## Abstract

We consider the singular boundary value problem -r(x)y′(x) + q(x)y(x) = f(x), x ε R lim_{|x|→∞} y(x) = 0, where f ∈ L_{p}(ℝ), p ∈ [1, ∞] (L_{∞}(ℝ) := C(ℝ)), r is a continuous positive function on ℝ, 0 ≤ q ∈ L_{1}^{loc}. 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 minimal requirements for r and q under which the above problem is correctly solvable in L_{p}(ℝ).

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

Pages (from-to) | 205-235 |

Number of pages | 31 |

Journal | Zeitschrift für Analysis und ihre Anwendungen |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - 2006 |

## Keywords

- Correct solvability
- First order linear differential equation