A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces

Atid Kangtunyakarn

doi:10.1186/1687-1812-2013-157

Abstract

In this paper we introduce a new mapping in a uniformly convex and 2-smooth Banach space to prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and two sets of solutions of variational inequality problems. Moreover, we also obtain a strong convergence theorem for a finite family of the set of solutions of variational inequality problems and the set of fixed points of a finite family of strictly pseudo-contractive mappings by using our main result.

Highlights

Throughout this paper, we use E and E∗ to denote a real Banach space and a dual space of E, respectively
Question How can we prove a strong convergence theorem for the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and the set of solutions of variational inequality problems in a uniformly convex and -uniformly smooth Banach space?
By using our main result, we obtain a strong convergence theorem for a finite family of the set of solutions of ( . ) and the set of fixed points of a finite family of strictly pseudo-contractive mappings

Summary

Introduction

Throughout this paper, we use E and E∗ to denote a real Banach space and a dual space of E, respectively. Mann’s iteration algorithm generated by a strict pseudo-contraction in a real -uniformly smooth Banach space as follows.

Results

Conclusion