The question of existence of the solution for an arbitrary system of non-linear equations $F(x)=0$ where $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is undecidable (following Existence and uniqueness of solutions to a system of non-linear equations)
Yet, one can come up with sufficient conditions for existence. For example, if the Jacobian is triangular with non-zero elements on the diagonal, and under appropriate monotonicity and continuity assumptions, the system can be solved by forward/backward substitution.
What is the best source for the (largest possible) list of such simple cases where the solution can be shown to exist?
