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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.
List of contents
Pre-Requisites of Fixed Points.- The Picard Iteration.- The Krasnoselskij Iteration.- The Mann Iteration.- The Ishikawa Iteration.- Other Fixed Point Iteration Procedures.- Stability of Fixed Point Iteration Procedures.- Iterative Solution of Nonlinear Operator Equations.- Error Analysis of Fixed Point Iteration Procedures.
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.
Additional text
From the reviews of the second edition:
"This is a very thoroughly revised and enlarged version of the author’s previous monograph with the same title … . The book is a surprisingly pleasant reading, a very well done elaboration … . I suggest it to all departments of numerical analysis both for researchers, teachers and students. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations." (László
L. Stachó, Acta Scientiarum Mathematicarum, Vol. 74, 2008)
"There is a huge amount of existence results for fixed points, very often from a purely theoretical viewpoint … . Specialists in the field will find the list of more than 1500 references particularly useful … this book may also serve as a guide to the literature." (Jürgen Appell, Zentralblatt MATH, Vol. 1165, 2009)
Report
From the reviews of the second edition:
"This is a very thoroughly revised and enlarged version of the author's previous monograph with the same title ... . The book is a surprisingly pleasant reading, a very well done elaboration ... . I suggest it to all departments of numerical analysis both for researchers, teachers and students. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations." (László L. Stachó, Acta Scientiarum Mathematicarum, Vol. 74, 2008)
"There is a huge amount of existence results for fixed points, very often from a purely theoretical viewpoint ... . Specialists in the field will find the list of more than 1500 references particularly useful ... this book may also serve as a guide to the literature." (Jürgen Appell, Zentralblatt MATH, Vol. 1165, 2009)
