Abstract
In this paper, we introduce and analyze a new general iterative scheme by the viscosity approximation method for finding the common element of the set of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set solutions of the variational inequality problems for an ξ-inverse-strongly monotone mapping in Hilbert spaces. We show that the sequence converge strongly to a common element of the above three sets under some parameters controlling conditions. The result extends and improves a recent result of Chang et al. (Nonlinear Anal. 70:3307–3319, 2009) and many others.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aslam Noor, M., Ottli, W.: On general nonlinear complementarity problems and quasi equilibria. Mathematics (Catania). 49, 313–331 (1994)
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 space. J. Math. Anal. Appl. 20, 197–228 (1967)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)
Ceng, L.C., Yao, J.C.: Iterative algorithms for generalized set-valued strong nonlinear mixed variational-like inequalities. J. Optim. Theory Appl. 124, 725–738 (2005)
Chang, S.S., Lee, H.W.J., Chan, C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009)
Colao, V., Marino, G., Xu, H.K.: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)
Deutsch, F., Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point set of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)
Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005)
Jaiboon, C., Kumam, P.: A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Fixed Point Theory Appl. 374815, 32 (2009)
Liu, F., Nashed, M.Z.: Regularization of nonlinear Ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998)
Marino, G., Xu, H.K.: A general iterative method for nonexpansive mapping in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)
Plubtieng, S., Punpaenga, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336(1), 455–469 (2007)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967)
Qin, X., Shang, M., Su, Y.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal. 69, 3897–3909 (2008)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Su, Y., Shang, M., Qin, X.: A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings. J. Appl. Math. Comput. 28, 283–294 (2008)
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)
Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 5, 387–404 (2001)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
Wangkeeree, R.: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fixed Point Theory Appl. 134148, 17 (2008)
Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)
Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)
Yamada, I.: The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithm for Feasibility and Optimization, pp. 473–504. Elsevier, Amsterdam (2001)
Yao, J.C., Chadli, O.: Pseudomonotone complementarity problems and variational in-equalities. In: Crouzeix, J.P., Haddjissas, N., Schaible, S. (eds.) Handbook of Generalized Convexity and Monotonicity, pp. 501–558. Kluwer Academic, Dordrecht (2005)
Yao, Y., Liou, Y.C., Yao, J.C.: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 64363, 12 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by The Commission on Higher Education under the project: “Fixed Point Theorem in Banach spaces and Metric spaces” Ministry of Education and Faculty of Science, King Mongkut’s University of Technology Thonburi.
Rights and permissions
About this article
Cite this article
Jaiboon, C., Kumam, P. A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings. J. Appl. Math. Comput. 34, 407–439 (2010). https://doi.org/10.1007/s12190-009-0330-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-009-0330-x
Keywords
- Nonexpansive mapping
- ξ-inverse-strongly monotone mapping
- Variational inequality problem
- Equilibrium problem
- Fixed points