## Abstract

Let X be a uniformly smooth Banach space and A be an m-accretive operator on X with A ^{-1} (0) ≠ θ. Assume that F: X → X is δstrongly accretive and λ-strictly pseudocontractive with δ+λ > 1. This article proposes hybrid viscosity approximation methods which combine viscosity approximation methods with hybrid steepest-descent methods. For each t∈ (0, 1) and each integer n≥0, let {x _{t,n}} be defined byx _{t,n} = tf(x _{t,n} + (1-t)[Jr _{n}x _{t,n}-θ F(Jr _{n}x _{t,n}] where f: X → X is a contractive map, {r _{n}}⊂[ε,∞) for some ε > 0 and {θ:t∈⊂(0, 1)}[0, 1) with lim _{t→}0θ _{t}/t=0. We deduce that as t→0, {x _{t,n}} converges strongly to a zero p of A, which is a unique solution of some variational inequality. On the other hand, given a point x _{0}∈ X and given sequences {λ _{n}}, {μ _{n}} in [0, 1], {α _{n}}, {β _{n}} in (0, 1], let the sequence {x _{n} be generated by {y _{n}= α _{n}x _{n}+(1 -α _{n})Jr _{n},x _{n+1} = β _{n}f (x _{n})+(1-β _{n}) [Jr _{n} -y _{n}-λμ _{n}μ _{n}F(Jr _{n} y _{n})], ∀n ≥ 0. It is proven that under appropriate conditions {x _{n}} converges strongly to the same zero p of A. The results presented here extend, improve and develop some very recent theorems in the literature to a great extent.

Original language | English |
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Pages (from-to) | 142-165 |

Number of pages | 24 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2012 |

Externally published | Yes |

## Keywords

- Contractive maps
- Hybrid viscosity approximation methods
- m-accretive operators
- Uniformly smooth Banach space
- Variational inequality