An asymptotic majorant for solutions of Sturm-liouville equations in L P(ℝ)

N. A. Chernyavskaya, L. A. Shuster

Research output: Contribution to journalArticlepeer-review

Abstract

Under certain assumptions on g(x), we obtain an asymptotic formula for computing integrals of the form F(x, α) = ∫-∞ g(t)αexp (- |∫xt g(ξ) dξ|) dt, α ∈ ℝ, as |x| → ∞. We use this formula to study the properties (as |x| → ∞) of the solutions of the correctly solvable equations in LP(ℝ), p ∈ [1, ∞], -y″(x) + q(x)y(x) = f(x), x ∈ ℝ, (1) where 0 ≤ q ∈ L1loc(ℝ), and f ∈ LP(ℝ). (Equation (1) is called correctly solvable in a given space L p(ℝ) if for any function f ∈ LP(ℝ) it has a unique solution y ∈ Lp(ℝ) and if the following inequality holds with an absolute constraint cp ∈ (0, ∞): ||y||L p,(ℝ) ≤ c(p)||f||Lp(ℝ), ∀f ∈ LP(ℝ).).

Original languageEnglish
Pages (from-to)87-114
Number of pages28
JournalProceedings of the Edinburgh Mathematical Society
Volume50
Issue number1
DOIs
StatePublished - 1 Feb 2007

Keywords

  • Asymptotic estimates
  • Asymptotic majorant for solutions
  • Sturm-Liouville equation

ASJC Scopus subject areas

  • General Mathematics

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