My title - Departamento de Matemática da Universidade do Minho

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Example (simple ODEs). Consider the simple ODE ẋ = v(t) with initial condition x(t 0 ) = x 0 .
Picard’s recipe, starting from the initial guess φ(t) = x 0 gives, already at the first step,
which is the solution we know.
∫ t
(Pφ) (t) = x 0 + v(s)ds
t 0
Example (the exponential). Suppose you want to solve ẋ = x with initial condition x(0) = 1.
You start with the guess φ(t) = 1, and then compute
(Pφ) (t) = 1 + t
(
P 2 φ ) (t) = 1 + t + 1 2 t2 ... (P n φ) (t) = 1 + t + 1 2 t2 + ... + 1 n! tn
Hence the sequence converges (uniformly on bounded intervals) to the Taylor series of the exponential
function
(P n φ) (t) → 1 + t + 1 2 t2 + ... + 1 n! tn + ... = e t ,
which is the solution we already knew.
Details of the proof of the Picard’s theorem. Choose a sufficiently small rectangular
neighborhood
I × B = [t 0 − ε, t 0 + ε] × B δ (x 0 )
around (t 0 , x 0 ), where B = B δ (x 0 ) denotes the closed ball with center x 0 and radius δ in X. There
follows from continuity of v that there exists K such that
|v(t, x)| ≤ K
for any (t, x) ∈ I × B. There follows from the local Lipschitz condition for v that there exists M
such that
|v(t, x) − v(t, y)| ≤ M|x − y|
for any t ∈ I and any x, y ∈ B. Now restrict, if needed, the (radius of the) interval I in such a
way to get both the inequalities Kε ≤ δ and Mε < 1. Let C be the space of continuous functions
t ↦→ φ(t) sending I into B. Equipped with the sup norm
‖φ − ϕ‖ = sup |φ(t) − ϕ(t)|
t∈I
this is a complete space. One verifies that the Picard’s map sends C into C, since
| (Pφ) (t) − x 0 | ≤
∫ t
Finally, given two functions φ, ϕ ∈ C, one sees that
| (Pφ) (t) − (Pϕ) (t)| ≤
∫ t
t 0
|v (s, φ(s)) |ds ≤ Kε ≤ δ.
t 0
|v (s, φ(s)) − v (s, ϕ(s)) |ds ≤ Mε sup
t∈I
|φ(t) − ϕ(t)|
hence ‖Pφ − Pϕ‖ < Mε‖φ − ϕ‖. Since Mε < 1, this proves that the Picard’s map is a contraction
and the fixed point theorem allows to conclude.
We may not be able to solve them! Last but not least, we must keep in mind that we are
not able to solve all equations. Actually, although we may prove the existence and the uniqueness
for large classes of equations, we are simply not able to explicitly integrate the really interesting
differential equations...
Ultimately we must recur to numerical methods to find approximate solutions and to qualitative
analysis