Theorems · Theorem · ordinary differential equations

ODE.FunSpace.exists_forall_closedBall_funSpace_dist_le_mul

∀ {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} [CompleteSpace E] (hf : IsPicardLindelof f t₀ x₀ a r L K),
  ∃ L',
    ∀ (x y : E) (hx : x ∈ Metric.closedBall x₀ ↑r) (hy : y ∈ Metric.closedBall x₀ ↑r) (α β : ODE.FunSpace t₀ x₀ r L),
      Function.IsFixedPt (ODE.FunSpace.next hf hx) α →
        Function.IsFixedPt (ODE.FunSpace.next hf hy) β → dist α β ≤ ↑L' * dist x y

The pointwise distance between any two integral curves α and β over their domains is bounded by a constant L' times the distance between their respective initial points. This is the result of taking the limit of dist_iterate_iterate_next_le_of_lipschitzWith as n → ∞. This implies that the local solution of a vector field is Lipschitz continuous in the initial condition.

Defined in

Mathlib.Analysis.ODE.PicardLindelof

Cited by

1 results in Mathlib

Foundations

Depth 277 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace