Nadezhkina, Natalia; Takahashi, Wataru Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. (English) Zbl 1143.47047 SIAM J. Optim. 16, No. 4, 1230-1241 (2006). The authors establish the strong convergence of an iteration scheme to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a Lipschitz-continuous and monotone mapping in a real Hilbert space. The iteration scheme is based on the well-known hybrid and extragradient methods. They also introduce an iteration scheme that converges strongly to a common fixed point of two mappings, one of which is nonexpansive and the other is Lipschitz-continuous and pseudocontractive. Reviewer: Naseer Shahzad (Jeddah) Cited in 8 ReviewsCited in 130 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47J20 Variational and other types of inequalities involving nonlinear operators (general) Keywords:hybrid method; extragradient method; fixed point; monotone mapping; nonexpansive mapping; strong convergence; variational inequality PDFBibTeX XMLCite \textit{N. Nadezhkina} and \textit{W. Takahashi}, SIAM J. Optim. 16, No. 4, 1230--1241 (2006; Zbl 1143.47047) Full Text: DOI