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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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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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
  • 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