### Abstract

Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R^{+}} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {L_{t}} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u_{0} ∈ K and t_{n} > 0 there exists a sequence {u_{n}} ⊂ K such that u_{n} = (1 - α_{n}) T (t_{n}) u_{n} + α_{n} u_{0}, for n ∈ N such that {α_{n}} ⊂ (0, 1) and L_{tn} - 1 < α_{n}, where t_{n} ∈ R^{+}. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that lim_{n → ∞} t_{n} = ∞, lim_{n → ∞} α_{n} = lim_{n → ∞} frac(L_{tn} - 1, α_{n}) = 0. Then the sequence {u_{n}} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {x_{n}} generated from x_{1} ∈ K by x_{n + 1} {colon equals} α_{n} u + (1 - α_{n}) T (t_{n}) x_{n}, n ≥ 1, converges to a fixed point of T.

Original language | English |
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Pages (from-to) | 2308-2315 |

Number of pages | 8 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 71 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - Sep 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics