Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rosenlicht fields

Author: John Shackell
Journal: Trans. Amer. Math. Soc. 335 (1993), 579-595
MSC: Primary 12H05; Secondary 26A12, 26E99, 34E99
MathSciNet review: 1085945
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Abstract: Let $ \phi $ satisfy an algebraic differential equation over $ {\mathbf{R}}$. We show that if $ \phi $ also belongs to a Hardy field, it possesses an asymptotic form which must be one of a restricted number of types. The types depend only on the order of the differential equation. For a particular equation the types are still more restricted. In some cases one can conclude that no solution of the given equation lies in a Hardy field, and in others that a particular asymptotic form is the only possibility for such solutions. This therefore gives a new method for obtaining asymptotic solutions of nonlinear differential equations. The techniques used are in part derived from the work of Rosenlicht in Hardy fields.

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Keywords: Hardy fields, asymptotic expansions, orders of growth
Article copyright: © Copyright 1993 American Mathematical Society