Asymptotic stability in the Schauder fixed point theorem

Mau-Hsiang Shih; Jinn-Wen Wu

Studia Mathematica (1998)

  • Volume: 131, Issue: 2, page 143-148
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Shih, Mau-Hsiang, and Wu, Jinn-Wen. "Asymptotic stability in the Schauder fixed point theorem." Studia Mathematica 131.2 (1998): 143-148. <http://eudml.org/doc/216570>.

@article{Shih1998,
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.},
author = {Shih, Mau-Hsiang, Wu, Jinn-Wen},
journal = {Studia Mathematica},
keywords = {fixed point; asymptotic stability; spectral radius; compact map; holomorphic map; normal family; Whitney smooth extension theorem; global asymptotic stability; Schauder fixed point theorem; holomorphic function},
language = {eng},
number = {2},
pages = {143-148},
title = {Asymptotic stability in the Schauder fixed point theorem},
url = {http://eudml.org/doc/216570},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Shih, Mau-Hsiang
AU - Wu, Jinn-Wen
TI - Asymptotic stability in the Schauder fixed point theorem
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 2
SP - 143
EP - 148
AB - 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.
LA - eng
KW - fixed point; asymptotic stability; spectral radius; compact map; holomorphic map; normal family; Whitney smooth extension theorem; global asymptotic stability; Schauder fixed point theorem; holomorphic function
UR - http://eudml.org/doc/216570
ER -

References

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  1. [1] G. R. Belitskiĭ and Yu. I. Lyubich, Matrix Norms and Their Applications, translated from the Russian by A. Iacob, Birkhäuser, 1988. 
  2. [2] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977. Zbl0368.47001
  3. [3] J. Dugundji, Topology, Allyn and Bacon, 1969. 
  4. [4] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969. Zbl0176.00801
  5. [5] T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland, 1980. Zbl0447.46040
  6. [6] L. A. Harris, Schwarz's lemma in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014-1017. Zbl0199.19401
  7. [7] R. B. Holmes, A formula for the spectral radius of an operator, Amer. Math. Monthly 75 (1968), 163-166. Zbl0156.38202
  8. [8] R. B. Kellogg, Uniqueness in the Schauder fixed point theorem, Proc. Amer. Math. Soc. 60 (1976), 207-210. Zbl0352.47025
  9. [9] V. Khatskevich and D. Shoiykhet, Differentiable Operators and Nonlinear Equations, Oper. Theory Adv. Appl. 66, Birkhäuser, 1994. Zbl0807.46047
  10. [10] J. Kitchen, Concerning the convergence of iterates to fixed points, Studia Math. 27 (1966), 247-249. Zbl0143.16601
  11. [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. Zbl0539.34010
  12. [12] W. Rudin, Function Theory in the Unit Ball of ℂ n , Springer, 1980. Zbl0495.32001
  13. [13] J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171-180. Zbl56.0355.01
  14. [14] J. T. Schwartz, Nonlinear Functional Analysis, Courant Inst. Math. Sci., New York Univ., 1964. 

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