Picard–Lindelöf theorem

The set of Picard - Lindelöf is in mathematics, in addition to the set of Peano, a fundamental set of the theory of existence of solutions of ordinary differential equations. It was first erected in 1890 by Ernst Leonard Lindelöf in an article for the solvability of differential equations. Around the same time also Émile Picard dealt with the gradual approximation of solutions. This Picarditeration, a fixed point in the sense of Banach's fixed point theorem, is the core of modern proof of this theorem.

Similar to the set of Peano also this sentence in several successive versions to be stated and proved.

If one possesses only once a (local) solution, one can conclude from this in a second step, the existence of a non- continuable solution. In this respect, the set of Picard - Lindelöf is the first step for the existence theory of a differential equation.

Remarks theoretical embedding: For the purposes of a scarce possible representation, it is sufficient from the continuity of the right-hand side f ( x, y) to the set of Peano ( possibly more ) to infer the existence of maximal solutions, and with the grönwallschen inequality ( see also below) the uniqueness of the solution. This path is usually not elected in introductory courses, since the set of Peano based on the set of Arzelà - Ascoli, while the set of Picard - Lindelöf with much more elementary means, such as the fixed point theorem of Banach, can be proved.

Problem

Be or be general or E be a real Banach space. In the simplest case. It can be any statements that are made and proved in this simplest case, transmitted by simply changing the notation to the general case. It must be replaced only by this, that the absolute value of the norm of the Banach space.

A differential equation for a function with values ​​in an equation of the form. The function of the right side is defined on a (open) region and has values ​​.

Often, the domain will be provided in the form of a vertical strip, then.

A continuously differentiable function on an interval is a (local) solution of the differential equation if and apply to all both.

The question now is whether it is to find a local solution of the differential equation over a target point, contains the domain and at the same time fulfilled.

The sentence in its versions

The conditions of 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 a "local Lipschitz continuity in the second variable ."

Global and local Lipschitz condition

Definition: Let and, where. It is said that a (global ) Lipschitz condition into the second variable satisfied when there is a constant, and such that for each point with the inequality

Applies.

Definition: Let and, where. It is said that a local Lipschitz condition into the second variable satisfied when there is a setting for each point on the limitation of a (global ) satisfies Lipschitz condition.

Remarks:

• The area surrounding the local Lipschitz condition can always be chosen as a sphere or cylinder, since there must be a subset of this form for each of their points in each open set. It denotes the open ball of radius around.
• Each continuously partial differentiable according to the second variable function with convex domain satisfies a local Lipschitz condition in the second variable, since, according to the mean value theorem

Local version of the theorem of Picard - Lindelöf

Be a Banach space, and with continuous and locally Lipschitz continuous in the second variable. referred to herein

The closed ball of radius. is

As well as

Then there exists a unique solution of the initial value problem

On the interval; it has values ​​.

Global version of the theorem of Picard - Lindelöf

It is a Banach space and a continuous function, which performs a global Lipschitz condition with respect to the second variable. Then, for every a global solution of the initial value problem

There are no other (local) solutions.