Abstract
We introduce a new iterative scheme for finding a common element of the set of solutions of the equilibrium problems, the set of solutions of variational inequality for a relaxed cocoercive mapping, and the set of fixed points of a nonexpansive mapping. The results presented in this paper extend and improve some recent results of Ceng and Yao (2008), Yao (2007), S. Takahashi and W. Takahashi (2007), Marino and Xu (2006), Iiduka and Takahashi (2005), Su et al. (2008), and many others.
Highlights
Throughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm ·, respectively, C is a nonempty closed and convex subset of H, and PC is the metric projection of H onto C
Suppose that x1 u ∈ C and {xn}, {yn} are given by yn PC xn − λnAxn, 1.17 xn 1 αnu βnxn γnSPC yn − λnAyn, where {αn}, {βn}, and {γn} are the sequences in 0, 1 and {λn} is a sequence in 0, 2α. They proved that the sequence {xn} defined by 1.17 converges strongly to common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings under some parameters controlling conditions
We prove that the sequence {xn} generated by the above iterative scheme converges strongly to a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequalities for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems 2.12, which solves another variational inequality γf q − Aq, q − P ≤ 0, ∀p ∈ F, 1.19 where F F S ∩ VI C, B ∩ EP F and is the optimality condition for the minimization problem minx∈F 1/2 Ax, x − h x, where h is a potential function for γf i.e., h x γf x for x ∈ H
Summary
Throughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm · , respectively, C is a nonempty closed and convex subset of H, and PC is the metric projection of H onto C. Suppose that x1 u ∈ C and {xn}, {yn} are given by yn PC xn − λnAxn , 1.17 xn 1 αnu βnxn γnSPC yn − λnAyn , where {αn}, {βn}, and {γn} are the sequences in 0, 1 and {λn} is a sequence in 0, 2α They proved that the sequence {xn} defined by 1.17 converges strongly to common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings under some parameters controlling conditions. Yao , Ceng and Yao 22 , Su et al 17 , and many others
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