nLab ordinary differential equation

Idea

An ordinary differential equation is a differential equation involving derivatives of a function with respect to one argument only, i.e. the function is on a manifold only of dimension $d = 1$. This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter. (Note that a higher-order differential equation can be turned into a system of first-order equations.)

Existence and uniqueness (Picard–Lindelöf theorem)

A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let $X$ be a Banach space; given $(t_0, y_0) \in \mathbb{R} \times X$ and $a, r \gt 0$, put $Q \coloneqq [t_0 - a, t_0 + a] \times B_r(y_0)$m where $B_r(y_0)$ is the closed ball in $X$ of radius $r$ about $y_0$.

Theorem

(Picard–Lindelöf) Suppose $f: Q \to X$ is a function satisfying the following conditions:

• (Continuity in $t$): Given any $x \in B_r(y_0)$, the function $f(-,x)$ (that is $t \mapsto f(t, x)$) is continuous from $[t_0 - a, t_0 + a]$ to $X$.

• (Lipschitz continuity in $y$): There is a Lipschitz constant $L$ such that

  $${\|f(t, x) - f(t, x')\|} \leq L{\|x - x'\|}$$

  for all $(t, x) \in Q$;

• (Boundedness): There is a constant $K$ such that $\sup_{(t, x) \in Q} \|f(t, x)\| \leq K$.

Then for any $c \leq \min(a, r/K)$, there exists exactly one? solution $y: [t_0 - c, t_0 + c] \to X$ to the initial value problem

$$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0 .$$

Proof sketch

We will define an infinite sequence of approximate solutions to the problem, prove that its limit exists, prove that this limit is an exact solution, and prove that this solution is unique.

1. The infinite sequence is given by Picard iteration: Starting with the given constant $y_0$, recursively define

   $$y_{n+1}(t) \coloneqq y_0 + \int_{t=t_0} f(t, y_n(t)) \,\mathrm{d}t ;$$

   that is, $y_{n+1}$ is that indefinite integral of $f(-,y_n)$ that takes the correct initial value. (To define $y_1$, use abuse of notation? to interpret $y_0(t)$ as $y_0$; that is, think of $y_0$ as a constant function.) To prove that this integral exists, use the continuity conditions and an inductive proof? that each $y_n$ is continuous to show that we are integrating a continuous function.

2. Thinking of Picard iteration as an operator between Banach spaces of continuous functions, use Lipschitz continuity and boundedness to show that the Banach fixed point theorem applies, so that the sequence $(y_1, y_2, \ldots)$ uniformly converges to a limit $y$.

3. Since uniform convergence of continuous functions behaves well with integration, the limiting instance of Picard iteration holds:

   $$y(t) = y_0 + \int_{t=t_0} f(t, y(t)) \,\mathrm{d}t .$$

   By differentiating with respect to $t$, and by evaluating at $t_0$, we confirm that $y$ is a solution of the initial value problem.

4. Given any putative solution $z$, apply Grönwall's inequality? to $z - y$ to prove that $z = y$.

See also

• inverse function theorem
• real quadratic function
• continuous square root
• Runge-Kutta method
• Chen iterated integral

References

• The English Wikipedia has a more detailed proof. (They assume that $X$ is $\mathbb{R}$, but this is not essential.)