  A wide variety of different (fixed-point) iterative methods for the solution
of nonlinear equations exists. In this work we will revisit a unified iteration
scheme in Hilbert spaces from our previous work that covers some prominent
procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration
methods). In combination with appropriate discretization methods so-called
(adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main
purpose of this paper is the derivation of an abstract convergence theory for
the unified ILG approach (based on general adaptive Galerkin discretization
methods) proposed in our previous work. The theoretical results will be tested
and compared for the aforementioned three iterative linearization schemes in
the context of adaptive finite element discretizations of strongly monotone
stationary conservation laws.
