Weight summability of solutions of the Sturm-Liouville equation

N. Chernyavskaya; L. Shuster

doi:10.1006/jdeq.1998.3517

Weight summability of solutions of the Sturm-Liouville equation

N. Chernyavskaya, L. Shuster

Department of Mathematics

Ben-Gurion University of the Negev

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

8 Scopus citations

Abstract

We consider an equation -y″(x)+q(x)y(x)=f(x), x∈R; (1) where f(x)∈L_p(R), p∈[1,∞] (L_∞(R):=C(R)), 1≤q(x)∈L^loc₁(R). We study requirements for a weight function r(x)∈L^loc_p(R) and for q(x) under which, for a given p∈[1,∞], regardless of f(x)∈L_p(R), the solution y(x)∈L_p(R) of Eq. (1) satisfies the inequalities: r(x)y(x)_p≤cf(x)_p, y″(x)_p+q(x)y(x)_p≤c f(x)_p, c=const.

Original language	English
Pages (from-to)	456-473
Number of pages	18
Journal	Journal of Differential Equations
Volume	151
Issue number	2
DOIs	https://doi.org/10.1006/jdeq.1998.3517
State	Published - 20 Jan 1999

Bibliographical note

Funding Information:
* Supported by the Israel Academy of sciences under Grant 431 95, -Supported by the Israel Academy of sciences under Grant 505 95.

Funding

* Supported by the Israel Academy of sciences under Grant 431 95, -Supported by the Israel Academy of sciences under Grant 505 95.

Funders	Funder number
Israel Academy of Sciences and Humanities	431 95, 505 95

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10.1006/jdeq.1998.3517

Cite this

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title = "Weight summability of solutions of the Sturm-Liouville equation",

abstract = "We consider an equation -y″(x)+q(x)y(x)=f(x), x∈R; (1) where f(x)∈Lp(R), p∈[1,∞] (L∞(R):=C(R)), 1≤q(x)∈Lloc1(R). We study requirements for a weight function r(x)∈Llocp(R) and for q(x) under which, for a given p∈[1,∞], regardless of f(x)∈Lp(R), the solution y(x)∈Lp(R) of Eq. (1) satisfies the inequalities: r(x)y(x)p≤cf(x)p, y″(x)p+q(x)y(x)p≤c f(x)p, c=const.",

author = "N. Chernyavskaya and L. Shuster",

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AU - Shuster, L.

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N2 - We consider an equation -y″(x)+q(x)y(x)=f(x), x∈R; (1) where f(x)∈Lp(R), p∈[1,∞] (L∞(R):=C(R)), 1≤q(x)∈Lloc1(R). We study requirements for a weight function r(x)∈Llocp(R) and for q(x) under which, for a given p∈[1,∞], regardless of f(x)∈Lp(R), the solution y(x)∈Lp(R) of Eq. (1) satisfies the inequalities: r(x)y(x)p≤cf(x)p, y″(x)p+q(x)y(x)p≤c f(x)p, c=const.

AB - We consider an equation -y″(x)+q(x)y(x)=f(x), x∈R; (1) where f(x)∈Lp(R), p∈[1,∞] (L∞(R):=C(R)), 1≤q(x)∈Lloc1(R). We study requirements for a weight function r(x)∈Llocp(R) and for q(x) under which, for a given p∈[1,∞], regardless of f(x)∈Lp(R), the solution y(x)∈Lp(R) of Eq. (1) satisfies the inequalities: r(x)y(x)p≤cf(x)p, y″(x)p+q(x)y(x)p≤c f(x)p, c=const.

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Weight summability of solutions of the Sturm-Liouville equation

Abstract

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