Let $\dot{\mathbf{y}} = \mathbf{f}(\mathbf{y}), \;\mathbf{f}: D \subset \mathbb{R}^d \to \mathbb{R}^d$ be an autonomous differential equation with $\mathbf{f}$ smooth. We define the averaged vector field one step method implicitely by
$$\mathbf{y}_1 = \mathbf{y}_0 + h \int_0^1 \mathbf{f}(\mathbf{y}_0+\tau (\mathbf{y}_1-\mathbf{y}_0)) \, \mathrm d\tau.$$
How can I show that for a sufficiently small $h > 0$, for every $\mathbf{y}_0 \in D$ there exists such a $\mathbf{y}_1$?
