Edelstein’s contractivity and attractors
HTML articles powered by AMS MathViewer
- by Ludvik Janos, Hwei Mei Ko and Kok Keong Tan PDF
- Proc. Amer. Math. Soc. 76 (1979), 339-344 Request permission
Abstract:
In this article an example is constructed to show that Theorem 1.1 of L. Janos [Canad. Math. Bull. 18 (1975), no. 5, 675-678] is false. A proper formulation is obtained as follows. Theorem. If $(X,\tau )$ is a metrizable topological space, $f:X \to X$ is continuous, and $a \in X$, then the following statements are equivalent: (1) There exists a metric d compatible with $\tau$ such that f is contractive with respect to d and the sequence $({f^n}(x))_{n = 1}^\infty$ converges to a for every $x \in X$. (2) The singleton {a} is an attractor for compact subsets under f. Furthermore, under this proper formulation, we show that Theorem 3.2 Janos [Proc. Amer. Math. Soc. 61 (1976), 161-175] and Theorem 2.3 Janos and J. L. Solomon [ibid. 71 (1978), 257-262], where the false Theorem 1.1 in [2] has been quoted in the original proofs, remain valid.References
- M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79. MR 133102, DOI 10.1112/jlms/s1-37.1.74
- Ludvik Janos, On the Edelstein contractive mapping theorem, Canad. Math. Bull. 18 (1975), no. 5, 675–678. MR 420589, DOI 10.4153/CMB-1975-118-8
- Ludvik Janos, On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc. 61 (1976), no. 1, 171–175 (1977). MR 425936, DOI 10.1090/S0002-9939-1976-0425936-3
- Ludvik Janos and J. L. Solomon, A fixed point theorem and attractors, Proc. Amer. Math. Soc. 71 (1978), no. 2, 257–262. MR 482716, DOI 10.1090/S0002-9939-1978-0482716-2 K. Kuratowski, Sur les espaces complètes, Fund. Math. 15 (1930), 301-309.
- Roger D. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349–375. MR 310719, DOI 10.1090/S0002-9947-1972-0310719-6
- B. N. Sadovskiĭ, Limit-compact and condensing operators, Uspehi Mat. Nauk 27 (1972), no. 1(163), 81–146 (Russian). MR 0428132
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 76 (1979), 339-344
- MSC: Primary 54H25; Secondary 54E35
- DOI: https://doi.org/10.1090/S0002-9939-1979-0537101-2
- MathSciNet review: 537101