Solvability in L_p of the Neumann problem for a singular non-homogeneous Sturm-Liouville equation

Consider the equation -(r(x)y′(x))′ + q(x)y(x)=f(x), x∈R (1) where f(x)∈L_s(R), s∈[1, ∞], r(x)>0, q(x)≥0 for x∈R, 1/r(x)∈L^loc₁(R), q(x)∈L^loc₁(R). The inversion problem for (1) is called regular in L_p if, uniformly in p∈[1, ∞] for any f(x)∈L_p(R), equation (1) has a unique solution y(x)∈L_p(R) of the form y(x) = ∫^∞_-∞ G(x, t)f(t)dt, x∈R with ∥y∥_p≤c∥f∥_p. Here G(x, t) is the Green function corresponding to (1) and c is an absolute constant. For a given s∈[1, ∞], necessary and sufficient conditions are obtained for assertions (2) and (3) to hold simultaneously: (2) the inversion problem for (1) is regular in L_p; (3) lim_|x|→∞ r(x)y′(x) = 0 for any f(x)∈L_s(R).