Theorems · Theorem · ordinary differential equations

ODE.FunSpace.exists_contractingWith_iterate_next

∀ {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 : E} {a r L K : NNReal} (hf : IsPicardLindelof f t₀ x₀ a r L K),
  ∃ n C, ∀ (x : E) (hx : x ∈ Metric.closedBall x₀ ↑r), ContractingWith C (ODE.FunSpace.next hf hx)^[n]

Some n-th iterate of next is a contracting map, and its associated Lipschitz constant is independent of the initial point.

Defined in

Mathlib.Analysis.ODE.PicardLindelof

Cited by

2 results in Mathlib

Foundations

Depth 275 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace