A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. (English) Zbl 1154.47053

Authors’ abstract: We introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for \(\alpha\)-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of Y.–H.Yao and J.–C.Yao [Appl.Math.Comput.186, No.2, 1551–1558 (2007; Zbl 1121.65064)].


47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H10 Fixed-point theorems
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.


Zbl 1121.65064
Full Text: DOI


[1] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 23-145 (1994) · Zbl 0888.49007
[2] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701
[3] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming using proximal-like algorithms, Math. Program, 78, 29-41 (1997) · Zbl 0890.90150
[4] Liu, F.; Nashed, M. Z., Regularization of nonlinear Ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6, 313-344 (1998) · Zbl 0924.49009
[5] Nadezhkina, N.; Takahashi, W., Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128, 191-201 (2006) · Zbl 1130.90055
[6] Osilike, M. O.; Igbokwe, D. I., Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl., 40, 559-567 (2000) · Zbl 0958.47030
[7] Suzuki, T., Strong convergence of krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without bochner integrals, J. Math. Anal. Appl., 305, 227-239 (2005) · Zbl 1068.47085
[8] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 311, 1, 506-515 (2007) · Zbl 1122.47056
[9] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118, 417-428 (2003) · Zbl 1055.47052
[10] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036
[11] Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060
[12] Yao, J.-C.; Chadli, O., Pseudomonotone complementarity problems and variational inequalities, (Crouzeix, J. P.; Haddjissas, N.; Schaible, S., Handbook of Generalized Convexity and Monotonicity (2005), Kluwer Academic), 501-558 · Zbl 1106.49020
[13] Yao, Y.; Yao, J.-C., On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., 186, 2, 1551-1558 (2007) · Zbl 1121.65064
[14] Zeng, L. C.; Schaible, S.; Yao, J. C., Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124, 725-738 (2005) · Zbl 1067.49007
[15] Zeng, L. C.; Yao, J. C., Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math., 10, 1293-1303 (2006) · Zbl 1110.49013
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.