In
mathematics – specifically, in
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s – the Picard–Lindelöf theorem gives a set of conditions under which an
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ...
has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and
uniqueness
Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison, or is remarkable, or unusual. When used in relation to humans, it is often in relation to a person's personality, or some specific characterist ...
theorem.
The theorem is named after
Émile Picard,
Ernst Lindelöf,
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as numbe ...
and
Augustin-Louis Cauchy.
Theorem
Let
be a closed rectangle with
. Let
be a function that is
continuous in
and
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
in
. Then, there exists some such that the initial value problem
has a unique solution
on the interval