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E-raamat: Approximate Solutions of Common Fixed-Point Problems

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Approximate Solutions of Common Fixed-Point ProblemsSuurem pilt
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This book presents results on the convergence behavior of algorithms which are known as vital tools for solving convex feasibility problems and common fixed point problems. The main goal for us in dealing with a known computational error is to find what approximate solution can be obtained and how many iterates one needs to find it. According to know results, these algorithms should converge to a solution. In this exposition, these algorithms are studied, taking into account computational errors which remain consistent in practice. In this case the convergence to a solution does not take place. We show that our algorithms generate a good approximate solution if computational errors are bounded from above by a small positive constant.



Beginning  with an introduction, this monograph moves on to study:

· dynamic string-averaging methods for common fixed point problems in a Hilbert space

· dynamic string methods for common fixed point problems in a metric space<

· dynamic string-averaging version of the proximal algorithm

· common fixed point problems in metric spaces

· common fixed point problems in the spaces with distances of the Bregman type

· a proximal algorithm for finding a common zero of a family of maximal monotone operators





· subgradient projections algorithms for convex feasibility problems in Hilbert spaces 

Arvustused

The title says it all: this book is a compilation of studies of algorithms for computing approximate solutions to the problem of finding common fixed points of several operators in the presence of computational errors. The perspective on the analysis of algorithms with fixed computational error is new, and the book is a tutorial on how to execute this analysis for dynamical string-averaging methods, which includes many classical algorithms as special cases. (Russell Luke, Mathematical Reviews, May, 2017)

The present book on fixed point topics focusses on the study of the convergence of iterative algorithms which are mainly intended to approximate solutions of common fixed point problems and of convex feasibility problems in the presence of computational errors. The book, including mainly original theoretical contributions of the author to the convergence analysis of the considered iterative algorithms, is addressed to researchers interested in fixed point theory and/or convex feasibility problems. (Vasile Berinde, zbMATH 1357.49007, 2017)

1 Introduction
1(12)
1.1 Common Fixed Point Problems in a Hilbert Space
1(3)
1.2 Proximal Point Algorithm
4(4)
1.3 Subgradient Projection Algorithms
8(5)
2 Dynamic String-Averaging Methods in Hilbert Spaces
13(36)
2.1 Preliminaries and the Main Result
13(5)
2.2 Proof of Theorem 2.1
18(10)
2.3 Asymptotic Behavior of Inexact Iterates
28(5)
2.4 Proof of Theorem 2.11
33(3)
2.5 Auxiliary Results
36(6)
2.6 A Convergence Result
42(2)
2.7 Asymptotic Behavior of Exact Iterates
44(5)
3 Iterative Methods in Metric Spaces
49(50)
3.1 The First Problem
49(4)
3.2 Proof of Theorem 3.1
53(4)
3.3 Proof of Theorem 3.3
57(2)
3.4 The Second Problem
59(7)
3.5 Proof of Theorem 3.5
66(3)
3.6 Proof of Theorem 3.7
69(3)
3.7 The Third Problem
72(8)
3.8 Proof of Theorem 3.5
80(3)
3.9 Proof of Theorem 3.16
83(5)
3.10 Proof of Theorem 3.18
88(3)
3.11 Proof of Theorem 3.21
91(2)
3.12 Proof of Theorem 3.22
93(1)
3.13 Generic Properties
94(5)
4 Dynamic String-Averaging Methods in Normed Spaces
99(54)
4.1 Preliminaries and the First Problem
99(6)
4.2 Proof of Theorem 4.1
105(7)
4.3 Proof of Theorem 4.3
112(6)
4.4 The Second Problem
118(10)
4.5 Proof of Theorem 4.5
128(8)
4.6 Proof of Theorem 4.6
136(8)
4.7 Proof of Theorem 4.8
144(9)
5 Dynamic String-Maximum Methods in Metric Spaces
153(46)
5.1 Preliminaries and Main Results
153(8)
5.2 Auxiliary Results
161(4)
5.3 Proof of Theorem 5.1
165(5)
5.4 Proof of Theorem 5.2
170(4)
5.5 Proof of Theorem 5.3
174(5)
5.6 Proof of Theorem 5.4
179(4)
5.7 Proof of Theorem 5.5
183(4)
5.8 Proof of Theorem 5.6
187(6)
5.9 Proof of Theorem 5.7
193(6)
6 Spaces with Generalized Distances
199(52)
6.1 Preliminaries and Main Results
199(6)
6.2 Auxiliary Results
205(8)
6.3 Proof of Theorem 6.1
213(4)
6.4 Proof of Theorem 6.2
217(5)
6.5 Proof of Theorem 6.3
222(5)
6.6 Proof of Theorem 6.4
227(3)
6.7 Proof of Theorem 6.5
230(7)
6.8 Proof of Theorem 6.6
237(7)
6.9 Proof of Theorem 6.7
244(7)
7 Abstract Version of CARP Algorithm
251(38)
7.1 Preliminaries and Main Results
251(9)
7.2 Auxiliary Results
260(2)
7.3 Proof of Theorem 7.1
262(3)
7.4 Proof of Theorem 7.2
265(6)
7.5 Proof of Theorem 7.3
271(8)
7.6 Proof of Theorem 7.4
279(10)
8 Proximal Point Algorithm
289(30)
8.1 Preliminaries and Main Results
289(9)
8.2 Auxiliary Results
298(4)
8.3 Proof of Theorem 8.1
302(4)
8.4 Proof of Theorem 8.2
306(3)
8.5 Proof of Theorem 8.3
309(2)
8.6 Proof of Theorem 8.5
311(1)
8.7 Proof of Theorem 8.8
311(2)
8.8 Proof of Theorem 8.9
313(1)
8.9 Proof of Theorem 8.15
314(5)
9 Dynamic String-Averaging Proximal Point Algorithm
319(22)
9.1 Preliminaries and Main Results
319(6)
9.2 Proof of Theorem 9.1
325(10)
9.3 Proof of Theorem 9.2
335(6)
10 Convex Feasibility Problems
341(44)
10.1 Iterative Methods in Infinite-Dimensional Spaces
341(3)
10.2 Proof of Theorem 10.3
344(2)
10.3 Iterative Methods in Finite-Dimensional Spaces
346(3)
10.4 Auxiliary Results
349(1)
10.5 Proof of Theorem 10.4
350(1)
10.6 Proof of Theorem 10.5
351(6)
10.7 Dynamic String-Averaging Methods in Infinite-Dimensional Spaces
357(3)
10.8 Proof of Theorem 10.11
360(6)
10.9 Dynamic String-Averaging Methods in Finite-Dimensional Spaces
366(1)
10.10 Proof of Theorem 10.12
367(2)
10.11 Problems in Finite-Dimensional Spaces with Computational Errors
369(1)
10.12 Proof of Theorem 10.13
370(10)
10.13 Extensions
380(5)
11 Iterative Subgradient Projection Algorithm
385(26)
11.1 Preliminaries
385(3)
11.2 The First Main Result
388(3)
11.3 The Second Main Result
391(2)
11.4 Proofs of Lemmas 11.3 and 11.5
393(2)
11.5 Proofs of Theorems 11.2 and 11.4
395(7)
11.6 The Third Main Result
402(2)
11.7 Auxiliary Results for Theorem 11.7
404(2)
11.8 Proof of Theorem 11.7
406(5)
12 Dynamic String-Averaging Subgradient Projection Algorithm
411(36)
12.1 Preliminaries and the First Main Result
411(5)
12.2 Proof of Theorem 12.1
416(11)
12.3 The Second Main Result
427(2)
12.4 Proof of Theorem 12.2
429(12)
12.5 The Third Main Result
441(2)
12.6 Proof of Theorem 12.3
443(4)
References 447(6)
Index 453
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