Picard–Lindelöf theorem
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In mathematics, the Picard–Lindelöf theorem or Picard's existence theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem
- <math>y'(t)=f(t,y(t)),\quad y(t_0)=y_0</math>
has exactly one solution if f is Lipschitz continuous in <math>y</math>, continuous in <math>t</math> as long as <math>y(t)</math> stays bounded.
A simple proof of existence of the solution is successive approximation: (also called Picard iteration)
Set
- <math>\varphi_0(t)=y_0 \,\!</math>
and
- <math>\varphi_i(t)=y_0+\int_{t_0}^{t}f(s,\varphi_{i-1}(s))\,ds.</math>
It can then be shown rather easily that the sequence of the <math>\varphi_i \,\!</math> (called the Picard iterates) is convergent and that the limit is a solution to the problem.
An application of Grönwall's lemma to <math>|\varphi(t)-\psi(t)|</math>, where <math>\varphi</math> and <math>\psi</math> are two solutions, shows that <math>\varphi(t)\equiv\psi(t)</math>, thus proving the uniqueness.
Related humor: Picard's existence theorem is not to be confused with Picard's resistance theorem, which states that resistance is futile.