### Abstract

Let D be an open subset of a real uniformly smooth Banach space E. Suppose T: D̄ → E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where D̄ denotes the closure of D. Then, it is proved that (i) D̄ ⊆ ℛ (I + r (I-T)) for every r > 0; (ii) for a given y_{0} ∈ D, there exists a unique path t → y_{t} ∈ D̄, t ∈ [0, 1], satisfying y_{t} := tTy_{t} + (1-t) y_{0}. Moreover, if F (T) ≠ ∅ or there exists y_{0} ∈ D such that the set K := {y ∈ D: T y = λ y + (1-λ) y_{0} for λ >1} is bounded, then it is proved that, as t → 1-, the path {y_{t}} converges strongly to a fixed point of T. Furthermore, explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of T.

Original language | English |
---|---|

Pages (from-to) | 67-77 |

Number of pages | 11 |

Journal | Fixed Point Theory and Applications |

Volume | 2005 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology
- Applied Mathematics

### Cite this

*Fixed Point Theory and Applications*,

*2005*(1), 67-77. https://doi.org/10.1155/FPTA.2005.67

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*Fixed Point Theory and Applications*, vol. 2005, no. 1, pp. 67-77. https://doi.org/10.1155/FPTA.2005.67

**Convergence theorems for fixed points of demicontinuous pseudocontractive mappings.** / Chidume, C. E.; Zegeye, H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence theorems for fixed points of demicontinuous pseudocontractive mappings

AU - Chidume, C. E.

AU - Zegeye, H.

PY - 2005/12/1

Y1 - 2005/12/1

N2 - Let D be an open subset of a real uniformly smooth Banach space E. Suppose T: D̄ → E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where D̄ denotes the closure of D. Then, it is proved that (i) D̄ ⊆ ℛ (I + r (I-T)) for every r > 0; (ii) for a given y0 ∈ D, there exists a unique path t → yt ∈ D̄, t ∈ [0, 1], satisfying yt := tTyt + (1-t) y0. Moreover, if F (T) ≠ ∅ or there exists y0 ∈ D such that the set K := {y ∈ D: T y = λ y + (1-λ) y0 for λ >1} is bounded, then it is proved that, as t → 1-, the path {yt} converges strongly to a fixed point of T. Furthermore, explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of T.

AB - Let D be an open subset of a real uniformly smooth Banach space E. Suppose T: D̄ → E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where D̄ denotes the closure of D. Then, it is proved that (i) D̄ ⊆ ℛ (I + r (I-T)) for every r > 0; (ii) for a given y0 ∈ D, there exists a unique path t → yt ∈ D̄, t ∈ [0, 1], satisfying yt := tTyt + (1-t) y0. Moreover, if F (T) ≠ ∅ or there exists y0 ∈ D such that the set K := {y ∈ D: T y = λ y + (1-λ) y0 for λ >1} is bounded, then it is proved that, as t → 1-, the path {yt} converges strongly to a fixed point of T. Furthermore, explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of T.

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U2 - 10.1155/FPTA.2005.67

DO - 10.1155/FPTA.2005.67

M3 - Article

VL - 2005

SP - 67

EP - 77

JO - Fixed Point Theory and Applications

JF - Fixed Point Theory and Applications

SN - 1687-1820

IS - 1

ER -