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Proceedings of the American Mathematical Society

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Note on the structure of fixed point sets of $ 1$-set-contractions


Author: W. V. Petryshyn
Journal: Proc. Amer. Math. Soc. 31 (1972), 189-194
DOI: https://doi.org/10.1090/S0002-9939-1972-0285944-9
MathSciNet review: 0285944
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Abstract | References | Additional Information

Abstract: Let $ X$ be a real Banach space, $ D$ a bounded open subset of $ X$, and $ T$ a demicompact $ 1$-set-contraction of the closure $ D$ into $ X$. It is shown that under certain conditions the set $ F(T)$ of fixed points of $ T$ in $ \bar D$ is a continuum (i.e., $ F(T)$ is a nonempty, compact and connected set).


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

  • [1] R. D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (to appear). MR 0312341 (47:903)
  • [2] W. V. Petryshyn, Structure of the fixed point sets of $ k$-set-contractions, Arch. Rational Mech. Anal. 40 (1971), 312-328. MR 0273480 (42:8358)
  • [3] M. Furi and A. Vignoli, A fixed point theorem in complete metric spaces, Boll. Un. Mat. Ital. (4) 4/5 (1969), 505-509. MR 0256378 (41:1034)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0285944-9
Keywords: Structure of fixed point sets, continuum, measure of noncompactness, $ 1$-set-contractions, condensing, demicompact, generalized degree
Article copyright: © Copyright 1972 American Mathematical Society

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