General System of Strongly Pseudomonotone Nonlinear Variational Inequalities Based on Projection Systems

Let K1 and K2, respectively, be non empty closed convex subsets of real Hilbert spaces H1 and H2. The Approximation − solvability of a generalized system of nonlinear variational inequality (SNV I) problems based on the convergence of projection methods is discussed. The SNVI problem is stated as follows: find an element (x∗, y∗) ∈ K1 × K2 such that 〈ρS(x∗, y∗), x− x∗〉 ≥ 0, ∀x ∈ K1 and for ρ > 0, 〈ηT (x∗, y∗), y − y∗〉 ≥ 0, ∀y ∈ K2 and for η > 0, where S : K1 ×K2 → H1 and T : K1 ×K2 → H2 are nonlinear mappings.