Weak and strong convergence theorems of implicit iteration process on Banach spaces

Lai-Jiu Lin,Chih-Sheng Chuang,Zenn-Tsun Yu

doi:10.1186/1687-1812-2011-96

Abstract

In this article, we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition (C). Next, we consider strong convergence theorem of an implicit-shrinking iterative process for two nonexpansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong convergence theorem concerning two nonexpansive mappings and the resolvent of a maximal monotone operator in a Banach space.

Highlights

Let E be a Banach space, and let C be a nonempty closed convex subset of E
A mapping T: C ® E is nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for every x, y Î C
A mapping T: C ® E is quasinonexpansive if F(T) = ∅ and ||Tx - y|| ≤ ||x - y|| for all x Î C and y Î F(T)

Summary

Introduction

Let E be a Banach space, and let C be a nonempty closed convex subset of E. (Here the mod N function takes values in I.) And they proved the weak convergence of process (1.3) to a common fixed point in the setting of a Hilbert space. Let E be a Banach space, C be a nonempty closed convex subset of E, and let T1, T2 : C ® C be two nonexpansive mappings, and let S: C ® C be a mapping which satisfy condition (C). We first consider the weak convergence theorems for the following implicit iterative process: x0 ∈ C chosen arbitrary, xn = anxn−1 + bnSxn−1 + cnT1xn + dnT2xn,. For this reason, we consider the following shrinking-implicit iterative processes and study the strong convergence theorem.

Cn Dn

Take any p