Necessary and sufficient conditions for the solvability of a problem of Hartman and Wintner |
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| Authors: | N. Chernyavskaya L. Shuster |
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| 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 ; Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel |
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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. |
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