Fixed point theorems on infinite dimensional manifolds

Abstract

Introduction. Let X be a differentiable manifold of finite or infinite dimension, f a continuous mapping of X into X. f is said to be a compact mapping if f(X) is relatively compact in X while f is said to be locally compact if each point x of X has a neighborhood Nx such that f(Nx) is relatively compact in X. We are concerned in the present paper with sufficient conditions for the existence of fixed points of compact and locally compact mappings of manifolds, especially those of infinite dimension. The questions which we wish to answer are the following: (1) If f is a compact self-mapping of an acyclic manifold X, does f have a fixed point in X? (2) Iff is a compact self-mapping of a manifold X, does the Lefschetz fixed point theorem hold for f ? (3) Letf be a locally compact mapping of a manifold X such that for some positive integer m, f'(X) is relatively compact in X. Then can we asser-t that f has a fixed point? For Banach spaces X, the affirmative answer to question (1) is the well-known Schauder fixed point theorem [21]. An affirmative result for question (3) for Banach spaces was obtained by the writer in [3]. For Banach spaces, question (2) is obliquely related to the Leray-Schauder theory of topological degree [20] for completely continuous displacements. The principal results of the present paper are affirmative answers to questions (1), (2), and (3) for a broad class of manifolds which includes all manifolds which can be imbedded as neighborhood retracts in Banach spaces. More specifically we have the following: