  We study Krasnoselskii-Mann style iterative algorithms for approximating
fixpoints of asymptotically weakly contractive mappings, with a focus on
providing generalised convergence proofs along with explicit rates of
convergence. More specifically, we define a new notion of being asymptotically
$\psi$-weakly contractive with modulus, and present a series of abstract
convergence theorems which both generalise and unify known results from the
literature. Rates of convergence are formulated in terms of our modulus of
contractivity, in conjunction with other moduli and functions which form
quantitative analogues of additional assumptions that are required in each
case. Our approach makes use of ideas from proof theory, in particular our
emphasis on abstraction and on formulating our main results in a quantitative
manner. As such, the paper can be seen as a contribution to the proof mining
program.
