Conditions for correct solvability of a simplest singular boundary value problem of general form. II

N. A. Chernyavskaya, L. A. Shuster

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We consider the singular boundary value problem -r(x)y′(x) + q(x)y(x) = f(x), x ∈ ℝ (1) lim y(x) = 0, (2) |x|→∞ where f ∈ Lp(ℝ), p ∈ [1, ∞] (L(ℝ):= C(ℝ)), r is a continuous positive function on ℝ, 0 ≤ q ∈ L1loc (ℝ). A solution of this problem is, by definition, any absolutely continuous function y satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space Lp(ℝ) if for any function f ∈ Lp(ℝ) it has a unique solution y ∈ L p(ℝ) and if the following inequality holds with an absolute constant cp ∈ (0, ∞): ||y||Lp(ℝ) ≤ cp||f||Lp(ℝ), ∀ f Lp(ℝ). We find a relationship between r, q, and the parameter p ∈ [1, ∞], which guarantees the correctly solvability of the problem (1) and (2) in L p(ℝ).

Original languageEnglish
Pages (from-to)439-458
Number of pages20
JournalZeitschrift für Analysis und ihre Anwendungen
Volume26
Issue number4
DOIs
StatePublished - 1 Jan 2007

Keywords

  • Correct solvability
  • First order linear differential equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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