I wondered if there is any more robust and faster way to solve a system of two equations with two variables than the NLsolve package.
The first equation is F(x,y)=0 and the second G(x,y)=0. Both of them are very complicated functions. However, what I know about them are as follows:

For any (x,y(x)) such that F(x,y_{F}(x))=0, y_{F}(x) is a decreasing function. In other words, if x increases, y must decrease to satisfy F(x,y)=0.

On the other hand, for G(x,y_{G}(x))=0, y_{G}(x) is an increasing function.
Given these two properties, I know that there can be at most one solution but not the existence of a solution. There are a few more parameters that can make the nonexistence of a solution.
Ideally, I could plot y_{F}(x) and y_{G}(x) and find the crossing point. Alternatively, I could use NLsolve package, but I have no clue how the algorithm searches for the root, especially when a solution does not exist.
Any comments or suggestions would be tremendously helpful.