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On continuity of fixed points of collectively condensing maps

Author: Martin Fan Cheng
Journal: Proc. Amer. Math. Soc. 63 (1977), 74-76
MSC: Primary 47H10; Secondary 54H25
MathSciNet review: 0435943
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Abstract: In this paper, we prove, in two parts, the following claim. Let X be a Banach space and $ \Lambda $ an arbitrary topological space. Suppose that $ T:\Lambda \times X \to X$ is collectively condensing; then the fixed point set $ S(\lambda ,y)$ has closed graph if and only if T is continuous in both $ \lambda $ and y.

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

  • [1] Zvi Artstein, On continuous dependence of fixed points of condensing maps, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 73–75. MR 0636801
  • [2] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581

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Keywords: Collective condensing, closed graph, upper-semicontinuity
Article copyright: © Copyright 1977 American Mathematical Society

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