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

Włodarczyk, Kazimierz

doi:10.1090/S0002-9939-1991-1072094-9

On the existence and uniqueness of fixed points for holomorphic maps in complex Banach spaces
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Proc. Amer. Math. Soc. 112 (1991), 983-987 Request permission

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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