### Abstract

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {k_{n}}_{n≥1} ⊂[1, ∞), limk_{n} = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {x_{n}}_{n≥1} is generated iteratively by x_{1} ∈ K, x_{n+1} = P((1-α_{n} x_{n}+α_{n}T(PT)^{n-1}x_{n}), n≥1, where {α_{n}}_{n≥1} ⊂ (0, 1) is such that ∈ < 1 - α_{n} < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑_{n≥1} (k_{n}^{2} - 1) < ∞ and T is completely continuous, strong convergence of {x_{n}} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {x_{n}} to some x* ∈ F(T) is obtained.

Original language | English |
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Pages (from-to) | 364-374 |

Number of pages | 11 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 280 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 15 2003 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*280*(2), 364-374. https://doi.org/10.1016/S0022-247X(03)00061-1

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*Journal of Mathematical Analysis and Applications*, vol. 280, no. 2, pp. 364-374. https://doi.org/10.1016/S0022-247X(03)00061-1

**Strong and weak convergence theorems for asymptotically nonexpansive mappings.** / Chidume, C. E.; Ofoedu, E. U.; Zegeye, Habz.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong and weak convergence theorems for asymptotically nonexpansive mappings

AU - Chidume, C. E.

AU - Ofoedu, E. U.

AU - Zegeye, Habz

PY - 2003/4/15

Y1 - 2003/4/15

N2 - Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.

AB - Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.

UR - http://www.scopus.com/inward/record.url?scp=0038730516&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038730516&partnerID=8YFLogxK

U2 - 10.1016/S0022-247X(03)00061-1

DO - 10.1016/S0022-247X(03)00061-1

M3 - Article

VL - 280

SP - 364

EP - 374

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -