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) is regarded as a perturbation of (2) , where the latter is nonoscillatory at infinity. The functions are assumed to be continuous real-valued, , whereas is continuous complex-valued. A problem of Hartman and Wintner regarding the asymptotic integration of (1) for large by means of solutions of (2) is studied. A new statement of this problem is proposed, which is equivalent to the original one if is real-valued. In the general case of 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.

**1.**S. Chen,*Asymptotic integrations of nonoscillatory second order differential equations*, Trans. Amer. Math. Soc.**327**(2) (1991). MR**92a:34057****2.**N. Chernyavskaya and L. Shuster,*Asymptotic integration of a nonoscillatory second order differential equation with a linear perturbation*, AMS PPS # 199508-34-001, preprint.**3.**P. Hartman,*Ordinary Differential Equations*, Wiley, New York, 1964. MR**30:1270****4.**P. Hartman and A. Wintner,*On non-oscillatory linear equations*, Amer. J. Math.**75**(1953), 717-730. MR**15:527c****5.**J. \'{S}im\'{s}a,*Asymptotic integration of a second order ordinary differential equation*, Proc. Amer. Math. Soc.**101**(1) (1987), 96-100. MR**89b:34129****6.**C.C. Titchmarsh,*The Theory of Functions*, Oxford, 1932.**7.**W.F. Trench,*Linear perturbations of a nonoscillatory second order equation*, Proc. Amer. Math. Soc.**97**(3) (1986), 423-428. MR**87g:34036**

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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:
https://doi.org/10.1090/S0002-9939-97-04186-5

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