## Abstract

The equation (1) (r(x)y′(x))′ = q(x)y(x) is regarded as a perturbation of (2) (r(x)z′(x))′ = qi(x)z(x), where the latter is nonoscillatory at infinity. The functions r(x), q1(x) are assumed to be continuous real-valued, r(x) > 0, whereas q(x) is continuous complex-valued. A problem of Hartinan and Wintner regarding the asymptotic integration of (1) for large x by means of solutions of (2) is studied. A new statement of this problem is proposed, which is equivalent to the original one if q(x) is real-valued. In the general case of (x) being complex-valued a criterion for the solvability of the HartmanWintner problem in the new formulation is obtained. The result improves upon the related theorems of Hartman and Wintner, Trench, Śimśa and some results of Chen.

Original language | English |
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Pages (from-to) | 3213-3228 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 125 |

Issue number | 11 |

DOIs | |

State | Published - 1 Jan 1997 |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics