Algorithm of a new variational inclusion problem and strictly pseudononspreading mapping with application

Wongvisarut Khuangsatung,Atid Kangtunyakarn

doi:10.1186/1687-1812-2014-209

Abstract

The purpose of this research is to modify the variational inclusion problems and prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problems in Hilbert space. By using our main result, we prove a strong convergence theorem involving a κ-quasi-strictly pseudo-contractive mapping in Hilbert space. We give a numerical example to support some of our results.

Highlights

Throughout this article, let H be a real Hilbert space with inner product ·, · and norm ·
In, Kangtunyakarn [ ] introduced an iterative algorithm for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inequality problems as follows: wn+ = αnu + βnPC I – λn(I – S) wn + γnSwn, ∀n ∈ N, and proved a strong convergence theorem of the sequence {wn} under suitable conditions of the parameters {αn}, {βn}, {γn}, and {λn}
In this paper, motivated by the research described above, we prove fixed point theory involving the modified variational inclusion and introduce iterative scheme for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem

Summary

Introduction

Throughout this article, let H be a real Hilbert space with inner product ·, · and norm ·. In , Kangtunyakarn [ ] introduced an iterative algorithm for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inequality problems as follows: wn+ = αnu + βnPC I – λn(I – S) wn + γnSwn, ∀n ∈ N, and proved a strong convergence theorem of the sequence {wn} under suitable conditions of the parameters {αn}, {βn}, {γn}, and {λn}. By using the same method as our main theorem, we prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space Applying such a problem, we have a convergence theorem associated with a nonspreading mapping.

Main result

Conclusion