We consider the singular boundary value problem -r(x)y′(x) + q(x)y(x) = f(x), x ε R lim|x|→∞ y(x) = 0, where f ∈ Lp(ℝ), p ∈ [1, ∞] (L∞(ℝ) := C(ℝ)), r is a continuous positive function on ℝ, 0 ≤ q ∈ L1loc. 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 ∈ L p(ℝ) it has a unique solution y ∈ Lp(ℝ) and if the following inequality holds with an absolute constant cp ∈ (0, ∞) : ||y||Lp(ℝ) ≤ cp||f||L p(ℝ), f ∈ Lp(ℝ). We find minimal requirements for r and q under which the above problem is correctly solvable in Lp(ℝ).
- Correct solvability
- First order linear differential equation
ASJC Scopus subject areas
- Applied Mathematics