Abstract
We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for a β‐inverse‐strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Su et al. (2008), Klin‐eam and Suantai (2009), Tan and Chang (2011), and some other authors.
Highlights
Let C be a closed convex subset of a real Hilbert space H with the inner product ·, · and the norm ·
We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for a β-inverse-strongly monotone mapping in a real Hilbert space
If F ≡ 0, the problem 1.1 is reduced into the mixed variational inequality of Browder type 1 for finding x ∈ C such that
Summary
Let C be a closed convex subset of a real Hilbert space H with the inner product ·, · and the norm ·. In 2005, Iiduka and Takahashi 17 introduced following iterative process for x0 ∈ C: xn 1 αnu 1 − αn SPC xn − λnAxn , ∀n ≥ 0, 1.21 where u ∈ C, {αn} ⊂ 0, 1 , and {λn} ⊂ a, b for some a, b with 0 < a < b < 2β They proved that under certain appropriate conditions imposed on {αn} and {λn}, the sequence {xn} generated by 1.21 converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly monotone mapping say x ∈ C which solve some variational inequality x − u, x − x ≥ 0, ∀x ∈ F S ∩ VI C, A. The purpose of this paper is to show that under some control conditions the sequence {xn} strongly converges to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space
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