Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space

Poom Kumam,Pongsakorn Sunthrayuth

doi:10.1186/1687-1812-2012-65

Abstract

In this article, we introduce iterative methods (implicit and explicit) for finding a common fixed point set of a countable family of strict pseudo-contractions, which is a unique solution of some variational inequality. Furthermore, we prove the strong convergence theorems of such iterative scheme in a q-uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping. The results presented in this article extend and generalize the corresponding results announced by Yamada and Ceng et al. from Hilbert spaces to Banach spaces. Mathematics Subject Classification 2010: 47H09; 47J05; 47J20; 65J15.

Highlights

Let X be a real Banach space, and X* be its dual space
Can we extend the iterative method of scheme (1.14) to a general iterative scheme define over the set of fixed points of a countable family of strict pseudocontractions
The purpose of this article is to give the affirmative answers to these questions mentioned above, motivated by Yamada [19], Tian [20] and Ceng et al [21], we introduce a general iterative method for finding a common fixed point set of a countable family of strict pseudo-contractions, which is a unique solution of some variational inequality

Summary

Introduction

Let X be a real Banach space, and X* be its dual space. Let U = {x Î X: ∥x∥ = 1}. In 2010, Tian [20] combined the iterative method (1.6) with the Yamada’s method (1.9) and considered a general iterative method for a nonexpansive mapping T as follows: xn+1 = αnγ f (xn) + (I − αnμF)Txn, ∀n ≥ 0 He proved that the sequence {xn} generated by (1.11) converges strongly to the unique solution of variational inequality (γ f − μF)x∗, x − x∗ ≤ 0, ∀x ∈ Fix(T). Ceng et al [21] introduced implicit and explicit iterative schemes for finding the fixed points of a nonexpansive mapping T on a nonempty, closed and convex subset C in a real Hilbert space H as follows: xt = PC[tγ Vxt + (I − tμF)Txt].

Main results

Implicit iteration scheme

Explicit iteration scheme