On the Stability of Krasnosel'ski\v{\i}-Mann Iterations

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.