Let $f(x,y)$ and $g(x,y)$ be two $\mathbb{R}^2\rightarrow\mathbb{R}$ functions, both strictly increasing in both arguments. Assume that they are well-behaved functions (continuous, differentiable, etc.)
Given these properties, I want to find a root to the following system of equations:
$$f(x,y)=f_0$$ $$g(x,y)=g_0$$
assuming that a solution exists and that it is unique.
Is there an efficient algorithm to iteratively bracket the root in increasingly smaller regions $\mathbb{R}^2$? That is, I am looking for a derivative free method that is an extension of the bisection method, for 2 variables, with convergence guaranteed.