### Abstract

Let K be a nonempty closed convex and bounded subset of a real Banach space E and T : K → K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {ε_{n}}, and asymptotically pseudocontractive with constant {k_{n}}, where {k_{n}} and {ε_{n}} satisfy certain mild conditions. Let a sequence {x_{n}} be generated from x_{1} ∈ K by x_{n+1} := (1 - λ_{n})x_{n} + λ_{n}T^{n}x_{n} - λ_{n}θ_{n}(x_{n} - x_{1}), for all integers n ≥ 1, where {λ_{n}} and {θ_{n}} are real sequences satisfying appropriate conditions, then ∥x_{n} - Tx_{n}∥ → 0 as n → ∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L < N(E)^{1/2} and has a uniformly Gâteaux differentiable norm, and T satisfies an additional mild condition, then {x_{n}} also converges strongly to a fixed point of T.

Original language | English |
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Pages (from-to) | 354-366 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 278 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 15 2003 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics