Abstract
Let be strict pseudo-contractions defined on a closed and convex subset of a real Hilbert space . We consider the problem of finding a common element of fixed point set of these mappings and the solution set of a system of equilibrium problems by parallel and cyclic algorithms. In this paper, new iterative schemes are proposed for solving this problem. Furthermore, we prove that these schemes converge strongly by hybrid methods. The results presented in this paper improve and extend some well-known results in the literature.
Highlights
Let H be a real Hilbert space with inner product ·, · and norm ·
In this paper, motivated by 5, 8–12, applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the fixed point set of a finite family of strict pseudocontractions and the solution set of the system of equilibrium problems 1.1 by the hybrid methods
We apply the hybrid methods to the parallel algorithm for finding a common element of the fixed point set of strict pseudocontractions and the solution set of the problem 1.1 in Hilbert spaces
Summary
Let H be a real Hilbert space with inner product ·, · and norm ·. Combettes and Hirstoaga 1 considered the following system of equilibrium problems: finding x ∈ C such that Fk x, y ≥ 0, ∀k ∈ Γ, ∀y ∈ C, 1.1 where Γ is an arbitrary index set. Acedo and Xu 9 considered the problem of finding a common fixed point of a finite family of strict pseudo-contractive mappings by the parallel and cyclic algorithms. Duan and Zhao 10 considered new hybrid methods for equilibrium problems and strict pseudocontractions. In this paper, motivated by 5, 8–12 , applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the fixed point set of a finite family of strict pseudocontractions and the solution set of the system of equilibrium problems 1.1 by the hybrid methods. 2 ωw xn {x : ∃xnj x} denotes the weak ω-limit set of {xn}
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