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

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