An iterative method of solution for equilibrium and optimization problems. (English) Zbl 1170.47047
Summary: We introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of these three sets. The results of this paper extend and improve the results of H.Iiduka and W.Takahashi [Nonlinear Anal., Theory Methods Appl.61, No.3 (A), 341–350 (2005; Zbl 1093.47058)] and S.Takahashi and W.Takahashi [J. Math.Anal.Appl.331, No.1, 506–515 (2007; Zbl 1122.47056)]. Using the above result, an iterative algorithm for the solution of an optimization problem is obtained.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
| 49J40 | Variational inequalities |
| 90C30 | Nonlinear programming |
Keywords:
nonexpansive mapping; inverse-strongly monotone mapping; variational inequality; equilibrium problem; optimization problem; fixed pointReferences:
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