Abstract
We introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a -inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesaro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang (2009), Peng and Yao (2009), Shimizu and Takahashi (1997), and some authors.
Highlights
Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by ·, · and ·, respectively and let C be a nonempty closed convex subset of H
In 2008, Peng and Yao 20 introduced an iterative algorithm based on extragradient method which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space
In this paper, motivated by the above results and the iterative schemes considered in 9, 18–20, we introduce a new iterative process below based on viscosity and Cesaro mean approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for a β-inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space
Summary
Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by ·, · and · , respectively and let C be a nonempty closed convex subset of H. In 2008, Peng and Yao 20 introduced an iterative algorithm based on extragradient method which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. In this paper, motivated by the above results and the iterative schemes considered in 9, 18–20 , we introduce a new iterative process below based on viscosity and Cesaro mean approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for a β-inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space. We extend and improve the corresponding results of Kumam and Katchang 9 , Peng and Yao 20 , Shimizu and Takahashi 18 and some authors
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