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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.

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)