## Abstract

We consider the equation -y″(x) + q(x)y(x) = f(x); x ∈ R (1) where f ∈ L^{loc} _{p} (R); p ∈ [1;1) and 0 q ∈2 L^{loc} _{1} (R): By a solution of (1) we mean any function y; absolutely continuous together with its derivative and satisfying (1) almost everywhere in R: Let positive and continuous functions μ(x) and θ;(x) for x ∈ R be given. Let us introduce the spaces L_{p}(R;μ) = {f∈2 ^{loc} _{p} (R) : //f//^{p}L_{p}p_{(R;μ)} = /∞-∞ /μ(x)f(x)jpdx < 1∞}. L_{p}(R;θ) = {f ∈ L^{loc} _{p} (R) : kfkp L_{p}_{(R;μ)} = /∞-∞ /θ(x)f(x)/^{p}dx <∞} : In the present paper, we obtain requirements to the functions f∈L_{p} and q under which 1) for every function f ∈ L_{p}(R;θ) there exists a unique solution (1) y ∈ L_{p}(R;μ) of (1); 2) there is an absolute constant c(p) ∈ (0;1) such that regardless of the choice of a function f ∈ L_{p}(R;θ) the solution of (1) satisfies the inequality //y//L_{p}(R;μ) c(p)kfkL_{p}(R;θ):.

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

Pages (from-to) | 1023-1052 |

Number of pages | 30 |

Journal | Communications on Pure and Applied Analysis |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - May 2018 |

### Bibliographical note

Publisher Copyright:© 2018 American Institute of Mathematical Sciences. All rights reserved.

## Keywords

- Admissible spaces
- Sturm-liouville equation