## Abstract

Let K be a closed convex subset of a reflexive Banach space X. We consider self-mappings of K which are bounded on bounded subsets of K and satisfy a relaxed form of nonexpansivity with respect to a given convex function f. The family of these operators is endowed with the topology of uniform convergence on bounded subsets of K. We show that "almost all" such operators T share the property that they have a fixed point z_{T} such that, for any x ∈ K, the orbit {T^{k}x}_{k=1}^{∞} converges weakly to Z_{T}. Here the meaning of "almost all" is in the sense of Baire's categories: the collection of all those operators which do not have this property is contained in a countable union of nowhere dense sets.

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

Pages (from-to) | 489-508 |

Number of pages | 20 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 24 |

Issue number | 5-6 |

DOIs | |

State | Published - 2003 |

## Keywords

- Bregman distance
- Fixed point
- Nonexpansivity of an operator with respect to a convex function
- Orbit
- Reflexive Banach space
- Total convexity
- Uniform convexity
- Weak convergence

## ASJC Scopus subject areas

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