Approximating the zeros of accretive operators by the Ishikawaiteration process

Zhou Haiyun,Jia Yuting

doi:10.1155/s1085337596000073

Abstract

Some strong convergence theorems are established for the Ishikawa iteration processes for accretive operators in uniformly smooth Banach spaces.

Highlights

Introduction and PreliminariesLet X be a real Banach space with a dual X∗ and normalized duality mapping J : X → 2X∗, defined byJx = {f ∈ X∗ : < f, x >= f x, f = x }, where < ·, · > denotes the generalized duality pairing
Some strong convergence theorems are established for the Ishikawa iteration processes for accretive operators in uniformly smooth Banach spaces
We prove that xnk+1 − x∗ ≤ 2ψ−1( Ax0 ) If not, xnk+1 − x∗ > 2ψ−1( Ax0 )

Summary

Introduction

Some strong convergence theorems are established for the Ishikawa iteration processes for accretive operators in uniformly smooth Banach spaces. ([29]) Let X be a uniformly smooth Banach space and let A : D(A) = X → X be a quasi-accretive, bounded operator which satisfies the condition (I). ([29]) Let X be a uniformly convex Banach space, D ⊂ X a nonempty closed convex subset of X, and T : D → D a quasi-nonexpansive mapping (that is, F (T ) = ∅ and T x − T y ≤ x − y for all x ∈ D and y ∈ F (T )).

Results

Conclusion