### 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 a non-self mapping which is asymptotically nonexpansive in the intermediate sense with F(T) := {x ∈ K : Tx = x} ≠ Ø. A demiclosed principle for T is proved. Moreover, if T is completely continuous, an iterative sequence {x_{n}} is constructed which converges strongly to some x* ∈ F(T). If T is not assumed to be completely continuous but the dual E* of E is assumed to have the Kadec-Klee property, then {x_{n}} converges weakly to some x* ∈ F(T). The operator P which plays a central role in our proofs is, in this case, the Banach space analogue of the proximity map in Hilbert spaces.

Original language | English |
---|---|

Pages (from-to) | 239-257 |

Number of pages | 19 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 25 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - May 1 2004 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization

### Cite this

*Numerical Functional Analysis and Optimization*,

*25*(3-4), 239-257. https://doi.org/10.1081/NFA-120039611

}

*Numerical Functional Analysis and Optimization*, vol. 25, no. 3-4, pp. 239-257. https://doi.org/10.1081/NFA-120039611

**Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense.** / Chidume, C. E.; Shahzad, Naseer; Zegeye, Habtu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense

AU - Chidume, C. E.

AU - Shahzad, Naseer

AU - Zegeye, Habtu

PY - 2004/5/1

Y1 - 2004/5/1

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 a non-self mapping which is asymptotically nonexpansive in the intermediate sense with F(T) := {x ∈ K : Tx = x} ≠ Ø. A demiclosed principle for T is proved. Moreover, if T is completely continuous, an iterative sequence {xn} is constructed which converges strongly to some x* ∈ F(T). If T is not assumed to be completely continuous but the dual E* of E is assumed to have the Kadec-Klee property, then {xn} converges weakly to some x* ∈ F(T). The operator P which plays a central role in our proofs is, in this case, the Banach space analogue of the proximity map in Hilbert spaces.

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 a non-self mapping which is asymptotically nonexpansive in the intermediate sense with F(T) := {x ∈ K : Tx = x} ≠ Ø. A demiclosed principle for T is proved. Moreover, if T is completely continuous, an iterative sequence {xn} is constructed which converges strongly to some x* ∈ F(T). If T is not assumed to be completely continuous but the dual E* of E is assumed to have the Kadec-Klee property, then {xn} converges weakly to some x* ∈ F(T). The operator P which plays a central role in our proofs is, in this case, the Banach space analogue of the proximity map in Hilbert spaces.

UR - http://www.scopus.com/inward/record.url?scp=3843131881&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3843131881&partnerID=8YFLogxK

U2 - 10.1081/NFA-120039611

DO - 10.1081/NFA-120039611

M3 - Article

AN - SCOPUS:3843131881

VL - 25

SP - 239

EP - 257

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 3-4

ER -