The modification of system of variational inequalities for fixed point theory in Banach spaces

Atid Kangtunyakarn

doi:10.1186/1687-1812-2014-123

Abstract

In this paper, we use methods different from extragradient methods to prove a strong convergence theorem for the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and the set of solutions of modification of a system of variational inequalities problems in a uniformly convex and 2-uniformly smooth Banach space. Applying the main result we obtain a strong convergence theorem involving two sets of solutions of variational inequalities problems introduced by Aoyama et al. (Fixed Point Theory Appl. 2006:35390, 2006, doi:10.1155/FPTA/2006/35390) in a uniformly convex and 2-uniformly smooth Banach space. We also give a numerical example to support our result.

Highlights

Let E be a real Banach space with its dual space E∗ and let C be a nonempty closed convex subset of E
Lemma . ([ ]) Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and let T be a nonexpansive mapping of C into itself with F(T) = ∅
Proof First, we show that QC(I – λAA) and QC(I – λBB) are nonexpansive mappings

Summary

Introduction

Let E be a real Banach space with its dual space E∗ and let C be a nonempty closed convex subset of E. ) and an element of the set of fixed points of two finite families of nonexpansive and strictly pseudocontractive mappings in a uniformly convex and -uniformly smooth Banach space. We obtain a strong convergence theorem involving two sets of solutions of variational inequalities problems introduced by Aoyama et al [ ] in a uniformly convex and -uniformly smooth Banach space. ([ ]) Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and let T be a nonexpansive mapping of C into itself with F(T) = ∅. ([ ]) 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).

Main results

Conclusion