Theorems · Theorem · ordinary differential equations

ODE.FunSpace.dist_iterate_next_iterate_next_le

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E → E} {tmin tmax : ℝ}
  {t₀ : ↑(Set.Icc tmin tmax)} {x₀ x : E} {a r L K : NNReal} (hf : IsPicardLindelof f t₀ x₀ a r L K)
  (hx : x ∈ Metric.closedBall x₀ ↑r) (α β : ODE.FunSpace t₀ x₀ r L) (n : ℕ),
  dist ((ODE.FunSpace.next hf hx)^[n] α) ((ODE.FunSpace.next hf hx)^[n] β) ≤
    (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ n / ↑n.factorial * dist α β

The n-th iterate of next is Lipschitz continuous with respect to FunSpace, with constant $(K \max(t_{\mathrm{max}}, t_{\mathrm{min}})^n / n!$.

Defined in

Mathlib.Analysis.ODE.PicardLindelof

Cited by

2 results in Mathlib

Foundations

Depth 274 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace