Solvability in Lp of the Neumann problem for a singular non-homogeneous Sturm-Liouville equation

N. Chernyavskaya, L. Shuster

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Consider the equation -(r(x)y′(x))′ + q(x)y(x)=f(x), x∈R (1) where f(x)∈Ls(R), s∈[1, ∞], r(x)>0, q(x)≥0 for x∈R, 1/r(x)∈Lloc1(R), q(x)∈Lloc1(R). The inversion problem for (1) is called regular in Lp if, uniformly in p∈[1, ∞] for any f(x)∈Lp(R), equation (1) has a unique solution y(x)∈Lp(R) of the form y(x) = ∫-∞ G(x, t)f(t)dt, x∈R with ∥y∥p≤c∥f∥p. Here G(x, t) is the Green function corresponding to (1) and c is an absolute constant. For a given s∈[1, ∞], necessary and sufficient conditions are obtained for assertions (2) and (3) to hold simultaneously: (2) the inversion problem for (1) is regular in Lp; (3) lim|x|→∞ r(x)y′(x) = 0 for any f(x)∈Ls(R).

Original languageEnglish
Pages (from-to)453-470
Number of pages18
JournalMathematika
Volume46
Issue number2
DOIs
StatePublished - 1 Jan 1999

ASJC Scopus subject areas

  • Mathematics (all)

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