Peano existence theorem

Theorem

Let ${\displaystyle D}$ be an open subset of ${\displaystyle \mathbb{R}\times \mathbb{R}}$ with ${\displaystyle f:D\to \mathbb{R}}$ a continuous function and ${\displaystyle y^{\prime }(t)=f\left(t,y(t)\right)}$ an explicit first-order differential equation defined on D, then every initial value problem ${\displaystyle y\left(t_0\right)=y_0}$ for f with ${\displaystyle (t_0,y_0)\in D}$ has a local solution ${\displaystyle z:I\to \mathbb{R}}$ where ${\displaystyle I}$ is a neighbourhood of ${\displaystyle t_0}$ in ${\displaystyle \mathbb{R}}$, such that ${\displaystyle z^{\prime }(t)=f\left(t,z(t)\right)}$ for all ${\displaystyle t\in I}$.

The solution need not be unique: one and the same initial value ${\displaystyle (t_0,y_0)}$ may give rise to many different solutions ${\displaystyle z}$.

Proof

By replacing ${\displaystyle y}$ with ${\displaystyle y-y_0}$, ${\displaystyle t}$ with ${\displaystyle t-t_0}$, we may assume ${\displaystyle t_0=y_0=0}$. As ${\displaystyle D}$ is open there is a rectangle ${\displaystyle R=[-t_1,t_1]\times [-y_1,y_1]\subset D}$.

Because ${\displaystyle R}$ is compact and ${\displaystyle f}$ is continuous, we have ${\displaystyle \underset{R}{sup}|f|\le C<\mathrm{\infty }}$ and by the Stone–Weierstrass theorem there exists a sequence of Lipschitz functions ${\displaystyle f_k:R\to \mathbb{R}}$ converging uniformly to ${\displaystyle f}$ in ${\displaystyle R}$. Without loss of generality, we assume ${\displaystyle \underset{R}{sup}|f_k|\le 2C}$ for all ${\displaystyle k}$.

We define Picard iterations ${\displaystyle y_{k,n}:I=[-t_2,t_2]\to \mathbb{R}}$ as follows, where ${\displaystyle t_2=min\{t_1,y_1/(2C)\}}$. ${\displaystyle y_{k,0}(t)\equiv 0}$, and ${\displaystyle y_{k,n+1}(t)={\int }_0^t{f}_k(t^{\prime },y_{k,n}(t^{\prime }))\mathrm{d}{t}^{\prime }}$. They are well-defined by induction: as

${\displaystyle \begin{array}{cc}|y_{k,n+1}(t)|& =\left|{\int }_0^t{f}_k(t^{\prime },y_{k,n}(t^{\prime }))\mathrm{d}{t}^{\prime }\right|\\ & \le {\int }_0^t|f_k(t^{\prime },y_{k,n}(t^{\prime }))|\mathrm{d}{t}^{\prime }\\ & \le |t|\underset{R}{sup}|f_k|\\ & \le t_2\cdot 2C\le y_1,\end{array}}$

${\displaystyle (t^{\prime },y_{k,n+1}(t^{\prime }))}$ is within the domain of ${\displaystyle f_k}$.

We have

${\displaystyle \begin{array}{cc}|y_{k,n+1}(t)-y_{k,n}(t)|& =\left|{\int }_0^t{f}_k(t^{\prime },y_{k,n}(t^{\prime }))-f_k(t^{\prime },y_{k,n-1}(t^{\prime }))\mathrm{d}{t}^{\prime }\right|\\ & \le {\int }_0^t|f_k(t^{\prime },y_{k,n}(t^{\prime }))-f_k(t^{\prime },y_{k,n-1}(t^{\prime }))|\mathrm{d}{t}^{\prime }\\ & \le L_k{\int }_0^t|y_{k,n}(t^{\prime })-y_{k,n-1}(t^{\prime })|\mathrm{d}{t}^{\prime },\end{array}}$

where ${\displaystyle L_k}$ is the Lipschitz constant of ${\displaystyle f_k}$. Thus for maximal difference ${\displaystyle M_{k,n}(t)=\underset{t^{\prime }\in [0,t]}{sup}|y_{k,n+1}(t^{\prime })-y_{k,n}(t^{\prime })|}$, we have a bound ${\displaystyle M_{k,n}(t)\le L_k{\int }_0^t{M}_{k,n-1}(t^{\prime })\mathrm{d}{t}^{\prime }}$, and

${\displaystyle \begin{array}{cc}{M}_{k,0}(t)& \le {\int }_0^t|f_k(t^{\prime },0)|\mathrm{d}{t}^{\prime }\\ & \le |t|\underset{R}{sup}|f_k|\le 2C|t|.\end{array}}$

By induction, this implies the bound ${\displaystyle M_{k,n}(t)\le 2C{L}_k^n|t{|}^{n+1}/(n+1)!}$ which tends to zero as ${\displaystyle n\to \mathrm{\infty }}$ for all ${\displaystyle t\in I}$.

The functions ${\displaystyle y_{k,n}}$ are equicontinuous as for ${\displaystyle -t_2\le t<t^{\prime }\le t_2}$ we have

${\displaystyle \begin{array}{cc}|y_{k,n+1}(t^{\prime })-y_{k,n+1}(t)|& \le {\int }_t^{t^{\prime }}|f_k(t^{″},y_{k,n}(t^{″}))|\mathrm{d}{t}^{″}\\ & \le |t^{\prime }-t|\underset{R}{sup}|f_k|\le 2C|t^{\prime }-t|,\end{array}}$

so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each ${\displaystyle k}$ there is a subsequence ${\displaystyle (y_{k,{\phi }_k(n)}{)}_{n\in \mathbb{N}}}$ converging uniformly to a continuous function ${\displaystyle y_k:I\to \mathbb{R}}$. Taking limit ${\displaystyle n\to \mathrm{\infty }}$ in

${\displaystyle \begin{array}{cc}\left|y_{k,{\phi }_k(n)}(t)-{\int }_0^x{f}_k(t^{\prime },y_{k,{\phi }_k(n)}(t^{\prime }))\mathrm{d}{t}^{\prime }\right|& =|y_{k,{\phi }_k(n)}(t)-y_{k,{\phi }_k(n)+1}(t)|\\ & \le M_{k,{\phi }_k(n)}(t_2)\end{array}}$

we conclude that ${\displaystyle y_k(t)={\int }_0^t{f}_k(t^{\prime },y_k(t^{\prime }))\mathrm{d}{t}^{\prime }}$. The functions ${\displaystyle y_k}$ are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence ${\displaystyle y_{\psi (k)}}$ converging uniformly to a continuous function ${\displaystyle z:I\to \mathbb{R}}$. Taking limit ${\displaystyle k\to \mathrm{\infty }}$ in ${\displaystyle y_{\psi (k)}(t)={\int }_0^t{f}_{\psi (k)}(t^{\prime },y_{\psi (k)}(t^{\prime }))\mathrm{d}{t}^{\prime }}$ we conclude that ${\displaystyle z(t)={\int }_0^{t}f(t^{\prime },z(t^{\prime }))\mathrm{d}{t}^{\prime }}$, using the fact that ${\displaystyle f_{\psi (k)}}$ are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus, ${\displaystyle z^{\prime }(t)=f(t,z(t))}$ in ${\displaystyle I}$.

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

${\displaystyle y^{\prime }={\left|y\right|}^{\frac{1}{2}}}$ on the domain ${\displaystyle \left[0,1\right].}$

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

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 ${\displaystyle \mathcal{H}}$: for an open subset ${\displaystyle D}$ of ${\displaystyle \mathbb{R}\times \mathcal{H}}$, the continuity of ${\displaystyle f:D\to \mathbb{R}}$ alone is insufficient for guaranteeing the existence of solutions for the associated initial value problem.