### Abstract

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {k_{n}}_{n≥1} ⊂[1, ∞), limk_{n} = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {x_{n}}_{n≥1} is generated iteratively by x_{1} ∈ K, x_{n+1} = P((1-α_{n} x_{n}+α_{n}T(PT)^{n-1}x_{n}), n≥1, where {α_{n}}_{n≥1} ⊂ (0, 1) is such that ∈ < 1 - α_{n} < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑_{n≥1} (k_{n}^{2} - 1) < ∞ and T is completely continuous, strong convergence of {x_{n}} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {x_{n}} to some x* ∈ F(T) is obtained.

Original language | English |
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Pages (from-to) | 364-374 |

Number of pages | 11 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 280 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 15 2003 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*280*(2), 364-374. https://doi.org/10.1016/S0022-247X(03)00061-1