### Abstract

We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods. We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.

Original language | English |
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Pages (from-to) | 1190-1204 |

Number of pages | 15 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 32 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

### Keywords

- Computer-assisted proof
- Fixed-point equation
- Local uniqueness
- Numerical verification
- Self-validated computing

### ASJC Scopus subject areas

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

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## Cite this

*Numerical Functional Analysis and Optimization*,

*32*(11), 1190-1204. https://doi.org/10.1080/01630563.2011.594348