A three-step iterative scheme for solving nonlinear ϕ-strongly accretive operator equations in Banach spaces

Safeer Hussain Khan,Nawab Hussain,Arif Rafiq

doi:10.1186/1687-1812-2012-149

Abstract

In this paper, we study a three-step iterative scheme with error terms for solving nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.

Highlights

Let K be a nonempty subset of an arbitrary Banach space X and X∗ be its dual space
(iii) T is called φ-strongly accretive if there exists j(x – y) ∈ J(x – y) and a strictly increasing function φ : [, ∞) → [, ∞) with φ( ) = such that for each x, y ∈ X, Tx – Ty, j(x – y) ≥ φ x – y x – y
We study a three-step iterative scheme with error terms for nonlinear φstrongly accretive operator equations in arbitrary real Banach spaces

Summary

Introduction

Let K be a nonempty subset of an arbitrary Banach space X and X∗ be its dual space. The symbols D(T), R(T) and F(T) stand for the domain, the range and the set of fixed points of T respectively (for a single-valued map T : X → X, x ∈ X is called a fixed point of T iff T(x) = x). (iii) T is called φ-strongly accretive if there exists j(x – y) ∈ J(x – y) and a strictly increasing function φ : [ , ∞) → [ , ∞) with φ( ) = such that for each x, y ∈ X, Tx – Ty, j(x – y) ≥ φ x – y x – y . Chidume and Osilike [ ] proved that each strongly pseudocontractive operator with a fixed point is strictly hemicontractive, but the converse does not hold in general.

Results

Conclusion