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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
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
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, Simsa and some results of Chen.


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Additional Information

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: nina@math.bgu.ac.il

L. Shuster
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04186-5
PII: S 0002-9939(97)04186-5
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
Article copyright: © Copyright 1997 American Mathematical Society