Title | Positive subharmonic solutions to nonlinear ODEs with indefinite weight |

Publication Type | Journal Article |

Year of Publication | 2018 |

Authors | Boscaggin, A, Feltrin, G |

Journal | Communications in Contemporary Mathematics |

Volume | 20 |

Pagination | 1750021 |

Abstract | We prove that the superlinear indefinite equation u″ + a(t)up = 0, where p > 1 and a(t) is a T-periodic sign-changing function satisfying the (sharp) mean value condition ∫0Ta(t)dt < 0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it). |

URL | https://doi.org/10.1142/S0219199717500213 |

DOI | 10.1142/S0219199717500213 |

## Positive subharmonic solutions to nonlinear ODEs with indefinite weight

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