Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations. (English) Zbl 0816.34040

Shackell, John

Ann. Inst. Fourier 45, No. 1, 183-221 (1995).

Summary: We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows. Let \(g\) be an element of a Hardy field which has an asymptotic series expansion in \(x\), \(e^ x\) and \(\lambda\), where \(\lambda\) tends to zero at least as rapidly as some negative power of \(\exp(e^ x)\). If \(\lambda\) actually occurs in the expansion, then \(g\) cannot satisfy a first-order algebraic differential equation over \({\mathbb{R}}(x)\).

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
26A12	Rate of growth of functions, orders of infinity, slowly varying functions
13N99	Differential algebra

Keywords:

asymptotic growth of Hardy-field solutions; algebraic differential equations; asymptotic series expansion