Peano's theorem

The Peano existence theorem is a proposition from the theory of ordinary differential equations . It gives a simple assumption under which the initial value problem has (at least) one local solution. This theorem was published in 1886 by mathematician Giuseppe Peano with an erroneous proof. In 1890 he provided correct evidence.

Compared to Picard-Lindelöf's theorem of existence and uniqueness, Peano's theorem of existence has the advantage that it has weaker requirements. Instead, he does not make any statements regarding the uniqueness of the solution.

Once you have a (local) solution, in a second step you can deduce from this the existence of a non-continuable solution . In this regard, Peano's theorem is a first step in existential theory of a differential equation.

formulation

Be a continuous function. Its domain is a comprehensive subset of . Here denote the closed sphere around with radius , i.e. H. ${\displaystyle F\colon G\to \mathbb {R} ^{n}}$ ${\displaystyle G}$ ${\displaystyle [a,b]\times {\overline {B}}\left(y_{0},R\right)}$ ${\displaystyle \mathbb {R} \times \mathbb {R} ^{n}}$ ${\displaystyle {\overline {B}}\left(y_{0},R\right)}$ ${\displaystyle y_{0}\in \mathbb {R} ^{n}}$ ${\displaystyle R>0}$

${\displaystyle {\overline {B}}\left(y_{0},R\right):=\{z\in \mathbb {R} ^{n}\mid \|z-y_{0}\|\leq R\}}$.

Then there is at least one local solution for every initial value problem of the differential equation . More precisely, this means that there is a and a continuously differentiable function that fulfills two conditions: ${\displaystyle \ y(a)=y_{0}}$ ${\displaystyle y'(t)=F(t,y(t))}$ ${\displaystyle \alpha >0}$ ${\displaystyle y\colon [a,a+\alpha ]\to \mathbb {R} ^{n}}$

• For everyone the point is in . ${\displaystyle t\in [a,a+\alpha ]}$ ${\displaystyle (t,y(t))}$ ${\displaystyle G}$
• The differential equation is fulfilled for all of them . ${\displaystyle t\in [a,a+\alpha ]}$ ${\displaystyle y'(t)=F(t,y(t))}$

Such a one can be specified exactly: On the closed and bounded set the continuous function has a maximum value, set ${\displaystyle \alpha >0}$ ${\displaystyle [a,b]\times {\overline {B}}\left(y_{0},R\right)}$ ${\displaystyle \|F\|}$

${\displaystyle M:=\max\{\|F(x,y)\|\mid (x,y)\in [a,b]\times {\overline {B}}\left(y_{0},R\right)\}}$.

This number is a bound on the slope of a possible solution. Choose now

${\displaystyle \alpha :=\min \left\{b-a,{\frac {R}{M}}\right\}>0\ .}$

Then there exists (at least) one solution to the initial value problem

${\displaystyle y'=F(x,y)\ ,\ y(a)=y_{0}}$

on the interval with values ​​in . ${\displaystyle [a,a+\alpha ]}$ ${\displaystyle {\overline {B}}\left(y_{0},R\right)}$

Note: Complex differential equations can be viewed analogously by considering the real and imaginary parts of a complex component as independent real components, i.e. i.e., by forgetting the complex multiplication with which to identify. ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {R} ^{2n}}$

For real Banach spaces

${\displaystyle X}$be a real Banach space and be continuous and compact . For each initial value there is one and a solution of the ordinary differential equation ${\displaystyle f\colon [0,T]\times X\to X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle \tau >0}$ ${\displaystyle x(\cdot )\in C^{1}([0,\tau ],X)}$

${\displaystyle x'(\cdot )=f(\cdot ,x(\cdot ))}$

with . ${\displaystyle x(0)=x_{0}}$

Comment: In the case , the compactness of . ${\displaystyle \dim(X)<\infty }$ ${\displaystyle f}$

Proof sketch of the finite-dimensional case

This theorem is proven in two parts. In the first step, Euler's polygon method is used to obtain special approximate solutions to this differential equation for each one , more precisely: A piece-wise, continuously differentiable function is constructed with which ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon }$ ${\displaystyle y_{\varepsilon }\in C([a,a+\alpha ];{\overline {B}}(y_{0},R))}$ ${\displaystyle y_{\varepsilon }(a)=y_{0}}$

${\displaystyle \|y_{\varepsilon }'(x)-F(x,y_{\varepsilon }(x))\|\leq \varepsilon }$

fulfilled in every differentiability point as well as the equality condition

${\displaystyle \|y(t)-y(s)\|\leq M|t-s|}$

for everyone . ${\displaystyle s,t\in [a,a+\alpha ]}$

In the second step, one shows with the help of Arzelà-Ascoli's theorem that there is a uniformly convergent subsequence . From its limit function one then shows that it is the integral equation ${\displaystyle (y_{\varepsilon _{j}})_{j\in \mathbb {N} }}$ ${\displaystyle y}$

${\displaystyle y(x)=y_{0}+\int _{a}^{x}F(s,y(s)){\rm {d}}s}$

Fulfills. From the fundamental theorem of analysis it follows that it is continuously differentiable and the differential equation is sufficient. ${\displaystyle y}$ ${\displaystyle \ y'(x)=F(x,y(x))}$

Proof sketch for real Banach spaces

We consider the corresponding Volterra integral equation for ${\displaystyle t\in [0,\tau ]}$

${\displaystyle x(t)=x_{0}+\int _{0}^{t}f(s,x(s))ds}$.

We define the operator

${\displaystyle T\colon C^{0}([0,\tau ],B_{1}(x_{0}))\to C^{0}([0,\tau ],B_{1}(x_{0})),x(\cdot )\mapsto x_{0}+\int _{0}^{\cdot }f(s,x(s))ds}$.

This operator is continuous with respect to the supremum norm , since it is compact and therefore restricted. Furthermore is . By means of the set of Arzela Ascoli , one can show that relatively compact with respect to the supremum in is. So T is a continuous function that maps a closed , convex subset into a compact subset . Thus T has at least one fixed point according to Schauder's Fixed Point Theorem . Each of these fixed points is the solution of the Volterra integral equation and thus the differential equation. ${\displaystyle {\overline {f([0,1]\times B_{2}(x_{0}))}}\subset X}$ ${\displaystyle \tau :=\min(1,(\sup _{[0,1]\times B_{2}(x_{0})}|f|)^{-1})}$ ${\displaystyle T(C^{0}([0,\tau ],B_{1}\left(x_{0}\right)))}$ ${\displaystyle C^{0}([0,\tau ],X)}$ ${\displaystyle K\subset X}$ ${\displaystyle C\subset K}$

Examples

The Peano existence theorem says nothing about the uniqueness. An example for this:

${\displaystyle y'(t)={\sqrt {|y(t)|}}}$with initial value . Ie an autonomous differential equation. She meets the requirements of Peano. The root function is bounded and continuous. A solution exists, but it is not clear-cut. ${\displaystyle y(0)=0}$ ${\displaystyle f(t,y(t))=f(y(t))={\sqrt {|y(t)|}}}$

${\displaystyle y_{1}(t)=0,y_{1}(0)=0}$and are fulfilled. But this also applies to and ${\displaystyle y_{1}'(t)=0={\sqrt {|0|}}={\sqrt {|y(t)|}}}$ ${\displaystyle y_{2}(t)={\frac {t^{2}}{4}},y_{2}(0)=0}$ ${\displaystyle y'_{2}(t)={\sqrt {\left|{\frac {t^{2}}{4}}\right|}}={\frac {t}{2}}=y_{2}'(t)}$

However, if the concept of continuity is expanded to include the so-called Lipschitz condition for the function , then a clearly defined solution exists. ${\displaystyle f}$

literature

• Herbert Amann: Ordinary differential equations . 2nd Edition. Gruyter - de Gruyter textbooks, Berlin / New York 1995, ISBN 3-11-014582-0 .
• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).