Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy.

Consider the initial value problem

Suppose $f$ is uniformly Lipschitz continuous in $y$ (meaning the Lipschitz constant can be taken independent of $t$ ) and continuous in $t$ . Then, for some value $ε > 0$ , there exists a unique solution $y (t)$ to the initial value problem on the interval $[t_0-\varepsilon, t_0+\varepsilon]$ .

Proof sketch

The proof relies on transforming the differential equation, and applying fixed-point theory. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation

A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration.

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Lindelöf's theorem

In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.

Statement of the theorem

Let Ω be a half-strip in the complex plane:

Suppose that ƒ is holomorphic (i.e. analytic) on Ω and that there are constants M, A and B such that

and

Then f is bounded by M on all of Ω:

Proof

Fix a point $\xi=\sigma+i\tau$ inside $\Omega$ . Choose $\lambda>-y_0$ , an integer $N>A$ and $y_1>\tau$ large enough such that $\frac {By_1^A}{(y_1 + \lambda)^N}\le \frac {M}{(y_0+\lambda)^N}$ . Applying maximum modulus principle to the function $g(z)=\frac {f(z)}{(z+i\lambda)^N}$ and the rectangular area $\{z \in \mathbb{C} | x_1 \leq \mathrm{Re} (z) \leq x_2 \text{ and } y_0 \leq \mathrm{Im} (z) \leq y_1 \}$ we obtain $|g(\xi)|\le \frac{M}{(y_0+\lambda)^N}$ , that is, $|f(\xi)|\le M\left(\frac{|\xi + \lambda|}{y_0+\lambda}\right)^N$ . Letting $\lambda \rightarrow +\infty$ yields $|f(\xi)| \le M$ as required.