ON THE EXISTENCE AND UNIQUENESS OF FIXED-POINTS FOR HOLOMORPHIC MAPS IN COMPLEX BANACH-SPACES

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., dist[F(X), E backward slash 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., dist[F(X), E backward slash X] = 0) whenever: (i) F is compact; (ii) F is continuous on XBAR and F(X)BAR subset-of XBAR; (iii) F has no fixed points on partial X; and (iv) for each x is-an-element-of X, 1 is not contained in the spectrum of DF(x).