Picard's F-trajectory Theorem

• categories: Real Analysis, Theorem

The F-Trajectory Theorem, often associated with Émile Picard, concerns the behavior of solutions to ordinary differential equations (ODEs) and their trajectories in the phase space. It provides insight into the uniqueness and existence of solutions based on initial conditions.

Statement of the Theorem

Consider the ordinary differential equation given by:

$$\frac{dy}{dx}=f(x,y)$$

where $f$ is a continuous function defined on some domain in the $xy$-plane. The F-Trajectory Theorem states:

1. If $f(x,y)$ is continuous in a region containing the point $(x_0,y_0)$, and $f$ satisfies a Lipschitz condition in the variable $y$ on this region, then there exists a unique solution $y(x)$ to the initial value problem:

   $$y(x_0)=y_0$$

2. The solution $y(x)$ is defined for some interval around $x_0$.

Intuition

• The theorem guarantees that if the function $f$ is well-behaved (continuous and satisfying the Lipschitz condition), then starting from an initial point $(x_0,y_0)$, there is a unique trajectory or path that the solution $y(x)$ will follow.

Example

Consider the ODE:

$$\frac{dy}{dx}=3y+2$$

Here, $f(x,y)=3y+2$ is continuous, and it satisfies the Lipschitz condition in $y$ because:

$$|f(x,y_1)-f(x,y_2)|=|3y_1+2-(3y_2+2)|=3|y_1-y_2|$$

Thus, according to the F-Trajectory Theorem, there exists a unique solution $y(x)$ that can be found given an initial condition.

Applications

• Existence and Uniqueness: The F-Trajectory Theorem is fundamental in establishing the existence and uniqueness of solutions to initial value problems in differential equations.