General iterative methods for monotone mappings and pseudocontractive mappings related to optimization problems

Jong Soo Jung

doi:10.1186/s13663-015-0451-x

Abstract

In this paper, we introduce two general iterative methods for a certain optimization problem of which the constrained set is the common set of the solution set of the variational inequality problem for a continuous monotone mapping and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. Under some control conditions, we establish the strong convergence of the proposed methods to a common element of the solution set and the fixed point set, which is the unique solution of a certain optimization problem. As a direct consequence, we obtain the unique minimum-norm common point of the solution set and the fixed point set.

Highlights

Let H be a real Hilbert space with the inner product ·, · and the induced norm ·
Under well-known control conditions on the sequence {αn} ⊂ [, ], they proved the strong convergence of the sequence {xn} generated by ( . ) to a point x ∈ Fix(S), which is the unique solution of the variational inequality (A – γ f )x, x – p ≤, ∀p ∈ Fix(S), which is the optimality condition for the optimization problem min Ax, x – h(x), x∈Fix(S)
Under different control conditions on the sequence {αn} ⊂ [, ] and the sequence {xn} generated by ( . ), he showed the strong convergence of the sequence {xn} to a point x ∈ Fix(T), which is the unique solution of the optimization problem min μ

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and the induced norm ·. ) coupled with the fixed point problem, many authors have introduced some iterative methods for finding an element of VI(C, F) ∩ Fix(S), where F is an α-inverse-strongly monotone mapping and {xt} converges strongly as t → to a point x ∈ VI(C, F), where is the unique solution of the optimization problem min μ x∈VI(C,F)

Results

Conclusion