A new approach to the extragradient method for nonlinear variational inequalities. Issue 4 (2000) - Verma, Ram U.
Abstract : The approximation-solvability of the following nonlinear variational inequality (NVI) problem is presented: Determine an elementx ∗ ∈ K such that〈 T ( x ∗ ), x − x ∗ 〉 ≥ 0   for   all   x ∈ K, whereT : K → H is a mapping from a nonempty closed convex subsetK of a real Hilbert spaceH intoH . The iterative procedure is characterized as a nonlinear variational inequality (for any arbitrarily chosen initial pointx 0 ∈ K )〈 ρ T ( P K [ x k − ρ T ( x k ) ] ) + x k + 1 − x k, x − x k + 1 〉 ≥ 0 for   all   x ∈ K   and   for   k ≥ 0, which is equivalent to a double projection formulax k + 1 = P K [ x k − ρ T ( P K [ x k − ρ T ( x k ) ] ) ], whereP K denotes the projection ofH ontoK .