# Asymptotic stability in the Schauder fixed point theorem

Studia Mathematica (1998)

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

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topShih, 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

top- [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. Zbl0368.47001
- [3] J. Dugundji, Topology, Allyn and Bacon, 1969.
- [4] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969. Zbl0176.00801
- [5] T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland, 1980. Zbl0447.46040
- [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] R. B. Holmes, A formula for the spectral radius of an operator, Amer. Math. Monthly 75 (1968), 163-166. Zbl0156.38202
- [8] R. B. Kellogg, Uniqueness in the Schauder fixed point theorem, Proc. Amer. Math. Soc. 60 (1976), 207-210. Zbl0352.47025
- [9] V. Khatskevich and D. Shoiykhet, Differentiable Operators and Nonlinear Equations, Oper. Theory Adv. Appl. 66, Birkhäuser, 1994. Zbl0807.46047
- [10] J. Kitchen, Concerning the convergence of iterates to fixed points, Studia Math. 27 (1966), 247-249. Zbl0143.16601
- [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] W. Rudin, Function Theory in the Unit Ball of ${\u2102}^{n}$, Springer, 1980. Zbl0495.32001
- [13] J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171-180. Zbl56.0355.01
- [14] J. T. Schwartz, Nonlinear Functional Analysis, Courant Inst. Math. Sci., New York Univ., 1964.

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