I have a "nice" function (vector field) $$\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and I need to find how many roots (zeros) it has in a certain domain (hopefully prove that it has at most one). I do not have the function in explicit form, but I can infer some properties such as being continuously differentiable, bounded (by individual vector components), and each component of the vector $\mathbf{f}$ changing monotonically along a corresponding axis.
Are there any theorems that could be helpful in this, i.e. proving that the system has no more than one root in some domain?
This is for an actual practical application related to physics. Anything that works for $n=2$ only is useful, but I'm also interested in $n>2$ (only small $n$ in practice, i.e. $3,4,5,6$).