Consider the IVP

$$\frac{d\mathbf{u}}{dt}=\mathbf{f}(t,\mathbf{u}),\mathbf{u}(0)=\mathbf{c}.$$

Suppose that $\mathbf{f}:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is continuous in its domain and @Lipschitz continuous in $\mathbf{u}$, i.e., there exists a constant such that

$$|\mathbf{f}(t,\mathbf{u})-\mathbf{f}(t,\mathbf{v})|\le L\Vert \mathbf{u}-\mathbf{v}\Vert ,\mathrm{\forall }(t,\mathbf{u}),(t,\mathbf{v})\in [0,T]\times {\mathbb{R}}^d.$$

Then it holds that:

1. Existence and Uniqueness: There exists a unique solution $\mathbf{u}(t)$ for any ${\mathbf{u}}_0\in {\mathbb{R}}^d.$
2. Continuous Dependence on initial data and right hand side: Suppose $\mathbf{r}:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is a bounded @perturbation of $\mathbf{f},$ i.e. there exists a constant such that $$\Vert \mathbf{r}(t,\mathbf{u})\Vert \le M,\mathrm{\forall }(t,\mathbf{u})\in [0,T]\times {\mathbb{R}}^d.$$ Further let $\hat{\mathbf{u}}(t)$ be the solution of the perturbed IVP $$\frac{d\hat{\mathbf{u}}}{dt}=\mathbf{f}(t,\hat{\mathbf{u}})+\mathbf{r}(t,\hat{\mathbf{u}}),\hat{\mathbf{u}}(0)=\hat{\mathbf{c}},$$ which we assume exists and is unique. Then we have $$\Vert \mathbf{u}(t)-\hat{\mathbf{u}}(t)\Vert \le e^{Lt}\Vert \mathbf{c}-\hat{\mathbf{c}}\Vert +\frac{M}{L}(e^{Lt}-1),\mathrm{\forall }t\in [0,T].$$