Strong convergence theorems for equilibrium problems and fixed point problems: A new iterative method, some comments and applications

Zhenhua He,Wei-Shih Du

doi:10.1186/1687-1812-2011-33

Abstract

In this paper, we introduce a new approach method to find a common element in the intersection of the set of the solutions of a finite family of equilibrium problems and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Under appropriate conditions, some strong convergence theorems are established. The results obtained in this paper are new, and a few examples illustrating these results are given. Finally, we point out that some 'so-called' mixed equilibrium problems and generalized equilibrium problems in the literature are still usual equilibrium problems.

Highlights

1 Introduction and preliminaries Throughout this paper, we assume that H is a real Hilbert space with zero vector θ, whose inner product and norm are denoted by 〈·, ·〉 and || · ||, respectively
In this paper, motivated by the preceding Example A, we introduce a new iterative algorithm for the problem of finding a common element in the set of solutions to the system of equilibrium problem and the set of fixed points of a nonexpansive mapping
Application (I) of Theorem 2.1 We will give an iterative algorithm for the following optimization problem with a nonempty common solution set: min x∈K

Summary

Then g is a

The sequences {xn} and {uin}, i = 1, 2, 3, defined by (2.1) all strongly converge to 0. The sequences {xn} and {uin}, for all i Î I, converge strongly to an element c = PΩg (c) Î Ω. If the above control coefficient sequences {an} ⊂ (0, 1) and {rn} ⊂ (0, +∞) satisfy all the restrictions in Theorem 2.1, the sequences {xn} and {un} converge strongly to an element c = PΩg(c) Î Ω, respectively. Application (I) of Theorem 2.1 We will give an iterative algorithm for the following optimization problem with a nonempty common solution set: min x∈K hi(x), i ∈ {1, 2, . {xn} and {uin}, defined by the algorithm (DH), converge strongly to a common fixed point of {A1, A2,..., Ak}, respectively.

For any

For each i

Conclusion