Download PDF - Asymptotic stability in the Schauder fixed point theoremAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Asymptotic stability in the Schauder fixed point theorem

Authors 1, 1

Affiliations

  1. Department of Mathematics, Chung Yuan University, Chung-Li, Taiwan 320

Abstract

This note presents a theorem which gives an answer to a conjecture which appears in the book Matrix Norms and Their Applications by Belitskiĭ and Lyubich and concerns the global asymptotic stability in the Schauder fixed point theorem. This is followed by a theorem which states a necessary and sufficient condition for the iterates of a holomorphic function with a fixed point to converge pointwise to this point.

Keywords

ENG
fixed point, asymptotic stability, spectral radius, compact map, holomorphic map, normal family, Whitney smooth extension theorem

Bibliography

  1. G. R. Belitskiĭ and Yu. I. Lyubich, Matrix Norms and Their Applications, translated from the Russian by A. Iacob, Birkhäuser, 1988.
  2. M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.
  3. J. Dugundji, Topology, Allyn and Bacon, 1969.
  4. H. Federer, Geometric Measure Theory, Springer, Berlin, 1969.
  5. T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland, 1980.
  6. L. A. Harris, Schwarz's lemma in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014-1017.
  7. R. B. Holmes, A formula for the spectral radius of an operator, Amer. Math. Monthly 75 (1968), 163-166.
  8. R. B. Kellogg, Uniqueness in the Schauder fixed point theorem, Proc. Amer. Math. Soc. 60 (1976), 207-210.
  9. V. Khatskevich and D. Shoiykhet, Differentiable Operators and Nonlinear Equations, Oper. Theory Adv. Appl. 66, Birkhäuser, 1994.
  10. J. Kitchen, Concerning the convergence of iterates to fixed points, Studia Math. 27 (1966), 247-249.
  11. Yu. I. Lyubich, A remark on the stability of complex dynamical systems, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (1983), 49-50 (in Russian); English transl.: Soviet Math. (Iz. VUZ) 10 (1983), 62-64.
  12. W. Rudin, Function Theory in the Unit Ball of ℂn, Springer, 1980.
  13. J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171-180.
  14. J. T. Schwartz, Nonlinear Functional Analysis, Courant Inst. Math. Sci., New York Univ., 1964.
Studia Mathematica
1998/131/2
Export to PBN, Opens in new tab
Pages:
143-148
Main language of publication
English
Received
1997-04-16
Accepted
1998-04-04
Published
1998
Exact and natural sciences