Soc. 2 (1972), 408-410. MR 0296778 (45:5837)
  • [2] L. Janos, On representations of self mappings, Proc. Amer. Math. Soc. 26 (1970), 529-533. MR 0270346 (42:5235)
  • [3] L. Janos and J. L. Solomon, A fixed-point theorem and attractors, Proc. Amer. Math. Soc. 71 (1978), 257-261. MR 0482716 (58:2772)
  • [4] A. D. Wallace, Inverses in Euclidean mobs, Math. J. Okayama Univ. 3 (1953), 23-28. MR 0062137 (15:933d)

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Keywords: Fixed points, nonexpansive mappings, retractions, strictly convex spaces
Article copyright: © Copyright 1980 American Mathematical Society

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