A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

Pongsakorn Sunthrayuth,Poom Kumam

doi:10.1155/2011/560671

Abstract

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

Highlights

Throughout this paper we denoted by Æ and Ê the set of all positive integers and all positive real numbers, respectively
We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping
Using the viscosity approximation method, Moudafi 6 introduced the iterative process for nonexpansive mappings see 3, 7 for further developments in both Hilbert and Banach spaces and proved that if H is a real Hilbert space, the sequence {xn} generated by the following algorithm: x0 ∈ C chosen arbitrarily, 1.4 xn 1 αnf xn 1 − αn Txn, ∀n ≥ 0, where f : C → C is a contraction mapping with constant α ∈ 0, 1 and {αn} ⊂ 0, 1 satisfies certain conditions, converges strongly to a fixed point of T in C which is unique solution x∗

Summary

Introduction

Throughout this paper we denoted by Æ and Ê the set of all positive integers and all positive real numbers, respectively. Using the viscosity approximation method, Moudafi 6 introduced the iterative process for nonexpansive mappings see 3, 7 for further developments in both Hilbert and Banach spaces and proved that if H is a real Hilbert space, the sequence {xn} generated by the following algorithm: x0 ∈ C chosen arbitrarily, 1.4 xn 1 αnf xn 1 − αn Txn, ∀n ≥ 0, where f : C → C is a contraction mapping with constant α ∈ 0, 1 and {αn} ⊂ 0, 1 satisfies certain conditions, converges strongly to a fixed point of T in C which is unique solution x∗

Results

Conclusion