### Abstract

Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T _{x} = x} ≠ Ø. An iterative sequence {x_{n}} is constructed for which ∥x_{n} - T_{xn}∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ Ø. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {x_{n}} converges strongly to a fixed point of T, Our iteration method is of independent interest.

Original language | English |
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Pages (from-to) | 831-840 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 132 |

Issue number | 3 |

Publication status | Published - Mar 1 2004 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*132*(3), 831-840.

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*Proceedings of the American Mathematical Society*, vol. 132, no. 3, pp. 831-840.

**Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps.** / Chidume, C. E.; Zegeye, H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

AU - Chidume, C. E.

AU - Zegeye, H.

PY - 2004/3/1

Y1 - 2004/3/1

N2 - Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T x = x} ≠ Ø. An iterative sequence {xn} is constructed for which ∥xn - Txn∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ Ø. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {xn} converges strongly to a fixed point of T, Our iteration method is of independent interest.

AB - Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T x = x} ≠ Ø. An iterative sequence {xn} is constructed for which ∥xn - Txn∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ Ø. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {xn} converges strongly to a fixed point of T, Our iteration method is of independent interest.

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UR - http://www.scopus.com/inward/citedby.url?scp=1442278169&partnerID=8YFLogxK

M3 - Article

VL - 132

SP - 831

EP - 840

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -