### Abstract

Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping J_{φ} for some gauge φ. Let T_{i} : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩_{i ≥ 1} F (T_{i}) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x_{0} ∈ K, let {x_{n}} be generated by the algorithm x_{n + 1} {colon equals} α_{n} f (x_{n}) + (1 - α_{n}) T_{n} (x_{n}), n ≥ 0, where f : K → K is a contraction mapping and {α_{n}} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {x_{n}} satisfies condition (A). Then it is proved that {x_{n}} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family T_{i}, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

Original language | English |
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Pages (from-to) | 2005-2012 |

Number of pages | 8 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 68 |

Issue number | 7 |

DOIs | |

Publication status | Published - Apr 1 2008 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Nonlinear Analysis, Theory, Methods and Applications*, vol. 68, no. 7, pp. 2005-2012. https://doi.org/10.1016/j.na.2007.01.027

**Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings.** / Zegeye, Habtu; Shahzad, Naseer.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2008/4/1

Y1 - 2008/4/1

N2 - Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

AB - Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

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U2 - 10.1016/j.na.2007.01.027

DO - 10.1016/j.na.2007.01.027

M3 - Article

VL - 68

SP - 2005

EP - 2012

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 7

ER -