An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces

Ming Tian

doi:10.1155/2011/173430

Abstract

We use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.

Highlights

Let H be a real Hilbert space and C a closed convex subset of H, and let φ be a bifunction of C × C into R, where R is the set of real numbers
A mapping T of C into itself is nonexpansive if Tx−Ty ≤ x−y, for all x, y ∈ C
In 2006, Marino and Xu 6 introduced the general iterative method and proved that for a given x0 ∈ H, the sequence {xn} is generated by the algorithm xn 1 αnγ f xn I − αnA Txn, n ≥ 0, 1.3 where T is a self-nonexpansive mapping on H, f is a contraction of H into itself with β ∈ 0, 1 and {αn} ⊂ 0, 1 satisfies certain conditions, and A is a strongly positive bounded linear operator on H and converges strongly to a fixed-point x∗ of T which is the unique solution to the following variational inequality: γ f − A x∗, x − x∗ ≤ 0, for x ∈ F T, and is the optimality condition for some minimization problem

Summary

Introduction

Let H be a real Hilbert space and C a closed convex subset of H, and let φ be a bifunction of C × C into R, where R is the set of real numbers. In 2006, Marino and Xu 6 introduced the general iterative method and proved that for a given x0 ∈ H, the sequence {xn} is generated by the algorithm xn 1 αnγ f xn I − αnA Txn, n ≥ 0, 1.3 where T is a self-nonexpansive mapping on H, f is a contraction of H into itself with β ∈ 0, 1 and {αn} ⊂ 0, 1 satisfies certain conditions, and A is a strongly positive bounded linear operator on H and converges strongly to a fixed-point x∗ of T which is the unique solution to the following variational inequality: γ f − A x∗, x − x∗ ≤ 0, for x ∈ F T , and is the optimality condition for some minimization problem.

Results

Conclusion