Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations

Shackell, John

Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 183-221.

Nous nous intéressons aux croissances asymptotiques des solutions des équations différentielles algébriques qui sont des éléments d’un corps de Hardy, c’est-à-dire des solutions qui n’ont aucune composante oscillatoire. Nous prouvons qu’un développement imbriqué d’une telle solution ne peut tendre plus vite vers zéro qu’une vitesse fixe déterminée par l’ordre de l’équation différentielle. Nous considérons aussi les développements en série asymptotique généralisée, et obtenons, par exemple, le résultat suivant.

Soit $g$ un élément d’un corps de Hardy qui a un développement en série avec fonctions de base $x$, $e^x$ et $\lambda$, où $\lambda$ tend vers zéro au moins aussi vite qu’une puissance négative de $exp\left(e^x\right)$. Si $\lambda$ apparaît vraiment dans le développement, il s’ensuit que $g$ ne peut satisfaire une équation différentielle du premier ordre sur $ℝ\left(x\right)$.

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 $exp\left(e^x\right)$. If $\lambda$ actually occurs in the expansion, then $g$ cannot satisfy a first-order algebraic differential equation over $ℝ\left(x\right)$.

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