A class of iterative algorithms and solvability of nonlinear variational inequalities involving multivalued mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented, Determine an element x* epsilon K and u* epsilon T(x*) such that (u*, x - x*) greater than or equal to 0 for all x epsilon K, where T : K --> P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x(0), y(0) epsilon T(y(0)) and nu(0) epsilon T(x(0)), we have (u(k) + x(k+1) - y(k), x - x(k+1)) greater than or equal to 0, For Allx epsilon K, for u(k) epsilon T(y(k)) and for k greater than or equal to 0, where (v(k) + y(k) - x(k), x - y(k)) greater than or equal to 0, For Allx epsilon K and for nu(k) epsilon T(x(k)).