### Abstract

Let X_{0} ⊂ X_{1} ⊂ ··· ⊂ X_{p} be Banach spaces with continuous injection of X_{k} into X_{k + 1} for 0 ≤ k ≤ p - 1, and with X_{0} dense in X_{p}. We seek a function u: [0, 1] → X_{0} such that its kth derivative u^{(k)}, k = 0, 1,..., p, is continuous from [0, 1] into x_{k}, and satisfies the initial condition u^{(k)}(0) = a_{k} ε{lunate} X_{k}. It is shown that such a function exists if and only if the initial values a_{0}, a_{1}, ..., a_{p} satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ≥ 2, the spaces X_{k} (k = 0, 1, ..., p) may or may not be such that the desired function exists for any given initial values a_{k} ε{lunate} X_{k}.

Original language | English (US) |
---|---|

Pages (from-to) | 328-335 |

Number of pages | 8 |

Journal | Journal of Functional Analysis |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1978 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

*Journal of Functional Analysis*,

*29*(3), 328-335. https://doi.org/10.1016/0022-1236(78)90035-6