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Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. (English) Zbl 1095.47049
In this very interesting paper, the authors prove that the strong convergence theorem of the method for nonexpansive mappings can be extended to convergence theorems of the method for the variational inequality problem over the fixed point set of a quasi-nonexpansive mapping. The main results can be regarded as generalizations of the convergence theorems of the method for nonexpansive mappings (cf.  among others, the results of F. Deutsch and I. Yamada [Numer. Funct. Anal. Optimization 19, No. 1–2, 33–56 (1998; Zbl 0913.47048)]).
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
90C25Convex programming
90C30Nonlinear programming
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
47J20Inequalities involving nonlinear operators
47N10Applications of operator theory in optimization, convex analysis, programming, economics
49J40Variational methods including variational inequalities