Pseudomonotone general mixed variational inequalities (Q1408339)

Let \(K\) be a nonempty closed convex subset of a Hilbert space \(H\), \(T,g:H\rightarrow H\) be two nonlinear operators and \(\varphi:X\rightarrow\mathbb{R}\cup\{+\infty\}.\) The author considers the ``general mixed variational inequality problem'', that is, the problem of finding \(u\in H\) such that \[ \langle T(u),g(v)-g(u)\rangle+\varphi(g(v))-\varphi(g(u))\geq 0, \] for all \(v\in H.\) Under the assumption that \(T\) is \(g\)-(Karamardian) pseudomonotone, the author presents convergence results for iterative methods based on the resolvent equation and the ``auxiliary principle technique''.