After presenting the Brouwer Fixed Point Theorem, Zeidler E. (1986) Nonlinear Functional Analysis and its Applications I - Fixed-Point Theorems, in p. 53-54, proves that (Proposition 2.8) a system of equations $$g_i(x)=0, \hskip 10pt \mbox{where } x=(x_1,...,x_N) \in \mathbb{R}^N , i=1,...,N$$ where $g_i$ are continuous functions, has a solution in the closed ball $\bar{U}(0,R)$ for some fixed radius $R$, if the following condition is satisfied $$\sum_{i=1}^N g_i(x) x_i \geq 0 \hskip 10pt \mbox{ for all } x \mbox{ such that } ||x||=R.$$
What I don't understand about this is how to "pair" the $g_i$'s with the $x_i$'s in order to take the inner product. Different pairings give different sums for the condition above. Does the condition mean that there exists a pairing of equations with unknowns that is positive, or that all pairings should give a positive inner product, or something else? I also have seen discussions about outward pointing rays and inward pointing rays, but I don't really understand them either.
Also on the same theme, how is this related to the Poincaré-Miranda theorem that is also about solutions of non-linear equations (of course, the BFT and the PMT are equivalent). There, each $g_i$ should be paired with the $x_i$ that makes the $g_i$ always negative on $x_i=-1$ and always positive at $x_i=1$. Is there a connection between the two conditions?
Thanks in advance!