Summary: The two-step algorithm is introduced and applied to the approximation solvability of a system of nonlinear variational inequalities in a Hilbert space setting. Let be a real Hilbert space and be a nonempty closed convex subset of . For arbitrarily chosen initial points , , compute sequences and such that
where is a nonlinear pseudococoercive mapping on , is the projection of onto , and for . Based on the two-step algorithm, we explore the approximation solvability of a system of nonlinear variational inequality problems: determine elements such that
where , .