A new class of iterative algorithms for approximation-solvability of nonlinear variational inequalities

Consider the convergence of the projection methods based on a new iterative algorithm for the approximation-solvability of the following class of nonlinear variational inequality (NVI) problems: find an element x* K such that T(x*),x − x* 0, for allx K, where T : K → H is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. The new iterative procedure is characterized as a nonlinear variational inequality (for any arbitrarily chosen initial point x0 K, and for constants > and β > 0) pT(yk) + xk+1 − yk,x − xk+1 0, for all x K, and for k 0; where βT(xk) + yk − xk, x − yk 0, for all x K. This nonlinear variational inequality type algorithm has an equivalent projection formula where for the projection PK onto K.