ISSN 1088-6826(online) ISSN 0002-9939(print)

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

Authors: N. Chernyavskaya and L. Shuster
Journal: Proc. Amer. Math. Soc. 125 (1997), 3213-3228
MSC (1991): Primary 34E10
DOI: https://doi.org/10.1090/S0002-9939-97-04186-5
MathSciNet review: 1443146
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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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