Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let $$F:D(F) \subseteq H \to H, K : D(K) \subseteq H \to H$$ be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u*,v*]∈D(F) × D(K) such that $$\left\{\begin{array}{lll}&\langle Fu^* - v^*, x - u^*\rangle \geq 0,\quad x \in D(F),\\ &\langle Kv^* + u^*, y - v^*\rangle \geq 0,\quad y \in D(K)\end{array}\right.$$ This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.