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Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

Part of: General theory in ordinary differential equations Qualitative theory

Published online by Cambridge University Press:  21 July 2014

N. A. Chernyavskaya  and
L. A. Shuster
N. A. Chernyavskaya
Affiliation:
Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel
L. A. Shuster
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel, ([email protected])
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Abstract

Consider the equation

where ƒ ∈ Lp(ℝ), p ∈ (1, ∞) and

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 Lp(ℝ), i.e.

(1) for any function , there exists a unique solution y ∈ Lp(ℝ) of (*);

(2) there exists an absolute constant c1(p) > 0 such that the solution y ∈ Lp(ℝ) of (*) satisfies the inequality

We study the following problem on the strengthening estimate (**). Let a non-negative function be given. We have to find minimal additional restrictions for θ under which the following inequality holds:

Here, y is a solution of (*) from the class Lp(ℝ), and c2 (p) is an absolute positive constant.

Keywords

linear differential equationssecond-order equationEveritt—Giertz problem

MSC classification

Primary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory
Secondary: 34A99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 1 , February 2015 , pp. 125 - 147
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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