a1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3
In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.
Let ℜ be an expansion of the real field (ℝ, +, ·).
A differentiable map F = (F 1,…, F 1): (a, b) → ℝ i is ℜ-Pfaffian if there exists G: ℝ1+l → ℝ l definable in ℜ such that F′(t) = G(t, F(t)) for all t ∈ (a, b) and each component function Gi : ℝ1+l → ℝ is independent of the last l − i variables (i = 1, …, l). If ℜ is o-minimal and F: (a, b) → ℝ l is ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝ l is ultimately ℜ-Pfaffian if there exists r ∈ ℝ such that the restriction F ↾(r, ∞) is ℜ-Pfaffian. (In general, ultimately abbreviates “for all sufficiently large positive arguments”.)
The structure ℜ is closed under asymptotic integration if for each ultimately non-zero unary (that is, ℝ → ℝ) function f definable in ℜ there is an ultimately differentiable unary function g definable in ℜ such that lim t→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and defines ex : ℝ → ℝ (Proposition 2).
Note that the above definitions make sense for expansions of arbitrary ordered fields.
(Received September 13 2000)
(Revised April 24 2001)
* Research supported by NSF Grants DMS-9896225 and DMS-9988855.
† Research supported by NSERC Grant OGP0009070 and in part by NSF Grant DMS-9988453.