I am interested in the existence of a fixed point of a vector function of the form $$\vec{f}(\vec{x}) = \vec{x}_0-\sum_{i=1}^K \alpha_i \vec{n}(\vec{x}),$$ where $\vec{n}(\cdot)$ are unit vectors.
In the simple case of one dimension and $K=1$, I know I could investigate $|f'(x)|=|\alpha||n'(x)| \leq K <1$, which ensures that $f$ is Lipschitz, hence a contraction, and so there exists a fixed point and we can obtain it via $x_n = f(x_{n-1})$ for any initial guess.
Is there a multivariate analog to this sort of condition? Well, now we would have to deal with Jacobians, and again in the case $K=1$, we have $D\vec{f}(\vec{x}) = -\alpha D\vec{n}(\vec{x})$. We would want a condition like $|\alpha| \|D\vec{n}(\vec{x})\|\leq K <1$ but it is not clear to me what norm to choose on the Jacobian. I am not familiar with a generalization of Lipschitz continuity to $\mathbb{R}^d$. A cursory glance at Wikipedia brought me to Rademacher's theorem which is relevant but not exactly what I need. On the page for Lipschitz continuity in particular they state "moreover, if $K$ is the best Lipschitz constant of $f$, then $\| D f ( x )\| \leq K$ whenever the total derivative $Df$ exists.[citation needed]". But I need the opposite direction (and the norms here are unspecified... but probably just the Euclidean?).