Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings. (English) Zbl 1225.47110
Summary: We introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Our results extend, for example, the recent result of V. Colao, G. Marino and H. -K. Xu [J. Math. Anal. Appl. 344, No. 1, 340–352 (2008; Zbl 1141.47040)] to systems of equilibrium problems.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 47H20 | Semigroups of nonlinear operators |
Keywords:
amenable semigroup; common fixed point; equilibrium problem; iterative algorithm; nonexpansive mapping; projection; variational inequality; viscosity approximationCitations:
Zbl 1141.47040References:
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