Projection methods, algorithms, and a new system of nonlinear variational inequalities

The convergence of projection methods is based on a new iterative algorithm for the approximation-solvability of the following system of nonlinear variational inequalities (SNVI): determine elements x*, y* ε K such that T(y*) + x* − y*, x − x* ≥ 0, for all x ε K and for > 0, and γT(x*) + y* − x*, x − y* ≥ 0, for all x K and for γ > 0, where T : K → H is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. This new class of generalized nonlinear variational inequalities reduces to standard class of nonlinear variational inequalities, which are widely studied and applied to various problems arising from mathematical sciences, optimization and control theory, and other related fields.