A criterion for correct solvability of the Sturm-Liouville equation in the space Lp(R)

N. Chernyavskaya; L. Shuster

doi:10.1090/S0002-9939-01-06145-7

A criterion for correct solvability of the Sturm-Liouville equation in the space L_p(R)

N. Chernyavskaya, L. Shuster

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

We consider an equation (1) - y″(x) + q(x) y(x) = f(x), x ∈ R, where f(x) ∈ L_p(R), p ∈ [1, ∞] (||f||_∞ := C(R)), and 0 ≤ q(x) ∈ L₁^loc(R). By a solution of equation (1), we mean any function y(x) such that y(x), y′(x) ∈ AC^loc(R), and equality (1) holds almost everywhere on R. In this paper, we obtain a criterion for the correct solvability of (1) in L_p(R), p ∈ [1, ∞].

Original language	English
Pages (from-to)	1043-1054
Number of pages	12
Journal	Proceedings of the American Mathematical Society
Volume	130
Issue number	4
DOIs	https://doi.org/10.1090/S0002-9939-01-06145-7
State	Published - 1 Jan 2002

Keywords

Correct solvability
Sturm-Liouville equation

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1090/S0002-9939-01-06145-7

Cite this

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title = "A criterion for correct solvability of the Sturm-Liouville equation in the space Lp(R)",

abstract = "We consider an equation (1) - y″(x) + q(x) y(x) = f(x), x ∈ R, where f(x) ∈ Lp(R), p ∈ [1, ∞] (||f||∞ := C(R)), and 0 ≤ q(x) ∈ L1loc(R). By a solution of equation (1), we mean any function y(x) such that y(x), y′(x) ∈ ACloc(R), and equality (1) holds almost everywhere on R. In this paper, we obtain a criterion for the correct solvability of (1) in Lp(R), p ∈ [1, ∞].",

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N2 - We consider an equation (1) - y″(x) + q(x) y(x) = f(x), x ∈ R, where f(x) ∈ Lp(R), p ∈ [1, ∞] (||f||∞ := C(R)), and 0 ≤ q(x) ∈ L1loc(R). By a solution of equation (1), we mean any function y(x) such that y(x), y′(x) ∈ ACloc(R), and equality (1) holds almost everywhere on R. In this paper, we obtain a criterion for the correct solvability of (1) in Lp(R), p ∈ [1, ∞].

AB - We consider an equation (1) - y″(x) + q(x) y(x) = f(x), x ∈ R, where f(x) ∈ Lp(R), p ∈ [1, ∞] (||f||∞ := C(R)), and 0 ≤ q(x) ∈ L1loc(R). By a solution of equation (1), we mean any function y(x) such that y(x), y′(x) ∈ ACloc(R), and equality (1) holds almost everywhere on R. In this paper, we obtain a criterion for the correct solvability of (1) in Lp(R), p ∈ [1, ∞].

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KW - Sturm-Liouville equation

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