# Gronwall’s lemma

If, for ${t}_{0}\le t\le {t}_{1}$, $\varphi (t)\ge 0$ and $\psi (t)\ge 0$ are continuous functions^{} such that the inequality^{}

$$\varphi (t)\le K+L{\int}_{{t}_{0}}^{t}\psi (s)\varphi (s)\mathit{d}s$$ |

holds on ${t}_{0}\le t\le {t}_{1}$, with $K$ and $L$ positive constants, then

$$\varphi (t)\le K\mathrm{exp}\left(L{\int}_{{t}_{0}}^{t}\psi (s)\mathit{d}s\right)$$ |

on ${t}_{0}\le t\le {t}_{1}$.

Title | Gronwall’s lemma |
---|---|

Canonical name | GronwallsLemma |

Date of creation | 2013-03-22 13:22:20 |

Last modified on | 2013-03-22 13:22:20 |

Owner | jarino (552) |

Last modified by | jarino (552) |

Numerical id | 4 |

Author | jarino (552) |

Entry type | Theorem |

Classification | msc 26D10 |

Synonym | Gronwall’s inequality |