Some results on the invertibility of nonlinear operators (Q1287039)

The author gives sufficient conditions for the invertibility of a nonlinear operator \(N\) in a Banach space \(X\). A critical idea is to approximate \(N\) by an invertible (usually, linear) operator \(L\), and to reduce the problem to the contraction mapping principle. Moreover, he introduces and studies certain ``measures of non-invertibility'' like \[ \nu(N)= \inf\{\| L^{-1}N-I\|^*: L\in{\mathcal L}(X)\}, \] where \(\| N\|^*\) denotes the smallest Lipschitz constant of \(N\).