General strongly nonlinear variational inequalities (Q1191799)

The authors consider the following variational inequality: Find \(u\in H\) such that \(g(u)\in K(u)\) and \[ \langle T(u),v-g(u)\rangle\geq\langle A(u),v-g(u)\rangle,\quad v\in K(u), \] where \(K(u)=m(u)+K\) and \(K\) is a closed convex subset of a Hilbert space \(H\). They assume that \(T\), \(g\), \(A\) and \(m\) are Lipschitz continuous operators on \(H\) and that \(T\) and \(g\) are strongly monotone. They prove that the iterates of an associated fixed point equation converge linearly to the solution of the variational inequality, if the Lipschitz respectively coercivity constants satisfy certain inequalities.