Theorems · Theorem · ordinary differential equations

ODE.FunSpace.exists_isFixedPt_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 x : E} {a r L K : NNReal} [CompleteSpace E] (hf : IsPicardLindelof f t₀ x₀ a r L K)
  (hx : x ∈ Metric.closedBall x₀ ↑r), ∃ α, Function.IsFixedPt (ODE.FunSpace.next hf hx) α

The map next has a fixed point in the space of curves. This will be used to construct a solution α : ℝ → E to the ODE.

Defined in

Mathlib.Analysis.ODE.PicardLindelof

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace