Theorems · Theorem · ordinary differential equations

ODE.FunSpace.dist_iterate_iterate_next_le_of_lipschitzWith

∀ {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 : ODE.FunSpace t₀ x₀ r L) {m : ℕ} {C : NNReal},
  LipschitzWith C (ODE.FunSpace.next hf hx)^[m] →
    ∀ (n : ℕ),
      dist α ((ODE.FunSpace.next hf hx)^[m] ^[n] α) ≤
        ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i.factorial) *
            ∑ i ∈ Finset.range n, ↑C ^ i) *
          dist α (ODE.FunSpace.next hf hx α)

Defined in

Mathlib.Analysis.ODE.PicardLindelof

Cited by

1 results in Mathlib

Foundations

Depth 276 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace