Weak and strong convergence theorems for a finite family of non-Lipschitzian nonself mappings in Banach spaces

Qianglian Huang,L Zhu

doi:10.1186/1687-1812-2013-170

Qianglian Huang, L Zhu

Open Access

https://doi.org/10.1186/1687-1812-2013-170

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Journal: Fixed Point Theory and Applications	Publication Date: Jul 1, 2013
Citations: 1	License type: CC BY 2.0

Affiliation: Yangzhou University

Abstract

In this paper, several weak and strong convergence theorems are established for a new modified iteration with errors for a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Mann-type, Ishikawa-type and Noor-type iterations are covered by this new iteration scheme. Our convergence theorems improve, unify and generalize many important results in the current literature.MSC:47H10, 47H09, 46B20.

Highlights

Fixed-point iteration processes for nonexpansive and asymptotically nonexpansive mappings including Mann-type and Ishikawa-type iterations have been studied extensively by many authors
Cho et al [ ], Xu and Noor [ ] studied weak and strong convergence theorems for the three-step Noor iterations with errors for asymptotically nonexpansive mappings in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable
Chidume et al [ ] gave the weak convergence theorem for such a mapping in a uniformly convex Banach space whose dual space has the Kadec-Klee property. As we know, it remains open whether the weak convergence theorem of a multi-step iteration process with errors for a finite family of continuous nonself mappings which are asymptotically nonexpansive in the intermediate sense holds in a uniformly convex Banach space which satisfies Opial’s condition or whose dual space has the Kadec-Klee property

Summary

Introduction

Fixed-point iteration processes for nonexpansive and asymptotically nonexpansive mappings including Mann-type and Ishikawa-type iterations have been studied extensively by many authors (see [ – ] and the references cited therein). In , Chidume and Bashir Ali [ ] introduced the iterative scheme for a finite family of asymptotically nonexpansive mappings and obtained the weak and strong convergence theorems in a Banach space whose dual space satisfies the Kadec-Klee property.

Results

Conclusion