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On the Existence and Uniqueness of Fixed Points for Holomorphic Maps in Complex Banach Spaces

Kazimierz Wlodarczyk
Proceedings of the American Mathematical Society
Vol. 112, No. 4 (Aug., 1991), pp. 983-987
DOI: 10.2307/2048643
Stable URL: http://www.jstor.org/stable/2048643
Page Count: 5
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On the Existence and Uniqueness of Fixed Points for Holomorphic Maps in Complex Banach Spaces
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Abstract

We consider the problem of the existence and uniqueness of fixed points in X of holomorphic maps F: X → X of bounded open convex sets X in complex Banach spaces E. As a result of the Earle-Hamilton theorem, the problem in the case where F(X) lies strictly inside X (i.e., $\operatorname{dist}\lbrack F(X), E\backslash X \rbrack > 0$) has a solution. In this article we show that this problem is also solved in the case where F(X) does not lie strictly inside X (i.e., $\operatorname{dist}\lbrack F(X), E\backslash X \rbrack = 0$) whenever: (i) F is compact; (ii) F is continuous on $\overline X$ and $F(\overline X) \subset \overline X$; (iii) F has no fixed points on ∂ X; and (iv) for each x ∈ X, 1 is not contained in the spectrum of DF(x).

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