## Abstract

Under certain assumptions on g(x), we obtain an asymptotic formula for computing integrals of the form F(x, α) = ∫_{-∞} ^{∞} g(t)^{α}exp (- |∫_{x}^{t} g(ξ) dξ|) dt, α ∈ ℝ, as |x| → ∞. We use this formula to study the properties (as |x| → ∞) of the solutions of the correctly solvable equations in L_{P}(ℝ), p ∈ [1, ∞], -y″(x) + q(x)y(x) = f(x), x ∈ ℝ, (1) where 0 ≤ q ∈ L_{1}^{loc}(ℝ), and f ∈ L_{P}(ℝ). (Equation (1) is called correctly solvable in a given space L _{p}(ℝ) if for any function f ∈ L_{P}(ℝ) it has a unique solution y ∈ L_{p}(ℝ) and if the following inequality holds with an absolute constraint c_{p} ∈ (0, ∞): ||y||L _{p},(ℝ) ≤ c(p)||f||L_{p}(ℝ), ∀f ∈ L_{P}(ℝ).).

Original language | English |
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Pages (from-to) | 87-114 |

Number of pages | 28 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 50 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2007 |

## Keywords

- Asymptotic estimates
- Asymptotic majorant for solutions
- Sturm-Liouville equation

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