### Abstract

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in Bleecker and Calcaterra (J Math Anal Appl, 248: 645-677, 2000). In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem, i. e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.

Original language | English (US) |
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Pages (from-to) | 231-256 |

Number of pages | 26 |

Journal | Journal of Dynamics and Differential Equations |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2013 |

### Keywords

- Arc fields
- Cauchy-Lipschitz Theorem
- Metric spaces
- Sum of arc fields

### ASJC Scopus subject areas

- Analysis