Fixed points of holomorphic maps in Banach spaces

Hayden, T. L.; Suffridge, T. J.

doi:10.1090/S0002-9939-1976-0417869-3

Fixed points of holomorphic maps in Banach spaces
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by T. L. Hayden and T. J. Suffridge PDF

Proc. Amer. Math. Soc. 60 (1976), 95-105 Request permission

Abstract:

This paper is concerned with the problem of existence of fixed points of continuous maps of the closed unit ball of a complex Banach space into itself which are holomorphic on the open unit ball. We show that if the Banach space is separable and reflexive and F is the map in question that for a.e. $\theta$ in $[0,2\pi ]$ the map ${e^{i\theta }}$ F has a fixed point. This result does not hold in general; hence, additional conditions are imposed which insure the existence of fixed points in every Banach space. Fixed points of some linear fractional maps are explicitly computed.

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