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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the existence and uniqueness of fixed points for holomorphic maps in complex Banach spaces


Author: Kazimierz Włodarczyk
Journal: Proc. Amer. Math. Soc. 112 (1991), 983-987
MSC: Primary 58C10; Secondary 46G20, 47H10, 58C30
MathSciNet review: 1072094
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Abstract: We consider the problem of the existence and uniqueness of fixed points in $ X$ of holomorphic maps $ F:X \to 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}[F(X),E\backslash X] > 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}[F(X),E\backslash X] = 0)$ whenever: (i) $ F$ is compact; (ii) $ F$ is continuous on $ \bar{X}$ and $ F(\bar X) \subset \bar X$; (iii) $ F$ has no fixed points on $ \partial X$; and (iv) for each $ x \in X$ , 1 is not contained in the spectrum of $ DF(x)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1072094-9
PII: S 0002-9939(1991)1072094-9
Keywords: Complex Banach spaces, holomorphic maps, bounded open convex sets, fixed points
Article copyright: © Copyright 1991 American Mathematical Society