Fixed point theory for nonlinear mappings in Banach spaces and applications

Atid Kangtunyakarn

doi:10.1186/1687-1812-2014-108

Abstract

The purpose of this research is to study a finite family of the set of solutions of variational inequality problems and to prove a convergence theorem for the set of such problems and the sets of fixed points of nonexpansive and strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. We also prove a fixed point theorem for finite families of nonexpansive and strictly pseudo-contractive mappings in the last section.

Highlights

Let E and E∗ be a Banach space and the dual space of E, respectively, and let C be a nonempty closed convex subset of E
Kangtunyakarn [ ] introduced a new mapping in uniformly convex and -smooth Banach spaces to prove a strong convergence theorem for finding a common element of the set of fixed points of finite families of nonexpansive and strictly pseudocontractive mappings and two sets of solutions of variational inequality problems as follows
From ( . ), we introduce the combination of variational inequality problems in Banach spaces as follows: to find a point x∗ ∈ C such that for some j(x – x∗) ∈ J(x – x∗), N

Summary

Introduction

Let E and E∗ be a Banach space and the dual space of E, respectively, and let C be a nonempty closed convex subset of E. Kangtunyakarn [ ] introduced a new mapping in uniformly convex and -smooth Banach spaces to prove a strong convergence theorem for finding a common element of the set of fixed points of finite families of nonexpansive and strictly pseudocontractive mappings and two sets of solutions of variational inequality problems as follows. ) we prove the convergence theorem for a finite family of the set of solutions of variational inequality problems and two sets of fixed points of nonlinear mappings in a Banach space. (See [ ]) Let C be a closed and convex subset of a real uniformly smooth Banach space E, and let T : C → C be a nonexpansive mapping with a nonempty fixed point F(T). Let C be a nonempty closed convex subset of a real smooth Banach space E.

Proof It is easy to see that

Tx γ