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).
- Correct solvability
- First order linear differential equation
ASJC Scopus subject areas
- Applied Mathematics