Diffeomorphisms with Banach space domains

The 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 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 {\it auxiliary scalar coercive functions}. As far as I know the use of such auxiliary functions in these questions is new. We find some first corollaries.