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).).
- Asymptotic estimates
- Asymptotic majorant for solutions
- Sturm-Liouville equation
ASJC Scopus subject areas
- Mathematics (all)