Theorems · Theorem · ordinary differential equations

ODE.FunSpace.dist_comp_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) (n : ℕ) (t : ↑(Set.Icc tmin tmax)) {α β : ODE.FunSpace t₀ x₀ r L},
  dist (((ODE.FunSpace.next hf hx)^[n] α).toFun t) (((ODE.FunSpace.next hf hx)^[n] β).toFun t) ≤
      (↑K * |↑t - ↑t₀|) ^ n / ↑n.factorial * dist α β →
    dist (f (↑t) (((ODE.FunSpace.next hf hx)^[n] α).toFun t)) (f (↑t) (((ODE.FunSpace.next hf hx)^[n] β).toFun t)) ≤
      ↑K ^ (n + 1) * |↑t - ↑t₀| ^ n / ↑n.factorial * dist α β

A key step in the inductive case of dist_iterate_next_apply_le

Defined in

Mathlib.Analysis.ODE.PicardLindelof

Cited by

1 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace