Necessary and sufficient conditions for the solvability of a problem of hartman and wintner

The equation (1) (r(x)y′(x))′ = q(x)y(x) is regarded as a perturbation of (2) (r(x)z′(x))′ = qi(x)z(x), where the latter is nonoscillatory at infinity. The functions r(x), q1(x) are assumed to be continuous real-valued, r(x) > 0, whereas q(x) is continuous complex-valued. A problem of Hartinan and Wintner regarding the asymptotic integration of (1) for large x by means of solutions of (2) is studied. A new statement of this problem is proposed, which is equivalent to the original one if q(x) is real-valued. In the general case of (x) being complex-valued a criterion for the solvability of the HartmanWintner problem in the new formulation is obtained. The result improves upon the related theorems of Hartman and Wintner, Trench, Śimśa and some results of Chen.