Global surjectivity of submersions via contractibility of the fibers

Abstract:

We give a sufficient condition for a ${C^1}$ submersion $F:X \to Y$, $X$ and $Y$ real Banach spaces, to be surjective with contractible fibers ${F^{ - 1}}(y)$. Roughly speaking, this condition "interpolates" two well-known but unrelated hypotheses corresponding to the two extreme cases: Hadamard’s criterion when $Y \simeq X$ and $F$ is a local diffeomorphism, and the Palais-Smale condition when $Y = \mathbb {R}$. These results may be viewed as a global variant of the implicit function theorem, which unlike the local one does not require split kernels. They are derived from a deformation theorem tailored to fit functionals with a norm-like nondifferentiability.