A fixed point theorem for holomorphic mappings in locally convex spaces

Abstract

IN 1921 Rrrr [l] proved that a holomorphic function mapping a nonempty region 0 in the complex plane into a compact subset has exactly one fixed point x E R and we have 1 f’(x)] -c 1. Using a theorem about separation properties of analytic sets in C”, due to Remmert and Stein [2], Hervt [3] extended this result to holomorphic functions of several variables. An analogon of Ritt’s theorem for complex spaces was given by Reiffen [4]. Earle and Hamilton [S] extended Ritt’s result to complex Banach spaces replacing the compactness condition by a boundary distance assumption which is much weaker than compactness in the case of an infinite dimensional space. In this paper we present an extension of Ritt’s fixed point theorem to complex sequentially complete locally convex Hausdorff spaces. Moreover we obtain results concerning the asymptotic behaviour of the operator and an estimate for the spectral radius of its derivative at the unique fixed point. Finally we indicate some corollaries of our main theorem.