Abstract
The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.
Highlights
Throughout this paper, we assume that H is a real Hilbert space with inner product and norm being denoted by ·, · and ·, respectively, 2H denoting the family of all subsets of H and leting C be a closed convex subset of H
A mapping S : C → C is called nonexpansive if Sx − Sy ≤ x − y, for all x, y ∈ C
It is assumed throughout the paper that S is a nonexpansive mapping such that F S / ∅
Summary
Throughout this paper, we assume that H is a real Hilbert space with inner product and norm being denoted by ·, · and · , respectively, 2H denoting the family of all subsets of H and leting C be a closed convex subset of H. Following from W-mappings, Peng and Yao 29 introduced iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings, and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. In this paper, motivated by the above results and the iterative schemes considered by Zhang et al in 6 , Peng et al in 22 , Peng and Yao in 29 , and Plubtieng and Sriprad in 23 , we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. The results presented in this paper extend and improve the results of Zhang et al 6 , Peng et al 22 , Peng and Yao 29 , Plubtieng and Sriprad 23 , and some authors
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