Uniqueness of monotone solutions to a non-linear Fredholm integral equation

In this paper the authors carry out a theoretical analysis of the simultaneous identification of thickness and refractive index profile for inhomogeneous thin films from indirect measurements. The problem leads to a non-linear integral equation of the first kind with smooth kernel, for which solutions are well-determined with the exception of rearrangements of function values. Paying attention to monotone solutions the authors present a uniqueness theorem that allows them to apply descriptive regularization in order to stabilize the problem.