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

We propose an abstract approach to prove local uniqueness and conditional H\"older 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 H\"older stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.