Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaces

Lawan Bulama Mohammed,Abba Auwalu,Afis Saliu

doi:10.1186/1687-1812-2013-202

Abstract

In this paper, we study synchronal and cyclic algorithms for finding a common fixed point of a finite family of strictly pseudocontractive mappings, which solve the variational inequality where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, is a positive integer, are arbitrary fixed constants, and are N-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real q-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors. MSC:47H06, 47H09, 47H10, 47J05, 47J20, 47J25.

Highlights

Let E be a real Banach space, and let E∗ be the dual of E
Where T is a nonexpansive mapping of H, f is a contraction, A is a linear bounded strongly positive operator, and {αn} is a sequence in (, ) satisfying the following conditions: (M ) limn→∞ αn = ; (M )
They proved that the sequence {xn} converges strongly to a fixed point xof T, which solves the variational inequality (γ f – A)x, x – x ≤, ∀x ∈ F(T)

Summary

Introduction

Let E be a real Banach space, and let E∗ be the dual of E. In , Marino and Xu [ ] considered the following general iterative method: Starting with an arbitrary initial point x ∈ H, define a sequence {xn} by xn+ = αnγ f (xn) + (I – αnA)Txn, n ≥ , where T is a nonexpansive mapping of H, f is a contraction, A is a linear bounded strongly positive operator, and {αn} is a sequence in ( , ) satisfying the following conditions: They proved that the sequence {xn} converges strongly to a fixed point xof T, which solves the variational inequality (γ f – A)x, x – x ≤ , ∀x ∈ F(T). Where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, N ≥ is a positive integer, γ , μ > are arbitrary fixed constants, and {Ti}Ni= are N -strict pseudocontractions defined on a closed convex subset C of a real q-uniformly smooth Banach space E whose norm is uniformly Gâteaux differentiable.

Synchronal algorithm

Cyclic algorithm

Conclusion