  We design an adaptive finite element method to approximate the solutions of
quasi-linear elliptic problems. The algorithm is based on a Ka\v{c}anov
iteration and a mesh adaptation step is performed after each linear solve. The
method is thus \emph{inexact} because we do not solve the discrete nonlinear
problems exactly, but rather perform one iteration of a fixed point method
(Ka\v{c}anov), using the approximation of the previous mesh as an initial
guess. The convergence of the method is proved for any \emph{reasonable}
marking strategy and starting from any initial mesh. We conclude with some
numerical experiments that illustrate the theory.
