Ordinary Differential Equations 12 | Picard–Lindelöf Theorem

Picard–Lindelöf’s existence result for ordinary differential equations hinges on turning an initial value problem into a fixed-point problem on a carefully chosen space of functions. For a system written as ẋ = V(t, x) with x(0) = x0, the key requirement is that V is locally Lipschitz in x. Under that condition, one can guarantee not only uniqueness (handled earlier), but also the existence of a solution—by applying the Banach fixed point theorem.

The construction starts with the integral form of the initial value problem. Define a map F on candidate functions α by F(α)(t) = x0 + ∫[0 to t] V(s, α(s)) ds. A fixed point α = F(α) corresponds exactly to a function satisfying the original initial value problem, because differentiating the integral equation recovers ẋ = V(t, x) and the initial condition is built in through the x0 term.

To use Banach’s theorem, the argument builds a complete metric space X of functions. Since solutions may only exist on a short time interval, the domain is restricted to t ∈ [−ε, ε]. The set X consists of continuous functions α : [−ε, ε] → R^N that satisfy the initial condition α(0) = x0 and take values in the domain where V is defined. The metric D on X is the supremum norm: for α, β ∈ X, D(α, β) = sup_{t ∈ [−ε, ε]} ||α(t) − β(t)||, using the standard norm in R^N. With boundedness ensured by choosing ε small enough (so functions don’t “blow up” on the interval), this function space becomes complete, meaning every Cauchy sequence converges within X.

The next step is proving F is a contraction on X. Take two candidate functions α and β. When comparing F(α)(t) and F(β)(t), the x0 terms cancel, leaving only the integral of V(s, α(s)) − V(s, β(s)). Using the triangle inequality to move the norm inside the integral, the integral is bounded by the length of the time interval times a supremum bound. Since t lies in [−ε, ε], the interval length contributes a factor at most ε. Then the local Lipschitz property of V supplies a constant L such that ||V(s, α(s)) − V(s, β(s))|| ≤ L ||α(s) − β(s)||. Combining these estimates yields D(F(α), F(β)) ≤ ε L · D(α, β). By choosing ε small enough so that εL < 1, F becomes a contraction.

Banach’s fixed point theorem then guarantees a unique fixed point α in X. That fixed point is the local solution to the initial value problem on [−ε, ε]. The final theorem statement: if V is locally Lipschitz and x0 lies in the domain U ⊂ R^N, then there exists ε > 0 and a unique solution α(t) to ẋ = V(t, x), x(0) = x0, valid at least for t in a neighborhood of 0. The method is notable because it produces existence by pure functional-analytic machinery, with the Lipschitz condition supplying the contraction needed for Banach’s theorem.