Ishikawa iterative sequence with errors for strongly pseudocontractive operators in arbitrary Banach spaces

Yongjin Li

doi:10.1155/s0161171204305272

Abstract

The Ishikawa iterative sequences with errors are studied for Lipschitzian strongly pseudocontractive operators in arbitrary real Banach spaces; some well‐known results of Chidume (1998) and Zeng (2001) are generalized.

Highlights

Let E be an arbitrary real Banach space with norm · and let E∗ be the dual space of E
An operator T : D(T ) ⊂ E → E is said to be accretive if the inequality x − y ≤ x − y + s(T x − T y) holds for every x, y ∈ D(T ) and for all s > 0
An operator T with domain D(T ) and range R(T ) in E is said to be a strong pseudocontraction if there exists t > 1 such that for all x, y ∈ D(T ) and r > 0, the following inequality holds: x − y ≤ (1 + r )(x − y) − r t(T x − T y)

Summary

Introduction

Let E be an arbitrary real Banach space with norm · and let E∗ be the dual space of E. An operator T with domain D(T ) and range R(T ) in E is said to be a strong pseudocontraction if there exists t > 1 such that for all x, y ∈ D(T ) and r > 0, the following inequality holds: x − y ≤ (1 + r )(x − y) − r t(T x − T y) . T is strongly pseudocontractive if and only if there exists k > 0 such that (I − T )x − (I − T )y, j(x − y) ≥ k x − y 2.

Results

Conclusion