In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at t ...
,
Ernst Lindelöf
Ernst is both a surname and a given name, the German, Dutch, and Scandinavian form of Ernest. Notable people with the name include:
Surname
* Adolf Ernst (1832–1899) German botanist known by the author abbreviation "Ernst"
* Anton Ernst (1975- ...
,
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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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