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= ( F1,…, F1): ( a, b) → ℝ iis ℜ-Pfaffianif there exists G: ℝ1+ l→ ℝ ldefinable 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− ivariables ( i= 1, …, l). If ℜ is o-minimal and F: ( a, b) → ℝ lis ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝ lis ultimately ℜ-Pfaffian if there exists r∈ ℝ such that the restriction F↾( r, ∞) is ℜ-Pfaffian. (In general, ultimatelyabbreviates “for all sufficiently large positive arguments”.)
The structure ℜ is
closed under asymptotic integrationif for each ultimately non-zero unary (that is, ℝ → ℝ) function fdefinable in ℜ there is an ultimately differentiable unary function gdefinable 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.