Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces

Dao-Jun Wen,Yi-An Chen

doi:10.1186/s13660-014-0528-9

Abstract

In this paper, we introduce a general iterative method for a split variational inclusion and nonexpansive semigroups in Hilbert spaces. We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Moreover, a numerical example is given, to illustrate our methods and results, which may be viewed as a refinement and improvement of the previously known results announced by many other authors.

Highlights

Let H and H be real Hilbert spaces with inner product ·, · and norm ·, respectively
Denote by Fix(T ) the common fixed point set of the semigroup T, i.e., Fix(T ) := {x ∈ H : T(s)x = x, ∀s > }
The fixed point problem of nonexpansive mappings and its iterative methods have become an attractive subject, and various algorithms have been developed for solving variational inequalities and equilibrium problems; see [ – ] and the references therein

Summary

Introduction

Let H and H be real Hilbert spaces with inner product ·, · and norm · , respectively. The fixed point problem of nonexpansive mappings and its iterative methods have become an attractive subject, and various algorithms have been developed for solving variational inequalities and equilibrium problems; see [ – ] and the references therein. Marino and Xu [ ] introduced the following general iterative methods to approximate a fixed point of a nonexpansive mapping: xn+ = αnγ f (xn) + (I – αnB)Txn,. To obtain a mean ergodic theorem of nonexpansive mappings, Shehu [ ] proposed an iterative method for nonexpansive semigroups, variational inclusions, and generalized equilibrium problems. In , Moudafi [ ] introduced the following split monotone variational inclusion problem: Find x∗ ∈ H such that. ) to the case of a split variational inclusion and the fixed point problem of a nonexpansive mapping.

Then the sequences

It follows from γ

Setting wn sn sn

From γ