existence and uniqueness of solution to Cauchy problem

Let

$$\{\begin{array}{cc}\dot{𝐱}=F(𝐱,t)\hfill & \\ 𝐱(t_0)={𝐱}_0\hfill & \end{array}$$

be a Cauchy problem, where $F:U\to ℝ$ is

• a continuous function of $n+1$ variables defined in a neighborhood $U\subseteq {ℝ}^{n+1}$ of $({𝐱}_0,t_0)$

• Lipschitz continuous with respect to the first $n$ variables (i.e. with respect to $𝐱$).

Then there exists a unique solution $𝐱:I\to {ℝ}^n$ of the Cauchy problem, defined in a neighborhood $I\subseteq ℝ$ of $t_0$.

Proof

Solving the Cauchy problem is equivalent to solving the following integral equation

$$x(t)=x(t_0)+{\int }_{t_0}^{t}F(𝐱(\tau ),\tau )d\tau$$

Let $X$ be the set of continuous functions $𝐟:[t_0-\delta ,t_0+\delta ]\to B({𝐱}_0,ϵ)$. We’ll assume $ϵ$ to be chosen such that the $B({𝐱}_0,ϵ)\subseteq U$ . In this ball, therefore, $F$ is Lipschitz continuous with respect to the first $n$ variable, in other words, there exists a real number $L$ such that

$$F(𝐱,t)-F(𝐲,t)\le L\parallel 𝐱-𝐲\parallel$$

for all points $𝐱,𝐲$ sufficiently near to ${𝐱}_0$.

Now let’s define the mapping $T:X\to X$ as follows

$$T𝐱:t\mapsto {𝐱}_0+{\int }_{t_0}^{t}F(𝐱(\tau ),\tau )d\tau$$

We make the following observations about $T$.

1. Since $F$ is continuous, $\parallel F\parallel$ attains a maximum value $M$ on the compact set $B({𝐱}_0,ϵ)\times [t_0\pm \delta ]$. But by hypothesis, $\parallel 𝐱(t)-{𝐱}_0\parallel \le ϵ$, hence

   $$\parallel 𝐱(t)-{𝐱}_0\parallel \le {\int }_{t_0}^t\parallel F(𝐱(\tau ),\tau )\parallel d\tau \le M(t-t_0)\le M\delta$$

   for all $t\in [t_0\pm \delta ]$.

2. The Lipschitz continuity of $F$ yields

   $$\parallel T𝐱(t)-T𝐲(t)\parallel \le {\int }_{t_0}^t\parallel F(𝐱(\tau ),\tau )-F(𝐲(\tau ),\tau )\parallel d\tau \le {\int }_{t_0}^{t}L\parallel 𝐱(\tau )-𝐲(\tau )\parallel d\tau \le L\delta d_{\mathrm{\infty }}(𝐱,𝐲)$$

If we choose these conditions ensure that

• $T(X)\subseteq X$, i.e. $T$ doesn’t send us outside of $X$.

• $T$ is a contraction mapping with respect to the uniform convergence metric $d_{\mathrm{\infty }}$ on $X$, i.e. there exists $\lambda \in ℝ$ such that for all $𝐱,𝐲\in X$,

  $$d_{\mathrm{\infty }}(T𝐱,T𝐲)\le \lambda d_{\mathrm{\infty }}(𝐱,𝐱)$$

In particular, the second point allows us to apply Banach’s theorem and define

$${𝐱}^{\star }=\underset{k\to \mathrm{\infty }}{lim}{T}^k{𝐱}_0$$

to find the unique fixed point of $T$ in $X$, i.e. the unique function which solves

$$T𝐱=𝐱\text{ in other words }𝐱(t)={𝐱}_0+{\int }_{t_0}^{t}F(𝐱(\tau ),\tau )d\tau$$

and which therefore locally solves the Cauchy problem.