The authors prove robustness results of three somewhat different versions (corresponding to three different settings) of Mann’s algorithm to obtain a fixed-point of a suitable mapping. We state the version in the first setting (Theorem 3.3) in extenso. Let be a uniformly convex Banach space, where either satisfies Opial’s property or where the dual of has the Kadec–Klee property. Let be a nonexpansive mapping having a nonempty set of fixed points. Starting from some point in , let the sequence be generated by the following perturbed Mann algorithm: , where and are sequences in and in , respectively, satisfying the following properties: and . Then the sequence converges weakly to a fixed point of .
In the second setting, the authors consider a nonexpansive map defined on a closed convex subset of a real Hilbert space (Theorem 4.1). Finally, in the third setting (Theorem 5.1), the map is an -accretive operator in a uniformly convex Banach space , where has the same additional properties as in the first setting.