Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A∗A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.