Abstract
We introduce a new iterative scheme based on both hybrid method and extragradient method for finding a common element of the solutions set of a system of equilibrium problems, the fixed points set of a nonexpansive mapping, and the solutions set of a variational inequality problems for a monotone and -Lipschitz continuous mapping in a Hilbert space. Some convergence results for the iterative sequences generated by these processes are obtained. The results in this paper extend and improve some known results in the literature.
Highlights
We always assume that H is a real Hilbert space with inner product ·, · and induced norm · and C is a nonempty closed convex subset of H, S : C → C is a nonexpansive mapping; that is, Sx − Sy ≤ x − y for all x, y ∈ C, PC denotes the metric projection of H onto C, and Fix S denotes the fixed points set of S
Peng and Yao 4 introduced a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions of problem 1.1, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping in a Hilbert space and obtained two strong convergence theorems
Peng et al 12 introduced a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed points set of a family of infinitely nonexpansive mappings, and the solution set of a variational inequality for a relaxed coercive mapping in a Hilbert space and obtained a strong convergence theorem
Summary
We always assume that H is a real Hilbert space with inner product ·, · and induced norm · and C is a nonempty closed convex subset of H, S : C → C is a nonexpansive mapping; that is, Sx − Sy ≤ x − y for all x, y ∈ C, PC denotes the metric projection of H onto C, and Fix S denotes the fixed points set of S. Peng and Yao 4 introduced a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions of problem 1.1 , the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping in a Hilbert space and obtained two strong convergence theorems. Peng et al 12 introduced a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed points set of a family of infinitely nonexpansive mappings, and the solution set of a variational inequality for a relaxed coercive mapping in a Hilbert space and obtained a strong convergence theorem. Extend, and improve those corresponding results in 8, 11 and the references therein
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