Criteria for the global invertibility of \(C^{1}\) functions between Banach spaces (Q697528)

A Hadamard type theorem for a \(C^1\)-function \(f\) in a Banach space with invertible derivatives is proved: If there is some \(0\leq \eta(t)\leq\inf_{\| x\|=t} \| f'(x)^{-1}\|^{-1}\) which is continuous or non-increasing with infinite integral, then \(f\) is globally invertible. Moreover, some more general versions for functions defined on balls are obtained, together with explicit lower estimates on the distance \(\| f(x_1)-f(x_0)\|\).