The implicit function theorem and the global methods of cohomology

Abstract

Let B be a real Banach space (possibly finite-dimensional) with the metric topology, let R” be m-dimensional Euclidean space, let 0 be a neighborhood of (0, 0) in R’” i; B that is homeomorphic to R” >: B, and let F: 0 -+ B be a continuous map. Assume F(0, 0) = 0. Let Q = {(h, U) E 0 1 F(X, U) = 01. Many analytical tools have been found useful in describing the local nature of B near the point (0, 0), foremost of which is the implicit function theorem. This note concerns a global conclusion for the implicit function theorem. The conclusion in this form has been found useful for studying stability and bifurcation problems. Let D,F(X, U) denote the differential of F with respect to U, that is, the Jacobian. The usual implicit function theorem assumes D,,F(h, U) exists and is nonsingular and continuous for all (h, U) in some neighborhood of (0,O). Th e conclusion is local. The point of this paper is that a global conclusion is possible with no additonal hypotheses. Weakening slightly the main hypothesis of the implicit function theorem, we assume here that D,F(O, 0) exists and is nonsingular. For B finite-dimensional, we consider all such (continuous) F. If B has infinite dimension, it is more usual to consider the fixed point set of a mapping. Thus, we assume F(A, u) = u ~ G(X, u) where (a) G(h, u): {A} n 0 -+ B is compact (i.e., takes bounded sets to sets with compact closure) for each h, and (b) G is locally uniformly continuous in h for bounded sets in B. Thus we are considering the fixed point set of a compact map which depends on a parameter. Conditions (a) and (b) are enough that G: 0 + B is compact, which is what we actually