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Iterative construction of fixed points of asymptotically nonexpansive mappings. (English) Zbl 0734.47036
Let T be a completely continuous and asymptotically non-expansive self- mapping (in the sense of Goebel and Kirk) of a nonempty closed bounded and convex subset of a Hilbert space. The author gives conditions under which a fixed point of T may be obtained as limit of the Mann-type iterates x n+1 =α n T n (x n )+(1-α n )x n · A parallel result is obtained for a new class of operators (called “asymptotically pseudocontractive”) whose iterates admit a universal Lipschitz constant.

MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties