*(English)*Zbl 1110.47057

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 $X$ be a uniformly convex Banach space, where either $X$ satisfies Opial’s property or where the dual of $X$ has the Kadec–Klee property. Let $T:X\to X$ be a nonexpansive mapping having a nonempty set of fixed points. Starting from some point $x\left(0\right)$ in $X$, let the sequence $x\left(n\right)$ be generated by the following perturbed Mann algorithm: $x(n+1)=(1-\alpha (n\left)\right)x\left(n\right)+\alpha \left(n\right)\left(Tx\right(n)+e(n\left)\right)$, where $\left\{\alpha \right(n\left)\right\}$ and $\left\{e\right(n\left)\right\}$ are sequences in $(0,1)$ and in $X$, respectively, satisfying the following properties: ${\Sigma}\alpha \left(n\right)(1-\alpha (n\left)\right)=\infty $ and ${\Sigma}\alpha \left(n\right)\parallel e\left(n\right)\parallel <\infty $. Then the sequence $\left\{x\right(n\left)\right\}$ converges weakly to a fixed point of $T$.

In the second setting, the authors consider a nonexpansive map $T$ 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 $m$-accretive operator in a uniformly convex Banach space $X$, where $X$ has the same additional properties as in the first setting.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

65J15 | Equations with nonlinear operators (numerical methods) |