## Abstract

We consider the equation (0.1) -(r(x)y′(x))′ + q(x)y(x) = f(x), x ∈ double-struck R sign, where r(x) > 0, q(x) ≥ 0 for x ∈ double-struck R sign, 1/r(x) ∈ L^{loc}_{1}(double-struck R sign), q(x) ∈ L^{loc}_{1} (double-struck R sign), f(x) ∈ L_{p}(double-struck R sign), p ∈ [1, ∞] (L_{∞}(double-struck R sign) := C(double-struck R sign)). We give necessary and sufficient conditions under which, regardless of p ∈ [1, ∞], the following statements hold simultaneously: I) For any f(x) ∈ L_{p}(double-struck R sign) Equation (0.1) has a unique solution y(x) ∈ L_{p}(double-struck R sign) where y(x) = (Gf)(x) ^{def}= ∫^{∞}_{-∞} G(x,t)f(t)dt, x ∈ double-struck R sign. II) The operator G : L_{p} (double-struck R sign) → L_{p}(double-struck R sign) is compact. Here G(x,t) is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm-Liouville operator.

Original language | English |
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Pages (from-to) | 33-53 |

Number of pages | 21 |

Journal | Mathematische Nachrichten |

Volume | 215 |

DOIs | |

State | Published - 2000 |

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