A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. (English) Zbl 1185.47079

Peng, Jian-Wen; Yao, Jen-Chih

Taiwanese J. Math. 12, No. 6, 1401-1432 (2008).

The authors define the generalized mixed equilibrium problem which has as particular cases various (generalized/mixed) equilibrium, variational and minimization problems. They prove the strong convergence of some sequences generated by certain iterative schemes, inspired by the extragradient and hybrid methods, under some conditions on the initial data. The common limit of these sequences is the nearest projection on a certain closed convex set. It is a fixed point of a nonexpansive map and usually the common solution of one or several (generalized/mixed) equilibrium and/or variational problems. An important role in their approach is played by the KKM multivalued maps, i.e., maps whose image of any finite set contains the convex hull of that finite set.

Reviewer: Cornel Pintea (Cluj-Napoca)

MSC:

47J25

Iterative procedures involving nonlinear operators

Keywords:

generalized mixed equilibrium problem; extragradient method; hybrid method; nonexpansive map; monotone mapping; variational inequality; strong convergence; fixed point