## Abstract

We consider the singular boundary value problem -r(x)y′(x) + q(x)y(x) = f(x), x ∈ ℝ (1) lim y(x) = 0, (2) |x|→∞ where f ∈ L_{p}(ℝ), p ∈ [1, ∞] (L_{∞}(ℝ):= C(ℝ)), r is a continuous positive function on ℝ, 0 ≤ q ∈ L_{1}^{loc} (ℝ). A solution of this problem is, by definition, any absolutely continuous function y satisfying the limit condition and almost everywhere the differential equation. This problem 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 constant c_{p} ∈ (0, ∞): ||y||L_{p}(ℝ) ≤ c_{p}||f||L_{p}(ℝ), ∀ f L_{p}(ℝ). We find a relationship between r, q, and the parameter p ∈ [1, ∞], which guarantees the correctly solvability of the problem (1) and (2) in L _{p}(ℝ).

Original language | English |
---|---|

Pages (from-to) | 439-458 |

Number of pages | 20 |

Journal | Zeitschrift für Analysis und ihre Anwendungen |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - 2007 |

## Keywords

- Correct solvability
- First order linear differential equation