# existence and uniqueness of solution to Cauchy problem

Let

 $\begin{cases}\mathbf{\dot{x}}=F(\mathbf{x},t)\\ \mathbf{x}(t_{0})=\mathbf{x}_{0}\end{cases}$

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

• a continuous function of $n+1$ variables defined in a neighborhood $U\subseteq\mathbb{R}^{n+1}$ of $(\mathbf{x}_{0},t_{0})$

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

Then there exists a unique solution $\mathbf{x}:I\to\mathbb{R}^{n}$ of the Cauchy problem, defined in a neighborhood $I\subseteq\mathbb{R}$ of $t_{0}$.

Proof

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

 $x(t)=x(t_{0})+\int_{t_{0}}^{t}F(\mathbf{x}(\tau),\tau)\mathrm{d}\tau$

Let $X$ be the set of continuous functions $\mathbf{f}:[t_{0}-\delta,t_{0}+\delta]\to B(\mathbf{x}_{0},\epsilon)$. We’ll assume $\epsilon$ to be chosen such that the $B(\mathbf{x}_{0},\epsilon)\subseteq U$ 11$B(\mathbf{x}_{0},\epsilon)$ denotes the closed ball $\{\mathbf{x}:\|\mathbf{x}_{0}-\mathbf{x}\|\leq\epsilon\}$. 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

 ${F(\mathbf{x},t)-F(\mathbf{y},t)}\leq L\|\mathbf{x}-\mathbf{y}\|$

for all points $\mathbf{x},\mathbf{y}$ sufficiently near to $\mathbf{x}_{0}$.

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

 $T\mathbf{x}:t\mapsto\mathbf{x}_{0}+\int_{t_{0}}^{t}F(\mathbf{x}(\tau),\tau)% \mathrm{d}\tau$

We make the following observations about $T$.

1. 1.

Since $F$ is continuous, $\|F\|$ attains a maximum value $M$ on the compact set $B(\mathbf{x}_{0},\epsilon)\times[t_{0}\pm\delta]$. But by hypothesis, $\|\mathbf{x}(t)-\mathbf{x}_{0}\|\leq\epsilon$, hence

 $\|\mathbf{x}(t)-\mathbf{x}_{0}\|\leq\int_{t_{0}}^{t}\|F(\mathbf{x}(\tau),\tau)% \|\mathrm{d}\tau\leq M(t-t_{0})\leq M\delta$

for all $t\in[t_{0}\pm\delta]$.

2. 2.

The Lipschitz continuity of $F$ yields

 $\|T\mathbf{x}(t)-T\mathbf{y}(t)\|\leq\int_{t_{0}}^{t}\|F(\mathbf{x}(\tau),\tau% )-F(\mathbf{y}(\tau),\tau)\|\mathrm{d}\tau\leq\int_{t_{0}}^{t}L\|\mathbf{x}(% \tau)-\mathbf{y}(\tau)\|\mathrm{d}\tau\leq L\delta d_{\infty}(\mathbf{x},% \mathbf{y})$

If we choose $\delta<\min\{1/L,\epsilon/M\}$ 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 $d_{\infty}$ on $X$, i.e. there exists $\lambda\in\mathbb{R}$ such that for all $\mathbf{x},\mathbf{y}\in X$,

 $d_{\infty}(T\mathbf{x},T\mathbf{y})\leq\lambda d_{\infty}(\mathbf{x},\mathbf{x})$

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

 $\mathbf{x}^{\star}=\lim_{k\to\infty}T^{k}\mathbf{x}_{0}$

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

 $T\mathbf{x}=\mathbf{x}\text{ in other words }\mathbf{x}(t)=\mathbf{x}_{0}+\int% _{t_{0}}^{t}F(\mathbf{x}(\tau),\tau)\mathrm{d}\tau$

and which therefore locally solves the Cauchy problem.

Title existence and uniqueness of solution to Cauchy problem ExistenceAndUniquenessOfSolutionToCauchyProblem 2013-03-22 16:54:45 2013-03-22 16:54:45 ehremo (15714) ehremo (15714) 20 ehremo (15714) Theorem msc 34A12