Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A fixed point theorem and attractors


Authors: Ludvik Janos and J. L. Solomon
Journal: Proc. Amer. Math. Soc. 71 (1978), 257-262
MSC: Primary 54H25
MathSciNet review: 0482716
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate attractors for compact sets by considering a certain quotient space. The following theorem is included. Let $ f:G \to G$, G a closed convex subset of a Banach space, f a mapping satisfying (i) there exists $ M \subset G$ which is an attractor for compact sets under f; (ii) the family $ \{ {f^n}\} _{n = 1}^\infty $ is equicontinuous. Then f has a fixed point.


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

  • [1] Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044. MR 0187120 (32:4574)
  • [2] J. Dugundgji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [3] Ludvik Janos, On representation of self-mappings, Proc. Amer. Math. Soc. 26 (1970), 529-533. MR 0270346 (42:5235)
  • [4] -, On the Edelstein contractive mapping theorem, Canad. Math. Bull. 18 (1975), 675-678. MR 0420589 (54:8603)
  • [5] W. A. Kirk, On fixed point theorems for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 0189009 (32:6436)
  • [6] Roger D. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349-375. MR 0310719 (46:9817)
  • [7] J. L. Solomon, A note on attractors for compact sets, Proc. Amer. Math. Soc. 65 (1977), 293-296. MR 0637100 (58:30584)

Similar Articles

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

Retrieve articles in all journals with MSC: 54H25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0482716-2
Keywords: Attractors for compact sets, equicontinuous family, quotient space, contractive mapping, retraction, fixed point
Article copyright: © Copyright 1978 American Mathematical Society