Diffeomorphisms with Banach space domains

Abstract

So f is a local diffeomorphism at every point by the inverse function theorem. The aim of this paper is to find a sufficient condition for f to be injective, and so a global diffeomorphism X + f(X) (theorem 2. l), and a sufficient condition for f to be bijective and so a global diffeomorphism onto Y (theorem 3.1). This last condition is also necessary in the particular case X = Y = R”. In these theorems the key role is played by nonnegative auxiliary scalar coercive functions, that is continuous mappings k: X --+ R, with k(x) + +a, as llxll -+ +a~. As far as we know the use of such auxiliary functions in these questions is new. First we find some corollaries to the aforementioned theorems. The author hopes that suitable auxiliary functions, adapted to particular problems, may lead to new consequences. In order to briefly discuss the results, and some of their relations with the literature, let us consider the case where X is a Hilbert space with scalar product “ “, and k E C’. This simplifies the formulas a little. However, our results will be formulated and proved for Banach spaces, where we will ask for k to be locally Lipschitz continuous and to have the right directional derivatives only; these last assumptions are not made to quibble about this matter, but are related to the nondifferentiability of the map x y ~Ix[[~ in general Banach spaces. In this paper we use the hypothesis that the operator norm 11 f I(x)-‘11 is bounded on bounded sets, i.e. SUP IIf’(‘II < +*, Vr:O<r<+w. (1.2) IL4 5 r