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) $(r(x)z'(x))'=q_1(x)z(x)$, where the latter is nonoscillatory at infinity. The functions $r(x), q_1(x)$ are assumed to be continuous real-valued, $r(x)>0$, whereas $q(x)$ is continuous complex-valued. A problem of Hartman 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 $q(x)$ being complex-valued a criterion for the solvability of the Hartman-Wintner problem in the new formulation is obtained. The result improves upon the related theorems of Hartman and Wintner, Trench, Simsa and some results of Chen.