Asymptotic stability in the Schauder fixed point theorem

• Autorzy: Mau-Hsiang Shih , Jinn-Wen Wu
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 47H10: Fixed-point theorems
• 34D20: Stability

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 131
• Numer: 2
• Strony: 143-148

Daty

wydano

1998

otrzymano

1997-04-16

poprawiono

1998-04-04

Twórcy

autor

Mau-Hsiang Shih

• Department of Mathematics, Chung Yuan University, Chung-Li, Taiwan 320

autor

Jinn-Wen Wu

• Department of Mathematics, Chung Yuan University, Chung-Li, Taiwan 320

Bibliografia

• [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 $ℂ^n$, 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.