## Abstract

We consider an equation(1)y^{″} (x) = q (x) y (x), x ∈ R, under the following assumptions on q:(2)0 ≤ q ∈ L_{1}^{loc} (R), underover(∫, - ∞, x) q (t) d t > 0, underover(∫, x, ∞) q (t) d t > 0 for all x ∈ R . Let v (respectively u) be a positive non-decreasing (respectively non-increasing) solution of (1) such thatv^{′} (x) u (x) - u^{′} (x) v (x) = 1, x ∈ R . These properties determine u and v up to mutually inverse positive constant factors, and the function ρ (x) = u (x) v (x), x ∈ R, is uniquely determined by q. In the present paper, we obtain an asymptotic formula for computing ρ (x) as | x | → ∞. As an application, under conditions (2), we study the behavior at infinity of solution of the Riccati equationz^{′} (x) + z (x)^{2} = q (x), x ∈ R .

Original language | English |
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Pages (from-to) | 998-1021 |

Number of pages | 24 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 334 |

Issue number | 2 |

DOIs | |

State | Published - 15 Oct 2007 |

## Keywords

- Asymptotics on the diagonal
- Green function
- Riccati equations
- Sturm-Liouville operator