Diffeomorphisms with Banach space domains

Our basic element is a $C^1$ mapping $f:X\to Y$, with $X,Y$ Banach spaces, and with derivative everywhere invertible. So $f$ is a local diffeomorphism at every point. The aim of this paper is to find a sufficient condition for $f$ to be injective, and so a global diffeomorphism $X\to f(X)$, and a sufficient condition for $f$ to be bijective and so a global diffeomorphism onto $Y$. This last condition is also necessary in the particular case $X=Y=\R^n$. In these theorems the key role is played by nonnegative auxiliary scalar coercive functions.