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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:XX 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-α(n))x(n)+α(n)(Tx(n)+e(n)), where {α(n)} and {e(n)} are sequences in (0,1) and in X, respectively, satisfying the following properties: Σα(n)(1-α(n))= and Σα(n)e(n)<. 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)