  Firstly, we invoke the weak convergence (resp. strong convergence) of
translated basic methods involving nonexpansive operators to establish the weak
convergence (resp. strong convergence) of the associated method with both
perturbation and approximation. Then we employ the technique obtaining the
result above to extend convergence results from the classic
Krasnosel'ski\v{\i}-Mann iterations to their relaxation variants for finding a
common fixed point of associated nonexpansive operators. At last, we show
applications on generalized proximal point algorithms for solving monotone
inclusion problems.
