### 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

### Cite this

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*Journal of Mathematical Analysis and Applications*, vol. 278, no. 2, pp. 354-366. https://doi.org/10.1016/S0022-247X(02)00572-3

**Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings.** / Chidume, C. E.; Zegeye, Hab.

Research output: Contribution to journal › Review article

TY - JOUR

T1 - Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings

AU - Chidume, C. E.

AU - Zegeye, Hab

PY - 2003/2/15

Y1 - 2003/2/15

N2 - 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 {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 - λn)xn + λnTnxn - λnθn(xn - x1), for all integers n ≥ 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then ∥xn - Txn∥ → 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 {xn} also converges strongly to a fixed point of T.

AB - 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 {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 - λn)xn + λnTnxn - λnθn(xn - x1), for all integers n ≥ 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then ∥xn - Txn∥ → 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 {xn} also converges strongly to a fixed point of T.

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U2 - 10.1016/S0022-247X(02)00572-3

DO - 10.1016/S0022-247X(02)00572-3

M3 - Review article

VL - 278

SP - 354

EP - 366

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -