Existence and convergence for fixed points of mappings of asymptotically nonexpansive type.

*(English)*Zbl 0747.47041This 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)

##### MSC:

47H10 | Fixed-point theorems |

47J25 | Iterative procedures involving nonlinear operators |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

##### Keywords:

mappings of asymptotically nonexpansive type; uniformly convex Banach space; fixed point theorem; weak convergence of the Picard approximations
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\textit{H.-K. Xu}, Nonlinear Anal., Theory Methods Appl. 16, No. 12, 1139--1146 (1991; Zbl 0747.47041)

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##### References:

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