Iterative approximation of fixed points / Vasile Berinde.

Format:

Book

Author/Creator:

Berinde, Vasile, 1955-

Series:

Lecture notes in mathematics, 0075-8434 ; 1912.

Lecture notes in mathematics, 0075-8434 ; 1912

Language:

English

Subjects (All):

Fixed point theory.

Iterative methods (Mathematics).

Physical Description:

xv, 322 pages ; 24 cm.

Edition:

Second revised and enlarged edition.

Place of Publication:

Berlin : Springer, 2007.

Summary:

The aim of this monograph is to give a unified introductory treatment of the most important iterative methods for constructing fixed points of nonlinear contractive type mappings. It summarizes the most significant contributions in the area by presenting, for each iterative method considered (Picard iteration, Krasnoselskij iteration, Mann iteration, Ishikawa iteration etc.), some of the most relevant, interesting, representative and actual convergence theorems. Applications to the solution of nonlinear operator equations as well as the appropriate error analysis of the main iterative methods, are also presented. Due to the explosive number of research papers on the topic (in the last 15 years only more than one thousand articles related to the subject were published), it was felt that such a monograph was imperatively necessary. The volume is useful for authors, editors, and reviewers. It introduces concrete criteria for evaluating and judging the plethora of published papers.

Contents:

1 Pre-Requisites of Fixed Points 3

1.1 The Background of Metrical Fixed Point Theory 3

1.2 Fixed Point Iteration Procedures 15

1.3 Fixed Point Formulation of Typical Functional Equations 19

2 The Picard Iteration 31

2.1 Banach's Fixed Point Theorem 31

2.2 Theorem of Nemytzki-Edelstein 34

2.3 Quasi-Nonexpansive Operators 36

2.4 Maia's Fixed Point Theorem 39

2.5 [phi]-Contractions 41

2.6 Generalized [phi]-Contractions 45

2.7 Weak Contractions 50

3 The Krasnoselskij Iteration 63

3.1 Nonexpansive Operators in Hilbert Spaces 63

3.2 Strictly Pseudocontractive Operators 70

3.3 Lipschitzian and Generalized Pseudocontractive Operators 71

3.4 Pseudo [phi]-Contractive Operators 77

3.5 Quasi Nonexpansive Operators 79

4 The Mann Iteration 89

4.1 The General Mann Iteration 89

4.2 Nonexpansive and Quasi-Nonexpansive Operators 93

4.3 Strongly Pseudocontractive Operators 104

5 The Ishikawa Iteration 113

5.1 Lipschitzian and Pseudo-Contractive Operators in Hilbert Spaces 114

5.2 Strongly Pseudo-Contractive Operators in Banach Spaces 117

5.3 Nonexpansive Operators in Banach Spaces Satisfying Opial's Condition 121

5.4 Quasi-Nonexpansive Type Operators 127

5.5 The Equivalence Between Mann and Ishikawa Iterations 131

6 Other Fixed Point Iteration Procedures 135

6.1 Mann and Ishikawa Iterations with Errors 135

6.2 Modified Mann and Ishikawa Iterations 139

6.3 Ergodic and Other Fixed Point Iteration Procedures 142

6.4 Perturbed Mann Iteration 145

6.5 Viscosity Approximation Methods 147

7 Stability of Fixed Point Iteration Procedures 157

7.1 Stability and Almost Stability of Fixed Point Iteration Procedures 157

7.2 Weak Stability of Fixed Point Iteration Procedures 162

7.3 Data Dependence of Fixed Points 166

7.4 Sequences of Applications and Fixed Points 172

8 Iterative Solution of Nonlinear Operator Equations 179

8.1 Nonlinear Equations in Arbitrary Banach Spaces 180

8.2 Nonlinear Equations in Smooth Banach Spaces 186

8.3 Nonlinear m-Accretive Operator Equations in Reflexive Banach Spaces 193

9 Error Analysis of Fixed Point Iteration Procedures 199

9.1 Rate of Convergence of Iterative Processes 200

9.2 Comparison of Some Fixed Point Iteration Procedures for Continuous Functions 202

9.3 Comparing Picard, Krasnoselskij and Mann Iterations in the Class of Lipschitzian Generalized Pseudocontractions 207

9.4 Comparing Picard, Mann and Ishikawa Iterations in a Class of Quasi Nonexpansive Maps 210

9.5 The Fastest Krasnoselskij Iteration for Approximating Fixed Points of Strictly Pseudo-Contractive Mappings 213

9.6 Empirical Comparison of Some Fixed Point Iteration Procedures 216.

Notes:

First edition published by Editura Efemeride, Baia Mare, Romania, 2002.

Revised thesis (doctoral) - Universitatea Babes-Bolyai, Cluj-Napoca.

Includes bibliographical references (pages [221]-304) and indexes.

ISBN:

9783540722335

3540722335

OCLC:

137248687