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 \(x_{n+1}=\alpha_ nT^ n(x_ n)+(1-\alpha_ n)x_ n.\) A parallel result is obtained for a new class of operators (called “asymptotically pseudocontractive”) whose iterates admit a universal Lipschitz constant.


47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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