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 p roblem of Hartman and Wintner regarding the asymptotic integration of (1) for large x by means of solutions of (2) is studied. A new stateme nt of this problem is proposed, which is equivalent to the original on e if q(x) is real-valued. In the general case of q(x) being complex-va lued 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 Ch en.