In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem

has exactly one solution if f is Lipschitz continuous in [itex]y[itex], continuous in [itex]t[itex] as long as [itex]y(t)[itex] stays bounded.

A simple proof of existence of the solution is successive approximation: (also called Picard iteration)

Set

[itex]\varphi_0(t)=y_0 \,\![itex]

and

[itex]\varphi_i(t)=y_0+\int_{t_0}^{t}f(s,\varphi_{i-1}(s))\,ds.[itex]

It can then be shown rather easily that the sequence of the [itex]\varphi_i \,\![itex] (called the Picard iterates) is convergent and that the limit is a solution to the problem.

An application of Grönwall's lemma to [itex]|\phi(t)-\psi(t)|[itex], where [itex]\phi[itex] and [itex]\psi[itex] are two solutions, shows that [itex]\phi(t)\equiv\psi(t)[itex], thus proving the uniqueness.

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