Picard-Lindelof's theorem

The Picard-Lindelöf theorem is in mathematics , in addition to the set of Peano , a fundamental theorem of the theory of the existence of solutions of ordinary differential equations . It was first set up in 1890 by Ernst Leonard Lindelöf in an article on the solvability of differential equations. At the same time, Émile Picard was also concerned with the step- by- step approximation of solutions. This Picarditeration , a fixed point iteration in the sense of Banach's Fixed Point Theorem, is the core of modern proofs of this theorem.

It is also referred to as Cauchy-Lipschitz's theorem (after Augustin-Louis Cauchy and Rudolf Lipschitz ) or Picard's existential theorem .

Similar to Peano's theorem, this theorem is formulated and proven in several versions that build on one another.

1. The local version says that every initial value problem for a differential equation can be solved uniquely in a certain neighborhood of , assuming the Lipschitz condition (see below) . The size of this environment strongly depends on the right side . ${\displaystyle y'(x)={\frac {\mathrm {d} y}{\mathrm {d} x}}=f(x,y),\quad y(a)=y_{a}}$ ${\displaystyle a}$ ${\displaystyle f(x,y)}$
2. The global version says that such an initial value problem, which satisfies a global Lipschitz condition on a vertical strip , has a unique solution over the entire interval . ${\displaystyle (x,y)\in [a,e]\times \mathbb {R} ^{n}}$ ${\displaystyle [a,e]}$

Once you have a (local) solution, in a second step you can infer the existence of a non-continuable solution . In this regard, Picard-Lindelöf's Theorem is the first step in the existence theory of a differential equation.

Remark on the theoretical embedding: In the interests of the briefest possible representation, it is sufficient to infer the existence of (possibly several) maximal solutions from the continuity of the right-hand side with Peano's theorem, and with Gronwall's inequality on the uniqueness of the solution. This path is usually not chosen in introductory courses, since Peano's theorem is based on the Arzelà-Ascoli theorem, while Picard-Lindelöf's theorem can be proven with much more elementary means, such as Banach's fixed point theorem. ${\displaystyle f(x,y)}$

Problem

Be or or be more generally a real Banach space . In the simplest case is . All statements that are made and proven in this simplest case can be transferred to the general case by simply changing the notation. It needs to only by replacing, d. H. the absolute amount through the norm of the Banach space. ${\displaystyle E=\mathbb {R} ^{n}}$ ${\displaystyle E=\mathbb {C} ^{n}}$ ${\displaystyle E}$ ${\displaystyle E=\mathbb {R} }$ ${\displaystyle \mathbb {R} ,|\cdot |}$ ${\displaystyle E,\|\cdot \|}$

A differential equation for a function with values ​​in is an equation of the form . The function of the right side is on a (open) area defined and has values in , . ${\displaystyle E}$ ${\displaystyle y'(x)=f(x,y(x))}$ ${\displaystyle f(x,y)}$ ${\displaystyle G\subset \mathbb {R} \times E}$ ${\displaystyle E}$ ${\displaystyle f\colon G\to E}$

Often the domain is assumed to be in the form of a vertical stripe, then it is . ${\displaystyle G}$ ${\displaystyle G=(a,b)\times E}$

A continuously differentiable function for an interval is a (local) solution of the differential equation if both and hold for all . ${\displaystyle y\colon I\to E}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle x\in I}$ ${\displaystyle (x,y(x))\in G}$ ${\displaystyle y'(x)=f(x,y(x))}$

The question is now whether a local solution of the differential equation can be found when a point is given , which contains the domain of definition and which simultaneously fulfills. ${\displaystyle (x_{0},y_{0})\in G}$ ${\displaystyle x_{0}}$ ${\displaystyle y(x_{0})=y_{0}}$

The sentence in its versions

The prerequisites for the sentence versions are always the continuity of the right-hand side and the existence of a Lipschitz condition. This Lipschitz condition is often described as “local Lipschitz continuity in the second variable”.

Global and local Lipschitz condition

Definition: Be and given. It is said that a (global) Lipschitz condition on in the second variable is fulfilled if there is a constant such that for each and points with the inequality ${\displaystyle U\subset \mathbb {R} \times E}$ ${\displaystyle f\colon U\to E}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle L\geq 0}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle y_{1},y_{2}\in E}$ ${\displaystyle (x,y_{1}),(x,y_{2})\in U}$

${\displaystyle \|f(x,y_{1})-f(x,y_{2})\|\leq L\|y_{1}-y_{2}\|}$

applies.

Definition: Be and given. It is said that a local Lipschitz condition on satisfies in the second variable if there is a neighborhood for every point on which the restriction of on satisfies a (global) Lipschitz condition. ${\displaystyle G\subset \mathbb {R} \times E}$ ${\displaystyle f\colon G\to E}$ ${\displaystyle f}$ ${\displaystyle G}$ ${\displaystyle (x,y)\in G}$ ${\displaystyle (x,y)\in U\subset G}$ ${\displaystyle f}$ ${\displaystyle U}$

Remarks:

• The neighborhood of the local Lipschitz condition can always be chosen as a sphere or cylinder , since in every open set there must be a subset of this shape for each of its points. The open sphere denotes the radius . ${\displaystyle U}$ ${\displaystyle [x-\delta ,x+\delta ]\times B(y,\delta )}$ ${\displaystyle B(y,\delta ):=\{z\in E\ |\ \|z-y\|<\delta \}}$ ${\displaystyle y}$ ${\displaystyle \delta }$
• Every function with a convex domain that is continuously partially differentiable according to the second variable also fulfills a local Lipschitz condition in the second variable, since according to the mean value theorem

${\displaystyle \|f(x,y_{2})-f(x,y_{1})\|\leq \sup _{t\in [0,1]}\|\partial _{y}f{\bigl (}x,y_{1}+t(y_{2}-y_{1}){\bigr )}\|\cdot \|y_{2}-y_{1}\|}$

with a suitable norm of derivation applies. As a continuous function, the norm of the derivative is locally restricted, from which the Lipschitz condition follows in the second variable.

Local version of Picard-Lindelöf's theorem

Be a Banach space , , with and continuous and locally Lipschitz continuous in the second variable. Referred to herein ${\displaystyle E}$ ${\displaystyle G\subset \mathbb {R} \times E}$ ${\displaystyle y_{0}\in E,R>0}$ ${\displaystyle [a,b]\times {\overline {B}}(y_{0},R)\subset G}$ ${\displaystyle f=f(x,y)\colon G\to E}$

${\displaystyle {\overline {B}}(y_{0},R):=\{z\in E\ |\ \|z-y_{0}\|\leq R\}}$

the completed sphere around with radius . Is ${\displaystyle y_{0}}$ ${\displaystyle R}$

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

and

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

then there is exactly one solution of the initial value problem

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

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

Global version of Picard-Lindelöf's theorem

Let it be a Banach space and a continuous function that satisfies a global Lipschitz condition with respect to the second variable. Then there is a global solution to the initial value problem for each ${\displaystyle E}$ ${\displaystyle f\colon [a,b]\times E\to E}$ ${\displaystyle y_{0}\in E}$ ${\displaystyle y\colon [a,b]\to E}$

${\displaystyle y'=f(\cdot ,y),\quad y(a)=y_{0}}$.

There are no other (local) solutions.

Web links

Wikibooks: Proof of Picard-Lindelöf's Theorem  - Learning and Teaching Materials

Individual evidence

1. ^ Cauchy-Lipschitz theorem. Encyclopedia of Mathematics, Springer.