## Abstract

Consider the equation -(r(x)y′(x))′ + q(x)y(x)=f(x), x∈R (1) where f(x)∈L_{s}(R), s∈[1, ∞], r(x)>0, q(x)≥0 for x∈R, 1/r(x)∈L^{loc}_{1}(R), q(x)∈L^{loc}_{1}(R). The inversion problem for (1) is called regular in L_{p} if, uniformly in p∈[1, ∞] for any f(x)∈L_{p}(R), equation (1) has a unique solution y(x)∈L_{p}(R) of the form y(x) = ∫^{∞}_{-∞} G(x, t)f(t)dt, x∈R with ∥y∥_{p}≤c∥f∥_{p}. Here G(x, t) is the Green function corresponding to (1) and c is an absolute constant. For a given s∈[1, ∞], necessary and sufficient conditions are obtained for assertions (2) and (3) to hold simultaneously: (2) the inversion problem for (1) is regular in L_{p}; (3) lim_{|x|→∞} r(x)y′(x) = 0 for any f(x)∈L_{s}(R).

Original language | English |
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Pages (from-to) | 453-470 |

Number of pages | 18 |

Journal | Mathematika |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1999 |

## ASJC Scopus subject areas

- General Mathematics

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