Abstract
The purpose of this paper is to study the strong convergence of a modified iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping, as well as the set of solutions of a mixed-equilibrium problem. Our results extend recent results of Takahashi and Takahashi (2007), Marino and Xu (2006), Combettes and Hirstoaga (2005), Iiduka and Takahashi (2005), and many others.
Highlights
Introduction and preliminariesLet H be a real Hilbert space whose inner product and norm are denoted by ·, · and ·, respectively
One can see that the variational inequality problem 1.1 is equivalent to some fixedpoint problems
∀u ∈ C, 1.14 xn 1 αnf xn 1 − αn T yn ∀n ≥ 1 for approximating a common element of the set of fixed points of a non-self nonexpansive mapping and the set of solutions of the equilibrium problem and obtained a strong convergence theorem in a real Hilbert space
Summary
The purpose of this paper is to study the strong convergence of a modified iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping, as well as the set of solutions of a mixed-equilibrium problem. Our results extend recent results of Takahashi and Takahashi 2007 , Marino and Xu 2006 , Combettes and Hirstoaga 2005 , Iiduka and Takahashi 2005 , and many others.
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