A new approach to the extragradient method for nonlinear variational inequalities. Issue 4 (2000)

• Record Type: Journal Article
• Authors: Verma, Ram U.
• Is Part Of: Journal of inequalities and applications. Volume 5:Issue 4(2000)
• Page Start: 407
• Page End: 418
• Publication Date: 2000
• Subjects: Extragradient method; g-α-cocoercive mapping; Double projection equation; Nonlinear quasivariational inequalities; Iterative algorithms; Expanding mappings; Inequalities (Mathematics) -- Periodicals; 512.97
• DOI: 10.1155/S1025583400000229
• Languages: English
• ISSNs: 1029-242X

Abstract

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 .