20 - Picard's Theorem

Picard's Theorem

Let $\mathbf{f}$ be continuous in neighbourhood $U\subset {\mathbb{R}}^{1+d}$ of $(x_0,{\mathbf{y}}_0)$ containing closed cylinder

$$R=\{(x,\mathbf{y}):x_0\le x\le X_m,||\mathbf{y}-{\mathbf{y}}_0||\le Y_M\}$$

where $X_M>x_0$ and $Y_M>0$ are constants

Suppose exists positive constant $L$ such that

$$||\mathbf{f}(x,\mathbf{y})-\mathbf{f}(x,\mathbf{z})||\le L||\mathbf{y}-\mathbf{z}||$$

holds whenever $(x,\mathbf{y})$ and $(x,\mathbf{z})$ lie in $R$

Let

$$M:=\max \{||\mathbf{f}(x,\mathbf{y})||:(x,\mathbf{y})\in R\}$$

Suppose

$$M(X_m-x_0)\le Y_M$$

Then there exists unique continuously differentiable function

$$[x_0,X_M]\ni nix\mapsto \mathbf{y}(x)\in {\mathbb{R}}^d$$

which is the solution to the IVP

Integral Form for Solution

If solution exists then, solution can be expressed by integral equation in form

$$\mathbf{y}(x)=\mathbf{y}(x_0)+{\int }_{x_0}^x\mathbf{f}(t,\mathbf{y}(t))dx$$

Existence of Solution for Finite Time

Picard’s Theorem only guarantees existence of solution only up to finite time $X_M$

Stability of Solutions with Respect to Initial Conditions

If IVP satisfies assumptions of Picard’s Theorem then

Solution is stable on bounded interval $[x_0,X]$
So if $\mathbf{y}:[x_0,X]\to D$ solves IVP

$${\mathbf{y}}^{\prime }(x)=\mathbf{f}(x,\mathbf{y}),\mathbf{y}(x_0)={\mathbf{y}}_0$$

and $\tilde{\mathbf{y}}:[x_0,X]\to D$ solves same ODE with perturbed initial condition ${\tilde{\mathbf{y}}}_0$ so

$${\tilde{\mathbf{y}}}^{\prime }(x)=\mathbf{f}(x,\tilde{\mathbf{y}}),\tilde{\mathbf{y}}(x_0)=\tilde{\mathbf{y}}0$$

Then

$$|||\mathbf{y}|(x)-\tilde{\mathbf{y}}(x)||\le e^{L(X-x_0)}||{\mathbf{y}}_0-{\tilde{\mathbf{y}}}_0||\forall x\in [x_0,X]$$