## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# 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
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).