Abstract
In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping. We obtain both weak and strong convergence theorems for the sequences generated by these processes in Hilbert spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ceng, L.C., Yao, J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)
Bigi, G., Castellani, M., Kassay, G.: A dual view of equilibrium problems. J. Math. Anal. Appl. 342, 17–26 (2008)
Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2006)
Su, Y., Shang, M., Qin, X.: An iterative method of solutions for equilibrium and optimization problems. Nonlinear Anal. (2007). doi:10.1016/j.na.2007.08.045
Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)
Plubtieng, S., Punpaeng, R.: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput. 197, 548–558 (2008)
Ceng, L.C., AI-Homidan, S., Ansari, Q.H., Yao, J.C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. (2008). doi: 10.1016/j.cam.2008.03.032
Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–346 (2007)
Zhou, H.Y.: Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 343, 546–556 (2008)
Peng, J.W., Yao, J.C.: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwan. J. Math. 12, 1401–1433 (2008)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Giannessi.
Rights and permissions
About this article
Cite this article
Peng, J.W. Iterative Algorithms for Mixed Equilibrium Problems, Strict Pseudocontractions and Monotone Mappings. J Optim Theory Appl 144, 107–119 (2010). https://doi.org/10.1007/s10957-009-9585-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-009-9585-5