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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
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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.

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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

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