Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex.
I am not sure about the variational inequality problem: find $x \in H$ s.t. $$ \langle f(x) - x, \ J(x-y) \rangle \geq 0$$ for all $y \in K \subset C \subset H$, where $J$ is the dual mapping.
How does this problem look like if $H = \mathbb{R}^n$?
Now assume that $f$ is decreasing (hence pseudocontractive, i.e., $\text{Id} - f$ is monotone).
Are there computational (iterative) methods to solve the variational inequality (say for $H = \mathbb{R}^n$)?