Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Edelstein's contractivity and attractors


Authors: Ludvik Janos, Hwei Mei Ko and Kok Keong Tan
Journal: Proc. Amer. Math. Soc. 76 (1979), 339-344
MSC: Primary 54H25; Secondary 54E35
MathSciNet review: 537101
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54H25, 54E35

Retrieve articles in all journals with MSC: 54H25, 54E35


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0537101-2
PII: S 0002-9939(1979)0537101-2
Keywords: Metrizable topological space, attractor for compact sets, contractive mapping, nonexpansive mapping, quotient space, Kuratowski measure of noncompactness, condensing map, fixed point
Article copyright: © Copyright 1979 American Mathematical Society