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)



Fixed points for some non-obviously contractive operators

Authors: E. De Pascale and L. De Pascale
Journal: Proc. Amer. Math. Soc. 130 (2002), 3249-3254
MSC (2000): Primary 47H10, 45D05
Published electronically: June 11, 2002
MathSciNet review: 1913003
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The use of K-normed spaces gives us the possibility to prove that a fixed point theorem due to B. Luo is equivalent to the Banach Contraction Principle. This confirms the conspiracy among fixed point theorems. Moreover the theorem of Lou is improved and extended to different contexts. A counterexample about the fixed points of the sum of a contraction and an integral operator is given. The usefulness of K-norm is tested on a Volterra integral equation as well.

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

  • 1. B. Lou,
    Fixed points for operators in a space of continuous functions and applications,
    Proc. Amer. Math. Soc. 127, 2259-2264 (1999). MR 99m:47065
  • 2. P. P. Zabrejko,
    K-metric and K-normed linear spaces: a survey,
    Collect. Math. 48, 825-859 (1997). MR 99a:46010
  • 3. I. J. Maddox,
    Elements of Functional Analysis,
    Cambridge University Press, Cambridge, 1988. MR 90k:46002
  • 4. I. Rosenholtz,
    Evidence of a conspiracy among fixed point theorems,
    Proc. Amer. Math. Soc. 53, 213-218 (1975). MR 53:4036
  • 5. D. Guo, V. Lakshmikantham,
    Nonlinear Problems in Abstract Cones,
    Academic Press, Boston (1988). MR 89k:47084
  • 6. L. Collatz,
    Functional Analysis and Numerical Mathematics,
    Academic Press: Boston, 1966. MR 34:4961
  • 7. M. Altman,
    Contractors and equations in pseudometric spaces,
    Boll. Un. Mat. Ital. 6, 376-384 (1972). MR 47:9376
  • 8. M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskij, V. Ya. Stetsenko,
    Approximate solution of operator equations,
    Nauka, Fitmatgiz, Moskow (1969); English translation: Wolters-Noordhoff Publishing, Groningen (1972). MR 52:6515

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H10, 45D05

Retrieve articles in all journals with MSC (2000): 47H10, 45D05

Additional Information

E. De Pascale
Affiliation: Dipartimento di Matematica, Universitá della Calabria, 87036 Rende (Cosenza), Italy

L. De Pascale
Affiliation: Dipartimento di Matematica Applicata “U.Dini”, Via Bonanno Pisano 25/B, 56126 Pisa, Italy

Keywords: Fixed points, iterative sequences, K-normed spaces, positive operators, normal cone, contraction operators
Received by editor(s): June 5, 2001
Published electronically: June 11, 2002
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society