Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems

ABSTRACT

In this paper, we introduce a hybrid iterative method for finding a common element of the set of common solutions of generalized mixed equilibrium problems and the set of common fixed points of an finite family of nonexpansive mappings. Furthermore, we show a strong convergence theorem under some mild conditions.

In this paper, we introduce a hybrid iterative method for finding a common element of the set of common solutions of generalized mixed equilibrium problems and the set of common fixed points of an finite family of nonexpansive mappings. Furthermore, we show a strong convergence theorem under some mild conditions.

KEYWORDS

Generalized Mixed Equilibrium Problem, Hybrid Iterative Scheme, Fixed Point, Nonexpansive Mapping, Strong Convergence

Generalized Mixed Equilibrium Problem, Hybrid Iterative Scheme, Fixed Point, Nonexpansive Mapping, Strong Convergence

Cite this paper

nullL. Chen and J. Huang, "Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1213-1220. doi: 10.4236/am.2011.210169.

nullL. Chen and J. Huang, "Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems,"

References

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[2] D. R. Sahu, N.-C. Wong and J.-C. Yao, “On Convergence Analysis of an Iterative Algorithm for Finding Common Solution of Generalized Mixed Equilibrium Problems and Fixed Point Problems,” Mathematical Inequlities and Applications, Article in Press.

[3] W. Takahashi and K. Shimoji, “Convergence Theorems for Nonexpansive Mappings and Feasibility Problems,” Mathematical and Computer Modelling, Vol. 32, No. 11, 2000, pp. 1463-1471. doi:10.1016/S0895-7177(00)00218-1

[4] L.-C. Ceng and J.-C. Yao, “A Hybrid Iterative Scheme for Mixed Equilibrium Problems and Fixed Point Problems,” Journal of Computational and Applied Mathematics, Vol. 214, No. 1, 2008, pp. 186-201. doi:10.1016/j.cam.2007.02.022

[5] H. K. Xu, “Viscosity Approximation for Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 298, No. 1, 2004, pp. 279-291. doi:10.1016/j.jmaa.2004.04.059

[6] T. Suzuki, “Strong Convergence of Krasnoselskii and Mann’s Type Se-quences for One-Parameter Nonexpansive Semigroups without Bochner Integrals,” Journal of Mathematical Analysis and Applications, Vol. 305, No. 1, 2005, pp. 227-239. doi:10.1016/j.jmaa.2004.11.017

[7] S. B. Nadle Jr., “Multi-Valued Contraction Mappings,” Pacific Journal of Mathematics, Vol. 30, 1969, pp. 475-488.

[1] W.-Y. Zeng, N.-J Huang and C.-W. Zhao, “Viscosity Approximation Methods for Generalized Mixed Equili-brium Problems and Fixed Points of a Sequence of Non-expansive Mappings,” Fixed Point Theory Applications, Vol. 2008, 2008, Article ID 714939.

[2] D. R. Sahu, N.-C. Wong and J.-C. Yao, “On Convergence Analysis of an Iterative Algorithm for Finding Common Solution of Generalized Mixed Equilibrium Problems and Fixed Point Problems,” Mathematical Inequlities and Applications, Article in Press.

[3] W. Takahashi and K. Shimoji, “Convergence Theorems for Nonexpansive Mappings and Feasibility Problems,” Mathematical and Computer Modelling, Vol. 32, No. 11, 2000, pp. 1463-1471. doi:10.1016/S0895-7177(00)00218-1

[4] L.-C. Ceng and J.-C. Yao, “A Hybrid Iterative Scheme for Mixed Equilibrium Problems and Fixed Point Problems,” Journal of Computational and Applied Mathematics, Vol. 214, No. 1, 2008, pp. 186-201. doi:10.1016/j.cam.2007.02.022

[5] H. K. Xu, “Viscosity Approximation for Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 298, No. 1, 2004, pp. 279-291. doi:10.1016/j.jmaa.2004.04.059

[6] T. Suzuki, “Strong Convergence of Krasnoselskii and Mann’s Type Se-quences for One-Parameter Nonexpansive Semigroups without Bochner Integrals,” Journal of Mathematical Analysis and Applications, Vol. 305, No. 1, 2005, pp. 227-239. doi:10.1016/j.jmaa.2004.11.017

[7] S. B. Nadle Jr., “Multi-Valued Contraction Mappings,” Pacific Journal of Mathematics, Vol. 30, 1969, pp. 475-488.