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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- N. Chernyavskaya
- Affiliation: Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel; Department of Agricultural Economics and Management, Hebrew University of Jerusalem, P.O.B. 12, Rehovot 76100, Israel
- Email: firstname.lastname@example.org
- L. Shuster
- Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel
- Received by editor(s): December 13, 1994
- Additional Notes: The authors were supported by the Israel Academy of Sciences under Grants 431/95 (first author) and 505/95 (second author).
- Communicated by: Hal L. Smith
- © Copyright 1997 American Mathematical Society
- 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