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^* K and u^* T(x^*) such that u^*, x – x^* 0 for all x Kwhere 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 x⁰, y⁰ K, u⁰ T(y⁰) and v⁰ T(x⁰), we have u^k + x^k+1 – y^k, x – x^k+1 0, x K, for u^k T(y^k) and for k 0where v^k + y^k – x^k , x – y^k 0, x K and for v^k T(x^k).