Assume one wants to find at least one zero of a mapping $F : X \rightarrow X$, but with the additional requirement that the zero has to lie in some (strict) given subset $C \subset X$ with $C$ closed. Strangely enough, I cannot not find any algorithm (aiming at) solving this problem (... except in the one-dimensional case). I might not be using the right keywords, or missing a fundamental issue.
To fix ideas, one can think of $X = \mathbb{R}^n$, and $C$ convex closed. One natural idea would be to use projected Newton iterates, i.e., under suitable regularity assumptions for $F$ and with $P_C$ the projection operator onto $C$, $$x_{n+1} = P_C\big(x_n - DF(x_n)^{-1}(F(x_n))\big).$$ These iterates are probably not guaranteed to converge to zeros in $C$. Indeed, fixed points $\bar x$ of this algorithm satisfy $\bar x = P_C(\bar x - DF(\bar x)^{-1}(F(\bar x)))$, or equivalently $$\forall y \in C, \quad \langle DF(\bar x)^{-1}(F(\bar x)), y-x\rangle \geq 0,$$ which in general yields $\bar x \in C$, but does not necessarily imply $F(\bar x) = 0$.
Does anyone have a reference about this problem, or some appropriate keywords to browse the literature?