## 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:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 90C25 Convex programming 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics 49J40 Variational inequalities

Zbl 0913.47048
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