Abstract

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 mapF= (F1,…,F1): (a, b) → ℝiisℜ-Pfaffianif there existsG: ℝ1+l→ ℝldefinable in ℜ such thatF′(t) =G(t, F(t)) for allt∈ (a, b) and each component functionGi: ℝ1+l→ ℝ is independent of the lastl−ivariables (i= 1, …,l). If ℜ is o-minimal andF: (a, b) → ℝlis ℜ-Pfaffian, then (ℜ,F) is o-minimal (Proposition 7). We say thatF: ℝ → ℝlis ultimately ℜ-Pfaffian if there existsr∈ ℝ such that the restrictionF↾(r, ∞) is ℜ-Pfaffian. (In general,ultimatelyabbreviates “for all sufficiently large positive arguments”.)The structure ℜ isclosed under asymptotic integrationif for each ultimately non-zero unary (that is, ℝ → ℝ) functionfdefinable in ℜ there is an ultimately differentiable unary functiongdefinable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and definesex: ℝ → ℝ (Proposition 2).Note that the above definitions make sense for expansions of arbitrary ordered fields.