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Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have been described in many publications. In this monograph we try to present the methods in a consolidated way. We introduce several classes of operators, examine their properties, define iterative methods generated by operators from these classes and present general convergence theorems. On this basis we discuss the conditions under which particular methods converge. A large part of the results presented in this monograph can be found in various forms in the literature (although several results presented here are new). We have tried, however, to show that the convergence of a large class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems.
List of contents
1 Introduction.- 2 Algorithmic Operators.- 3 Convergence of Iterative Methods.- 4 Algorithmic Projection Operators.- 5 Projection methods.
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From the reviews:
“Cegielski provides us with a very carefully written monograph on solving convex feasibility (and more general fixed point) problems. … Cegielski’s monograph can serve as an excellent source for an upper-level undergraduate or graduate course. … researchers in this area now have a valuable source of recent results on projection methods to which the author contributed considerably in his work over the past two decades. In summary, I highly recommend this book to anyone interested in projection methods, their generalizations and recent developments.” (Heinz H. Bauschke, Mathematical Reviews, July, 2013)
“This book is mainly concerned with iterative methods to obtain fixed points. … this book is an excellent introduction to various aspects of the iterative approximation of fixed points of nonexpansive operators in Hilbert spaces, with focus on their important applications to convex optimization problems. It would be an excellent text for graduate students, and, by the way the material is structured and presented, it will also serve as a useful introductory text for young researchers in this field.” (Vasile Berinde, Zentralblatt MATH, Vol. 1256, 2013)
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From the reviews:
"Cegielski provides us with a very carefully written monograph on solving convex feasibility (and more general fixed point) problems. ... Cegielski's monograph can serve as an excellent source for an upper-level undergraduate or graduate course. ... researchers in this area now have a valuable source of recent results on projection methods to which the author contributed considerably in his work over the past two decades. In summary, I highly recommend this book to anyone interested in projection methods, their generalizations and recent developments." (Heinz H. Bauschke, Mathematical Reviews, July, 2013)
"This book is mainly concerned with iterative methods to obtain fixed points. ... this book is an excellent introduction to various aspects of the iterative approximation of fixed points of nonexpansive operators in Hilbert spaces, with focus on their important applications to convex optimization problems. It would be an excellent text for graduate students, and, by the way the material is structured and presented, it will also serve as a useful introductory text for young researchers in this field." (Vasile Berinde, Zentralblatt MATH, Vol. 1256, 2013)
