### Abstract

Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x_{1} ∈ E define the sequence x_{n} ∈ E by x_{n+i} := x_{n} - α_{n}Ax_{n}, n ≥1, where {α_{n}} is a positve real sequence satisfying the following conditions: (i) Σα_{n} = ∞; (ii) lim α_{n} = 0. For x* ∈ N (A) := {x ∈ E : Ax = 0}, assume that σ := inf_{n∈N0} ψ(∥x_{n+1}-x*∥)/∥x_{n+1}-x*∥ > 0 and that ∥Ax_{n+1} - Ax_{n}∥ → 0, where N_{0} := {n ∈ N (the set of all positive integers): x_{n+1} ≠ x*} and ψ : [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

Original language | English |
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Pages (from-to) | 2467-2478 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 131 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*131*(8), 2467-2478. https://doi.org/10.1090/S0002-9939-02-06769-2

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*Proceedings of the American Mathematical Society*, vol. 131, no. 8, pp. 2467-2478. https://doi.org/10.1090/S0002-9939-02-06769-2

**Approximation methods for nonlinear operator equations.** / Chidume, C. E.; Zegeye, H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximation methods for nonlinear operator equations

AU - Chidume, C. E.

AU - Zegeye, H.

PY - 2003/8/1

Y1 - 2003/8/1

N2 - Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x1 ∈ E define the sequence xn ∈ E by xn+i := xn - αnAxn, n ≥1, where {αn} is a positve real sequence satisfying the following conditions: (i) Σαn = ∞; (ii) lim αn = 0. For x* ∈ N (A) := {x ∈ E : Ax = 0}, assume that σ := infn∈N0 ψ(∥xn+1-x*∥)/∥xn+1-x*∥ > 0 and that ∥Axn+1 - Axn∥ → 0, where N0 := {n ∈ N (the set of all positive integers): xn+1 ≠ x*} and ψ : [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

AB - Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x1 ∈ E define the sequence xn ∈ E by xn+i := xn - αnAxn, n ≥1, where {αn} is a positve real sequence satisfying the following conditions: (i) Σαn = ∞; (ii) lim αn = 0. For x* ∈ N (A) := {x ∈ E : Ax = 0}, assume that σ := infn∈N0 ψ(∥xn+1-x*∥)/∥xn+1-x*∥ > 0 and that ∥Axn+1 - Axn∥ → 0, where N0 := {n ∈ N (the set of all positive integers): xn+1 ≠ x*} and ψ : [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

UR - http://www.scopus.com/inward/record.url?scp=0042745484&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042745484&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-02-06769-2

DO - 10.1090/S0002-9939-02-06769-2

M3 - Article

AN - SCOPUS:0042745484

VL - 131

SP - 2467

EP - 2478

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 8

ER -