### Abstract

Let K be a closed convex subset of a Banach space E and let T : K → E be a continuous weakly inward pseudocontractive mapping. Then for t ∈ (0, 1), there exists a sequence {y_{t}} ⊂ K satisfying y_{t} = (1 - t)f(y_{t}) + tT(y_{t}), where f ∈ Π_{K} {colon equals} {f : K → K, a contraction with a suitable contractive constant}. Suppose further that F(T) ≠ ∅ and E is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that {y_{t}} converges strongly to a fixed point of T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of T and hence to a solution of certain variational inequality is constructed provided that T is Lipschitzian.

Original language | English |
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Pages (from-to) | 538-546 |

Number of pages | 9 |

Journal | Applied Mathematics and Computation |

Volume | 185 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Mathematics and Computation*,

*185*(1), 538-546. https://doi.org/10.1016/j.amc.2006.07.063

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*Applied Mathematics and Computation*, vol. 185, no. 1, pp. 538-546. https://doi.org/10.1016/j.amc.2006.07.063

**Viscosity approximation methods for pseudocontractive mappings in Banach spaces.** / Zegeye, Habtu; Shahzad, Naseer; Mekonen, Tefera.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Viscosity approximation methods for pseudocontractive mappings in Banach spaces

AU - Zegeye, Habtu

AU - Shahzad, Naseer

AU - Mekonen, Tefera

PY - 2007/2/1

Y1 - 2007/2/1

N2 - Let K be a closed convex subset of a Banach space E and let T : K → E be a continuous weakly inward pseudocontractive mapping. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t)f(yt) + tT(yt), where f ∈ ΠK {colon equals} {f : K → K, a contraction with a suitable contractive constant}. Suppose further that F(T) ≠ ∅ and E is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that {yt} converges strongly to a fixed point of T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of T and hence to a solution of certain variational inequality is constructed provided that T is Lipschitzian.

AB - Let K be a closed convex subset of a Banach space E and let T : K → E be a continuous weakly inward pseudocontractive mapping. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t)f(yt) + tT(yt), where f ∈ ΠK {colon equals} {f : K → K, a contraction with a suitable contractive constant}. Suppose further that F(T) ≠ ∅ and E is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that {yt} converges strongly to a fixed point of T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of T and hence to a solution of certain variational inequality is constructed provided that T is Lipschitzian.

UR - http://www.scopus.com/inward/record.url?scp=33846953549&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846953549&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2006.07.063

DO - 10.1016/j.amc.2006.07.063

M3 - Article

VL - 185

SP - 538

EP - 546

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 1

ER -