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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 $x(0)$ in $X$, let the sequence $x(n)$ be generated by the following perturbed Mann algorithm: $x(n+1)=(1-\alpha(n))x(n) +\alpha(n) (Tx (n)+e(n))$, where $\{\alpha(n)\}$ and $\{e(n)\}$ are sequences in $(0,1)$ and in $X$, respectively, satisfying the following properties: $\Sigma\alpha(n)(1-\alpha(n))=\infty$ and $\Sigma\alpha(n)\|e(n)\|< \infty$. Then the sequence $\{x(n)\}$ 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:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)
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References:
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