## Abstract

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_{χ} of X. An element x of X is called fixed point of R iff its membership degree to R_{χ} is at least equal to the membership degree to R_{χ} of any y ε{lunate} X, i.e. R_{χ}(χ)≥ R_{χ}(y)(∀y ε{lunate} X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_{χ} has a 'complementary function'. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.

Original language | English |
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Pages (from-to) | 191-207 |

Number of pages | 17 |

Journal | Fuzzy Sets and Systems |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1982 |

Externally published | Yes |

## Keywords

- Eaves' fixed point algorithm
- Fixed point
- Fuzzy mapping
- Fuzzy sets
- Kuhn's fixed point algorithm
- Linear function relative to a triangulation
- Triangulation

## ASJC Scopus subject areas

- Logic
- Artificial Intelligence