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

HTML articles powered by AMS MathViewer

- 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.## References

- Shao Zhu Chen,
*Asymptotic integrations of nonoscillatory second order differential equations*, Trans. Amer. Math. Soc.**327**(1991), no. 2, 853–865. MR**1028756**, DOI 10.1090/S0002-9947-1991-1028756-7 - N. Chernyavskaya and L. Shuster,
*Asymptotic integration of a nonoscillatory second order differential equation with a linear perturbation*, AMS PPS # 199508-34-001, preprint. - Philip Hartman,
*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0171038** - Sam Perlis,
*Maximal orders in rational cyclic algebras of composite degree*, Trans. Amer. Math. Soc.**46**(1939), 82–96. MR**15**, DOI 10.1090/S0002-9947-1939-0000015-X - Jaromír Šimša,
*Asymptotic integration of a second order ordinary differential equation*, Proc. Amer. Math. Soc.**101**(1987), no. 1, 96–100. MR**897077**, DOI 10.1090/S0002-9939-1987-0897077-X - C.C. Titchmarsh,
*The Theory of Functions*, Oxford, 1932. - William F. Trench,
*Linear perturbations of a nonoscillatory second order equation*, Proc. Amer. Math. Soc.**97**(1986), no. 3, 423–428. MR**840623**, DOI 10.1090/S0002-9939-1986-0840623-1

## 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