Ordinary Differential Equations 17 | Picard–Lindelöf Theorem (General and Special Version) [dark]

The Picard–Lindelöf theorem’s guarantee of existence and uniqueness for ordinary differential equations extends beyond time-independent (autonomous) systems to time-dependent (nonautonomous) ones—without changing the core proof strategy. For a nonautonomous initial value problem driven by a vector field W(t, x), the key requirement becomes a Lipschitz condition in the state variable x, applied locally in time–space neighborhoods around the initial point (t0, x0). With that adjustment, the same fixed-point machinery yields a unique solution on some time interval, and solutions can be extended to maximal ones.

In the nonautonomous setting, the system is defined on a domain in R^{N+1} (time plus state space), with W mapping into R^N. The discussion emphasizes that the Lipschitz property need not hold globally; it suffices to hold on any compact region inside the domain. Concretely, for points (t, x) and (t, y) that share the same time coordinate, the inequality ||W(t, x) − W(t, y)|| ≤ L_K ||x − y|| must hold throughout a chosen neighborhood (often represented as a generalized rectangle around (t0, x0)). The constant L_K may depend on the neighborhood, but not on the particular x and y within it. Under this locally Lipschitz-in-x condition, the initial value problem has a unique solution defined on an interval around t0.

The proof approach mirrors the earlier autonomous case: it uses the Banach fixed point theorem on a carefully constructed space of candidate functions. The fixed-point map f takes a function α and returns the integral formulation of the differential equation, replacing the autonomous vector field V(x) with the nonautonomous field W(t, x). Specifically, the map builds α(t) from x0 plus the integral from t0 to t of W(s, α(s)) ds. The only substantive change is swapping V for W and invoking the Lipschitz condition tailored to the second argument of W.

A second, more application-friendly “special version” strengthens the Lipschitz requirement to obtain global solutions. Here, W is continuous on the full domain R × R^N, and a global Lipschitz bound holds for all x, y in R^N, uniformly over time slices restricted to any finite interval [−T, T]. The constant may depend on T, but the bound must work for every x and y across the entire state space. This stronger control prevents blow-up over time and yields a unique solution defined for all t ∈ R.

To make the fixed-point argument work on larger time intervals, the proof modifies the metric on the function space. Instead of the plain supremum norm, it introduces an exponentially weighted distance using the Lipschitz constant L_T, scaling points farther from the origin in time so the fixed-point map becomes a contraction. With this weighted metric, the contraction estimate goes through, Banach’s theorem produces a unique fixed point, and because the construction works for arbitrarily large T, the solution extends to the whole real line. The result is a practical criterion for global existence and uniqueness, setting up the next step toward linear differential equations.