The analysis of dynamical systems, whether in physics or mathematics, frequently begins with an ordinary differential equation (ODE). A primary question is whether a solution to such an equation exists and, if it does, whether it is unique given a specified initial state.
The Picard–Lindelöf theorem, also known as the Cauchy–Lipschitz theorem, provides a definitive answer to this question under certain reasonable conditions on the function defining the differential equation.
We consider an open set D \subset \mathbb{R} \times \mathbb{R}^n and a continuous function \mathbf f: D \to \mathbb{R}^n. We are interested in the initial value problem (IVP) for a system of first-order ordinary differential equations:
\frac{\mathrm d \mathbf y}{\mathrm d t} = \mathbf f(t, \mathbf y(t))
with the initial condition:
\mathbf y(t_0) = \mathbf y_0
where (t_0, \mathbf y_0) is a point in D.
The function \mathbf f is said to be Lipschitz continuous in its second argument on D if there exists a constant L > 0, the Lipschitz constant, such that for all (t, \mathbf v_1) and (t, \mathbf v_2) in D, the following inequality holds:
\|\mathbf f(t, \mathbf v_1) - \mathbf f(t, \mathbf v_2)\| \le L \|\mathbf v_1 - \mathbf v_2\|
Here, \|\cdot\| denotes a norm on \mathbb{R}^n.
Theorem: let \mathbf f: D \to \mathbb{R}^n be a continuous function on an open set D \subset \mathbb{R} \times \mathbb{R}^n, and let \mathbf f be Lipschitz continuous in its second argument on D. For any point (t_0, \mathbf y_0) \in D, there exists an interval I = [t_0 - \alpha, t_0 + \alpha] for some \alpha > 0 such that the initial value problem admits a unique solution \mathbf y: I \to \mathbb{R}^n.
Proof: the proof hinges on reformulating the differential equation as an integral equation and then employing the Banach Fixed-Point Theorem on a suitable complete metric space. The strategy is to construct a sequence of functions, known as Picard iterations, that converges to the unique solution.
Integral Equation Reformulation
Operator and Metric Space Definition
Banach Fixed-Point Theorem Utilization
A function \mathbf y(t) is a solution to the initial value problem if it satisfies the differential equation and the initial condition. By integrating the differential equation from t_0 to t, we obtain:
\int_{t_0}^t \frac{\mathrm d \mathbf y}{\mathrm d s} \, \mathrm d s = \int_{t_0}^t \mathbf f(s, \mathbf y(s)) \, \mathrm d s
Applying the fundamental theorem of calculus to the left side and using the initial condition \mathbf y(t_0) = \mathbf y_0, we arrive at:
\mathbf y(t) - \mathbf y(t_0) = \int_{t_0}^t \mathbf f(s, \mathbf y(s)) \, \mathrm d s
This leads to the following integral equation:
\mathbf y(t) = \mathbf y_0 + \int_{t_0}^t \mathbf f(s, \mathbf y(s)) \, \mathrm d s
A continuous function \mathbf y(t) is a solution to this integral equation if and only if it is a differentiable function that solves the original IVP.
The integral formulation has the advantage of directly incorporating the initial condition and transforming the problem of finding a solution into a fixed-point problem.
Since (t_0, \mathbf y_0) \in D and D is open, there exists a closed cylindrical domain R centered at (t_0, \mathbf y_0) fully contained in D.
Let us define this cylinder as:
R = [t_0 - a, t_0 + a] \times \overline{B(\mathbf y_0, b)} = \{ (t, \mathbf v) \mid |t - t_0| \le a, \|\mathbf v - \mathbf y_0\| \le b \} \subset D
for some a, b > 0. Here, \overline{B(\mathbf y_0, b)} is the closed ball of radius b centered at \mathbf y_0.
Since \mathbf f is continuous on the compact set R, it is bounded on R. Let M be a bound for \mathbf f:
M = \sup_{(t, \mathbf v) \in R} \|\mathbf f(t, \mathbf v)\|
We will seek a solution within a specific space of functions.
Let I_\alpha = [t_0 - \alpha, t_0 + \alpha] for some \alpha > 0 to be determined. Let C(I_\alpha, \overline{B(\mathbf y_0, b)}) be the set of all continuous functions \mathbf g: I_\alpha \to \overline{B(\mathbf y_0, b)}.
We equip this space with the supremum norm:
\|\mathbf g\|_\infty = \sup_{t \in I_\alpha} \|\mathbf g(t)\|
The metric on this space is d(\mathbf g_1, \mathbf g_2) = \|\mathbf g_1 - \mathbf g_2\|_\infty.
The space (C(I_\alpha, \overline{B(\mathbf y_0, b)}), d) is a complete metric space.
Now, we define the Picard operator T on the space of functions C(I_\alpha, \overline{B(\mathbf y_0, b)}).
For a function \mathbf g in this space, T(\mathbf g) is a new function whose value at t is given by:
(T\mathbf g)(t) = \mathbf y_0 + \int_{t_0}^t \mathbf f(s, \mathbf g(s)) \, \mathrm d s
A solution to the integral equation is a function \mathbf y such that T\mathbf y = \mathbf y.
This is a fixed point of the operator T.
We must show that, for a suitable choice of \alpha, the operator T satisfies two conditions:
First, let’s ensure that if \mathbf g \in C(I_\alpha, \overline{B(\mathbf y_0, b)}), then T\mathbf g is also in this space.
For any t \in I_\alpha, we examine the norm of (T\mathbf g)(t) - \mathbf y_0:
\|(T\mathbf g)(t) - \mathbf y_0\| = \left\| \int_{t_0}^t \mathbf f(s, \mathbf g(s)) \, \mathrm d s \right\| \le \left| \int_{t_0}^t \|\mathbf f(s, \mathbf g(s))\| \, \mathrm d s \right|
Since \mathbf g(s) remains in the ball \overline{B(\mathbf y_0, b)}, the pair (s, \mathbf g(s)) is in R, then, \|\mathbf f(s, \mathbf g(s))\| \le M:
\|(T\mathbf g)(t) - \mathbf y_0\| \le M |t - t_0| \le M\alpha
To ensure that T\mathbf g maps into the ball \overline{B(\mathbf y_0, b)}, we require its values to satisfy \|(T\mathbf g)(t) - \mathbf y_0\| \le b.
This condition is satisfied if we impose the constraint M\alpha \le b. Therefore, we choose \alpha \le \frac{b}{M}.
Now, we show that T is a contraction.
Let \mathbf g_1 and \mathbf g_2 be two functions in our space. We consider the distance between their images under T:
\begin{aligned} \|(T\mathbf g_1)(t) - (T\mathbf g_2)(t)\| &= \left\| \int_{t_0}^t (\mathbf f(s, \mathbf g_1(s)) - \mathbf f(s, \mathbf g_2(s))) \, \mathrm d s \right\| \\ &\le \left| \int_{t_0}^t \|\mathbf f(s, \mathbf g_1(s)) - \mathbf f(s, \mathbf g_2(s))\| \, \mathrm d s \right| \end{aligned}
Using the Lipschitz condition on \mathbf f:
\begin{aligned} \|(T\mathbf g_1)(t) - (T\mathbf g_2)(t)\| &\le \left| \int_{t_0}^t L \|\mathbf g_1(s) - \mathbf g_2(s)\| \, \mathrm d s \right| \\ &\le \left| \int_{t_0}^t L \sup_{u \in I_\alpha} \|\mathbf g_1(u) - \mathbf g_2(u)\| \, \mathrm d s \right| \\ &\le L \|\mathbf g_1 - \mathbf g_2\|_\infty |t - t_0| \\ &\le L\alpha \|\mathbf g_1 - \mathbf g_2\|_\infty \end{aligned}
Taking the supremum over all t \in I_\alpha on the left side, we get:
\|T\mathbf g_1 - T\mathbf g_2\|_\infty \le L\alpha \|\mathbf g_1 - \mathbf g_2\|_\infty
For T to be a contraction, we require the constant L\alpha to be strictly less than 1.
This gives us a second constraint on \alpha: L\alpha < 1, or \alpha < \frac{1}{L}.
By choosing \alpha such that \alpha < \min\left(a, \frac{b}{M}, \frac{1}{L}\right), we ensure that the operator T maps the complete metric space C(I_\alpha, \overline{B(\mathbf y_0, b)}) to itself and is a contraction mapping.
The Banach fixed-point theorem states that if (X, d) is a non-empty complete metric space and T: X \to X is a contraction mapping, then T has a unique fixed point in X.
We have established that our space C(I_\alpha, \overline{B(\mathbf y_0, b)}) is complete and that the operator T is a contraction on this space for a suitably small \alpha > 0.
The Banach Fixed-Point Theorem therefore guarantees the existence of a unique function \mathbf y \in C(I_\alpha, \overline{B(\mathbf y_0, b)}) such that T\mathbf y = \mathbf y.
This fixed point is the unique solution to the integral equation on the interval I_\alpha.
The proof of the Banach Fixed-Point Theorem tells us that the fixed point can be found by starting with any function \mathbf y_0(t) \in C(I_\alpha, \overline{B(\mathbf y_0, b)}) and iterating the operator T.
The canonical choice is the constant function \mathbf y_0(t) = \mathbf y_0. The sequence of Picard iterations is defined as:
\begin{aligned} & \mathbf y_0(t) = \mathbf y_0 \\ & \mathbf y_{k+1}(t) = (T\mathbf y_k)(t) = \mathbf y_0 + \int_{t_0}^t \mathbf f(s, \mathbf y_k(s)) \, \mathrm d s \end{aligned}
The theorem guarantees that this sequence:
\{\mathbf y_k\}_{k=0}^\infty
converges uniformly on I_\alpha to the unique solution \mathbf y(t).
This completes the proof of existence and uniqueness on the local interval I_\alpha.
The uniqueness is a direct consequence of the uniqueness of the fixed point: if there were two distinct solutions, \mathbf y_1 and \mathbf y_2, on I_\alpha, they would both be fixed points of T, which contradicts the uniqueness established by the Banach fixed-point theorem, and therefore the solution is unique on this interval.