Theorem of Cauchy-Lipschitz

The theorem of Cauchy-Lipschitz ensures the local existence and the unicity of the solution of a differential equation. Stated by Augustin Louis Cauchy in 1820, it is Rudolf Lipschitz which will give him its final form in 1868. In many countries, the most current name is that of theorem of Picardy-Lindelöf , of the name of the mathematicians Emile Picard and Ernst Lindelöf.

Theorem

Elementary statement

That is to say F a function of two real variables to actual values:

where $U \ subset \ mathbb R$ is a real line interval, and $I \ subset \ mathbb R$ another interval of this same line. Let us consider the first order differential equation:

One supposes moreover than the differential equation is subjected to the initial condition: $\ displaystyle X (t_0) = x_0$ , where $t_0 \ in I$ and $x_0 \ in U$ .

If the function F is continuous and K - Lipschitzienne in X , i.e if F checks the condition of Lipschitz :

then there exists one and only one solution X (T) of the definite differential equation for all $t \ in J$ , $J \ subset I$ being an interval centered on t₀ , checking the initial condition given.

This theorem is to be brought closer to the concept of Déterminisme in traditional physics: if a system follows a law of evolution given (the differential equation), the same causes (initial conditions) produce the same effects.

Notice

The theorem of Cauchy-Lipschitz provides a local existence : there exists one and only one solution X (T) which is a priori defined only for moments T located in an interval J centered on t₀ . The question of the maximum prolongation of this solution, i.e of its total existence , treats well within the framework of the study of the differential equations for times T complex . This maximum prolongation is related to the presence of Singularité S. One must in particular with Paul Painlevé of important contributions on this subject.

General statement

That is to say E a Space of Banach of size finished on ℝ, O open of E X ℝ, F an application of O in E :

continuous on O and locally lipschitzienne in the first variable on O , i.e the function F checks the condition of Lipschitz :

where K is a constant. Let us consider the first order differential equation:

Then:

the maximum approaches of the differential equation are defined on open intervals of ℝ;

the graphs of the maximum approaches form a partition of O;

any solution of the differential equation is the restriction of one and only one approach maximum of the equation.

Extension to the partial derivative equations

The theorem of Cauchy-Lipschitz ensuring the existence and the unicity of the solution of a differential equation admits an extension to the partial derivative equations: the Theorem of Cauchy-Kovalevskaïa.

Differential equation
dynamic Determinism
System
Theory of chaos
Theorem of Cauchy-Kovalevskaïa