Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. (English) Zbl 0747.47041

This article deals with mappings \(T:C\to C\) of asymptotically nonexpansive type of a nonempty closed convex subset in a uniformly convex Banach space \(X\) or, in other words, with mappings for which \[ \varlimsup_{n\to\infty}\sup_{y\in C}(\| T^ nx-T^ ny\|- \| x-y\|)\leq 0. \] The main result is a fixed point theorem for such mappings and a theorem on the weak convergence of the Picard approximations to a fixed point.
Reviewer: P.Zabreiko (Minsk)


47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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