Iterative Methods for Nonexpansive Type Mappings

Abdul Rahim Khan; Hafiz Fukhar-ud-din

doi:10.1016/B978-0-12-804295-3.50006-1

Iterative Methods for Nonexpansive Type Mappings

Abdul Rahim Khan^*
, Hafiz Fukhar-ud-din

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

The classical fixed point theorem of Goebel and Kirk for a nonexpansive mapping on a uniformly convex Banach space and a CAT(0) space is presented. The exact value of a fixed point for certain mappings cannot be found analytically. In order to find its approximate value, the iterative construction of fixed points becomes essential. In this chapter, fixed point (common fixed point) problems for nonexpansive (asymptotically quasi-nonexpansive) mappings in Banach spaces and some important classes of a metric space are studied; in particular, results on weak convergence, ?-convergence and strong convergence of explicit and multistep schemes of nonexpansive type mappings to a common fixed point in the context of uniformly convex Banach spaces, CAT(0) spaces, hyperbolic spaces and convex metric spaces are presented.

Original language	English
Title of host publication	Fixed Point Theory and Graph Theory
Subtitle of host publication	Foundations and Integrative Approaches
Publisher	Elsevier Inc.
Pages	231-285
Number of pages	55
ISBN (Electronic)	9780128043653
ISBN (Print)	9780128042953
DOIs	https://doi.org/10.1016/B978-0-12-804295-3.50006-1
State	Published - 10 Jun 2016

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc. All rights reserved.

Keywords

Asymptotically nonexpansive mappings
CAT(0) spaces
Fixed points
Generalized asymptotically nonexpansive mappings
Geodesic segment
Iterative methods
Nonexpansive type mappings
Poincare metric
Uniformly convex Banach spaces
Viscosity method

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1016/B978-0-12-804295-3.50006-1

Cite this

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AB - The classical fixed point theorem of Goebel and Kirk for a nonexpansive mapping on a uniformly convex Banach space and a CAT(0) space is presented. The exact value of a fixed point for certain mappings cannot be found analytically. In order to find its approximate value, the iterative construction of fixed points becomes essential. In this chapter, fixed point (common fixed point) problems for nonexpansive (asymptotically quasi-nonexpansive) mappings in Banach spaces and some important classes of a metric space are studied; in particular, results on weak convergence, ?-convergence and strong convergence of explicit and multistep schemes of nonexpansive type mappings to a common fixed point in the context of uniformly convex Banach spaces, CAT(0) spaces, hyperbolic spaces and convex metric spaces are presented.

KW - Asymptotically nonexpansive mappings

KW - CAT(0) spaces

KW - Fixed points

KW - Generalized asymptotically nonexpansive mappings

KW - Geodesic segment

KW - Iterative methods

KW - Nonexpansive type mappings

KW - Poincare metric

KW - Uniformly convex Banach spaces

KW - Viscosity method

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DO - 10.1016/B978-0-12-804295-3.50006-1

M3 - Chapter

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SN - 9780128042953

SP - 231

EP - 285

BT - Fixed Point Theory and Graph Theory

PB - Elsevier Inc.

ER -

Iterative Methods for Nonexpansive Type Mappings

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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