## Abstract

Consider the equation -(r(x))y′(x))′ + q(x)y(x) = f(x), x ∈ ℝ, (∗) where f ∈ Lp(ℝ), p ∈ (1, ∞) and r > 0, q ≥ 0, 1/r ∈ L_{1}^{loc}(ℝ), q ∈L_{1}^{loc}(ℝ) lim _{|d|→∞}∫_{x-d}^{x} dt/r(t) ∫_{x-d}^{x} q(t) dt = ∞ ∀x ∈ ℝ. By a solution of (∗), we mean any function y absolutely continuous together with (ry′) and satisfying (∗) almost everywhere on ℝ. In addition, we assume that (∗) is correctly solvable in the space L_{p} (ℝ), i.e. (1) for any function f ∈ L_{p}^{loc}(ℝ), there exists a unique solution y ∈ L_{p}(ℝ) of (∗); (2) there exists an absolute constant c_{1}(p) > 0 such that the solution y ∈ L_{p}(ℝ) of (∗) satisfies the inequality ∥y∥L_{p}(ℝ) ≤ c_{1}(p)∥f∥L_{p}(ℝ) ∀f ∈ L_{p}(ℝ). (∗∗) We study the following problem on the strengthening estimate (∗∗). Let a non-negative function θ ∈ L_{1}^{loc}(ℝ) be given. We have to find minimal additional restrictions for θ under which the following inequality holds: ∥θy∥L_{p}(ℝ) ≤ c_{2}(p)∥f∥L_{p}(ℝ) ∀f ∈ L_{p}(ℝ). Here, y is a solution of (∗) from the class L_{p}(ℝ), and c_{2}(p) is an absolute positive constant.

Original language | English |
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Pages (from-to) | 125-147 |

Number of pages | 23 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2015 |

### Bibliographical note

Publisher Copyright:© 2014 The Edinburgh Mathematical Society.

## Keywords

- Everitt-Giertz problem
- linear differential equations
- second-order equation