Algorithms for solving common fixed point problems
In this textbook, the author studies approximate solutions of common fixed point and convex feasibility problems in the presence of perturbations. In order to to serve the purpose, he considers a number of algorithms, which are known as important tools for the purpose. The author discusses both summable perturbations and nonsummable perturbations. The book contains eight chapters. Chapter 1 is an introduction. In Chapter~2, iterative methods in metric spaces are studied. The dynamic string-averaging methods for common fixed point problems in normed spaces are analysed in Chapter~3. Dynamic string methods for common fixed point problems in a metric space are introduced and studied in Chapter~4. Chapter~5 is devoted to the study of the convergence of an abstract version of the algorithm which is known in the literature as component averaged row projections or CARP. In Chapter~6, the author addresses a proximal algorithm for finding a common zero of a family of maximal monotone operators. In Chapter~7, the results of Chapter~6 are extended for a dynamic string-averaging version of the proximal algorithm. In Chapter~8, subgradient projection algorithms for convex feasibility problems are studied for infinite-dimensional Hilbert spaces. Theorems 2.1 and 3.1 here are known, but all other results are new. Three things, in our opinion, are important in approximating (common) fixed points: the underlying space, the iterative algorithm, and operators (or mappings) being used. The book examines different underlying spaces such as metric spaces, normed spaces, and Hilbert spaces; different algorithms such as the dynamic string-averaging method, dynamic string-maximum method, component-averaged row projections or CARP, proximal point algorithm, its generalization dynamic string-averaging proximal point method, and inexact subgradient projection algorithms; different mappings such as nonexpansive-type mappings and maximal monotone operators. Let $A$ be the underlying space, $B$ the iterative algorithm and $C$ the mapings used. Then the problem addressed in this book (as suggested by the introductions of various chapters) may be more or less formulated as follows: We study the convergence of $B$ to approximate common fixed points of \(C\) under the presence of perturbations in the space $A$. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact algorithm generates an approximate solution if perturbations are summable. We also show that if the perturbations are sufficiently small, then the inexact algorithm produces approximate solutions. The book appears like a bunch of roses of different colours; one may be tempted to say it is a ``Jack of all trades but master of none''. The book shows how to modify and solve the problem in the presence of different tools. Most of chapters are independent of each other. After reading Chapter~1 carefully, the reader may easily decide on which chapter (or part) of the book fits his or her taste or requirement. In our opinion, if someone wishes to master the technique of handling the same idea in different settings, this book can be a good start, provided he or she has a sufficient background knowledge.
scientific article; zbMATH DE number 6856394