Peano existence theorem

Peano first published the theorem in 1886 with an incorrect proof.{{cite journal |first=G. |last=Peano |title=Sull'integrabilità delle equazioni differenziali del primo ordine |journal=Atti Accad. Sci. Torino |volume=21 |year=1886 |pages=437–445 |url=https://archive.org/stream/attidellaraccade21real#page/436/mode/2up/search/peano }} In 1890 he published a new correct proof using successive approximations.{{cite journal |first=G. |last=Peano |title=Demonstration de l'intégrabilité des équations différentielles ordinaires |journal=Mathematische Annalen |volume=37 |issue=2 |year=1890 |pages=182–228 |doi=10.1007/BF01200235 |s2cid=120698124 }}

Let $D$ be an open subset of $\backslash mathbb\{R\}\backslash times\backslash mathbb\{R\}$ with $f\backslash colon\; D\; \backslash to\; \backslash mathbb\{R\}$

a continuous function and $y\text{'}(t)\; =\; f\backslash left(t,\; y(t)\backslash right)$

a continuous, explicit first-order differential equation defined on D, then every initial value problem $y\backslash left(t\_0\backslash right)\; =\; y\_0$

for f with $(t\_0,\; y\_0)\; \backslash in\; D$

has a local solution $z\backslash colon\; I\; \backslash to\; \backslash mathbb\{R\}$

where $I$ is a neighbourhood of $t\_0$ in $\backslash mathbb\{R\}$,

such that $z\text{'}(t)\; =\; f\backslash left(t,\; z(t)\backslash right)$ for all $t\; \backslash in\; I$.{{harv|Coddington|Levinson|1955|p=6}}

The solution need not be unique: one and the same initial value $(t\_0,\; y\_0)$ may give rise to many different solutions $z$.

By replacing $y$ with $y-y\_0$, $t$ with $t-t\_0$, we may assume $t\_0=y\_0=0$. As $D$ is open there is a rectangle $R=[-t\_1,t\_1]\backslash times[-y\_1,y\_1]\backslash subset\; D$.

Because $R$ is compact and $f$ is continuous, we have and by the Stone–Weierstrass theorem there exists a sequence of Lipschitz functions $f\_k:R\backslash to\backslash mathbb\{R\}$ converging uniformly to $f$ in $R$. Without loss of generality, we assume $\backslash textstyle\backslash sup\_R|f\_k|\backslash le2C$ for all $k$.

We define Picard iterations $y\_\{k,n\}:I=[-t\_2,t\_2]\backslash to\backslash mathbb\{R\}$ as follows, where $t\_2=\backslash min\backslash \{t\_1,y\_1/(2C)\backslash \}$. $y\_\{k,0\}(t)\backslash equiv0$, and $\backslash textstyle\; y\_\{k,n+1\}(t)=\backslash int\_0^t\; f\_k(t\text{'},y\_\{k,n\}(t\text{'}))\backslash ,\backslash mathrm\{d\}t\text{'}$. They are well-defined by induction: as

:

$(t\text{'},y\_\{k,n+1\}(t\text{'}))$ is within the domain of $f\_k$.

We have

:

where $L\_k$ is the Lipschitz constant of $f\_k$. Thus for maximal difference $\backslash textstyle\; M\_\{k,n\}(t)=\backslash sup\_\{t\text{'}\backslash in[0,t]\}|y\_\{k,n+1\}(t\text{'})-y\_\{k,n\}(t\text{'})|$, we have a bound $\backslash textstyle\; M\_\{k,n\}(t)\backslash le\; L\_k\; \backslash int\_0^t\; M\_\{k,n-1\}(t\text{'})\backslash ,\backslash mathrm\{d\}t\text{'}$, and

:

By induction, this implies the bound $M\_\{k,n\}(t)\backslash le\; 2CL\_k^n|t|^\{n+1\}/(n+1)!$ which tends to zero as $n\backslash to\backslash infty$ for all $t\backslash in\; I$.

The functions $y\_\{k,n\}$ are equicontinuous as for

:,y_{k,n}(t))|\,\mathrm{d}t''\\&\textstyle\le|t'-t|\sup_R|f_k|\le 2C|t'-t|,\end{aligned}

so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each $k$ there is a subsequence $(y\_\{k,\backslash varphi\_k(n)\})\_\{n\backslash in\backslash mathbb\{N\}\}$

converging uniformly to a continuous function $y\_k:I\backslash to\backslash mathbb\{R\}$. Taking limit

$n\backslash to\backslash infty$ in

:

we conclude that $\backslash textstyle\; y\_k\; (t)\; =\; \backslash int\_0^t\; f\_k(t\text{'},\; y\_k(t\text{'}))\; \backslash ,\; \backslash mathrm\{d\}t\text{'}$. The functions $y\_k$ are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence $y\_\{\backslash psi(k)\}$ converging uniformly to a continuous function $z:I\backslash to\backslash mathbb\{R\}$. Taking limit $k\backslash to\backslash infty$ in $\backslash textstyle\; y\_\{\backslash psi(k)\}(t)=\backslash int\_0^t\; f\_\{\backslash psi(k)\}(t\text{'},y\_\{\backslash psi(k)\}(t\text{'}))\backslash ,\backslash mathrm\{d\}t\text{'}$ we conclude that $\backslash textstyle\; z(t)=\backslash int\_0^t\; f(t\text{'},z(t\text{'}))\backslash ,\backslash mathrm\{d\}t\text{'}$, using the fact that $f\_\{\backslash psi(k)\}$ are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus, $z\text{'}(t)\; =\; f(t,\; z(t))$ in $I$.

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

:$y\text{'}\; =\; \backslash left\backslash vert\; y\backslash right\backslash vert^\{\backslash frac\{1\}\{2\}\}$ on the domain $\backslash left[0,\; 1\backslash right].$

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 $y(0)=0$, either $y(t)=0$ or $y(t)=t^2/4$. The transition between $y=0$ and $y=(t-C)^2/4$ can happen at any $C$.

The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.

The Peano existence theorem cannot be straightforwardly extended to a general Hilbert space $\backslash mathcal\{H\}$: for an open subset $D$ of $\backslash mathbb\{R\}\backslash times\; \backslash mathcal\{H\}$, the continuity of $f\backslash colon\; D\; \backslash to\; \backslash mathbb\{R\}$ alone is insufficient for guaranteeing the existence of solutions for the associated initial value problem.{{Cite journal |last=Yorke |first=J. A. |author-link=James A. Yorke |year=1970 |title=A continuous differential equation in Hilbert space without existence |url=http://www.math.kobe-u.ac.jp/~fe/xml/mr0264196.xml |journal=Funkcjalaj Ekvacioj |volume=13 |pages=19–21 |mr=0264196}}

{{Reflist}}

• {{cite journal |first=W. F. |last=Osgood |title=Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung |journal=Monatshefte für Mathematik |volume=9 |year=1898 |pages=331–345 |doi=10.1007/BF01707876 |s2cid=122312261 |url=https://zenodo.org/record/2150462 }}
• {{cite book |last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations |url=https://archive.org/details/theoryofordinary00codd |url-access=registration | publisher=McGraw-Hill | location=New York | year=1955 }}
• {{cite book |last1=Murray |first1=Francis J. |last2=Miller |first2=Kenneth S. |title=Existence Theorems for Ordinary Differential Equations |publisher=Krieger |location=New York |edition=Reprint |year=1976 |orig-year=1954 }}
• {{cite book |last=Teschl |first=Gerald |author-link=Gerald Teschl |title=Ordinary Differential Equations and Dynamical Systems |publisher=American Mathematical Society |location=Providence |year=2012 |isbn=978-0-8218-8328-0 |url=https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ }}

Category:Augustin-Louis Cauchy

Category:Theorems in mathematical analysis

Category:Ordinary differential equations