  We prove convergence with optimal algebraic rates for an adaptive finite
element method for nonlinear equations with strongly monotone operator. Unlike
prior works, our analysis also includes the iterative and inexact solution of
the arising nonlinear systems by means of the Picard iteration. Using nested
iteration, we prove, in particular, that the number of of Picard iterations is
uniformly bounded in generic cases, and the overall computational cost is
(almost) optimal. Numerical experiments confirm the theoretical results.
