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Robustness of Mann’s algorithm for nonexpansive mappings. (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: $x\left(n+1\right)=\left(1-\alpha \left(n\right)\right)x\left(n\right)+\alpha \left(n\right)\left(Tx\left(n\right)+e\left(n\right)\right)$, where $\left\{\alpha \left(n\right)\right\}$ and $\left\{e\left(n\right)\right\}$ are sequences in $\left(0,1\right)$ and in $X$, respectively, satisfying the following properties: ${\Sigma }\alpha \left(n\right)\left(1-\alpha \left(n\right)\right)=\infty$ and ${\Sigma }\alpha \left(n\right)\parallel e\left(n\right)\parallel <\infty$. Then the sequence $\left\{x\left(n\right)\right\}$ 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)