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 the study of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after
Giuseppe Peano and
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
, is a fundamental
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
which guarantees the
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia' ...
of solutions to certain
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 o ...
s.
History
Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.
Theorem
Let
be an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
subset of
with
a continuous function and
a
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 g ...
,
explicit
Explicit refers to something that is specific, clear, or detailed. It can also mean:
* Explicit knowledge
Explicit knowledge (also expressive knowledge) is knowledge that can be readily articulated, codified, stored and accessed. It can be expres ...
first-order differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
defined on ''D'', then every initial value problem
for ''f'' with
has a local solution
where
is a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of
in
,
such that
for all
.
The solution need not be unique: one and the same initial value
may give rise to many different solutions
.
Proof
By replacing
with
,
with
, we may assume
. As
is open there is a rectangle
.
Because
is compact and
is continuous, we have
and by the
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the ...
a sequence of
Lipschitz Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
functions
converging
uniformly to
in
. Without loss of generality, we assume
for all
.
We define
Picard iterations as follows, where
.
, and
. They are well-defined by induction: as
:
is within the domain of
.
We have
:
where
is the Lipschitz constant of
. Thus for maximal difference
, we have a bound
, and
:
By induction, this implies the bound
which tends to zero as
for all
.
The functions
are
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
as for