Ordinary Differential Equations 12 | Picard–Lindelöf Theorem [dark version]

The Picard–Lindelöf theorem guarantees not just that an initial value problem for an ordinary differential equation has a solution, but that—under a local Lipschitz condition—that solution is unique. The key move is turning the differential equation into a fixed-point problem: candidate solutions are functions, and the “integral form” of the ODE becomes a mapping on a carefully chosen space of functions. Once that mapping is shown to be a contraction, Banach’s fixed point theorem delivers existence (and, together with earlier uniqueness work, the full Picard–Lindelöf result).

Start with the standard initial value problem written as ẋ = V(x) with x(0) = x0, where V is locally Lipschitz. Uniqueness had already been established earlier, so the focus here is existence. The method rewrites the ODE in integral form: any solution α should satisfy

F(α)(t) = x0 + ∫[0 to t] V(α(s)) ds.

A fixed point of F—meaning F(α) = α—corresponds exactly to a function α that solves the initial value problem.

To apply Banach’s fixed point theorem, the construction needs two ingredients: a complete metric space and a contraction mapping on it. The metric space is built from continuous functions α: R → R^N, but restricted to a small time interval [−ε, ε] because solutions may only exist locally. The functions in the space are continuous on that interval and satisfy the initial condition α(0) = x0. The distance between two functions α and β is defined using the supremum norm:

D(α, β) = sup_{t in [−ε, ε]} ||α(t) − β(t)||,

where ||·|| is the standard norm on R^N. With boundedness ensured by choosing ε small enough (so functions don’t “blow up”), this function space becomes complete.

The contraction estimate comes from comparing F(α) and F(β). The initial value x0 cancels when taking differences, leaving only an integral of V(α(s)) − V(β(s)). Using the triangle inequality to move the norm inside the integral, the integral is bounded by the “area of a rectangle”: the interval length |t| times the supremum of the integrand. Since |t| ≤ ε on the chosen interval, the bound simplifies to something proportional to ε. Finally, the local Lipschitz property of V supplies a constant L such that

||V(α(s)) − V(β(s))|| ≤ L ||α(s) − β(s)||.

Putting these steps together yields

D(F(α), F(β)) ≤ ε L · D(α, β).

Choosing ε small enough makes εL < 1, turning F into a contraction. Banach’s fixed point theorem then guarantees a fixed point in the space, which is the desired local solution to the initial value problem.

The theorem’s final statement: if V is locally Lipschitz on an open set U ⊂ R^N and x0 ∈ U, then there exists ε > 0 such that the initial value problem has a unique solution α on [−ε, ε]. The existence proof hinges on the contraction mapping built from the integral form of the ODE and the completeness of the function space under the supremum metric.