Necessary and sufficient conditions for the solvability of a problem of Hartman and Wintner

Chernyavskaya, N.; Shuster, L.

doi:10.1090/S0002-9939-97-04186-5

Necessary and sufficient conditions for the solvability of a problem of Hartman and Wintner
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by N. Chernyavskaya and L. Shuster PDF

Proc. Amer. Math. Soc. 125 (1997), 3213-3228 Request permission

Abstract:

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, Śimśa and some results of Chen.

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