A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. (English) Zbl 1127.47053
Summary: We introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of [G.Marino and H.K.Xu, J. Math.Anal.Appl.318, No.1, 43–52 (2006; Zbl 1095.47038); S.Takahashi and W.Takahashi, ibid.331, No.1, 506–515 (2007; Zbl 1122.47056)], and many others.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 49J40 | Variational inequalities |
| 47H10 | Fixed-point theorems |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 90C47 | Minimax problems in mathematical programming |
Keywords:
equilibrium problem; viscosity approximation method; nonexpansive mapping; minimization problem; fixed pointReferences:
| [1] | Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007 |
| [2] | Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150 |
| [3] | Moudafi, A., Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241, 46-55 (2000) · Zbl 0957.47039 |
| [4] | Opial, Z., Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 561-597 (1967) · Zbl 0179.19902 |
| [5] | Marino, G.; Xu, H. K., A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl., 318, 43-52 (2006) · Zbl 1095.47038 |
| [6] | Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331, 1, 506-515 (2007) · Zbl 1122.47056 |
| [7] | Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060 |
| [8] | Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063 |
| [9] | Xu, H. K., Iterative algorithms for nonlinear operators, J. London Math. Soc., 66, 240-256 (2002) · Zbl 1013.47032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
