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A note on the extension of contractive mappings

Author: J. L. Solomon
Journal: Proc. Amer. Math. Soc. 40 (1973), 319-322
MSC: Primary 54C20; Secondary 54H25
MathSciNet review: 0322784
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Abstract: In this note a question posed by J. D. Bryant and L. F. Guseman, Jr. is answered. It is shown that for dense subsets of finite dimensional Banach spaces, the iterative test being conclusive and the contractive extension property are not equivalent. If $ (X,d)$ is a complete locally compact and chainable-metric space, a characterization of these dense subsets for which the iterative test is conclusive is given.

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

  • [1] Jack Bryant and L. F. Guseman, Jr., Extensions of contractive mappings and Edelstein's iterative test, Canad. Math. Bull. (to appear).
  • [2] M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79. MR 0133102
  • [3] P. R. Meyers, Minimum number of subsets to distinguish individual elements, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 21–22. MR 0210604
  • [4] Sam B. Nadler Jr., A note on an iterative test of Edelstein, Canad. Math. Bull. 15 (1972), 381–386. MR 0312483

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Keywords: Fixed point, contractive map, contractive extension property, iterative test
Article copyright: © Copyright 1973 American Mathematical Society