General approximation solvability of a system of strongly gg-rr-pseudomonotonic nonlinear variational inequalities and projection methods

Let KK be nonempty closed convex subsets of a real Hilbert space HH. Approximation solvability of a generalized system of nonlinear strongly gg-rr-pseudomonotonic variational inequality (SNVI) problems based on the convergence of projection methods is presented as follows: find elements x∗,y∗∈Kx∗,y∗∈K (and hence g(x∗),g(y∗)∈Kg(x∗),g(y∗)∈K) such that 〈ρT(y∗)+g(x∗)−g(y∗),g(x)−g(x∗)〉≥0,∀g(x)∈K and for ρ>0,〈ηT(x∗)+g(y∗)−g(x∗),g(x)−g(y∗)〉≥0,∀g(x)∈K and for η>0, where T:K→HT:K→H and g:K→Kg:K→K are nonlinear mappings.