Theorem regarding the existence of a solution to a differential equation.
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In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.
History
Peano first published the theorem in 1886 with an incorrect proof.[1] In 1890 he published a new correct proof using successive approximations.[2]
Theorem
Let
be an open subset of
with
a continuous function and
a continuous, explicit first-order differential equation defined on D, then every initial value problem
for f with
has a local solution
where
is a neighbourhood of
in
, such that
for all
.[3]
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 there exists a sequence of Lipschitz 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
![{\displaystyle {\begin{aligned}|y_{k,n+1}(x)|&\leq \textstyle \left|\int _{0}^{x}|f_{k}(x',y_{k,n}(x'))|\,\mathrm {d} x'\right|\\&\leq \textstyle |x|\sup _{R}|f_{k}|\\&\leq x_{2}\cdot 2C\leq y_{1},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f96079e9f8fe7620c046eb08280a1bc86a3c0e90)
is within the domain of
.
We have
![{\displaystyle {\begin{aligned}|y_{k,n+1}(x)-y_{k,n}(x)|&\leq \textstyle \left|\int _{0}^{x}|f_{k}(x',y_{k,n}(x'))-f_{k}(x',y_{k,n-1}(x'))|\,\mathrm {d} x'\right|\\&\leq \textstyle L_{k}\left|\int _{0}^{x}|y_{k,n}(x')-y_{k,n-1}(x')|\,\mathrm {d} x'\right|,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b67486a7319b455b505e539d471bd7aa6b7380)
where
is the Lipschitz constant of
. Thus for maximal difference
, we have a bound
, and
![{\displaystyle {\begin{aligned}M_{k,0}(x)&\leq \textstyle \left|\int _{0}^{x}|f_{k}(x',0)|\,\mathrm {d} x'\right|\\&\leq |x|\textstyle \sup _{R}|f_{k}|\leq 2C|x|.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/633e8a2d3435a7efab3690070a33918b9184268c)
By induction, this implies the bound
which tends to zero as
for all
.
The functions
are equicontinuous as for
we have
![{\displaystyle {\begin{aligned}|y_{k,n+1}(x')-y_{k,n+1}(x)|&\leq \textstyle \int _{x}^{x'}|f_{k}(x'',y_{k,n}(x''))|\,\mathrm {d} x''\\&\textstyle \leq |x'-x|\sup _{R}|f_{k}|\leq 2C|x'-x|,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5e7dfc021495d0518ee4c9c8af141983c0f454)
so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each
there is a subsequence
converging uniformly to a continuous function
. Taking limit
in
![{\displaystyle {\begin{aligned}\textstyle \left|y_{k,\varphi _{k}(n)}(x)-\int _{0}^{x}f_{k}(x',y_{k,\varphi _{k}(n)}(x'))\,\mathrm {d} x'\right|&=|y_{k,\varphi _{k}(n)}(x)-y_{k,\varphi _{k}(n)+1}(x)|\\&\leq M_{k,\varphi _{k}(n)}(x_{2})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3bda10b86931a2f032ead7bae19dca0e8d76c1d)
we conclude that
. The functions
are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence
converging uniformly to a continuous function
. Taking limit
in
we conclude that
, using the fact that
are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus,
in
.
Related theorems
The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ordinary differential equation
on the domain ![{\displaystyle \left[0,1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7151a24c48a590784c4900dd9c909fe6f3c42285)
According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at
, either
or
. The transition between
and
can happen at any
.
The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.
Notes
- ^ Peano, G. (1886). "Sull'integrabilità delle equazioni differenziali del primo ordine". Atti Accad. Sci. Torino. 21: 437–445.
- ^ Peano, G. (1890). "Demonstration de l'intégrabilité des équations différentielles ordinaires". Mathematische Annalen. 37 (2): 182–228. doi:10.1007/BF01200235. S2CID 120698124.
- ^ (Coddington & Levinson 1955, p. 6)
References
- Osgood, W. F. (1898). "Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung". Monatshefte für Mathematik. 9: 331–345. doi:10.1007/BF01707876. S2CID 122312261.
- Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill.
- Murray, Francis J.; Miller, Kenneth S. (1976) [1954]. Existence Theorems for Ordinary Differential Equations (Reprint ed.). New York: Krieger.
- Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.