### Abstract

Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F (T) ≠ ∅ and f : K → K be a contraction. Then for t ∈ (0, 1), there exists a sequence {y_{t}} ⊂ K satisfying y_{t} = (1 - t) f (y_{t}) + tT (y_{t}). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {y_{t}} converges strongly to a fixed point p of T such that p is the unique solution in F (T) to a certain variational inequality. Moreover, if {T_{i}, i = 1, 2, ..., r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {T_{i}, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family T_{i}, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.

Original language | English |
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Pages (from-to) | 155-163 |

Number of pages | 9 |

Journal | Applied Mathematics and Computation |

Volume | 191 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Cite this

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*Applied Mathematics and Computation*, vol. 191, no. 1, pp. 155-163. https://doi.org/10.1016/j.amc.2007.02.072

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

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2007/8/1

Y1 - 2007/8/1

N2 - Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F (T) ≠ ∅ and f : K → K be a contraction. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t) f (yt) + tT (yt). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {yt} converges strongly to a fixed point p of T such that p is the unique solution in F (T) to a certain variational inequality. Moreover, if {Ti, i = 1, 2, ..., r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {Ti, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family Ti, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.

AB - Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F (T) ≠ ∅ and f : K → K be a contraction. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t) f (yt) + tT (yt). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {yt} converges strongly to a fixed point p of T such that p is the unique solution in F (T) to a certain variational inequality. Moreover, if {Ti, i = 1, 2, ..., r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {Ti, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family Ti, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.

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UR - http://www.scopus.com/inward/citedby.url?scp=34547660273&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2007.02.072

DO - 10.1016/j.amc.2007.02.072

M3 - Article

VL - 191

SP - 155

EP - 163

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 1

ER -