Abstract
We first consider an auxiliary problem for the generalized mixed vector equilibrium problem with a relaxed monotone mapping and prove the existence and uniqueness of the solution for the auxiliary problem. We then introduce a new iterative scheme for approximating a common element of the set of solutions of a generalized mixed vector equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings. The results presented in this paper can be considered as a generalization of some known results due to Wang et al. (2010).
Highlights
Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively
We introduce a new iterative scheme for approximating a common element of the set of solutions of a generalized mixed vector equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings
We denote the set of all fixed points of S by F(S), that is, F(S) = {z ∈ X : z = Sz}
Summary
Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. Shan and Huang [20] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem, and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping in Hilbert spaces. By using the result for the auxiliary problem, we introduce a new iterative scheme for finding a common element of the set of solutions of a generalized mixed vector equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings and obtain a strong convergence theorem. The results presented in this paper improve and generalize some known results of Wang et al [9]
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