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We consider the singular boundary value problem −r(x)y ′ (x) + q(x)y(x) = f (x), x ∈ R lim |x|→∞ y(x) = 0, where f ∈ L p (R), p ∈ [1, ∞] (L ∞ (R) := C(R)), r is a continuous positive function for x ∈ R, q ∈ L loc 1 (R), q ≥ 0. A solution of this problem is, by definition, any absolutely continuous function y satisfying the limit condition and almost(More)
We consider an equation y ′′ (x) = q(x)y(x), x ∈ R (1) under the following assumptions on q(x) : 0 ≤ q(x) ∈ L loc 1 (R), x −∞ q(t)dt > 0, ∞ x q(t)dt > 0 for all x ∈ R. (2) Let v(x) (resp. u(x)) be a positive non-decreasing (resp. non-increasing) solution of (1) such that v ′ (x)u(x) − u ′ (x)v(x) = 1, x ∈ R. These properties determine u(x) and v(x) up to(More)