Solutions of Fixed Point Problems with Computational Errors (Springer Optimization and Its Applications, 210) [2024 ed.] 3031508785, 9783031508783

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Springer Optimization and Its Applications  210

Alexander J. Zaslavski

Solutions of Fixed Point Problems with Computational Errors

Springer Optimization and Its Applications Volume 210

Series Editors Panos M. Pardalos My T. Thai

, University of Florida, Gainesville, FL, USA

, CSE Building, University of Florida, Gainesville, FL, USA

Advisory Editors Roman V. Belavkin, Faculty of Science and Technology, Middlesex University, London, UK John R. Birge, University of Chicago, Chicago, IL, USA Sergiy Butenko, Texas A&M University, College Station, TX, USA Vipin Kumar, Dept Comp Sci & Engg, University of Minnesota, Minneapolis, MN, USA Anna Nagurney, Isenberg School of Management, University of Massachusetts Amherst, Amherst, MA, USA Jun Pei, School of Management, Hefei University of Technology, Hefei, Anhui, China Oleg Prokopyev, Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, USA Steffen Rebennack, Karlsruhe Institute of Technology, Karlsruhe, BadenWürttemberg, Germany Mauricio Resende, Amazon (United States), Seattle, WA, USA Tamás Terlaky, Lehigh University, Bethlehem, PA, USA Van Vu, Department of Mathematics, Yale University, New Haven, CT, USA Michael N. Vrahatis, Mathematics Department, University of Patras, Patras, Greece Guoliang Xue, Ira A. Fulton School of Engineering, Arizona State University, Tempe, AZ, USA Yinyu Ye, Stanford University, Stanford, CA, USA Honorary Editor Ding-Zhu Du, University of Texas at Dallas, Richardson, TX, USA

Aims and Scope Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences. The series Springer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS.

Alexander J. Zaslavski

Solutions of Fixed Point Problems with Computational Errors

Alexander J. Zaslavski Department of Mathematics Technion, Israel Institute of Technology Haifa, Israel

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-031-50878-3 ISBN 978-3-031-50879-0 (eBook) https://doi.org/10.1007/978-3-031-50879-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

In this book, we study approximate solutions of star-shaped feasibility problems in the presence of perturbations. A star-shaped problem is to find a point which belongs to the intersection of a given finite family of closed subsets of a Hilbert space which are star-shaped at a common point. Our goal is to show the convergence of algorithms, which are known as important tools for solving convex feasibility problems and common fixed point problems. We also study algorithms based on unions of nonexpansive maps, inconsistent convex feasibility problems and split common fixed point problems. The study of feasibility problems and common fixed point has recently been a rapidly growing area of research. This is due not only to theoretical achievements in this area, but also because of numerous applications to engineering and, in particular, to computed tomography and radiation therapy planning. In the book we consider a number of algorithms used for solving convex feasibility problems and common fixed point problems. According to the results known in the literature, these algorithms should converge to a solution. But it is clear that in practice it is sufficient to find a good approximate solution instead of constructing a minimizing sequence. In our book Approximate Solutions of Common Fixed Point Problems (Springer, 2016), we analyzed these algorithms and showed that almost all exact iterates generated by them are approximate solutions. Moreover, we obtained an estimate of the number of iterates which are not approximate solutions. This estimate depends on the algorithm but does not depend on the starting point. In our more recent monograph Algorithms for Solving Common Fixed Point Problems (Springer, 2018), we generalize these results for perturbed algorithms in the case when perturbations are summable. These generalizations are important because such results find interesting applications and are important ingredients in superiorization and perturbation resilience of algorithms. The superiorization methodology works by taking an iterative algorithm, investigating its perturbation resilience, and then using proactively such perturbations in order to “force” the perturbed algorithm to do in addition to its original task something useful. In this book, we extend the results of these two books to the class of star-shaped feasibility problems. For this class of feasibility problems, we apply iterative methods, the Cimmino algorithm and dynamic string-averaging algorithms which are usually used for v

vi

Preface

convex feasibility problems and analyze approximate solutions in the presence of computational errors. We show that the methods mentioned above generate a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this. It should be mentioned that the study of convex feasibility problems is based on the existence of a projection operator on a convex, closed set. For a star-shaped set, such a projection mapping does not exist. This makes the situation more difficult and less understood. The monograph contains eight chapters. Chap. 1 is an introduction. In Chap. 2, we study iterative methods for star-shaped feasibility problems. These problems are analyzed in Chap. 3 using Cimmino algorithm. The dynamic string-averaging methods for star-shaped feasibility problems are analyzed in Chap. 4. Methods with remotest set control are discussed in Chap. 5. In Chap. 6, we analyze iterative algorithms, which can be described in terms of a structured set-valued operator. Namely, at every point in the ambient space, it is assumed that the value of the operator can be expressed as a finite union of values of single-valued quasinonexpansive operators. For such algorithms, it is shown their global convergence for an arbitrary starting point. We also obtain an approximate solution of the fixed point problem in the presence of computational errors and show that the iterative method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this. In Chap. 7, we study the method of cyclic projections for inconsistent convex feasibility problems in a Hilbert space under the presence of computational errors. Again we show that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. In Chap. 8, we study split common fixed point problems in a Hilbert space under the presence of computational errors. Rishon LeZion, Israel July 31, 2023

Alexander J. Zaslavski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Star-Shaped Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms Based on Unions of Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . . . . Inconsistent Convex Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Split Common Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 10 17 21

2

Iterative Methods in a Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inexact Iterates with Summable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Second Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Third Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Star-Shaped Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 29 39 50 57 69 76

3

The Cimmino Algorithm in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Cimmino Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inexact Cimmino Iterates with Summable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Result with Nonsummable Errors for the Cimmino Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Second Result with Nonsummable Errors for the Cimmino Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Third Result with Nonsummable Errors for the Cimmino Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cimmino Algorithm for Almost Star-Shaped Feasibility Problems . . .

85 85 96

4

Dynamic String-Averaging Methods in Hilbert Spaces . . . . . . . . . . . . . . . . . . Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inexact Iterates with Summable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Second Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108 113 125 131 137 137 140 149 164 175 vii

viii

Contents

The Third Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Almost Star-Shaped Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5

Methods with Remotest Set Control in a Hilbert Space . . . . . . . . . . . . . . . . . Exact Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inexact Iterates with Summable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Second Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Third Result with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Star-Shaped Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213 213 218 227 231 241 245

6

Algorithms Based on Unions of Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . Global Convergence of Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krasnosel’ski-Mann Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cimmino Type Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Convergence of Iterates with Summable Errors . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cimmino Algorithm with Summable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iterates with Nonsummable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Auxiliary Results for Theorem 6.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cimmino Algorithm with Nonsummable Errors. . . . . . . . . . . . . . . . . . . . . . . . Auxiliary Results for Theorem 6.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Behavior of Iterates Without Compactness Assumptions . . . . . . . . . . . . . . Inexact Iterates Without Compactness Assumptions . . . . . . . . . . . . . . . . . . . . . . . . Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251 254 255 257 259 261 263 266 267 270 272 273 276 279 287 293 300 308 310 316 322 329

7

Inconsistent Convex Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 337 339 343 347 351

8

Split Common Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Contents

ix

Proof of Theorem 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 The Split Common Fixed Point Problem with Several Maps . . . . . . . . . . . . . . . 363 Proof of Theorem 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Chapter 1

Introduction

In this book we study approximate solutions of star-shaped feasibility problems in the presence of perturbations. A star-shaped problem is to find a point which belongs to the intersection of a given finite family of closed subsets of a Hilbert space which are star-shaped at a common point. Our goal is to show the convergence of algorithms, which are known as important tools for solving feasibility problems and common fixed point problems. Some of these algorithms are discussed is this chapter. We also discuss algorithms based on unions of nonexpansive maps, inconsistent convex feasibility problems and split common fixed point problems.

Star-Shaped Feasibility Problems In our book we use the following notation. Let .(X, ) be a Hilbert space equipped with an inner product . which induce the norm ||x|| = 1/2 , x ∈ C.

.

For every .x ∈ X and every .r > 0 set B(x, r) = {y ∈ X : ||x − y|| ≤ r}.

.

For each .x ∈ X and each nonempty set .C ⊂ X define d(x, C) = inf{||x − y|| : y ∈ C}.

.

Denote by Card.(A) the cardinality of a set A and suppose that the sum over empty set is zero.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_1

1

2

1 Introduction

For each .z ∈ R 1 set Lz⎦ = max{i : i is an integer and i ≤ z}.

.

Suppose that m is a natural number, .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets and for each .i ∈ {1, . . . , m} and each .x ∈ X define Pi (x) := {v ∈ Ci : ||x − v|| ≤ ||x − y|| for every y ∈ Ci }.

.

Note that in general these sets can be empty but they are nonempty if the space X is finite-dimensional or the sets .Ci , .i = 1, . . . , m are convex. Set C = ∩m i=1 Ci .

.

We study the following feasibility problem Find x ∈ C.

.

If all the sets .Ci , .i = 1, . . . , m are convex this problem is called as the convex feasibility problem. In this case for each .i ∈ {1, . . . , m}, the operator .Pi : X → X is nonexpansive which means that ||Pi (x) − Pi (y)|| ≤ ||x − y||, x, y ∈ X

.

and moreover ||Pi (x) − y)||2 + ||Pi (x) − x||2 ≤ ||x − y||2

.

(1.1)

for each .x ∈ X and each .y ∈ Ci . It should be mentioned that there has been a lot of research activity regarding nonexpansive mappings and mappings satisfying Eq. (1.1). See, for example, [1, 3, 11, 21, 22, 32, 34, 35, 36, 38, 73, 91, 92, 105, 106, 117, 122, 132, 163, 165] and references cited therein. This activity stems from Banach’s classical theorem [12] concerning the existence of a unique fixed point for a strict contraction. It also concerns the convergence of (inexact) iterates of a nonexpansive mapping to one of its fixed points. Since that seminal result, many developments have taken place in this field including, in particular, studies of feasibility [2, 6, 8, 16, 17, 18, 19, 20, 37, 45, 46, 49, 52, 53, 54, 56, 57, 61, 72, 78, 87, 94, 101, 109, 110, 115, 116, 121, 123, 125, 126, 130, 142, 147, 148, 150, 151, 156, 158, 159, 160, 161], common fixed point problems [15, 48, 58, 59, 60, 66, 75, 80, 81, 82, 127, 128, 129, 131, 137, 138, 139, 141, 143, 153, 157, 162, 163, 164, 165], nonlinear operator theory [5, 13, 14, 50, 84, 85, 132, 135, 136, 144, 166, 167, 168, 169, 170, 171], variational inequalities [7, 24, 25, 27, 28, 39, 40, 41, 42, 43, 44, 55, 86, 89, 90, 95, 96, 97, 98, 104, 107, 108, 111, 112, 114, 118, 119, 120, 140, 145, 152] and

Star-Shaped Feasibility Problems

3

proximal point methods [4, 9, 10, 23, 26, 29, 33, 65, 68, 76, 83, 100, 102, 103, 113, 124, 133, 134, 146, 149, 154, 155] which find important applications in engineering, medical and the natural sciences [30, 31, 47, 51, 70, 71, 74, 77, 88, 93, 99]. It should be mentioned that by (1.1) projection algorithms for solving convex feasibility property produce iterates which are called in the literature as Fejer monotone sequences. In our study we do not assume the convexity of the sets .Ci , .i = 1, . . . , m but we made the following assumption. Assume that z∗ ∈ ∩m i=1 Ci

.

and that the following assumption holds: (A1) for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

In other words all the sets .Ci , .i = 1, . . . , m are star-shaped at .z∗ . We call our problem as the star-shaped feasibility problem which is quite natural. For this class of feasibility problems we apply iterative methods, the Cimmino algorithm and dynamic string-averaging algorithms which are usually used for convex feasibility problems and analyze approximate solutions of the problem in the presence of computational errors. We show that the methods mentioned above generate a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this. It should be mentioned that the operators .Pi , .i = 1, . . . , m in the case of star-shaped feasibility problems do not exist. This makes the situation more difficult and less understood. In this section we apply the iterative methods and discuss some of the results of Chap. 2. ¯ Fix a natural number .N. Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer j , {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

.

First we consider the case when .Pi (x) = / ∅ for each .i ∈ {1, . . . , m} and each x ∈ X. Every .r ∈ R generates the following algorithm: Initialization: select an arbitrary .x0 ∈ X.

.

4

1 Introduction

Iterative step: given a current iteration point .xk calculate the next iteration point xk+1 by

.

xk+1 ∈ Pr(k+1) (xk ).

.

The following result describes the behavior of exact iterates of our algorithm. Theorem 1.1 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X. Let M > 0,

.

||z∗ || < M,

.

ɛ ∈ (0, 1), .r ∈ R, .{xi }∞ i=0 ⊂ X,

.

||x0 || ≤ M

.

and let for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi ).

.

Then Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ })

.

.

¯ N¯ + 1)2 M 2 ɛ −2 . < 4N(

We say that .C1 , . . . , Cm have a bounded regularity property (BRP for short) [163] if for each .M > 0 and each . ɛ > 0 there exists .δ > 0 such that for each .x ∈ B(0, M) satisfying d(x, Ci ) ≤ δ, i = 1, . . . , m

.

the inequality .d(x, C) ≤ ɛ holds. Theorem 1.2 Assume that BRP holds and .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X. Let .M > max{1, ||z∗ ||}, . ɛ ∈ (0, 1). Then there exists a natural number Q such that for each .r ∈ R and each .{xi }∞ i=0 ⊂ X satisfying ||x0 || ≤ M

.

and for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi )

.

Star-Shaped Feasibility Problems

5

the inequality Card({i ∈ {0, 1, . . . } : d(xi , C) > ɛ }) < Q

.

holds. Theorem 1.3 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X and that the following property holds: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M, L > 1, .Δ0 ∈ (0, 1), .||z∗ || < M, for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, Ci ) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

hold, . ɛ ∈ (0, Δ0 ) and Q > 16N¯ (N¯ + 1)2 M 2 ɛ −2 L2

.

be an integer. Assume that .r ∈ R and .{xi }∞ i=0 ⊂ X satisfies ||x0 || ≤ M

.

and for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi ).

.

Then there exists an integer .j ∈ {0, . . . , Q} such that ||z − xi || < ɛ for all integers i > j.

.

From now we do not assume anymore that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X. Every .r ∈ R generates the following algorithm. Assume that .{Δi }∞ i=0 ⊂ (0, 1] satisfies ∞ Σ .

i=0

1/2

Δi

< ∞.

6

1 Introduction

Initialization: select an arbitrary .x0 ∈ X. Iterative step: given a current iteration point .xk calculate the next iteration point .xk+1 such that B(xk+1 , Δk ) ∩ {ξ ∈ Cr(k+1) : ||xk − ξ || ≤ ||xk − η|| + Δk : η ∈ Cr(k+1) } /= ∅.

.

The following theorems describes the behavior of inexact iterates with computational errors. Theorem 1.4 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1), .{Δi }∞ i=0 ⊂ (0, 1],

.

∞ Σ .

1/2

Δi

< ∞,

i=0

M1 = 8M + 64(

∞ Σ

.

1/2

Δi )2 + 4,

i=0

M2 = 4M + 32(2M1 + 1) 2

.

3/2

(

∞ Σ

1/2

Δi ),

i=0

p¯ is a natural number and that

.

Δi < ɛ (N¯ + 1)−1 for all integers i ≥ p. ¯

.

Let .r ∈ R, .{xi }∞ i=0 ⊂ X, ||x0 || ≤ M

.

and let for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } /= ∅.

.

Then ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} ≥ ɛ })

.

.

¯ N¯ + 1)2 M2 ɛ −2 . ≤ p¯ + N(

Star-Shaped Feasibility Problems

7

Theorem 1.5 Assume that .M > max{1, ||z∗ ||}, . ɛ ∈ (0, 1), .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

i=0

M1 = 8M + 64(

∞ Σ

.

1/2

Δ i )2 + 4

i=0

and that the bounded regularity property holds. Then there exists a natural number Q such that for each .r ∈ R, each .{xi }∞ i=0 ⊂ X satisfying ||x0 || ≤ M

.

and for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } /= ∅

.

the inequalities ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({j ∈ {0, 1, . . . } : d(xj , C) > ɛ }) < Q

.

hold and lim d(xi , C) = 0.

.

i→∞

Theorem 1.6 Assume that the bounded regularity property and the following property hold: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

i=0

M1 = 8M + 64(

∞ Σ

.

i=0

1/2

Δi )2 + 4,

8

1 Introduction

ɛ ∈ (0, 1). Then there exists a natural number Q such that for each .r ∈ R and each {xi }∞ i=0 ⊂ X satisfying

. .

||x0 || ≤ M

.

and for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } /= ∅

.

the inequality ||xi || ≤ 2M1 , i = 0, 1, . . .

.

holds, there exists .z¯ ∈ C such that ||z − xi || ≤ ɛ for each integer i ≥ 2Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Theorem 1.7 Let M > max{1, ||z∗ ||},

.

¯ −2 (4M + 4)−1 , 0 < δ ≤ 2−1 (2N)

.

1/2 1/2 ¯ ɛ 0 = (32(4M + 4)3/2 N(2δ) )

.

and a natural number .n0 satisfy n0 ≥ 1 + L16M 2 ɛ 0−2 ⎦.

.

Assume that r ∈ R,

.

x0 ∈ B(θ, M)

.

and that .{xi }∞ i=1 ⊂ X satisfies for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} /= ∅.

.

Then there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(θ, 3M + 1), i = 0, . . . , q N¯ ,

.

||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N¯ .

.

Star-Shaped Feasibility Problems

9

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies the equation above, then for each ¯ and all .s = 1, . . . , m, i = q N¯ , . . . , (q + 1)N,

.

d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

It should be mentioned that in Theorem 1.7 .δ is the computational error made by our computer system, we obtain a point z such that .d(z, Cs ) ≤ (N¯ + 2) ɛ 0 and in order to obtain this point we need .(n0 − 1)N¯ iterations. It is not difficult to see that . ɛ 0 = c1 δ 1/4 and .n0 = Lc2 δ −1/2 ⎦ + 1, where .c1 and .c2 are positive constants depending on M. Theorem 1.8 Assume that the bounded regularity property holds, . ɛ ¯ ∈ (0, 1), M¯ > max{1, ||z∗ ||},

.

¯ C ⊂ B(0, M),

.

M > M¯ + 1, .Δ0 ∈ (0, 1) and that the following property holds:

.

(i) for each .ξ ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .η ∈ Cs ∩ B(ξ, Δ0 ), αξ + (1 − α)η ∈ Cs .

.

Let . ɛ ∈ (0, ɛ ). ¯ Then there exist .δ ∈ (0, ɛ ) and a natural number .n1 such that for each .r ∈ R, each x0 ∈ B(0, M)

.

and each .{xi }∞ i=1 ⊂ X satisfying for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} = / ∅

.

the inequality ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

is true and d(xi , C) ≤ ɛ for all integers i ≥ n1 .

.

We have already mentioned that Eq. (1.1) plays a important role in the study of convex feasibility problems. This equation hold for exact projection algorithms but in [163, 165] we deduced from (1.1) its version which is true for inexact projections and allows us to study the convex feasibility problem with computational errors. In

10

1 Introduction

the case of the star-shaped feasibility problem the situation is more difficult and less understood because a projection operator on a star-shaped set in a general Hilbert space does not exist. We can obtain only its approximation. Fortunately, we prove Lemma 2.3 which gives us an analog of Eq. (1.1) for approximate projections on a star-shaped set. This lemma is our main tool in the study of approximate solutions of the star-shaped feasibility problem. In Chap. 3 the results stated above are extended for the Cimmino algorithm introduced in [69]. They are extended for the dynamic string-averaging algorithm in Chap. 4 and for methods with remotest set control in Chap. 5. It should be mentioned that string-averaging algorithms were first introduced by Censor et al. in [53] for solving a convex feasibility problem, when a given collection of sets is divided into blocks and the algorithms operate in such a manner that all the blocks are processed in parallel. Iterative methods for solving common fixed point problems is a special case of dynamic string-averaging methods with only one block. Iterative methods, the Cimmono algorithm and dynamic string-averaging methods are important tools for solving common fixed point problems in a Hilbert space [6, 8, 17, 20, 21, 30, 31, 36, 38, 45, 46, 47, 48, 49, 50, 53, 54, 56, 58, 60, 61, 63, 64, 66, 69, 71, 72, 75, 77, 78, 87, 93, 94, 122, 126, 129, 148, 150, 153]. The results of Chaps. 2–5 are new.

Algorithms Based on Unions of Nonexpansive Maps In Chap. 6 we analyze iterative algorithms, which can be described in terms of a structured set-valued operator. Namely, at every point in the ambient space, it is assumed that the value of the operator can be expressed as a finite union of values of single-valued quasi-nonexpansive operators. For such algorithms it is shown their global convergence for an arbitrary starting point. An analogous result is also proved for the Krasnosel’ski-Mann iterations. Our main goal is also to obtain an approximate solution of the fixed point problem in the presence of computational errors. We show that the iterative method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this. Suppose that .(X, ρ) is a metric space and that .C ⊂ X is its nonempty, closed set. For every point .x ∈ X and every positive number r define B(x, r) = {y ∈ X : ρ(x, y) ≤ r}.

.

For each .x ∈ X and each nonempty set .D ⊂ X set ρ(x, D) = inf{ρ(x, y) : y ∈ D}.

.

Algorithms Based on Unions of Nonexpansive Maps

11

For every operator .S : C → C set Fix(S) = {x ∈ C : S(x) = x}.

.

Fix θ ∈ C.

.

Suppose that the following assumption holds: (A1) For each .M > 0 the set .B(θ, M) ∩ C is compact. Assume that m is a natural number, .Ti : C → C, .i = 1, . . . , m are continuous operators and that the following assumption holds: (A2) For every natural number .i ∈ {1, . . . , m}, every point .z ∈ Fix(Ti ), every point .x ∈ C and every .y ∈ C \ Fix(Ti ), we have ρ(z, Ti (x)) ≤ ρ(z, x)

.

and ρ(z, Ti (y)) < ρ(z, y).

.

Note that operators satisfying (A2) are called paracontractions [85]. Assume that for every point .x ∈ X, a nonempty set φ(x) ⊂ {1, . . . , m}

.

is given. In other words, φ : X → 2{1,...,m} \ {∅}.

.

Suppose that the following assumption holds: (A3) For each .x ∈ C there exists .δ > 0 such that for each .y ∈ B(x, δ) ∩ C, φ(y) ⊂ φ(x).

.

Define T (x) = {Ti (x) : i ∈ φ(x)}

.

for each .x ∈ C, F¯ (T ) = {z ∈ C : Ti (z) = z, i = 1, . . . , m}

.

12

1 Introduction

and F (T ) = {z ∈ C : z ∈ T (z)}.

.

Assume that F¯ (T ) /= ∅.

.

We study the asymptotic behavior of sequences of iterates .xt+1 ∈ F (xt ), .t = 0, 1, . . . . In particular we are interested in their convergence to a fixed point of T . This iterative algorithm was introduced in [144] which also contains its application to sparsity constrained minimisation. It should be mentioned that in the case when .φ(x) = {1, . . . , m} for each .x ∈ X our fixed point problem was already studied in the literature. Without this assumption the situation became more difficult and less understood. The following result, which is proved in section “Proof of Theorem 6.2”, shows that almost all iterates of our set-valued mappings are approximated solutions of the corresponding fixed point problem. It was obtained in [169]. Theorem 1.9 Assume that .M > 0, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) /= ∅.

.

Then there exists an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfy ρ(x0 , θ ) ≤ M

.

and xt+1 ∈ F (xt ) for each integer t ≥ 0

.

the inequality ρ(xt , θ ) ≤ 3M

.

holds for all integers .t ≥ 0, Card({t ∈ {0, 1, . . . , } : ρ(xt , xt+1 ) > ɛ }) ≤ Q

.

and .limt→∞ ρ(xt , xt+1 ) = 0. The following global convergence result was also obtained in [169]. Theorem 1.10 Assume that a sequence .{xt }∞ t=0 ⊂ C and that for each integer .t ≥ 0,

Algorithms Based on Unions of Nonexpansive Maps

13

xt+1 ∈ F (xt ).

.

Then there exist x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 φ(xt ) ⊂ φ(x∗ )

.

and if an integer .i ∈ φ(xt ) satisfies .xt+1 = Ti (xt ), then Ti (x∗ ) = x∗ .

.

The theorem above generalizes the main result of [144] which establishes a local convergence of the iterative algorithm for iterates starting from a point belonging to a neighborhood of a strong fixed point belonging to the set .F¯ (T ). The following result shows that almost all inexact iterates of our set-valued mappings with summable computational errors are approximated solutions of the corresponding fixed point problem. Many results of this type are collected in [163, 165]. Theorem 1.11 Assume that .M > 0, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) /= ∅,

.

{rt }∞ t=0 ⊂ (0, ∞),

.

∞ Σ .

rt < ∞.

t=0

Then there exists an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfy ρ(x0 , θ ) ≤ M

.

and B(xt+1 , rt ) ∩ F (xt ) /= ∅

.

14

1 Introduction

for each integer .t ≥ 0 the inequality ρ(xt , θ ) ≤ 3M +

∞ Σ

.

rj

j =0

holds for all integers .t ≥ 0, Card({t ∈ {0, 1, . . . , } : ρ(xt , xt+1 ) > ɛ }) ≤ Q

.

and .limt→∞ ρ(xt , xt+1 ) = 0. The following result shows the global convergence of inexact iterates of our setvalued mappings with summable computational errors. Theorem 1.12 Assume that .{rt }∞ t=0 ⊂ (0, ∞), ∞ Σ .

rt < ∞.

t=0

a sequence .{xt }∞ t=0 ⊂ C and that for each integer .t ≥ 0, B(xt+1 , rt ) ∩ F (xt ) /= ∅.

.

Then there exist x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 φ(xt ) ⊂ φ(x∗ )

.

and if an integer .i ∈ φ(xt ) satisfies .ρ(xt+1 , Ti (xt )) ≤ rt , then Ti (x∗ ) = x∗ .

.

In the next result we consider inexact iterates with nonsummable errors. Theorem 1.13 Assume that .M > 1, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) /= ∅.

.

Then there exist .δ ∈ (0, ɛ ) and an integer .Q > 2 such that for each sequence {xi }∞ i=Q ⊂ C which satisfy

.

ρ(x0 , θ ) ≤ M

.

Algorithms Based on Unions of Nonexpansive Maps

15

and for each .t = 0, . . . , Q − 1, ρ(xt+1 , T (xt )) ≤ δ

.

there exists an integer .p ∈ [2, Q) such that ρ(xt , θ ) ≤ 3M + 1, t = 1, . . . , p

.

and ρ(xp , xp+1 ) ≤ ɛ .

.

Our previous results stated in this section are obtained under assumptions (A1)– (A3). In particular in (A1) we assume that bounded, closed subsets of C are compact. In Chap. 6 we will extend these results for self-mappings of any closed subset of an arbitrary metric space. Assume that .m ≥ 1 is an integer, .Ti : C → C, .i = 1, . . . , m, .c¯ ∈ (0, 1] and that for each .i ∈ {1, . . . , m}, each .z ∈ Fix(Ti ) and each .y ∈ C, ρ(z, y)2 − ρ(z, Ti (y))2 ≥ cρ(y, ¯ Ti (y))2 .

.

Note that the equation above holds for many nonlinear mappings [163, 165] including projections on closed, convex sets in a Hilbert space. Assume that for each .x ∈ X, φ(x) ⊂ {1, . . . , m}

.

is given. In other words, φ : X → 2{1,...,m} \ {∅}.

.

Define T (x) = {Ti (x) : i ∈ φ(x)}

.

for each .x ∈ C, F¯ (T ) = {z ∈ C : Ti (z) = z, i = 1, . . . , m}

.

and F (T ) = {x ∈ C : x ∈ T (x)}.

.

16

1 Introduction

Assume that F¯ (T ) /= ∅.

.

Fix θ ∈ C.

.

The following result which shows that almost all iterates of our set-valued mappings are approximated solutions of the corresponding fixed point problem was obtained in [166]. Many results of this type are collected in [163, 165]. Theorem 1.14 Assume that .M > 0, . ɛ ∈ (0, 1], F¯ (T ) ∩ B(θ, M) /= ∅

.

and that Q = L4M 2 c¯−1 ɛ −2 ⎦.

.

Then for each sequence .{xi }∞ i=0 ⊂ C which satisfies ρ(θ, x0 ) ≤ M

.

and xi+1 ∈ T (xi ), i = 0, 1, . . .

.

the inequality Card({i ∈ {0, 1, . . . , } : min{ρ(xi , Tj (xi )) : j ∈ φ(xi )} ≥ ɛ }) ≤ Q

.

holds. Theorem 1.15 Assume that .M > 0, . ɛ ∈ (0, 1), F¯ (T ) ∩ B(θ, M) /= ∅,

.

an integer Q satisfies Q > 8M 2 c¯−1 ɛ −2 ,

.

δ ∈ (0, 1) satisfies

.

δ ≤ (4M + 1)−1 ɛ 2 c/2, ¯

.

Inconsistent Convex Feasibility Problems

17

a sequence .{xi }∞ i=0 ⊂ C satisfies ρ(θ, x0 ) ≤ M

.

and that for each integer .i ≥ 0, B(xi+1 , δ) ∩ T (xi ) /= ∅.

.

Then there exists a nonnegative integer .j ≤ Q − 1 such that B(xj , ɛ ) ∩ T (xj ) /= ∅.

.

Inconsistent Convex Feasibility Problems In Chap. 7 we study the method of cyclic projections for inconsistent convex feasibility problems in a Hilbert space under the presence of computational errors. We show that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterations one needs for this. The convex feasibility problem is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the convex feasibility problem is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent feasibility problems exists and is developed in the following two directions (See [2, 18, 19, 52, 57, 62, 67, 70, 74, 94, 101, 123] and the references mentioned therein.). One direction is to define other solution concepts which can be used for inconsistent feasibility problems such as proximity function minimization, wherein a proximity function measures in some way the total violation of all constraints. The second direction analyzed the behavior of algorithms which are used to solve a consistent feasibility problems, when they are applied to inconsistent problems. In Chap. 7 the second direction is chosen. The results presented here were obtained in [168]. Let .(X, ) be a real Hilbert space equipped with an inner product . which induces a complete norm .|| · || in X. For each nonempty convex closed set .C ⊂ X and each .x ∈ X, there exists a unique point .PC (x) ∈ C such that ||PC (x) − x|| = inf{||y − x|| : y ∈ X}.

.

The mapping .PC is called the metric projection on C. Denote by I the identity operator in X. For each mapping .S : X → X, define 0 1 i+1 = S ◦ S i for all integers .i ≥ 0. .S = I , .S = S and .S

18

1 Introduction

A mapping .T : X → X is firmly nonexpansive [19, 79] if for all .x, y ∈ X, ||T (x) − T (y)||2 + ||(I − T )(x) − (I − T )(y)||2 ≤ ||x − y||2 .

.

We will use the following well-known fact. Proposition 1.16 ([19, 79]) Assume that .C ⊂ X is a nonempty closed and convex set. Then the mapping .PC is firmly nonexpansive. More information on firmly nonexpansive mappings can be found in [92]. Assume that .Ci , .i = 1, . . . , m, where .m ≥ 2 is an integer, are closed and convex sets and .Pi = PCi , .i = 1, . . . , m are the corresponding metric projections. We consider the case when .

∩m i=1 Ci = ∅.

If .m = 2, then the solution of our inconsistent feasibility problem is a pair of points .xi ∈ Ci , .i = 1, 2 such that ||x1 − x2 || = inf{||y1 − y2 || : yi ∈ Ci , i = 1, 2}.

.

This condition is equivalent to one in the general case below for .m = 2. In the general case the solution is a finite sequence .x¯i ∈ Ci , .i = 1, . . . , m such that Pi+1 (x¯i ) = x¯i+1 , i = 1, . . . , m − 1, P1 (x¯m ) = x¯1 .

.

It turns out that under certain mild assumptions, the cyclic projections method converges to the solution of our inconsistent problem. More precisely, for each integer .n ≥ 0, set i(n) = n(mod m) + 1

.

and for a given .x0 ∈ X, define xn+1 = Pi(n) (xn ) for all integers n ≥ 0.

.

It was shown in [94] (see also Theorem 2.4 of [62]) that if at least one of the sets .Ci , i = 1, . . . , m is bounded, then there exists a finite sequence .x¯i ∈ Ci , .i = 1, . . . , m such that

.

Pi+1 (x¯i ) = x¯i+1 , i = 1, . . . , m − 1, P1 (x¯m ) = x¯1

.

Inconsistent Convex Feasibility Problems

19

and xkm+i+1 − xkm+i → x¯i+1 − x¯i , i = 1, . . . , m − 1, xkm+1 → x¯1

.

weakly in X, as .k → ∞. In Chap. 7 based on [168] we study the cyclic projections algorithm taking into account computational errors which are always present 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. Clearly, in practice it is sufficient to find a good approximate solution instead of constructing a minimizing sequence. On the other hand in practice computations induce numerical errors and if one uses methods in order to solve feasibility problems, these methods usually provide only approximate solutions of the problems. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterations one needs for this. A finite sequence .{x¯i }m i=1 ⊂ X is considered as an approximate solution of our problems if Pi+1 (x¯i ) = x¯i+1 , i = 1, . . . , m − 1, ||P1 (x¯m ) − x¯1 || ≤ ɛ ,

.

where . ɛ > 0 is a small number. Under the presence of a computational error .δ, a finite sequence .{x¯i }m i=1 ⊂ X is an approximate solution of our problems if ||Pi+1 (x¯i ) − x¯i+1 || ≤ δ, i = 1, . . . , m − 1, ||P1 (x¯m ) − x¯1 || ≤ ɛ ,

.

where the computational error .δ is essentially smaller than . ɛ . As a matter of fact, . ɛ depends on .δ and mappings .Pi , .i = 1, . . . , m. We assume that .

∩m i=1 Ci = ∅.

Assume that x¯i ∈ Ci , i = 1, . . . , m,

.

x¯i+1 = Pi+1 (x¯i ), i = 1, . . . , m − 1, x¯1 = P1 (x¯m ).

.

We mentioned before that such finite sequence exists if at least one of the sets .Ci , i = 1, . . . , m is bounded. Set

.

Pm+1 = P1 , x¯m+1 = x¯1 .

.

The following theorem is the main result of Chap. 7.

20

1 Introduction

Theorem 1.17 Let .M > 0, . ɛ ∈ (0, 1], ||x¯i || ≤ M, i = 1, . . . , m,

.

a nonnegative number .δ satisfy δ < (8m)−1 (2M + 2)−1 ɛ 2

.

and n0 = L8(2M + 1)2 ɛ −2 ⎦ + 1.

.

Assume that a sequence .{xk }∞ k=0 ⊂ X satisfies x0 ∈ B(0, M)

.

and that for each integer .k ≥ 0, xk+1 ∈ B(Pi(k) (xk ), δ).

.

Then the following assertions hold. 1. There exists an integer .k ∈ {0, . . . , n0 } such that for all .i = 1, . . . , m, ||xkm+i+1 − xkm+i − (x¯i+1 − x¯i )|| ≤ ɛ ,

.

if an integer .j ≥ 0 satisfies .j < k, then .

max{||xj m+i+1 − xj m+i − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ

and ||xj m+1 − x¯1 || ≤ 2M + 1, j = 0, . . . , k.

.

Moreover, ||xkm+1 − x(k+1)m+1 || ≤ ɛ m.

.

2. Assume that 16 ɛ m(4M + 2) ≤ 1,

.

k ∈ {0, . . . , n0 },

.

||xkm+1 − x(k+1)m+1 || ≤ ɛ m

.

Split Common Fixed Point Problems

21

and that for all integers j satisfying .0 ≤ j < k, ||xj m+1 − x(j +1)m+1 || > ɛ m.

.

Then ||xkm+1 − x¯1 || ≤ 2M + 1

.

and for all .i = 1, . . . , m, ||xkm+i+1 − xkm+i − (x¯i+1 − x¯i )|| ≤ 4( ɛ m(4M + 2))1/2 .

.

Theorem 1.17 can be used in a very easy way. We need just to find the smallest integer .k ∈ {0, . . . , n0 } for which ||xkm+1 − x(k+1)m+1 || ≤ ɛ m.

.

Then .{xi }(k+1)m i=km+1 is our approximate solution.

Split Common Fixed Point Problems In Chap. 8 we study split common fixed point problems in a Hilbert space under the presence of computational errors. We show that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterations one needs for this. Assume that for .i = 1, 2, .(Hi , ) be Hilbert spaces equipped with inner products which induce the norm ||x|| = 1/2 , x ∈ Hi , i = 1, 2.

.

For each linear bounded operator .A : H1 → H2 denote by .A∗ : H2 → H1 its dual operator. Assume that .U : H1 → H1 , .T : H2 → H2 are mappings .A : H1 → H2 are linear bounded operators. Recall that for each .S : X → X, where X is a nonempty set, .Fix(S) = {x ∈ X : S(x) = x}. Consider the problem Find x ∗ ∈ Fix(U ) such that Ax ∗ ∈ Fix(T ).

.

Set F = {x ∈ Fix(U ); Ax ∈ Fix(T )}.

.

22

1 Introduction

This split common fixed point problem was introduced and studied in [59]. It was also analyzed in [8, 45, 60, 88, 97, 101, 120, 121, 125, 127, 130, 138, 139, 142, 143, 145, 147, 148, 150, 151, 153]. In Chap. 8 we use the algorithm of [59] taking into account computational errors produced by our computer system. Let .(H, ) be Hilbert spaces equipped with the inner product .. Denote by .I : H → H the identity operator. For each .x, y ∈ H define H (x, y) = {u ∈ H : ≤ 0}.

.

An operator .T : H → H is called a directed operator [59] if Fix(T ) ⊂ H (x, T (x)) for all x ∈ H

.

or equivalently if .g ∈ Fix(T ), then ≤ 0 for all x ∈ H.

.

Clearly, if .T : H → H is a directed operator and .Fix(T ) /= ∅, then Fix(T ) = ∩x∈X H (x, T (x))

.

and it is a closed, convex set and for each .λ ∈ [0, 1] the mapping .I + λ(T − I ) is also directed. For each .x ∈ Hi , .i = 1, 2 and each .r > 0 set B(x, r) = {y ∈ Hi : ||x − y|| ≤ r}.

.

Proposition 1.18 ([147]) Let .T : H → H be a mapping. Then the following statements are equivalent. (i) T is directed; (ii) .||x − T (x)||2 ≤ for each .z ∈ Fix(T ) and each .x ∈ H ; (iii) .||z − T (z)||2 ≤ ||x − z||2 − ||x − T (x)||2 for each .z ∈ Fix(T ) and each .x ∈ H . Assume that .T , U are directed mappings and that γ ∈ (0, 2||A||−2 ).

.

The following algorithm was studied in [59]. Initialization: select an arbitrary .x0 ∈ H1 . Iterative step: for an integer .k ≥ 0, given a current iteration point .xk ∈ H1 calculate the next iteration point xk+1 = U (xk + γ A∗ (T − I )(Axk )).

.

Split Common Fixed Point Problems

23

Now we consider this algorithm under the presence of computational errors. Assume that δU , δT , δA ∈ [0, 1].

.

Initialization: select an arbitrary .x0 ∈ H1 . Iterative step: for an integer .k ≥ 0, given a current iteration point .xk ∈ H1 calculate .zk ∈ H2 such that ||zk − (T − I )A(xk )|| ≤ δT ,

.

yk ∈ H1 such that

.

||yk − γ A∗ zk || ≤ δA

.

and .xk+1 ∈ H1 such that ||xk+1 − U (xk + yk )|| ≤ δU .

.

Here .δT is a computational error when we calculate .(T − I )A(xk ), .δA is a computational error when we calculate .γ A∗ zk and .δU is a computational error when we calculate .U (xk + yk ). In Chap. 8 we prove the following result which shows that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. For known computational errors, it shows what approximate solution can be obtained and how many iterations one needs for this. Theorem 1.19 Assume that .M ≥ 1, {x ∈ H1 : U (x) = x, T (Ax) = Ax} ∩ B(0, M) /= ∅,

.

a positive number . ɛ satisfies ɛ ≥ max{4δT , 4δU , 4δA , (32(min{1, γ (2 − γ ||A||2 )−1 })1/2

.

.

× [4M(2δU + δA + γ ||A||δT )(γ ||A||δT + δA )2 + δU2 + 2δU (γ ||A||δT + δA )]1/2 }, Δ = 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )},

.

∞ ∞ x0 ∈ H1 ∩ B(0, M), .{xk }∞ k=0 ⊂ H1 , .{yk }k=0 ⊂ H1 , .{zk }k=0 ⊂ H2 and that for each integer .k ≥ 0,

.

||zk − (T − I )(Axk )|| ≤ δT ,

.

||yk − γ A∗ zk || ≤ δA ,

.

24

1 Introduction

||xk+1 − U (xk + yk )|| ≤ δU .

.

Then there exists a nonnegative integer .n ≤ 4M 2 Δ−1 such that for each integer .k ∈ {0, . . . , n} \ {n}, .

.

max{||xk + yk − xk+1 ||, ||zk ||} ≥ ɛ /2, max{||xn + yn − xn+1 ||, ||zn ||} < ɛ /2, ||xk || ≤ 3M, k = 0, . . . , n

.

If .n ≥ 0 is an integer and .

max{||xn + yn − xn+1 ||, ||zn ||} < ɛ /2,

then ||yn || ≤ 2−1 ɛ (1 + γ ||A||),

.

||T (Axn ) − Axn || < ɛ ,

.

||U (xn + yn ) − (xn + yn )|| < ɛ .

.

By Theorem 1.19, there is a nonnegative integer .n ≤ 4M 2 Δ−1 such that ||T (Axn ) − Axn || < ɛ

.

and there is .ξ ∈ B(xn , 2−1 ɛ (1 + γ ||A||)) such that ||U (ξ ) − ξ || < ɛ .

.

This .xn is an approximate solution of our split common fixed point problem. The results of Chap. 8 are new.

Chapter 2

Iterative Methods in a Hilbert space

In this chapter we study the convergence of iterative methods for solving star-shaped feasibility problems in a Hilbert space. Our main goal is to obtain an approximate solution of the problem in the presence of computational errors. We show that the iterative method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

Preliminaries Let .(X, ) be a Hilbert space equipped with an inner product . which induce the norm ||x|| = 1/2 , x ∈ X.

.

For every .x ∈ X and every .r > 0 set B(x, r) = {y ∈ X : ||x − y|| ≤ r}.

.

For each .x ∈ X and each nonempty set .C ⊂ X define d(x, C) = inf{||x − y|| : y ∈ C}.

.

Denote by Card.(A) the cardinality of a set A and suppose that the sum over empty set is zero.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_2

25

26

2 Iterative Methods in a Hilbert space

Proposition 2.1 Let .z ∈ X, .y, x ∈ X, .y /= z, .x ∈ / {y, z}, ||x − y|| ≤ ||x − (tz + (1 − t)y)|| for all t ∈ [0, 1].

.

Then ≥ 0,

.

||z − x||2 ≥ ||z − y||2 + ||x − y||2 .

.

Proof Define φ(t) = ||tz + (1 − t)y − x||2 , t ∈ R 1 .

.

Clearly, for each .t ∈ R 1 , φ(t) = ||t (z − y) + y − x||2 = t 2 ||z − y||2 + ||y − x||2 + 2t

.

and φ(0) ≤ φ(t), t ∈ [0, 1].

.

Hence 0 ≤ lim (φ(t) − φ(0))t −1 = 2

.

t→0+

and ||z − x||2 − ||z − y||2 − ||x − y||2

.

= −2||y||2 − 2 + 2 + 2

.

.

= 2 ≥ 0.

Proposition 2.1 is proved. It is easy to see that the following proposition holds. Proposition 2.2 Let .x, y ∈ X. Then ||x + y||2 + ||x − y||2 = 2||x||2 + 2||y||2 .

.

The next lemma plays a crucial role in our study.

U ∩

Preliminaries

27

Lemma 2.3 Assume that z∗ ∈ D ⊂ X, x, x˜ ∈ X, δ ∈ (0, 1],

.

ξ ∈ D ∩ B(x, ˜ δ),

(2.1)

||ξ − x|| ≤ d(x, D) + δ,

(2.2)

{αz∗ + (1 − α)ξ : α ∈ [0, 1]} ⊂ D.

(2.3)

.

.

.

Then ||z∗ − x|| ˜ ≤ ||z∗ − x|| + 2(2δ(2||x − z∗ || + 1))1/2 ,

.

||z∗ − x||2 ≥ ||z∗ − x|| ˜ 2 + ||x − x|| ˜ 2

.

.

− 8(2δ(2||x − z∗ || + 1))1/2 ||z∗ − x|| − 16δ(2||x − z∗ || + 1).

Proof Set Ω = {αz∗ + (1 − α)ξ : α ∈ [0, 1]}.

(2.4)

ˆ .ξ ∈ Ω

(2.5)

||x − ˆ ξ || = d(x, Ω).

(2.6)

||x − ˆ ξ || ≤ ||x − ξ || ≤ ||x − ˆ ξ || + δ.

(2.7)

.

Clearly, there exists

such that .

By (2.2)–(2.6), .

Proposition 2.1 and Eqs. (2.4)–(2.6) imply that ||z∗ − ˆ ξ ||2 + ||x − ˆ ξ ||2 ≤ ||x − z∗ ||2 .

.

(2.8)

It follows (2.4) and (2.5) that 2−1 (ξ + ˆ ξ ) ∈ Ω.

.

Proposition 2.2 and Eqs. (2.5)–(2.7) and (2.9) imply that ||ξ − ˆ ξ ||2 = ||x − ˆ ξ − (x − ξ )||2

.

(2.9)

28

2 Iterative Methods in a Hilbert space

= 2||x − ˆ ξ ||2 + 2||x − ξ ||2 − ||x − ˆ ξ + x − ξ ||2

.

= 2||x − ˆ ξ ||2 + 2||x − ξ ||2 − 4||x − 2−1 (ˆ ξ + ξ )||2

.

.

≤ 2||x − ˆ ξ ||2 + 2||x − ξ ||2 − 4||x − ˆ ξ ||2 .

.

≤ 2||x − ξ ||2 − 2||x − ˆ ξ ||2

= 2(||x − ξ || − ||x − ˆ ξ ||)(||x − ξ || + ||x − ˆ ξ ||) .

≤ 2δ(2||x − ˆ ξ || + δ) ≤ 2δ(2x − z∗ || + 1), ||ξ − ˆ ξ || ≤ (2δ(2||x − z∗ || + 1))1/2 .

.

(2.10)

By (2.1) and (2.10), ||x˜ − ˆ ξ || ≤ ||x˜ − ξ || + ||ξ − ˆ ξ ||

.

.

≤ δ + (2δ(2||x − z∗ || + 1))1/2 ≤ 2(2δ(2||x − z∗ || + 1))1/2 .

(2.11)

In view of (2.8) and (2.11), ||z∗ − x|| ˜ ≤ ||z∗ − ˆ ξ || + ||ˆ ξ − x|| ˜

.

.

≤ ||z∗ − x|| + 2(2δ(2||x − z∗ || + 1))1/2 .

(2.12)

In view of (2.12), ||z∗ − x|| ˜ 2 ≤ (||z∗ − x|| + 2(2δ(2||x − z∗ || + 1))1/2 )2

.

.

≤ ||z∗ − x||2 + 4(2δ(2||x − z∗ || + 1))1/2 ||z∗ − x|| .

+ 8δ(2||x − z∗ || + 1).

(2.13)

Equations (2.8) and (2.11) imply that |||z∗ − x|| ˜ 2 − ||z∗ − ˆ ξ ||2 |

.

.

= |||z∗ − x|| ˜ − ||z∗ − ˆ ξ |||(||z∗ − x|| ˜ + ||z∗ − ˆ ξ ||) .

.

.

≤ ||x˜ − ˆ ξ ||(2||z∗ − ˆ ξ || + ||x˜ − ˆ ξ ||)

≤ 2(2δ(2||x − z∗ || + 1))1/2 (2||z∗ − x|| + 2(2δ(2||x − z∗ || + 1))1/2 )

≤ 2(2δ(2||x − z∗ || + 1))1/2 (2||z∗ − x||) + 8δ(2||x − z∗ || + 1).

(2.14)

Exact Iterates

29

It follows from (2.8) and (2.11) that |||x − x|| ˜ 2 − ||x − ˆ ξ ||2 |

.

.

= |||x − x|| ˜ − ||x − ˆ ξ |||(||x − x|| ˜ + ||x − ˆ ξ ||) .

.

≤ ||x˜ − ˆ ξ ||(2||x − ˆ ξ || + ||x˜ − ˆ ξ ||)

≤ 2(2δ(2||x − z∗ || + 1))1/2 (2||z∗ − x|| + 2(2δ(2||x − z∗ || + 1))1/2 ).

(2.15)

It follows from (2.8), (2.14) and (2.15) that ||z∗ − x||2 ≥ ||x − ˆ ξ ||2 + ||ˆ ξ − z∗ ||2

.

.

.

≥ ||z∗ − x|| ˜ 2 + ||x − x|| ˜ 2

− 8(2δ(2||x − z∗ || + 1))1/2 ||z∗ − x|| − 16δ(2||x − z∗ || + 1). U ∩

Lemma 2.3 is proved.

Exact Iterates Suppose that m is a natural number, .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets and for each .i ∈ {1, . . . , m} and each .x ∈ X define Pi (x) := {v ∈ Ci : ||x − v|| ≤ ||x − y|| for every y ∈ Ci }.

.

(2.16)

Note that in general these sets can be empty but they are nonempty if the space X is finite-dimensional or the sets .Ci , .i = 1, . . . , m are convex. Set C = ∩m i=1 Ci .

.

We study the following feasibility problem Find x ∈ C.

.

If all the sets .Ci , .i = 1, . . . , m are convex this problem is called as the convex feasibility problem. In our study we do not assume the convexity of the sets .Ci , .i = 1, . . . , m but we made the following assumption. Assume that z∗ ∈ ∩m i=1 Ci

.

30

2 Iterative Methods in a Hilbert space

and that the following assumption holds: (A1) for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

In other words all the sets .Ci , .i = 1, . . . , m are star-shaped at .z∗ . We call our problem as the star-shaped feasibility problem which is quite natural. Proposition 2.1 and assumption (A1) imply the following result. Proposition 2.4 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X. Then for each .i ∈ {1, . . . , m}, each .x ∈ X and each .y ∈ Pi (x), ||z∗ − x||2 ≥ ||z∗ − y||2 + ||x − y||2 .

.

¯ Fix a natural number .N. Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer j , {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

.

(2.17)

Every .r ∈ R generates the following algorithm: Initialization: select an arbitrary .x0 ∈ X. Iterative step: given a current iteration point .xk calculate the next iteration point .xk+1 by xk+1 ∈ Pr(k+1) (xk ).

.

The following result describes the behavior of exact iterates of our algorithm. Theorem 2.5 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X. Let M > 0,

.

||z∗ || < M,

.

ɛ ∈ (0, 1), .r ∈ R, .{xi }∞ i=0 ⊂ X,

.

||x0 || ≤ M

.

and let for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi ).

.

Then Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ })

.

.

¯ N¯ + 1)2 M 2 ɛ −2 . < 4N(

(2.18)

Exact Iterates

31

Proof Proposition 2.4 and (2.18) imply that ||z∗ − xi ||2 − ||z∗ − xi+1 ||2 ≥ ||xi − xi+1 ||2 .

.

(2.19)

It follows from (2.19) and our assumptions that for each natural number Q, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xQ ||2

.

.

=

Q−1 Σ

[||z∗ − xi || − ||z∗ − xi+1 ||2 ]

i=0

.

≥

Q−1 Σ

||xi − xi+1 ||2

i=0 .

≥ ɛ 2 (N¯ + 1)−2 Card({i ∈ {0, . . . , Q − 1} : ||xi − xi+1 || ≥ ɛ (N¯ + 1)−1 }).

Since the relation above holds for every natural number Q we conclude that Card({i ∈ {0, 1, . . . } : ||xi −xi+1 || ≥ ɛ (N¯ +1)−1 }) ≤ 4M 2 (N¯ +1)2 ɛ −2 .

.

(2.20)

Set E0 = {i ∈ {0, 1, . . . } : ||xi − xi+1 || ≥ ɛ (N¯ + 1)−1 }.

.

(2.21)

In view of (2.20) and (2.21), Card(E0 ) ≤ 4(N¯ + 1)2 M 2 ɛ −2 .

.

(2.22)

Set E1 = {i ∈ {0, 1, . . . } : [i, i + N¯ − 1] ∩ E0 /= ∅}.

.

(2.23)

By (2.22) and (2.23), ¯ ¯ N¯ + 1)2 M 2 ɛ −2 . Card(E1 ) ≤ NCard(E 0 ) ≤ 4N(

.

(2.24)

Let a nonnegative integer j satisfies j /∈ E1 .

(2.25)

[j, j + N¯ − 1] ∩ E0 = ∅.

(2.26)

.

In view of (2.23) and (2.25), .

32

2 Iterative Methods in a Hilbert space

Equations (2.21) and (2.26) imply that for each .i ∈ {j, . . . , j + N¯ − 1}, ||xi − xi+1 || < ɛ (N¯ + 1)−1 .

.

(2.27)

¯ In view of (2.27), for each pair of integers .i1 , i2 ∈ {j, . . . , j + N}, ¯ N¯ + 1)−1 . ||xi1 − xi2 || < ɛ N(

.

(2.28)

Let .s ∈ {1, . . . , m}. By (2.17), there is ¯ j0 ∈ {j + 1, . . . , j + N}

(2.29)

s = r(j0 ).

(2.30)

.

such that .

It follows from (2.27) and (2.29) that ||xj0 −1 − xj0 || < ɛ (N¯ + 1)−1 .

.

(2.31)

It follows from (2.16), (2.18), (2.30) and (2.31) that xj0 ∈ Pr(j0 ) (xj0 −1 ) = Ps (xj0 −1 ) ∈ Cs

.

and d(xj0 −1 , Cs ) < ɛ (N¯ + 1)−1 .

.

(2.32)

Equations (2.28), (2.29) and (2.32) imply that d(xj , Cs ) ≤ ||xj − xj0 −1 || + d(xj0 −1 , Cs ) < ɛ .

.

Thus d(xj , Cs ) < ɛ , s = 1, . . . , m

.

for each .j ∈ {0, 1, . . . , } \ E1 . Theorem 2.5 is proved.

U ∩

Corollary 2.6 Assume that the assumptions of Theorem 2.5 hold. Then there exist a nonnegative integer j < 4M 2 ɛ −2 (N¯ + 1)2 N¯

.

Exact Iterates

33

such that .

max{d(xj , Cs ) : s = 1, . . . , m} ≤ ɛ .

Corollary 2.7 Assume that the assumptions of Theorem 2.5 hold. Then .

lim max{d(xj , Cs ) : s = 1, . . . , m} = 0.

j →∞

We say that .C1 , . . . , Cm have a bounded regularity property (BRP for short) [163] if for each .M > 0 and each . ɛ > 0 there exists .δ > 0 such that for each .x ∈ B(0, M) satisfying d(x, Ci ) ≤ δ, i = 1, . . . , m

.

the inequality .d(x, C) ≤ ɛ holds. We say that .C1 , . . . , Cm have a bounded linear regularity property (BLRP for short) if for each .M > 0 there exist .L, r0 > 0 such that for each .x ∈ B(0, M) satisfying d(x, Ci ) ≤ r0 , i = 1, . . . , m

.

the inequality d(x, C) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Proposition 2.8 Let .r ∈ (0, 1], B(z∗ , r) ⊂ C, ||z∗ || < M

.

(2.33)

and the following property hold: (P1) for each .z ∈ B(z∗ , r), each .i ∈ {1, . . . , m} and each .x ∈ Ci , αz + (1 − α)x ∈ Ci .

.

Then for each .x ∈ X satisfying ||x|| ≤ M, max{d(x, Ci ) : i = 1, . . . , m} < 2−1 r

.

the inequality d(x, C) ≤ 4Mr −1 max{d(x, Ci ) : i = 1, . . . , m}

.

holds.

(2.34)

34

2 Iterative Methods in a Hilbert space

Proof Assume that x ∈ B(0, M)

.

(2.35)

and (2.34) holds. We assume that .x /∈ C. Set α = 2r −1 max{d(x, Ci ) : i = 1, . . . , m} ∈ (0, 1)

.

(2.36)

(see (2.34)). Fix a positive number γ ≤ αr/4.

(2.37)

xi ∈ Ci

(2.38)

||x − xi || ≤ d(x, Ci ) + γ .

(2.39)

.

For each .i ∈ {1, . . . , m} there exists .

such that .

Let .i ∈ {1, . . . , m}. Property (P1) and (2.38) imply that B(αz∗ + (1 − α)xi , αr) = αB(z∗ , r) + (1 − α)xi ⊂ Ci .

.

(2.40)

It follows from (2.36), (2.37) and (2.39) that ||αz∗ + (1 − α)x − (αz∗ + (1 − α)xi )|| = (1 − α)||x − xi || ≤ (1 − α)d(x, Ci ) + γ

.

.

≤ (1 − α)d(x, Ci ) + 4−1 αr ≤ d(x, Ci ) + 4−1 αr < αr.

(2.41)

By (2.40) and (2.41), αx∗ + (1 − α)x ∈ Ci , i = 1, . . . , m.

.

(2.42)

In view of (2.35), ||αz∗ + (1 − α)x − x|| = α||z∗ − x|| ≤ 2αM.

.

By (2.36), (2.42), d(x, C) ≤ 2αM ≤ 4Mr −1 max{d(x, Ci ) : i = 1, . . . , m}.

.

Proposition 2.8 is proved.

U ∩

Exact Iterates

35

Theorem 2.9 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X and that (BRP) holds. Let .M > max{1, ||z∗ ||}, . ɛ ∈ (0, 1). Then there exists a natural number Q such that for each .r ∈ R and each .{xi }∞ i=0 ⊂ X satisfying ||x0 || ≤ M

(2.43)

xi+1 ∈ Pr(i+1) (xi )

(2.44)

.

and for each integer .i ≥ 0, .

the inequality Card({i ∈ {0, 1, . . . } : d(xi , C) > ɛ }) < Q

.

holds. Proof Property (BPP) implies that there exists . ɛ 0 ∈ (0, ɛ ) such that the following property holds: (i) for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ ɛ 0 , i = 1, . . . , m

.

the inequality .d(x, C) ≤ ɛ holds. Choose a natural number ¯ Q > 4M 2 ɛ 0−2 (N¯ + 1)2 N.

.

(2.45)

Assume that .r ∈ R, .{xi }∞ i=0 ⊂ X and (2.43) and (2.44) hold for all integers .i ≥ 0. Proposition 2.4 and (2.43) and (2.44) imply that for each integer .i ≥ 0, ||xi+1 − z∗ || ≤ ||xi − z∗ || ≤ ||x0 − z∗ || ≤ 2M,

.

||xi || ≤ 3M, i = 0, 1, . . . .

.

(2.46)

It follows from Theorem 2.5 and (2.43)–(2.45) that Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ 0 })

.

.

¯ N¯ + 1)2 M 2 ɛ −2 < Q. ≤ 4N( 0

Property (i) and (2.46) imply that {j ∈ {0, 1, . . . } : d(xj , C) > ɛ }

.

(2.47)

36

2 Iterative Methods in a Hilbert space .

⊂ {j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ 0 }.

Together with (2.47) this implies that Card({j ∈ {0, 1, . . . } : d(xj , C) > ɛ }) ≤ 4N¯ (N¯ + 1)2 M 2 ɛ 0−2 < Q.

.

U ∩

Theorem 2.9 is proved.

Corollary 2.10 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X and that (BRP) holds. Let .M > max{1, ||z∗ ||}, . ɛ ∈ (0, 1) and let a natural number Q be as guaranteed by Theorem 2.9. Let .r ∈ R and .{xi }∞ i=0 ⊂ X satisfy ||x0 || ≤ M

.

and for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi ).

.

Then there exists a nonnegative integer .j ∈ {0, . . . , Q} such that d(xj , C) ≤ ɛ

.

and .

lim d(xi .C) = 0.

i→∞

Theorem 2.11 Let .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X, .M > max{1, ||z∗ ||}, .Δ0 ∈ (0, 1), .L > 1 and let the following property hold: (i) for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, C) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Assume that . ɛ ∈ (0, Δ0 ], .r ∈ R, .{xi }∞ i=0 ⊂ X satisfies ||x0 || ≤ M

.

and for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi ).

.

Exact Iterates

37

Then Card({i ∈ {0, 1, . . . } : d(xi , C) > ɛ })

.

≤ 4N¯ (N¯ + 1)2 M 2 ɛ −2 L2 .

.

Proof Theorem 2.5 and our assumptions imply that Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ L−1 })

.

¯ N¯ + 1)2 M 2 ɛ −2 L2 . < 4N(

.

(2.48)

Proposition 2.4 implies that for each integer .i ≥ 0, ||z∗ − xi+1 || ≤ ||z∗ − xi || ≤ ||z∗ − x0 || ≤ 2M, ||xi || ≤ 3M.

.

(2.49)

Assume that i ∈ {0, 1, . . . }, d(xi , Cs ) ≤ ɛ L−1 , s = 1, . . . , m.

.

Together with property (i), (5.49) and the inequality . ɛ ≤ Δ this implies that d(xi , Cs ) ≤ Δ0 , i = 1, . . . , m,

.

d(xi , C) ≤ L · ɛ L−1 = ɛ .

.

Together with (5.48) this implies that Card({i ∈ {0, 1, . . . } : d(xi , C) > ɛ }) ≤ 4N¯ (N¯ + 1)2 M 2 ɛ −2 L2 .

.

U ∩

Theorem 2.11 is proved.

Theorem 2.12 Assume that .Pi (x) /= ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X, the bounded regularity property and the following property hold: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > 0, . ɛ ∈ (0, 1). Then there exists a natural number Q such that for each r ∈ R and each .{xi }∞ i=0 ⊂ X satisfying

.

||x0 || ≤ M

.

(2.50)

38

2 Iterative Methods in a Hilbert space

and for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi )

.

(2.51)

there exists .z¯ ∈ C such that ||z − xi || for each integer i ≥ Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Corollary 2.10 implies that that there exists a natural number Q such that the following property holds: (ii) for each .r ∈ R and each .{xi }∞ i=0 ⊂ X satisfying (2.50) and (2.51) there exists a nonnegative integer .j ≤ Q such that d(xj , C) ≤ ɛ /2.

.

(2.52)

Assume that .r ∈ R and .{xi }∞ i=0 ⊂ X satisfies (2.50) and (2.51). Property (ii), (2.50) and (2.51) imply that there exists .j ∈ {0, . . . , Q} such that (2.52) holds. There exists z0 ∈ C

(2.53)

||xj − z0 || ≤ ɛ .

(2.54)

.

such that .

Proposition 2.4 with .z∗ = z0 and (2.51), (2.53) and (2.54) imply that for all integers i ≥ j,

.

||xi − z0 || ≤ ɛ .

.

Since . ɛ is any element of .(0, 1) and the set C is closed the sequence .{xi }∞ i=0 converges to a point of C. Theorem 2.12 is proved. U ∩ Theorem 2.13 Assume that .Pi (x) = / ∅ for each .i ∈ {1, . . . , m} and each .x ∈ X and that the following property holds: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M, L > 1, .Δ0 ∈ (0, 1), .||z∗ || < M, for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

Inexact Iterates with Summable Errors

39

the inequality d(x, Ci ) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

hold, . ɛ ∈ (0, Δ0 ) and Q > 16N¯ (N¯ + 1)2 M 2 ɛ −2 L2

.

be an integer. Assume that .r ∈ R and .{xi }∞ i=0 ⊂ X satisfies ||x0 || ≤ M

.

and for each integer .i ≥ 0, xi+1 ∈ Pr(i+1) (xi ).

.

Then there exists an integer .j ∈ {0, . . . , Q} such that ||z − xi || < ɛ for all integers i > j.

.

Proof Theorem 2.11 and our assumptions imply that there exists a nonnegative integer .j ≤ Q such that d(xj , C) ≤ ɛ /2.

.

Hence there exists .z ∈ C such that ||z − xj || < ɛ .

.

Proposition 2.4 with .z∗ = z, property (i) and our assumptions imply that ||z − xi || ≤ ||z − xj || < ɛ for each integer i ≥ j.

.

U ∩

Theorem 2.13 is proved.

Inexact Iterates with Summable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in section “Exact Iterates” holds.

40

2 Iterative Methods in a Hilbert space

(A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

¯ Fix a natural number .N. Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer j , {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

.

The following theorem describes the behavior of inexact iterates with computational errors. Theorem 2.14 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1), .{Δi }∞ i=0 ⊂ (0, 1],

.

∞ Σ .

1/2

Δi

< ∞,

(2.55)

i=0

M1 = 8M + 64(

∞ Σ

.

1/2

Δi )2 + 4,

(2.56)

i=0

M2 = 4M 2 + 32(2M1 + 1)3/2 (

∞ Σ

.

1/2

Δi ),

(2.57)

i=0

p¯ is a natural number and that

.

Δi < ɛ (N¯ + 1)−1 for all integers i ≥ p. ¯

.

(2.58)

Let .r ∈ R, .{xi }∞ i=0 ⊂ X, ||x0 || ≤ M

.

(2.59)

and let for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } /= ∅. (2.60)

.

Inexact Iterates with Summable Errors

41

Then ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} ≥ ɛ })

.

.

¯ N¯ + 1)2 M2 ɛ −2 . ≤ p¯ + N(

Proof Let .i ≥ 0 be an integer. Lemma 2.3 applied with .δ = Δi , .x = xi , .x˜ = xi+1 , D = Cr(i+1) imply that

.

||z∗ − xi+1 || ≤ ||z∗ − xi || + 2(2Δi (2||xi − z∗ || + 1))1/2 ,

.

(2.61)

||z∗ − xi ||2 ≥ ||z∗ − xi+1 ||2 + ||xi − xi+1 ||2

.

.

− 8(2Δi (2||xi − z∗ || + 1))1/2 ||z∗ − xi || − 16Δi (2||xi − z∗ || + 1).

(2.62)

By (2.59) and (2.61), ||z∗ − x0 || ≤ 2M,

(2.63)

||z∗ − x1 || ≤ 2M + 2(2Δ0 (2M + 1))1/2 .

(2.64)

.

.

We show that for each .k = 1, 2, . . . , ||z∗ − xk || ≤ 2M + 2

.

k−1 Σ (2Δi (2M1 + 1))1/2 .

(2.65)

i=0

Equations (2.55) and (2.56) imply that 2M + 2

.

∞ ∞ Σ Σ 1/2 1/2 (2Δi (2M1 + 1))1/2 ≤ 2M + 6 Δ i M1 i=0

.

1/2

≤ M1 /4 + 6M1

i=0 ∞ Σ

1/2

Δi

≤ 4−1 M1 + 6M1 M1 /8 = M1 . 1/2

1/2

(2.66)

i=0

In view of (2.56) and (2.64), Eq. (2.65) holds for .k = 1. Assume that .k ≥ 1 is an integer and (2.65) holds. By (2.61), (2.65) and (2.66), ||z∗ − xk+1 || ≤ ||z∗ − xk || + 2(2Δk (2||xk − z∗ || + 1))1/2

.

.

≤ 2M + 2

k−1 Σ (2Δi (2M1 + 1))1/2 + 2(2Δk (2M1 + 1))1/2 . i=0

42

2 Iterative Methods in a Hilbert space

Thus our assumption holds for .k + 1 too. Therefore by induction we showed that (2.65) holds for all natural numbers k. Equations (2.65) and (2.66) imply that for all natural numbers k ||z∗ − xk || ≤ M1 , ||xk || ≤ 2M1 .

(2.67)

.

By (2.62) and (2.67), for each integer .k ≥ 0, ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ||xk − xk+1 ||2

.

.

.

− 8(2Δk )1/2 (2M1 + 1)3/2 − 16Δk (2M1 + 1) 1/2

≥ ||z∗ − xk+1 ||2 + ||xk − xk+1 ||2 − 32Δk (2M1 + 1)3/2 .

(2.68)

It follows from (2.63) and (2.68) that for each natural number Q, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xQ ||2

.

.

=

Q−1 Σ

[||z∗ − xi || − ||z∗ − xi+1 ||2 ]

i=0

.

≥

Q−1 Σ

||xi − xi+1 ||2 − 32(2M1 + 1)3/2

i=0

∞ Σ

1/2

Δi

i=0

and 4M + 32(2M1 + 1) 2

.

3/2

∞ Σ

1/2 Δi

i=0 .

≥

Q−1 Σ

||xi − xi+1 ||2

i=0

≥ ɛ 2 (N¯ + 1)−1 Card({i ∈ {0, . . . , Q − 1} : ||xi − xi+1 || ≥ ɛ (N¯ + 1)−1 }).

Since the relation above holds for every natural number Q we conclude that Card({i ∈ {0, 1, . . . } : ||xi − xi+1 || ≥ ɛ (N¯ + 1)−1 })

.

.

≤ ɛ −2 (N¯ + 1)2 (4M 2 + 32(2M1 + 1)3/2

∞ Σ

Δi ) ≤ ɛ −2 (N¯ + 1)2 M2 . 1/2

(2.69)

i=0

Set E0 = {i ∈ {0, 1, . . . } : ||xi − xi+1 || ≥ ɛ (N¯ + 1)−1 }.

.

(2.70)

Inexact Iterates with Summable Errors

43

In view of (2.69) and (2.70), Card(E0 ) ≤ ɛ −2 (N¯ + 1)2 M2 .

.

(2.71)

Set E1 = {i ∈ {0, 1, . . . } : [i, i + N¯ − 1] ∩ E0 /= ∅}.

.

(2.72)

By (2.71) and (2.72), ¯ N¯ + 1)2 M ɛ −2 . Card(E1 ) ≤ N¯ Card(E0 ) ≤ N(

.

(2.73)

Let a nonnegative integer j satisfies j /∈ E1 , j ≥ p. ¯

(2.74)

[j, j + N¯ − 1] ∩ E0 = ∅.

(2.75)

.

In view of (2.72) and (2.74), .

Equations (2.70) and (2.75) imply that for each .i ∈ {j, . . . , j + N¯ − 1}, ||xi − xi+1 || < ɛ (N¯ + 1)−1 .

.

(2.76)

¯ In view of (2.76), for each pair of integers .i1 , i2 ∈ {j, . . . , j + N}, ¯ N¯ + 1)−1 . ||xi1 − xi2 || < ɛ N(

(2.77)

¯ s ∈ {1, . . . , m}. i ∈ {j, . . . , j + N},

(2.78)

.

Let .

By the definition of .R, there is ¯ j0 ∈ {j + 1, . . . , j + N}

(2.79)

s = r(j0 ).

(2.80)

.

such that .

It follows from (2.77) and (2.79) that ¯ ||xj0 −1 − xj0 || < ɛ (N¯ + 1)−1 N.

.

(2.81)

44

2 Iterative Methods in a Hilbert space

It follows from (2.60) and (2.80) that d(xj0 , Cs ) = d(xj0 , Cr(j0 ) ) ≤ Δj0 .

.

(2.82)

It follows from (2.58), (2.74), (2.79), (2.81) and (2.82) that ¯ N¯ + 1)−1 + Δj0 < ɛ . d(xj , Cs ) ≤ ||xj − xj0 || + d(xj0 , Cs ) < ɛ N(

.

Thus d(xj , Cs ) < ɛ , s = 1, . . . , m

.

for each .j ∈ {p, ¯ p¯ + 1, . . . , } \ E1 . Together with (2.73) this completes the proof of Theorem 2.14 is proved. U ∩ Corollary 2.15 Assume that the assumptions of Theorem 2.14 hold. Then there exist a nonnegative integer j ≤ p¯ + M2 ɛ −2 (N¯ + 1)2 N¯

.

such that d(xj , Cs ) < ɛ , s = 1, . . . , m.

.

Corollary 2.16 Assume that the assumptions of Theorem 2.14 hold. Then .

lim max{d(xj , Cs ) : s = 1, . . . , m} = 0.

j →∞

Theorem 2.17 Assume that .M > max{1, ||z∗ ||}, . ɛ ∈ (0, 1), .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

(2.83)

i=0

M1 = 8M + 64(

∞ Σ

.

1/2

Δ i )2 + 4

(2.84)

i=0

and that the bounded regularity property holds. Then there exists a natural number Q such that for each .r ∈ R, each .{xi }∞ i=0 ⊂ X satisfying ||x0 || ≤ M

.

(2.85)

and for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } = / ∅ (2.86)

.

Inexact Iterates with Summable Errors

45

the inequalities ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({j ∈ {0, 1, . . . } : d(xj , C) > ɛ }) < Q

.

hold and .

lim d(xi , C) = 0.

i→∞

Proof Set M2 = 4M 2 + 32(2M1 + 1)3/2

∞ Σ

.

1/2

Δi .

(2.87)

i=0

Property (BRP) implies that there exists . ɛ 0 ∈ (0, ɛ ) such that the following property holds: (i) for each .x ∈ B(0, 2M1 ) satisfying d(x, Ci ) ≤ ɛ 0 , i = 1, . . . , m

.

the inequality .d(x, C) ≤ ɛ holds. In view of (2.84), there exists an integer .p¯ ≥ 1 such that Δi < ɛ 0 (N¯ + 1)−1 for all integers i ≥ p. ¯

.

(2.88)

Fix a natural number ¯ −2 (N¯ + 1)2 M2 . Q ≥ p¯ + N ɛ 0

.

(2.89)

Assume that .r ∈ R, .{xi }∞ i=0 ⊂ X and (2.85) and (2.86) hold for all integers .i ≥ 0. Theorem 2.14 and Eqs. (2.85), (2.86) and (2.89) imply that ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} ≥ ɛ 0 }) ≤ Q.

.

Assume that .j ≥ 0 is an integer and d(xj , C) > ɛ .

.

46

2 Iterative Methods in a Hilbert space

Property (i) and (2.89) imply that .

max{d(xj , Cs ) : s = 1, . . . , m} > ɛ 0

and Card({j ∈ {0, 1, . . . } : max{d(xj , C) > ɛ })

.

.

≤ Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ 0 }) < Q.

Since . ɛ is any positive element of .(0, 1) we conclude that .

lim d(xi , C) = 0.

i→∞

U ∩

Theorem 2.17 is proved. Theorem 2.18 Let .M > max{1, ||z∗ ||}, .Δ ∈ (0, 1), .L > ∞ Σ .

1/2

Δi

1, .{Δi }∞ i=0

⊂ (0, 1],

< ∞,

i=0

M1 = 8M + 64(

∞ Σ

.

1/2

Δi )2 + 4,

(2.90)

i=0

M2 = 4M 2 + 32(2M1 + 1)3/2

∞ Σ

.

1/2

Δi ,

(2.91)

i=0

ɛ ∈ (0, Δ), .p¯ be a natural number,

.

Δi < ɛ (N¯ + 1)L−1 for every integer i ≥ p¯

.

and let the following property hold: (i) for each .x ∈ B(0, 2M1 ) satisfying d(x, Ci ) ≤ Δ, i = 1, . . . , m

.

the inequality d(x, Ci ) ≤ L max{d(xi , Cs ) : s = 1, . . . , m}

.

is true.

(2.92)

Inexact Iterates with Summable Errors

47

Assume that .r ∈ R, .{xi }∞ i=0 ⊂ X, ||x0 || ≤ M

.

(2.93)

and that for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } = / ∅. (2.94)

.

Then ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

¯ N¯ + 1)2 . Card({i ∈ {0, 1, . . . } : d(xi , C) > ɛ }) ≤ p¯ + M2 ɛ −2 L2 N(

.

Proof Theorem 2.14 and (2.89)–(2.94) imply that ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

(2.95)

Card({i ∈ {0, 1, . . . } : max{d(xi , Cs ) : s = 1, . . . , m} ≥ ɛ L−1 })

.

.

¯ −2 (N¯ + 1)2 M2 . ≤ Q := p¯ + N ɛ

(2.96)

Let E = {i ∈ {0, 1, . . . } : max{d(xi , Cs ) : s = 1, . . . , m} ≥ ɛ L−1 }.

.

(2.97)

In view of (2.96) and (2.97), Card(E) ≤ Q.

.

(2.98)

Assume that .i ∈ {0, 1, . . . } \ E. By (2.95) and (2.97), d(xi , Cs ) < ɛ L−1 , s = 1, . . . , m, ||xi || ≤ 2M1 .

.

Property (i) and (2.99) imply that d(xi , C) ≤ L max{d(xi , Cs ) : s = 1, . . . , m} < ɛ

.

and {i ∈ {0, 1, . . . } : d(xi , C) > ɛ } ⊂ E

.

(2.99)

48

2 Iterative Methods in a Hilbert space

and in view of (2.98), Card({i ∈ {0, 1, . . . } : d(xi , C) > ɛ }) ≤ Q.

.

U ∩

Theorem 2.18 is proved.

Theorem 2.19 Assume that the bounded regularity property and the following property hold: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

(2.100)

i=0

M1 = 8M + 64(

∞ Σ

.

1/2

Δi )2 + 4,

(2.101)

i=0

ɛ ∈ (0, 1). Then there exists a natural number Q such that for each .r ∈ R and each {xi }∞ i=0 ⊂ X satisfying

. .

||x0 || ≤ M

(2.102)

.

and for each integer .i ≥ 0, B(xi+1 , Δi ) ∩ {ξ ∈ Cr(i+1) : ||xi − ξ || ≤ ||xi − η|| + Δi : η ∈ Cr(i+1) } = / ∅ (2.103)

.

the inequality ||xi || ≤ 2M1 , i = 0, 1, . . .

.

holds, there exists .z¯ ∈ C such that ||z − xi || ≤ ɛ for each integer i ≥ 2Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Theorem 2.17 implies that there exists a natural number Q such that 4(8M1 + 8)1/2

∞ Σ

.

i=Q

1/2

Δi

< ɛ /2

(2.104)

Inexact Iterates with Summable Errors

49

and the following property holds: (iii) for each .r ∈ R and each .{xi }∞ i=0 ⊂ X satisfying (2.102), (2.103) for each integer .i ≥ 0, the inequality ||xi || ≤ 2M1 , i = 0, 1, . . .

.

holds and there is .j ∈ {Q, . . . , 2Q} for which d(xj , C) < ɛ /2.

.

Assume that .r ∈ R, .{xi }∞ i=0 ⊂ X satisfy (2.102), (2.103) for each integer .i ≥ 0. Property (ii), (2.102) and (2.103) imply that ||xi || ≤ 2M1 , i = 0, 1, . . .

(2.105)

k ∈ {Q, . . . , 2Q}

(2.106)

.

and there is .

for which d(xk , C) < ɛ /2.

.

Clearly, there exists z¯ ∈ C

(2.107)

||xk − z¯ || < ɛ /2.

(2.108)

.

such that .

By (2.105), (2.107) and (2.108), ||¯z|| ≤ 2M1 + 1.

.

Assume that an integer .q ≥ k. Property (i), (2.103), (2.105), by the relation above and Lemma 2.3 applied with x = xq , x˜ = xq+1 , δ = Δq , z∗ = z¯ , D = Cr(q+1)

.

imply that ||¯z − xq+1 || ≤ ||¯z − xq || + 2(2Δq (2||xq − z¯ || + 1))1/2

.

.

1/2

≤ ||¯z − xq || + 4Δq (8M1 + 3)1/2 .

50

2 Iterative Methods in a Hilbert space

Together with (2.104), (2.106) and (2.108) this implies that for each integer .q > k, ||¯z − xq || ≤ ||¯z − xk || + 4(8M1 + 3)1/2

q−1 Σ

.

1/2

Δi

i=k

.

< ɛ /2 + 4(8M1 + 3)1/2

∞ Σ

1/2

Δi

< ɛ .

i=Q

Thus ||¯z − xq || < ɛ

.

for each integer .q ≥ 2Q. Since . ɛ is any element of .(0, 1) and the set C is closed the sequence .{xi }∞ U ∩ i=0 converges to a point of C. Theorem 2.19 is proved.

The First Result with Nonsummable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in section “Exact Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

¯ Fix a natural number .N. Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer j , {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

(2.109)

M > max{1, ||z∗ ||},

(2.110)

.

Theorem 2.20 Let .

The First Result with Nonsummable Errors

51

δ ∈ (0, (N¯ + 1)−1 ],

(2.111)

ɛ 0 ≥ (32δ(4M + 4)N¯ )1/2

(2.112)

.

.

and a natural number .n0 satisfy n0 ≥ 1 + L64M 2 ɛ 0−2 ⎦ .

.

(2.113)

Assume that r ∈ R,

.

x0 ∈ B(θ, M)

.

(2.114)

and that .{xi }∞ i=1 ⊂ X satisfies for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || = d(xi−1 , Cr(i) } /= ∅.

.

(2.115)

Then there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(θ, 3M + 1), i = 0, . . . , q N¯ ,

(2.116)

¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

(2.117)

.

.

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (2.117), then for each .i = q N¯ , . . . , (q + 1)N¯ and all .s = 1, . . . , m, d(xi , Cs ) ≤ (N¯ + 1) ɛ 0 .

.

Note that in Theorem 2.20 we assume that for each natural number i, we have {ξ ∈ Cr(i) : ||xi−1 − ξ || = d(xi−1 , Cr(i) } /= ∅.

.

This assumption holds if the space X is finite-dimensional or all the sets .Ci , i = 1, . . . , m are convex. It should be mentioned that in Theorem 2.20 .δ is the computational error made by our computer system, we obtain a point z such that ¯ + 1) ɛ 0 and in order to obtain this point we need .(n0 − 1)N¯ iterations. .d(z, Cs ) ≤ (N It is not difficult to see that . ɛ 0 = c1 δ 1/2 and .n0 = Lc2 δ −1 ⎦ + 1, where .c1 and .c2 are positive constants depending on M. Applying by induction Theorem 2.20 we obtain the following result. .

Theorem 2.21 Suppose that . ɛ ¯ ∈ (0, 1), M¯ > max{1, ||z∗ ||},

.

¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

52

2 Iterative Methods in a Hilbert space

¯ Let .M > M, 0 < δ ≤ (N¯ + 1)−1 ,

.

¯ 1/2 ≤ ɛ 0 < ɛ ¯ (N¯ + 1)−1 (32δ(4M + 4)N)

.

and let a natural number .n0 satisfies n0 ≥ L642M 2 ɛ 0−2 ⎦ + 1.

.

Assume that r ∈ R,

.

x0 ∈ B(θ, M), {xi }∞ i=1 ⊂ X

.

and that for each natural number i, (2.115) holds. Then xi ∈ B(θ, 3M + 1) for all integers i ≥ 0

.

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 0 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

¯ ¯ . . . , (qp + 1)N, and that for each integer .p ≥ 0 and each .i = qp N, d(xi , Cs ) ≤ ɛ 0 (N¯ + 1), s = 1, . . . , m.

.

Proof of Theorem 2.20 Assume that s is a natural number and that for each integer k ∈ [0, s],

.

.

¯ > ɛ 0 . max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N}

(2.118)

By (2.110) and (2.114), ||x0 − z∗ || ≤ 2M.

.

(2.119)

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

.

(2.120) U ∩

We prove the following auxiliary result.

The First Result with Nonsummable Errors

53

Lemma 2.22 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ 2M + iδ.

.

(2.121)

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + δ

.

and ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i − z∗ ||2 + 2δ(2M + 1).

.

If .||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 , then ||xk N¯ +i+1 − z∗2 || − ||xk N¯ +i − z∗ ||2 ≤ −8−1 ɛ 02 .

.

Proof In view of (2.111), there exists ξ ∈ Cr(k N¯ +i+1) ∩ B(xk N¯ +i+1 , δ)

(2.122)

||ξ − xk N¯ +i || = d(xk N¯ +i , Cr(k N¯ +i+1) ).

(2.123)

.

such that .

Proposition 2.1, assumption (A1), (2.122) and (2.123) imply that ||z∗ − ξ ||2 + ||xk N¯ +i − ξ ||2 ≤ ||z∗ − xk N¯ +i ||2 .

.

(2.124)

In view of (2.122) and (2.124), ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − ξ || + ||ξ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + δ.

.

(2.125)

Equations (2.121) and (2.125) imply that ||z∗ − xk N¯ +i+1 || ≤ 2M + (i + 1)δ.

.

(2.126)

It follows from (2.111), (2.121) and (2.125) that ||xk N¯ +i+1 − z∗ ||2

.

.

≤ ||xk N¯ +i − z∗ ||2 + 2δ(2M + 1).

(2.127)

54

2 Iterative Methods in a Hilbert space

Assume that ||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 .

.

(2.128)

By (2.112), (2.122) and (2.128), ||xk N¯ +i − ξ ||

.

≥ ||xk N¯ +i − xk N¯ +i+1 || − ||xk N¯ +i+1 − ξ || > ɛ 0 − δ ≥ ɛ 0 /2.

.

(2.129)

By (2.124) and (2.129), ||z∗ − xk N¯ +i ||2 ≥ ||z∗ − ξ ||2 + ||ξ − xk N¯ +i ||2 ≥ ||z∗ − ξ ||2 + ɛ 02 /4.

.

(2.130)

It follows from (2.111), (2.122), (2.124) and (2.126) that |||z∗ − ξ ||2 − ||z∗ − xk N¯ +i ||2 |

.

.

≤ ||ξ − xk N¯ +i ||(||z∗ − ξ || + ||z∗ − xk N¯ +i ||) ≤ δ(4M + 4).

(2.131)

In view of (2.112), (2.130) and (2.131), ||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ − ξ ||2 + δ(4M + 4)

.

.

≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 /4 + δ(4M + 4) ≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 /8.

This completes the proof of Lemma 2.22.

U ∩

It follows from (2.111), (2.120) and Lemma 2.22 applied by induction that for all .i = 0, . . . , N¯ − 1, ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + δ,

.

||z∗ − xk N¯ +i+1 || ≤ 2M + δ N¯ ,

.

||z∗ − xk N¯ +i || ≤ 2M + 1, i = 0, . . . , N¯

.

(2.132)

and if ||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 , then

.

||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 /8.

.

(2.133)

The First Result with Nonsummable Errors

55

Lemma 2.22 and (2.112), (2.118) and (2.133) imply that 2 2 ||z∗ − x(k+1)N) ¯ || − ||z∗ − xk N¯ ||

.

.

=

¯ −1 N Σ

(||z∗ − xk N¯ +i+1 ||2 − ||z∗ − xk N¯ +i ||2 )

i=0 .

≤ − ɛ 02 /8 + 2δ(2M + 1)N¯ ≤ − ɛ 02 /16.

(2.134)

Thus we have shown that the following property holds: (P1) if an integer .k ∈ [0, s] satisfies .||z∗ − xk N¯ || ≤ 2M, then ¯ . . . , (k + 1)N¯ , ||z∗ − xi || ≤ 2M + 1, i = k N,

.

||z∗ − x(k+1)N¯ ||2 − ||z∗ − xk N¯ ||2 ≤ −16−1 ɛ 02 .

.

By (2.120) and property (P1), ¯ ||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N,

.

and (2.135) holds for all .k = 0, . . . , s. In view (2.119) and (2.135), 16−1 ɛ 02 (s + 1) ≤

.

s Σ (||z∗ − xk N¯ ||2 − ||z∗ − x(k+1)N¯ ||2 ) k=0

.

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 64M 2 ɛ 0−2 M 2 .

.

Thus we have shown that the following property holds: (P2) If an integer .s ≥ 1 and for each integer .k ∈ [0, s] (2.118) holds, then s ≤ 64M 2 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , (s + 1).

.

(2.135)

56

2 Iterative Methods in a Hilbert space

Property (P2) implies that there exists an integer .q ∈ [0, n0 − 1] such that for each integer k satisfying .0 ≤ k < q, .

max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 ;

.

max{||xi − xi−1 || : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 , ||xq N¯ − z∗ || ≤ 2M,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , q N¯ ,

.

||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(2.136)

¯ there exists In view of (2.115), for all .i = q N¯ + 1, . . . , (q + 1)N, ξi ∈ Cr(i)

.

(2.137)

||xi − ξi || ≤ δ, ||xi−1 − ξi || = d(xi−1 , Cr(i) ).

(2.138)

such that .

By (2.136) and (2.138), for each .i = {q N¯ + 1, . . . , (q + 1)N¯ }, ||xi−1 − ξi || ≤ ||xi−1 − xi || + ||xi − ξi || ≤ ɛ 0 + δ.

.

(2.139)

Let ¯ . . . , (q + 1)N} ¯ i ∈ {q N,

.

and s ∈ {1, . . . , m}.

.

Relation () implies that there exists j ∈ {q N¯ + 1, . . . , (q + 1)N¯ }

.

such that s = r(j ).

.

(2.140)

The Second Result with Nonsummable Errors

57

By (2.139), ||xj −1 − ξj || ≤ ɛ 0 + δ.

.

(2.141)

Equations (2.136) and (2.140) imply that ¯ 0. ||xi − xj −1 || ≤ N ɛ

(2.142)

ξj ∈ Cr(j ) = Cs .

(2.143)

.

By (2.137) and (2.140), .

It follows from (2.112), (2.141) and (2.142) that ||xi − ξj || ≤ (N¯ + 1) ɛ 0 .

.

Together with (2.143) this implies that d(xi , Cs ) ≤ (N¯ + 1) ɛ 0 .

.

U ∩

Theorem 2.20 is proved.

The Second Result with Nonsummable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in section “Exact Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

¯ Fix a natural number .N.

58

2 Iterative Methods in a Hilbert space

Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer j , {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

(2.144)

M > max{1, ||z∗ ||},

(2.145)

.

¯ −2 (4M + 4)−1 , 0 < δ ≤ 2−1 (2N)

(2.146)

1/2 1/2 ¯ ɛ 0 = (32(4M + 4)3/2 N(2δ) )

(2.147)

.

Theorem 2.23 Let .

.

and a natural number .n0 satisfy n0 ≥ 1 + L16M 2 ɛ 0−2 ⎦ .

.

(2.148)

Assume that r ∈ R,

.

x0 ∈ B(θ, M)

.

(2.149)

and that .{xi }∞ i=1 ⊂ X satisfies for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} /= ∅.

.

(2.150)

Then there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(θ, 3M + 1), i = 0, . . . , q N¯ ,

(2.151)

¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

(2.152)

.

.

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (2.152), then for each .i = ¯ and all .s = 1, . . . , m, q N¯ , . . . , (q + 1)N, d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

It should be mentioned that in Theorem 2.23 .δ is the computational error made by our computer system, we obtain a point z such that .d(z, Cs ) ≤ (N¯ + 2) ɛ 0 and in order to obtain this point we need .(n0 − 1)N¯ iterations. It is not difficult to see that . ɛ 0 = c1 δ 1/4 and .n0 = Lc2 δ −1/2 ⎦ + 1, where .c1 and .c2 are positive constants depending on M.

The Second Result with Nonsummable Errors

59

Proof Assume that s is a natural number s such that for each integer .k ∈ [0, s], .

¯ > ɛ 0 . max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N}

(2.153)

By (2.145) and (2.149), ||x0 − z∗ || ≤ 2M.

.

(2.154)

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

.

(2.155) U ∩

We prove the following auxiliary result. Lemma 2.24 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ 2i(2δ)1/2 (4M + 4)1/2 + 2M.

.

(2.156)

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + 2(2δ)1/2 (4M + 4)1/2

.

and ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i − z∗ ||2 + 4(2δ)1/2 (4M + 3)1/2 (2M + 2).

.

If .||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 , then ||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ≤ −2−1 ɛ 02 .

.

Proof Assumption (A1), (2.150) and Lemma 2.3 applied with x = xk N¯ +i , x˜ = xk N¯ +i+1 , D = Cr(k N¯ +i+1)

.

imply that ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + 2(2δ(2||z∗ − xk N¯ +i || + 1))1/2 ,

.

(2.157)

||z∗ − xk N¯ +i ||2 ≥ ||z∗ − xk N¯ +i+1 ||2 + ||xk N¯ +i − xk N¯ +i+1 ||2

.

.

− 32δ 1/2 (2||z∗ − xk N¯ +i || + 1)3/2 .

(2.158)

60

2 Iterative Methods in a Hilbert space

In view of (2.146) and (2.156), ||z∗ − xk N¯ +i || ≤ 2M + 1.

.

(2.159)

By (2.157)–(2.159), ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + 2(2δ(4M + 3))1/2 ,

.

(2.160)

||z∗ − xk N¯ +i ||2

.

.

≥ ||z∗ − xk N¯ +i+1 ||2 + ||xk N¯ +i − xk N¯ +i+1 ||2 − 32δ 1/2 (4M + 3)3/2 .

(2.161)

It follows from (2.146), (2.159) and (2.160), ||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ − xk N¯ +i ||2

.

.

+ 4(2δ(4M + 3))1/2 (||z∗ − xk N¯ +i || + 2(2δ(4M + 3))1/2 ) .

≤ ||z∗ − xk N¯ +i ||2 + 4(2δ(4M + 3))1/2 (2M + 2).

Assume that ||xk N¯ +i+1 − xk N¯ +i )|| > ɛ 0 .

.

By (2.147), (2.158), (2.159) and by the relation above, ||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 + 32δ 1/2 (4M + 3)1/2

.

.

≤ ||z∗ − xk N¯ +i+1 ||2 − ɛ 02 /2.

Lemma 2.24 is proved. It follows from (2.154) and Lemma 2.24 applied by induction that for all .i = 0, . . . , N¯ − 1, ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + 2(2δ(4M + 4))1/2 ,

.

||z∗ − xk N¯ +i+1 || ≤ 2M + (i + 1)(2(2δ(4M + 4))1/2 )

.

.

¯ ≤ 2M + N(2(2δ(4M + 4))1/2 ).

(2.162)

Together with (2.146) and (2.154) this implies that ||z∗ − xk N¯ +i || ≤ 2M + 1, i = 0, . . . , N¯ .

.

(2.163)

The Second Result with Nonsummable Errors

61

Lemma 2.24 and Eq. (2.147), (2.153) and () imply that 2 2 ||z∗ − x(k+1)N) ¯ || − ||z∗ − xk N¯ ||

.

.

=

¯ −1 N Σ

(||z∗ − xk N¯ +i+1 ||2 − ||z∗ − xk N¯ +i ||2 )

i=0

≤ − ɛ 02 /2 + 4N¯ ((2δ(4M + 3))1/2 )(2M + 2) ≤ − ɛ 02 /4.

.

Thus we have shown that the following property holds: (P1) if an integer .k ∈ [0, s] satisfies .||z∗ − xk N¯ || ≤ 2M, then ¯ . . . , (k + 1)N, ¯ ||z∗ − xi || ≤ 2M + 1, i = k N,

(2.164)

||z∗ − x(k+1)N¯ ||2 − ||z∗ − xk N¯ ||2 ≤ −4−1 ɛ 02 .

(2.165)

.

.

By (2.155), (2.164), (2.165) and property (P1), ¯ ||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N,

.

and (2.165) holds for all .k = 0, . . . , s. In view (2.154) and (2.165), 4−1 ɛ 02 (s + 1) ≤

.

s Σ (||z∗ − xk N¯ ||2 − ||z∗ − x(k+1)N¯ ||2 ) k=0

.

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 ɛ 0−2 M 2 .

.

Thus we have shown that the following property holds: (P2) If an integer .s ≥ 0 and for each integer .k ∈ [0, s] (2.153) holds, then s < 16M 2 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , (s + 1).

.

62

2 Iterative Methods in a Hilbert space

Property (P2) and (2.148) imply that there exists an integer .q ∈ [0, n0 − 1] such that for each integer k satisfying .0 ≤ k < q, .

¯ > ɛ 0 , max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N}

(2.166)

.

max{||xi − xi−1 || : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 .

(2.167)

Property (P2) and (2.145), (2.154), (2.166) imply that ||xq N¯ − z∗ || ≤ 2M,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , q N¯ ,

.

||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(2.168)

In view of (2.150), for all ¯ i = q N¯ + 1, . . . , (q + 1)N,

(2.169)

ξi ∈ Cr(i) ∩ B(xi , δ).

(2.170)

.

there exists .

¯ By (2.168) and (2.170), for each .i ∈ {q N¯ + 1, . . . , (q + 1)N}, ||xi−1 − ξi || ≤ ||xi−1 − xi || + ||xi − ξi || ≤ ɛ 0 + δ.

.

(2.171)

Let i ∈ {q N¯ , . . . , (q + 1)N¯ }, s ∈ {1, . . . , m}.

.

(2.172)

Relations (2.144) and (2.172) imply that there exists j ∈ {q N¯ + 1, . . . , (q + 1)N¯ }

(2.173)

s = r(j ).

(2.174)

.

such that .

The Second Result with Nonsummable Errors

63

By (2.168), (2.171) and (2.173) ||xj −1 − ξj || ≤ ɛ 0 + δ.

.

(2.175)

Equations (2.169), (2.173) and (2.174) imply that ||xi − xj −1 || ≤ N¯ ɛ 0 , ξj ∈ Cr(j ) = Cs .

.

(2.176)

It follows from (2.141), (2.175) and (2.176) that ||xi − ξj || ≤ (N¯ + 1) ɛ 0 + δ ≤ (N¯ + 2) ɛ 0

.

and d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

U ∩

Theorem 2.23 is proved. Applying by induction Theorem 2.23 we obtain the following result. Theorem 2.25 Suppose that . ɛ ¯ ∈ (0, 1), M > max{1, ||z∗ ||},

.

¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

¯ Let .M > M, ¯ −1 (4M + 4)−1 , 0 < δ ≤ 2−1 (2N)

.

2(2δ/N¯ )1/2 (16(4M + 4))3/2 (N¯ + 2)1/2 ≤ ɛ < ɛ ¯ ,

.

and let a natural number .n0 satisfies n0 ≥ L16M 2 ɛ −2 (N¯ + 2)2 ⎦ + 1.

.

Assume that r ∈ R,

.

x0 ∈ B(θ, M) and {xi }∞ i=1 ⊂ X

.

satisfy (2.150) for each natural number i. Then xi ∈ B(θ, 3M + 1) for all integers i ≥ 0

.

64

2 Iterative Methods in a Hilbert space

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 0 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

¯ ¯ . . . , (qp + 1)N, and that for each integer .p ≥ 0 and each .i = qp N, d(xi , Cs ) ≤ ɛ , s = 1, . . . , m.

.

Theorem 2.25 implies the following two results. Theorem 2.26 Suppose that . ɛ ¯ ∈ (0, 1), M¯ > max{1, ||z∗ ||},

.

¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

¯ Let .M > M, ¯ −1 (4M + 4)−1 , 0 < δ¯ ≤ 2−1 (2N)

.

¯ 1/2 (16(4M + 4))3/2 (N¯ + 2). ɛ ¯ ≥ 4(2δ¯N)

.

Assume that ¯ r ∈ R, {δi }∞ i=1 ⊂ (0, δ],

.

.

lim δi = 0,

i→∞

x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X

.

satisfy for each natural number i, B(xi , δi ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δi } /= ∅.

.

Then xi ∈ B(0, 3M + 1), i = 0, 1, . . . ,

.

.

lim inf max{d(xi , Cs ) : s = 1, . . . , m} = 0. i→∞

The Second Result with Nonsummable Errors

65

Theorem 2.27 Suppose that the assumptions of Theorem 2.25 hold and that the bounded regularity property holds. Then lim inf d(xi , C) = 0.

.

i→∞

Theorem 2.23 implies the following result. Theorem 2.28 Assume that the bounded regularity property holds, M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1). Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each r ∈ R, each

. .

x0 ∈ B(0, M),

.

each .{xi }∞ i=1 ⊂ X satisfying for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} = / ∅

.

there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(θ, 3M + 1), i = 0, . . . , q N¯ ,

.

d(xi , C) ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N¯ .

.

Theorem 2.28 implies the following result. Theorem 2.29 Assume that the bounded regularity property holds, M¯ > max{1, ||z∗ ||},

.

¯ C ⊂ B(0, M),

.

M > M¯ + 1, . ɛ ∈ (0, 1). Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each .r ∈ R, each

.

x0 ∈ B(0, M),

.

each .{xi }∞ i=1 ⊂ X satisfying for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) + δ}) /= ∅

.

the inequality ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

66

2 Iterative Methods in a Hilbert space

is true and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 0 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

¯ and that for each integer .p ≥ 0 and each .i = (qp − 1)N¯ , . . . , (qp )N, d(xi , C) ≤ ɛ .

.

Theorem 2.30 Assume that the bounded regularity property holds, . ɛ ¯ ∈ (0, 1), M¯ > max{1, ||z∗ ||},

.

¯ C ⊂ B(0, M),

.

(2.177)

M > M¯ + 1, .Δ0 ∈ (0, 1) and that the following property holds:

.

(i) for each .ξ ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .η ∈ Cs ∩ B(ξ, Δ0 ), αξ + (1 − α)η ∈ Cs .

.

Let . ɛ ∈ (0, ɛ ). ¯ Then there exist .δ ∈ (0, ɛ ) and a natural number .n1 such that for each .r ∈ R, each x0 ∈ B(0, M)

.

and each .{xi }∞ i=1 ⊂ X satisfying for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} /= ∅

.

the inequality ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

is true and d(xi , C) ≤ ɛ for all integers i ≥ n1 .

.

Proof We may assume without loss of generality that ɛ < Δ0 /2.

.

(2.178)

The Second Result with Nonsummable Errors

67

By Theorem 2.29, there exist .δ0 ∈ (0, ɛ ) and a natural number .n0 such that the following property holds: (ii) for each .r ∈ R, each x0 ∈ B(0, M),

.

each .{xi }∞ i=1 ⊂ X satisfying for each natural number i, B(xi , δ0 ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ0 } = / ∅

.

the inequality ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

is true and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 0 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

¯ and that for each integer .p ≥ 0 and each .i = (qp − 1)N¯ , . . . , (qp )N, d(xi , C) ≤ ɛ /4.

.

Choose .δ ∈ (0, δ0 ) satisfying ¯ 0 (6δ)1/2 < ɛ /6. 2Nn

(2.179)

¯ n1 = n0 N.

(2.180)

r ∈ R, x0 ∈ B(0, M)

(2.181)

.

Set .

Assume that .

and .{xi }∞ i=1 ⊂ X satisfies (2.178) for each integer .i ≥ 1. Property (ii), (2.178) and (2.181) imply that ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

(2.182)

68

2 Iterative Methods in a Hilbert space

is true and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 0 ≤ q0 ≤ n0 ,

(2.183)

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

(2.184)

.

.

¯ and that for each integer .p ≥ 0 and each .i = (qp − 1)N¯ , . . . , (qp )N, d(xi , C) ≤ ɛ /4.

.

(2.185)

In order to complete the proof it is sufficient to show that d(xi , C) ≤ ɛ

.

¯ Assume the contrary. Then there exists an integer for all integers .i ≥ q0 N. j > q0 N¯

.

such that d(xj , C) > ɛ .

.

(2.186)

There exists an integer .p ≥ 0 such that ¯ qp N¯ ≤ j < (qp+1 )N.

.

(2.187)

In view of (2.185), d(xqp N¯ , C) ≤ ɛ /4

.

and there exists z0 ∈ C

(2.188)

||xqp N¯ − z0 || < ɛ /3.

(2.189)

.

such that .

Assume that .i ∈ {qp N¯ , . . . , (qp+1 )N¯ − 1} satisfies 1/2 ¯ ||xi − z0 || ≤ 2(i − qp N)(6δ) + ɛ /3.

.

(2.190)

¯ By (2.179), (2.184) (In view of (2.189), inequality (2.190) holds for .i = qp N.) and (2.190), ||z0 − xi || ≤ ɛ /6 + ɛ /3 < ɛ /2 < Δ0 /4.

.

(2.191)

The Third Result with Nonsummable Errors

69

It follows from (2.178) that there exists ξ ∈ Cr(i+1)

.

(2.192)

such that ||xi+1 − ξ || ≤ δ, ||xi − ξ || ≤ d(xi , Cr(i+1) ) + δ.

.

(2.193)

It follows from (2.188), (2.191) and (2.193) that d(xi , Cr(i+1) ) ≤ ||xi − z0 ||, ||xi − ξ || ≤ ||xi − z0 || + δ,

.

||z0 − ξ || ≤ ||z0 − xi || + ||xξ || ≤ 2||xi − z0 || + δ < 2 ɛ < Δ0 .

.

Equations (2.188), (2.191)–(2.193), (2.196), property (i) and Lemma 2.3 applied with D = Cr(i+1) , z∗ = z0 , x = xi , x˜ = xi+1

.

imply that ||z∗ − xi+1 || ≤ ||z∗ − xi || + 2(6δ)1/2

.

.

1/2 ¯ ≤ ɛ /3 + 2(i + 1 − qp N)(6δ) .

¯ . . . , qp+1 N¯ this implies that for all .i = Thus (2.190) holds for all .i = qp N, ¯ ¯ qp N, . . . , qp+1 N, ¯ 0 (6δ)1/2 < ɛ . ||xi − z0 || ≤ ɛ /3 + 2Nn

.

Together with (2.179), (2.184) and (2.187) this implies that ||xj − z0 || < ɛ .

.

This contradicts (2.186). The contradiction we have reached completes the proof of Theorem 2.30. U ∩

The Third Result with Nonsummable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets, .δC ∈ (0, 1]. Set C = ∩m i=1 Ci .

.

70

2 Iterative Methods in a Hilbert space

Assume that the following assumption holds. (A2) There exists z∗ ∈ X

.

and for each .s ∈ {1, . . . , m} there exists z∗s ∈ B(z∗ , δC ) ∩ Cs

.

(2.194)

such that for each .x ∈ Cs and each .α ∈ [0, 1], αz∗s + (1 − α)x ∈ Cs .

.

Clearly, assumption (A2) is weaker than its prototype (A1). In (A1) we assume that all the sets .Ci , .i = 1, . . . , m are star-shaped at the same point. In (A2) we suppose that for each .i ∈ {1, . . . , m} the set .Ci is star-shaped at .z∗i belonging to a .δC -neighborhood of a fixed point .z∗ ∈ X. Fix a natural number .N¯ ≥ m. Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer j , {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

.

(2.195)

Theorem 2.31 Let M > max{1, ||z∗ ||},

.

¯ C + 2(2δ(4M + 8))1/2 ) ≤ 1, 0 < δ ≤ 8−1 (N¯ + 1)−1 (2M + 4)−1 , N(2δ

.

(2.196)

ɛ 0 ≥ max{32N¯ δ(4M + 4), [(64δ)1/2 (4M + 7)3/2 + 2δC (4M + 7)1/2 ](2N)1/2 }, (2.197)

.

a natural number .n0 satisfy n0 ≥ 1 + L16M 2 ɛ 0−2 ⎦ .

.

(2.198)

Assume that r ∈ R,

.

x0 ∈ B(0, M)

.

(2.199)

The Third Result with Nonsummable Errors

71

and that .{xi }∞ i=1 ⊂ X satisfies for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} /= ∅.

.

(2.200)

Then there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(θ, 3M + 3), i = 0, . . . , q N¯ ,

.

¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(2.201)

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (2.201), then for each .i = ¯ and all .s = 1, . . . , m, q N¯ , . . . , (q + 1)N, d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

Proof Assume that s is a natural number and for each integer .k ∈ [0, s], .

¯ > ɛ 0 . max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N}

(2.202)

By (2.199), ||x0 − z∗ || ≤ 2M.

.

(2.203)

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M + 2.

(2.204)

Δ = 2δC + 2(2δ(4M + 7))1/2 .

(2.205)

.

Set .

In view of (2.196) and (2.205), ΔN¯ ≤ 1.

.

We prove the following auxiliary result.

(2.206) U ∩

Lemma 2.32 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ 2M + 2 + iΔ.

.

(2.207)

72

2 Iterative Methods in a Hilbert space

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + Δ,

.

||xk N¯ +i+1 − z∗ ||2

.

.

≤ ||xk N¯ +i − z∗ ||2 − ||xk N¯ +i − xk N¯ +i+1 ||2 + 32δ 1/2 (4M + 7)3/2 + 2δC (4M + 7).

If .||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 , then ||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ≤ −2−1 ɛ 02 .

.

Proof Equations (2.206) and (2.207) imply that ||xk N¯ +i − z∗ || ≤ 2M + 3.

.

(2.208)

In view of (2.200), there exists ξ ∈ Cr(k N¯ +i+1) ∩ B(xk N¯ +i+1 , δ)

(2.209)

||ξ − xk N¯ +i || ≤ d(xk N¯ +i , Cr(k N¯ +i+1 ) + δ.

(2.210)

.

such that .

Equations (2.194) and (2.205)–(2.207) imply that ||xk N¯ +i − z∗r(k N¯ +i+1) || ≤ ||xk N¯ +i − z∗ || + δ0

.

.

¯ ≤ 2M + 3. ≤ 2M + 2 + iΔ + δC ≤ 2M + 2 + NΔ

(2.211)

Assumption (A2), (2.194), (2.209)–(2.211) and Lemma 2.3 applied with x = xk N¯ +i , x˜ = xk N¯ +i+1 , D = Cr(k N¯ +i+1) , z∗ = z∗r(k N¯ +i+1)

.

imply that ||z∗r(k N¯ +i+1) −xk N¯ +i+1 || ≤ ||z∗r(k N¯ +i+1) −xk N¯ +i ||+2(2δ(4M +7))1/2 ,

.

(2.212)

||z∗r(k N¯ +i+1) − xk N¯ +i ||2 ≥ ||z∗r(k N¯ +i+1) − xk N¯ +i+1 ||2 + ||xk N¯ +i − xk N¯ +i+1 ||2

.

.

− 32δ 1/2 (4M + 7)3/2 .

(2.213)

The Third Result with Nonsummable Errors

73

In view of (2.194), (2.205) and (2.212), ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − z∗r(k N¯ +i+1) || + ||z∗r(k N¯ +i+1) − xk N¯ +i+1 ||

.

.

.

≤ δC + ||z∗r(k N¯ +i+1) − xk N¯ +i || + 2(2δ(4M + 7))1/2

≤ δC + 2(2δ(4M + 7))1/2 + ||z∗ − xk N¯ +i || + ||z∗r(k N¯ +i+1) − z∗ || ≤ 2δC + 2(2δ(4M + 7))1/2 + ||z∗ − xk N¯ +i ||

.

≤ ||z∗ − xk N¯ +i || + Δ.

(2.214)

||xk N¯ +i+1 − z∗ || ≤ 2M + 2 + (i + 1)Δ.

(2.215)

.

By (2.207) and (2.214), .

Let j ∈ {k N¯ + i, . . . , k N¯ + i + 1}.

.

It follows from (2.194), (2.207) and (2.215) that |||z∗r(k N¯ +i+1) − xj ||2 − ||z∗ − xj ||2 |

.

.

= |||z∗r(k N¯ +i+1) − xj || − ||z∗ − xj |||(||z∗r(k N¯ +i+1) − xj || + ||z∗ − xj ||) ≤ ||z∗ − z∗r(k N¯ +i+1) ||(4M + 7) ≤ δC (4M + 7).

.

(2.216)

Equations (2.213) and (2.216) imply that ||z∗ − xk N¯ +i ||2 ≥ ||z∗r(k N¯ +i+1) − xk N¯ +i ||2 − δC (4M + 7)

.

.

≥ ||z∗r(k N¯ +i+1) − xk N¯ +i+1 ||2 + ||xk N¯ +i − xk N¯ +i+1 ||2 .

.

− 32δ 1/2 (4M + 7)3/2 − δC (4M + 7)

≥ ||z∗ − xk N¯ +i+1 ||2 + ||xk N¯ +i − xk N¯ +i+1 ||2 .

− 32δ 1/2 (4M + 7)3/2 − 2δC (4M + 7).

(2.217)

||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 .

(2.218)

Assume that .

74

2 Iterative Methods in a Hilbert space

By (2.197), (2.217) and (2.218), ||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 + 32δ 1/2 (4M + 7)3/2 + 2δC (4M + 7)

.

.

≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 /2. U ∩

Lemma 2.32 is proved.

It follows from (2.196), (2.197), (2.202), (2.204), (2.205) and Lemma 2.32 applied by induction that for all .i = 0, . . . , N¯ − 1, ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + 2δC + 2(2δ(4M + 7))1/2 ,

.

||z∗ − xk N¯ +i+1 || ≤ 2M + 2 + (i + 1)[2δC + 2(2δ(4M + 7))1/2 ]

.

.

¯ C + 2(2δ(4M + 7))1/2 ) ≤ 2M + 3 ≤ 2M + 2 + N(2δ

(2.219)

and 2 2 ||z∗ − x(k+1)N) ¯ || − ||z∗ − xk N¯ ||

.

.

=

¯ −1 N Σ

(||z∗ − xk N¯ +i+1 ||2 − ||z∗ − xk N¯ +i ||2 )

i=0 .

1/2 ¯ ≤ − ɛ 02 /2 + N(32δ (4M + 7))3/2 + 2δC (4M + 7) ≤ − ɛ 02 /4.

(2.220)

Thus we have shown that the following property holds: (P1) if an integer .k ∈ [0, s] satisfies .||z∗ − xk N¯ || ≤ 2M + 2, then ¯ . . . , (k + 1)N¯ , ||z∗ − xi || ≤ 2M + 3, i = k N,

.

||z∗ − x(k+1)N¯ ||2 − ||z∗ − xk N¯ ||2 ≤ −4−1 ɛ 02 .

.

By (2.203) and property (P1), ||xj − z∗ || ≤ 2M + 3, j = 0, . . . , (s + 1)N¯ ,

.

and (2.221) holds for all .k = 0, . . . , s. In view (2.203) and (2.221), 4−1 ɛ 02 (s + 1) ≤

.

s Σ (||z∗ − xk N¯ ||2 − ||z∗ − x(k+1)N¯ ||2 ) k=0

(2.221)

The Third Result with Nonsummable Errors

.

75

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 ɛ 0−2 M 2 .

.

Thus we have shown that the following property holds: (P2) If an integer .s ≥ 0 and for each integer .k ∈ [0, s] (2.202) holds, then s ≤ 16M 2 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 3, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , (s + 1).

.

Property (P2) and (2.198) imply that there exists an integer .q ∈ [0, n0 − 1] such that for each integer k satisfying .0 ≤ k < q, .

max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 ,

.

max{||xi − xi−1 || : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 , ||xq N¯ − z∗ || ≤ 2M,

.

||xj − z∗ || ≤ 2M + 3, j = 0, . . . , q N¯ ,

.

||xj || ≤ 3M + 3, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(2.222)

In view of (2.200), for each .i ∈ {q N¯ + 1, . . . , (q + 1)N¯ }, there exists ξi ∈ Cr(i)

(2.223)

||xi − ξi || ≤ δ.

(2.224)

.

such that .

By (2.222) and (2.224), for each .i = {q N¯ + 1, . . . , (q + 1)N¯ }, ||xi−1 − ξi || ≤ ||xi−1 − xi || + ||xi − ξi || ≤ ɛ 0 + δ.

.

(2.225)

76

2 Iterative Methods in a Hilbert space

Let i ∈ {q N¯ , . . . , (q + 1)N¯ }, s ∈ {1, . . . , m}.

.

(2.226)

Relation (2.195) implies that there exists j ∈ {q N¯ + 1, . . . , (q + 1)N¯ }

(2.227)

s = r(j ).

(2.228)

.

such that .

By (2.222), (2.225)–(2.227), ¯ 0. ||xj −1 − ξj || ≤ ɛ 0 + δ, ||xi − xj −1 || ≤ N ɛ

.

It follows from (2.223), (2.227) and (2.228) that ||xi − ξj || ≤ (N¯ + 2) ɛ 0

.

and ξj ∈ Cr(j ) = Cs , d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

Theorem 2.29 is proved.

Almost Star-Shaped Feasibility Problems Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

(2.229)

z∗ ∈ B(0, M),

(2.230)

B(0, M) ∩ Cs /= ∅, s = 1, . . . , m

(2.231)

.

Assume that .M > 0, .δM ∈ (0, 1]. .

.

and that the following assumption holds. (A3) For each .s ∈ {1, . . . , m}, .x ∈ B(0, 5M + 2) ∩ Cs and each .α ∈ [0, 1], d(αz∗ + (1 − α)x, Cs ) < δM .

.

(2.232)

Almost Star-Shaped Feasibility Problems

77

Assumption (A3) and (2.230) imply that for each .s ∈ {1, . . . , m} there exists z∗s ∈ Cs

(2.233)

||z∗ − z∗s || < δM .

(2.234)

||z∗s || ≤ M + δM ≤ M + 1, s = 1, . . . , m.

(2.235)

.

such that .

In view of (2.230) and (2.234), .

We study the following feasibility problem Find z ∈ C

.

which is called, in view of (A3), as an almost star-shaped feasibility problem. Fix a natural number .N¯ ≥ m. Denote by .R the set of all mappings .r : {1, 2, . . . , } → {1, . . . , m} such that for each integer .j ≥ 1, {1, . . . , m} ⊂ {r(j ), . . . , r(j + N¯ − 1)}.

(2.236)

δ ∈ (0, 1],

(2.237)

4N¯ (2δ + 2δM )1/2 (4M + 4)1/2 ≤ 1,

(2.238)

ɛ 0 ≥ (64N¯ (δ + δM )1/2 (4M + 4)3/2 )1/2 ,

(2.239)

.

Theorem 2.33 Let .

.

.

a natural number .n0 satisfy n0 ≥ 1 + L16M 2 ɛ 0−2 ⎦ .

.

(2.240)

Assume that r ∈ R,

.

x0 ∈ B(0, M)

.

(2.241)

and that .{xi }∞ i=1 ⊂ X satisfies for each natural number i, B(xi , δ) ∩ {ξ ∈ Cr(i) : ||xi−1 − ξ || ≤ d(xi−1 , Cr(i) ) + δ} /= ∅.

.

(2.242)

78

2 Iterative Methods in a Hilbert space

Then there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(θ, 3M + 1), i = 0, . . . , q N¯ ,

.

¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(2.243)

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (2.243), then for each .i = ¯ and all .s = 1, . . . , m, q N¯ , . . . , (q + 1)N, d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

Proof Assume that s is a natural number and that for each integer .k ∈ [0, s], .

¯ > ɛ 0 . max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N}

(2.244)

By (2.230) and (2.241), ||x0 − z∗ || ≤ 2M.

.

(2.245)

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

(2.246)

Δ = 4(2(δ + δM )1/2 (4M + 4))1/2 .

(2.247)

.

Set .

In view of (2.238) and (2.247), ΔN¯ ≤ 1.

.

We prove the following auxiliary result.

(2.248) U ∩

Lemma 2.34 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ 2M + iΔ.

.

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + Δ,

.

(2.249)

Almost Star-Shaped Feasibility Problems

79

||xk N¯ +i+1 − z∗ ||2

.

.

≤ ||xk N¯ +i − z∗ ||2 − ||xk N¯ +i − xk N¯ +i+1 ||2 + 32(δ + δM )1/2 (4M + 4)3/2 .

If .||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 , then ||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ≤ −2−1 ɛ 02 .

.

Proof Equations (2.248) and (2.249) imply that ||xk N¯ +i − z∗ || ≤ 2M + 1.

.

(2.250)

In view of (2.242), there exists ξ ∈ Cr(k N¯ +i+1) ∩ B(xk N¯ +i+1 , δ)

(2.251)

||ξ − xk N¯ +i || ≤ d(xk N¯ +i , Cr(k N¯ +i+1) ) + δ.

(2.252)

.

such that .

Equations (2.233), (2.234), (2.247)–(2.249), and (2.252) imply that ||ξ − xk N¯ +i || ≤ ||xk N¯ +i − z∗r(k N¯ +i+1) || + δ ≤ ||xk N¯ +i − z∗ || + δ + δM

.

.

≤ 2M + (i + 1)Δ ≤ 2M + 1.

(2.253)

Equations (2.230), (2.250) and (2.53) imply that ||ξ || ≤ 2M + 1 + ||xk N¯ +i || ≤ 2M + 1 + 3M + 1.

.

(2.254)

Let E0 = {αz∗ + (1 − α)ξ : α ∈ [0, 1]}.

(2.255)

y ∈ E0 .

(2.256)

.

Fix .

Assumption (A3) and Eqs. (2.251), (2.254) and (2.255) imply that there exists ˆ y ∈ Cr(k N¯ +i+1)

.

(2.257)

80

2 Iterative Methods in a Hilbert space

such that ||y − ˆ y || < δM .

.

By (2.252), (2.257) and (2.258), ||xk N¯ +i − y|| ≥ ||xk N¯ +i − ˆ y || − ||ˆ y − y|| > ||xk N¯ +i − ˆ y || − δM

.

.

≥ d(xk N¯ +i , Cr(k N¯ +i+1) ) − δM ≥ ||ξ − xk N¯ +i || − δ − δM

and ||ξ − xk N¯ +i || ≤ ||xk N¯ +i − y|| + δ + δM for each y ∈ E0 .

.

(2.258)

Equations (2.251), (2.255), (2.258) and Lemma 2.3 applied with D = E0 , x = xk N¯ +i , x˜ = xk N¯ +i+1

.

imply that ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + 2(2(δ + δM )(4M + 4))1/2 ,

.

(2.259)

||z∗ − xk N¯ +i ||2 ≥ ||z∗ − xk N¯ +i+1 ||2 + ||xk N¯ +i − xk N¯ +i+1 ||2

.

.

− 32(δ + δM )1/2 (4M + 4)3/2 .

(2.260)

In view of (2.247) and (2.259), ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + Δ.

.

(2.261)

Assume that ||xk N¯ +i+1 − xk N¯ +i || > ɛ 0 .

.

By (2.239), (2.260) and by the relation above, ||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ − xk N¯ +i ||2 − ɛ 02 /2.

.

Lemma 2.34 is proved. It follows from (2.238), (2.239), (2.244), (2.246) and Lemma 2.34 applied by induction that for all .i = 0, . . . , N¯ − 1, ||z∗ − xk N¯ +i+1 || ≤ ||z∗ − xk N¯ +i || + 4[2(δM + δ)(4M + 4)]1/2 ,

.

Almost Star-Shaped Feasibility Problems

81

||z∗ − xk N¯ +i+1 || ≤ 2M + 4(i + 1)[2(δM + δ)(4M + 4)]1/2

.

≤ 2M + 4N¯ [2(δM + δ)(4M + 4)]1/2 ≤ 2M + 1

.

and 2 2 ||z∗ − x(k+1)N) ¯ || − ||z∗ − xk N¯ ||

.

.

=

¯ −1 N Σ

(||z∗ − xk N¯ +i+1 ||2 − ||z∗ − xk N¯ +i ||2 )

i=0 .

≤ − ɛ 02 /2 + 4 · 32N¯ (δ + δM )1/2 (4M + 4)3/2 ≤ − ɛ 02 /4.

Thus we have shown that the following property holds: (P1) if an integer .k ∈ [0, s] satisfies .||z∗ − xk N¯ || ≤ 2M, then ¯ . . . , (k + 1)N¯ , ||z∗ − xi || ≤ 2M + 1, i = k N,

.

||z∗ − x(k+1)N¯ ||2 − ||z∗ − xk N¯ ||2 ≤ −4−1 ɛ 02 .

.

By (2.240) and property (P1), ||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

and () holds for all .k = 0, . . . , s. In view (2.245) and (), s Σ + 1) ≤ (||z∗ − xk N¯ ||2 − ||z∗ − x(k+1)N¯ ||2 )

−1 2 .4 ɛ 0 (s

k=0 .

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 ɛ 0−2 .

.

Thus we have shown that the following property holds: (P2) If an integer .s ≥ 0 and for each integer .k ∈ [0, s] (2.244) holds, then s ≤ 16M 2 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , s + 1.

.

(2.262)

82

2 Iterative Methods in a Hilbert space

Property (P2) and (2.240) imply that there exists an integer .q ∈ [0, n0 − 1] such that for each integer k satisfying .0 ≤ k < q, .

max{||xi − xi−1 || : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 ,

.

max{||xi − xi−1 || : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 , ||xq N¯ − z∗ || ≤ 2M,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , q N¯ ,

.

||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ ||xi − xi−1 || ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(2.263)

In view of (2.242), for each .i ∈ {q N¯ + 1, . . . , (q + 1)N¯ }, there exists ξi ∈ Cr(i)

.

(2.264)

||xi − ξi || ≤ δ, ||xi−1 − ξi || ≤ d(xi−1 , Cr(i) ) + δ.

(2.265)

such that .

¯ By (2.263) and (2.265), for each .i ∈ {q N¯ + 1, . . . , (q + 1)N}, ||xi−1 − ξi || ≤ ||xi−1 − xi || + ||xi − ξi || ≤ ɛ 0 + δ.

.

(2.266)

Let i ∈ {q N¯ , . . . , (q + 1)N¯ }, s ∈ {1, . . . , m}.

.

Relation (2.236) implies that there exists j ∈ {q N¯ + 1, . . . , (q + 1)N¯ }

(2.267)

s = r(j ).

(2.268)

||xj −1 − ξj || ≤ ɛ 0 + δ, ||xi − xj −1 || ≤ ɛ 0 .

(2.269)

.

such that .

By (2.263), (2.266) and (2.267), .

Almost Star-Shaped Feasibility Problems

83

It follows from (2.263), (2.264) and (2.266) that ξj ∈ Cr(j ) = Cs , ||xi − ξj || ≤ (N¯ + 1) ɛ 0 + ɛ 0

.

and d(xi , Cs ) ≤ (N¯ + 2) ɛ 0 .

.

Theorem 2.33 is proved.

U ∩

Chapter 3

The Cimmino Algorithm in a Hilbert Space

In this chapter we study the convergence of the Cimmino algorithm for solving star-shaped feasibility problems in a Hilbert space. Our main goal is to obtain an approximate solution of the problem in the presence of computational errors. We show that the Cimmino algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

Exact Cimmino Iterates Let .(X, ) be a Hilbert space equipped with an inner product . which induce the norm ||x|| = 1/2 , x ∈ C.

.

For every .x ∈ X and every .r > 0 set B(x, r) = {y ∈ X : ||x − y|| ≤ r}.

.

For each .x ∈ X and each nonempty set .C ⊂ X define d(x, C) = inf{||x − y|| : y ∈ C}.

.

Suppose that m is a natural number, .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets and that for each .i ∈ {1, . . . , m} and each .x ∈ X Pi (x) := {v ∈ Ci : ||x − v|| ≤ ||x − y|| for every y ∈ Ci }

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_3

85

86

3 The Cimmino Algorithm in a Hilbert Space

is nonempty. This assumption holds if X is finite-dimensional or all bounded closed subsets of .Ci , .i = 1, . . . , m} are compact or all these sets are convex. Set C = ∩m i=1 Ci .

.

We study the following feasibility problem Find x ∈ C.

.

Assume that z∗ ∈ ∩m i=1 Ci

.

and the following assumption holds: (A1) for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

(3.1)

Fix ˆ ∈ (0, m−1 ]. Δ

.

We consider the following Cimmino algorithm. Initialization: select an arbitrary .x0 ∈ X. Iterative step: given a current iteration point .xk choose ˆ i = 1, . . . , m wk,i ≥ Δ,

.

satisfying m Σ .

wk,i = 1

i=1

and calculate the next iteration point .xk+1 by xk+1 ∈

m Σ

.

wk,i Pi (xk ).

i=1

Theorem 3.1 Assume that M > ||z∗ ||,

.

(3.2)

Exact Cimmino Iterates

87

ɛ ∈ (0, 1], .{xk }∞ k=0 ⊂ X,

.

ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

(3.3)

||x0 || ≤ M,

(3.4)

.

.

and that for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.5)

i=1

and xk+1 ∈

m Σ

.

wk,i Pi (xk ).

(3.6)

i=1

Then ||xk || ≤ 3M, k = 0, 1, . . . ,

.

ˆ −1 . Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} ≥ ɛ }) < 4M 2 ɛ −2 Δ

.

Proof By (3.6), for each integer .k ≥ 0 and each .i ∈ {1, . . . , m} there exists yk,i ∈ Pi (xk )

(3.7)

.

such that xk+1 =

m Σ

.

wk,i yk,i .

(3.8)

i=1

Proposition 2.4 and (3.7) imply that for each integer .k ≥ 0 and each .i ∈ {1, . . . , m}, ||z∗ − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xk − yk,i ||2 .

.

(3.9)

It follows from (3.3), (3.5), (3.7)–(3.9) and the convexity of .|| · ||2 that for each integer .k ≥ 0, ||z∗ − xk+1 ||2 = ||z∗ −

m Σ

.

wk,i yk,i ||2

i=1

.

≤

m Σ i=1

wk,i ||z∗ − yk,i ||2

88

3 The Cimmino Algorithm in a Hilbert Space

.

≤

m Σ (||z∗ − xk ||2 − ||xk − yk,i ||2 )wk,i i=1

.

ˆ ≤ ||z∗ − xk ||2 − Δ

m Σ

||xk − yk,i ||2

i=1 .

.

ˆ max{||xk − yk,i || : i = 1, . . . , m}2 ≤ ||z∗ − xk ||2 − Δ

ˆ max{d(xk , Ci ) : i = 1, . . . , m}2 . ≤ ||z∗ − xk ||2 − Δ

(3.10)

In view of (3.4), for each integer .k ≥ 0, ||z∗ − xk || ≤ ||z∗ − x0 || ≤ 2M

.

and ||xk || ≤ 3M.

.

By (3.4) and (3.10), for each natural number Q, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xQ ||2

.

.

=

Q−1 Σ

[||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ]

k=0 .

.

ˆ max{d(xk , Ci ) : i = 1, . . . , m}2 ≥Δ

ˆ 2 Card({k ∈ {0, . . . , Q − 1} : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ }) ≥ Δɛ

and Card({k ∈ {0, . . . , Q − 1} :

.

.

ˆ −1 . max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ }) ≤ 4M 2 ɛ −2 Δ

Since the relation above holds for every natural number Q we conclude that ˆ −1 . Card({k ∈ {0, 1, . . . , } : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ }) ≤ 4M 2 ɛ −2 Δ

.

Theorem 3.1 is proved.

U ∩

Exact Cimmino Iterates

89

Corollary 3.2 Assume that the assumptions of Theorem 3.1 hold. Then there exist a nonnegative integer ˆ −1 k < 4M 2 ɛ −2 Δ

.

such that d(xk , Cs ) ≤ ɛ , s = 1, . . . , m.

.

Corollary 3.3 Assume that the assumptions of Theorem 3.1 hold. Then .

lim max{d(xj , Cs ) : s = 1, . . . , m} = 0.

j →∞

We continue to use the bounded regularity property (BRP for short) and the bounded regularity property (BLRP for short) introduced in Chap. 2. Theorem 3.4 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1] and that the bounded regularity property holds. Then there exists a natural number Q such that for each .{xk }∞ k=0 ⊂ X, each

.

ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

which satisfy ||x0 || ≤ M,

.

(3.11)

and for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.12)

i=1

and xk+1 ∈

m Σ

.

wk,i Pi (xk )

i=1

the inequality Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) < Q

.

holds.

(3.13)

90

3 The Cimmino Algorithm in a Hilbert Space

Proof Property (BPP) implies that there exists .ɛ 0 ∈ (0, ɛ ) such that the following property holds: (i) for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ ɛ 0 , i = 1, . . . , m

.

the inequality .d(x, C) ≤ ɛ holds. Choose a natural number ˆ −1 . Q > 4M 2 ɛ 0−2 Δ

.

(3.14)

Assume that .{xk }∞ k=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

satisfy (3.11) and (3.12), (3.13) for each integer .k ≥ 0. It follows from Theorem 3.1 and (3.14) that ||xk || ≤ 3M, k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} ≥ ɛ 0 }) ≤ Q.

.

(3.15) (3.16)

Property (i) and (3.15) imply that {k ∈ {0, 1, . . . } : d(xk , C) > ɛ }

.

.

⊂ {k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 }.

Together with (3.16) this implies that Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) ≤ Q.

.

U ∩

Theorem 3.4 is proved. Corollary 3.5 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1], the bounded regularity property holds and that a natural number Q is as guaranteed by Theorem 3.4. Assume that .{xk }∞ k=0 ⊂ X,

.

ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

Exact Cimmino Iterates

91

satisfy (3.11) and (3.12), (3.13) for each integer .k ≥ 0. Then there exists a nonnegative integer .k ∈ {0, . . . , Q} such that d(xk , C) ≤ ɛ

.

and .

lim d(xj , C) = 0.

j →∞

Theorem 3.6 Let .M > max{1, ||z∗ ||}, .Δ0 ∈ (0, 1), .L > 1 and let the following property hold: for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, C) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Assume that .ɛ ∈ (0, Δ0 ], .{xk }∞ k=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

satisfy ||x0 || ≤ M,

.

and for each integer .k ≥ 0, m Σ .

wk,i = 1

i=1

and xk+1 ∈

m Σ

.

wk,i Pi (xk ).

i=1

Then ˆ −1 ɛ −2 L2 . Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) < 4M 2 Δ

.

92

3 The Cimmino Algorithm in a Hilbert Space

Proof Theorem 3.1 implies that ||xk || ≤ 3M, k = 0, 1, . . . ,

.

(3.17)

Card({k ∈ {0, 1, . . . , } : max{d(xk , Cs ) : s = 1, . . . , m}) ≥ ɛ L−1 })

.

.

ˆ −1 L2 . ≤ 4M 2 ɛ −2 Δ

(3.18)

By (3.17) and our assumptions, if .k ∈ {0, 1, . . . } and d(xk , Cs ) ≤ ɛ L−1 , s = 1, . . . , m,

.

then d(xk , C) ≤ ɛ

.

and {k ∈ {0, 1, . . . } : d(xk , C) > ɛ }

.

.

⊂ {k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} > L−1 ɛ }.

Together with (3.18) this implies that ˆ −1 L2 . Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) ≤ 4M 2 ɛ −2 Δ

.

U ∩

Theorem 3.6 is proved.

Theorem 3.7 Assume that the bounded regularity property and the following property hold: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .ɛ ∈ (0, 1). Then there exists a natural number Q such that for each .{xk }∞ k=0 ⊂ X, each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

which satisfy ||x0 || ≤ M

.

(3.19)

Exact Cimmino Iterates

93

and for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.20)

i=1

and xk+1 ∈

m Σ

.

wk,i Pi (xk )

(3.21)

i=1

there exists .z¯ ∈ C such that ||¯z − xk || for each integer k ≥ Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Corollary 3.5 implies that that there exists a natural number Q such that the following property holds: (ii) for each .{xk }∞ k=0 ⊂ X, each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

satisfying (3.19)–(3.21) for each integer .k ≥ 0 there exists a nonnegative integer k ∈ {0, . . . , Q} such that

.

d(xk , C) ≤ ɛ /2.

.

Assume that .{xk }∞ k=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

and that (3.19)–(3.21) hold for each integer .k ≥ 0. Property (ii) implies that there exists z∈C

.

such that ||xk − z|| < ɛ .

.

We show that for each integer .j ≥ k, ||xj − z|| < ɛ .

.

(3.22)

94

3 The Cimmino Algorithm in a Hilbert Space

Assume that .j ≥ q is an integer and ||xj − z|| < ɛ .

(3.23)

yi ∈ Pi (xj ), i = 1, . . . , m

(3.24)

.

By (3.21), there exist .

such that xj +1 =

m Σ

.

wj,i yj,i .

(3.25)

i=1

Proposition 2.4 applied with .z∗ = z, property (i), (3.22) and (3.24) imply that for each .i ∈ {1, . . . , m}, ||z − yi,j || ≤ ||z − xj ||.

.

Together with (3.20), (3.23), (3.25) and the convexity of the norm this implies that ||z − xj +1 || = ||z −

m Σ

.

wj,i yj,i || ≤

i=1

m Σ

wj,i ||z − yj,i || ≤ ||z − xj || < ɛ .

i=1

Since .ɛ is any element of .(0, 1) and the set C is closed the sequence .{xi }∞ i=0 converges to a point of C. Theorem 3.7 is proved. U ∩ Theorem 3.8 Assume that the following property holds: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .L > 1, .Δ0 ∈ (0, 1) and the following property holds: (ii) for each .x ∈ B(0, 5M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, Ci ) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Assume .ɛ ∈ (0, Δ0 ). .{xk }∞ k=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

Exact Cimmino Iterates

95

satisfy ||x0 || ≤ M,

.

and for each integer .k ≥ 0, m Σ .

wk,i = 1,

i=1

and xk+1 ∈

m Σ

.

wk,i Pi (xk ).

i=1

Then there exists .limk→∞ xk ∈ C and ˆ −1 ɛ −2 L2 . d(xk , C) < ɛ for each integer k > 16M 2 Δ

.

Proof Theorem 3.6 implies that there exists a nonnegative integer ˆ −1 ɛ −2 L2 j ≤ 16M 2 Δ

.

such that d(xj , C) ≤ ɛ /2.

.

We show that for each integer .k ≥ j d(xk , C) < ɛ .

.

Assume that an integer .k ≥ j and d(xk , C) < ɛ .

.

For each .i ∈ {1, . . . , m}, there exists yi ∈ Pi (xk )

.

such that xk+1 =

m Σ

.

i=1

wk,i yi .

96

3 The Cimmino Algorithm in a Hilbert Space

Proposition 2.4, property (i) and the relation above imply that ||z − xk+1 || = ||z −

m Σ

.

wk,i yi || ≤

i=1

m Σ

wk,i ||z − yi || ≤ ||z − xk || < ɛ .

i=1

U ∩

Theorem 3.8 is proved.

Inexact Cimmino Iterates with Summable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in section “Exact Cimmino Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

(3.26)

ˆ ∈ (0, m−1 ]. Δ

(3.27)

.

Fix .

For each .i ∈ {1, . . . , m}, each .x ∈ X and each .ɛ > 0 set Pi,ɛ (x) = {y ∈ Ci : ||x − y|| ≤ d(x, Ci ) + ɛ }.

.

(3.28)

The following theorem describes the behavior of inexact iterates with computational errors. Theorem 3.9 Assume that M > max{1, ||z∗ ||},

.

(3.29)

ɛ ∈ (0, 1), .{Δi }∞ i=0 ⊂ (0, 1],

.

∞ Σ .

i=0

1/2

Δi

< ∞,

(3.30)

Inexact Cimmino Iterates with Summable Errors

97

M1 = 8M + 162 (

∞ Σ

.

1/2

Δj )2 < ∞,

(3.31)

j =0

M2 = 4M 2 + 64(2M1 + 1)3/2

∞ Σ

.

1/2

Δi ,

(3.32)

i=0

{xk }∞ k=0 ⊂ X,

.

ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

(3.33)

||x0 || ≤ M,

(3.34)

.

.

for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.35)

i=1

and B(xk+1 , Δk ) ∩

m Σ

.

wk,i Pi,Δk (xk ) = / ∅.

(3.36)

i=1

Then ||xk || ≤ M1 + M, k = 0, 1, . . . ,

.

ˆ −1 . Card({k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} ≥ ɛ }) ≤ M 2 ɛ −2 Δ

.

Proof Let .k ≥ 0 be an integer. By (3.36), for each .i ∈ {1, . . . , m} there exists yk,i ∈ Pi,Δk (xk )

.

(3.37)

such that ||xk+1 −

m Σ

.

wk,i yk,i || ≤ Δk .

(3.38)

i=1

By (3.28) and (3.37), for each .i ∈ {1, . . . , m}, yk,i ∈ Ci ,

(3.39)

||xk − yk,i || ≤ ||xk − η|| + Δk , for each η ∈ Ci .

(3.40)

.

.

98

3 The Cimmino Algorithm in a Hilbert Space

Let .i ∈ {1, . . . , m}. Assumption (A1), (3.26), (3.39), (3.40) and Lemma 2.3 applied with .δ = Δk , .x = xk , .x˜ = yk,i , .D = Ci imply that ||z∗ − yk,i || ≤ ||z∗ − xk || + 2(2Δk (2||x − z∗ || + 1))1/2 ,

.

(3.41)

||z∗ − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xi − yk,i ||2

.

.

− 8(2Δk (2||xk − z∗ || + 1))1/2 ||z∗ − xk || − 16Δk (2||xk − z∗ || + 1).

(3.42)

By (3.33), (3.35), (3.38) and the convexity of the norm, ||z∗ − xk+1 ||

.

.

m Σ

≤ ||z∗ −

wk,i yk,i || + ||

i=1

.

≤

m Σ

wk,i yk,i − xk+1 ||

i=1

m Σ

wk,i ||z∗ − yk,i || + Δk

i=1 .

≤ ||z∗ − xk || + 2(2Δk (2||x − z∗ || + 1))1/2 + Δk .

≤ ||z∗ − xk || + 4(2Δk (2||x − z∗ || + 1))1/2 .

(3.43)

Equation (3.31) implies that 2M + 4

.

∞ ∞ Σ Σ 1/2 1/2 (2Δj (2M1 + 1))1/2 ≤ 2M + 12 Δ j M1 j =0

.

j =0

≤ M1 /4 + 12M1 M1 16−1 ≤ M1 . 1/2

1/2

(3.44)

We show that for each .k = 1, 2, . . . , k−1 Σ .||z∗ − xk || ≤ 2M + 4 (2Δi (2M1 + 1))1/2 .

(3.45)

i=0

In view of (3.29) and (3.34), Eq. (3.45) holds for .k = 1. Assume that .k ≥ 1 is an integer and (3.45) holds. By (3.43)–(3.45), ||z∗ − xk+1 || ≤ ||z∗ − xk || + 4(2Δk (2M1 + 1))1/2

.

.

≤ 2M + 4

k−1 Σ (2Δi (2M1 + 1))1/2 + 4(2Δk (2M1 + 1))1/2 . i=0

Inexact Cimmino Iterates with Summable Errors

99

Thus our assumption holds for .k + 1 too. Therefore by induction we showed that (3.45) holds for all natural numbers k. Equations (3.44) and (3.45) imply that for all natural numbers k ||z∗ − xk || ≤ M1 .

(3.46)

.

By (3.41)–(3.46), for each integer .k ≥ 0 and each .i ∈ {1, . . . , m}, ||z∗ − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xk − yk,i ||2

.

.

1/2

− 32Δk (2M1 + 1)3/2 ,

(3.47)

||z∗ − yk,i || ≤ ||z∗ − xk || + 2(2Δk (2M1 + 1))1/2 .

.

(3.48)

Equations (3.44), (3.45) and (3.47) imply that for each integer .k ≥ 0 and each i ∈ {1, . . . , m},

.

||z∗ − yk,i || ≤ M1 .

(3.49)

.

It follows from (3.35), (3.49) and the convexity of the norm .||·|| that for each integer k ≥ 0,

.

||z∗ −

m Σ

.

wk,i yk,i || ≤

i=1

m Σ

wk,i ||z∗ − yk,i || ≤ M1 .

(3.50)

i=1

By (3.33), (3.35), (3.39), (3.47) and the convexity of the function .|| · ||2 , for each integer .k ≥ 0, ||z∗ −

m Σ

.

i=1

.

≤

m Σ

wk,i yk,i ||2 ≤

m Σ

wk,i ||z∗ − yk,i ||2

i=1

1/2

wk,i (||z∗ − xk ||2 − ||xk − yk,i ||2 ) + 32Δk (2M1 + 1)3/2

i=1 .

.

1/2

≤ ||z∗ − xk ||2 + 32Δk (2M1 + 1)3/2

ˆ max{||xk − yk,i ||2 : i = 1, . . . , m} −Δ 1/2

.

≤ ||z∗ − xk ||2 + 32Δk (2M1 + 1)3/2

.

ˆ max{d(xk , Ci ) : i = 1, . . . , m}2 . −Δ

(3.51)

100

3 The Cimmino Algorithm in a Hilbert Space

Let .k ≥ 0 be an integer. By (3.38), (3.46) and (3.50) imply that |||z∗ − xk+1 ||2 − ||z∗ −

m Σ

.

wk,i yk,i ||2 |

i=1

.

≤ |||z∗ − xk || − ||z∗ −

m Σ

wk,i yk,i |||

i=1

.

× (||z∗ − xk || + ||z∗ −

m Σ

wk,i yk,i ||)

i=1

.

≤ 2M1 ||xk+1 −

m Σ

wk,i yk,i || ≤ 2Δk M1 .

(3.52)

i=1

In view of (3.51) and (3.52), ||z∗ − xk+1 ||2 ≤ ||z∗ −

m Σ

.

wk,i yk,i ||2 + 2Δk M + 1

i=1

.

ˆ max{d(xk , Ci ) : i = 1, . . . , m}2 . ≤ ||z∗ − xk ||2 + 64Δk (2M1 + 1)3/2 − Δ (3.53) 1/2

It follows from (3.29), (3.34) and (3.53) that for each natural number Q, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xQ ||2

.

.

=

Q−1 Σ

[||z∗ − xk || − ||z∗ − xk+1 ||2 ]

k=0

.

≥

Q−1 Σ

ˆ max{d(xk , Ci ) : i = 1, . . . , m}2 − 64Δ (2M1 + 1)3/2 ) (Δ k 1/2

k=0

and M2 ≥ 4M 2 +

∞ Σ

.

k=0

1/2

64Δk (2M1 + 1)3/2

Inexact Cimmino Iterates with Summable Errors

≥

.

Q−1 Σ

101

ˆ max{d(xk , Ci ) : i = 1, . . . , m}2 ) (Δ

k=0 .

ˆ 2 Card({k ∈ {0, . . . , Q − 1} : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ }) ≥ Δɛ

and Card({k ∈ {0, . . . , Q − 1} : max{d(xk , Ci ) : i = 1, . . . , m}) ≥ ɛ })

.

.

ˆ −1 ɛ −2 M2 . ≤Δ

(3.54)

Since (3.54) holds for every natural number Q we conclude that Card({k ∈ {0, 1, . . . } : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ })

.

.

ˆ −1 ɛ −2 M2 . ≤Δ U ∩

Theorem 3.9 is proved.

Corollary 3.10 Assume that the assumptions of Theorem 3.9 hold. Then there exist a nonnegative integer ˆ −1 j ≤ M2 ɛ −2 Δ

.

such that d(xj , Cs ) < ɛ , s = 1, . . . , m.

.

Corollary 3.11 Assume that the assumptions of Theorem 3.9 hold. Then .

lim max{d(xj , Cs ) : s = 1, . . . , m} = 0.

j →∞

Theorem 3.12 Assume that .M > max{1, ||z∗ ||}, .ɛ ∈ (0, 1), .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

i=0

M1 = 8M + 162 (

∞ Σ

.

1/2

Δ i )2 ,

i=0

M2 = 4M 2 + 64(2M1 + 1)3/2

∞ Σ

.

i=0

1/2

Δi

102

3 The Cimmino Algorithm in a Hilbert Space

and that the bounded regularity property holds. Then there exists a natural number Q such that for each .{xi }∞ i=0 ⊂ X and each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

which satisfy ||x0 || ≤ M,

.

(3.55)

for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.56)

i=1

and B(xk+1 , Δk ) ∩

m Σ

.

wk,i Pi,Δk (xk ) = / ∅

(3.57)

i=1

the inequalities ||xk || ≤ M1 + M, k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) ≤ Q

.

hold and .

lim d(xk , C) = 0.

k→∞

Proof Property (BPP) implies that there exists .ɛ 0 ∈ (0, ɛ ) such that the following property holds: (i) for each .x ∈ B(0, 2M1 ) satisfying d(x, Ci ) ≤ ɛ 0 , i = 1, . . . , m

.

the inequality .d(x, C) ≤ ɛ holds. Assume that .{xi }∞ i=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

and (3.55)–(3.57) hold for .k = 0, 1, . . . . Theorem 3.9 implies that ||xi || ≤ M1 + M, i = 0, 1, . . . ,

.

Inexact Cimmino Iterates with Summable Errors

103

ˆ −1 . Card({k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} ≥ ɛ 0 }) ≤ M2 ɛ 0−2 Δ

.

Together with property (i) this implies that Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ })

.

.

≤ Card({k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 }) ˆ −1 . ≤ M2 ɛ 0−2 Δ

.

U ∩

Theorem 3.12 is proved. Theorem 3.13 Let .M > max{1, ||z∗ ||}, .Δ ∈ (0, 1), .L > 1, .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

i=0

M1 = 8M + 162 (

∞ Σ

.

1/2

Δ i )2 ,

i=0

M2 = 4M + 64(2M1 + 1) 2

.

3/2

∞ Σ

1/2

Δi ,

i=0

ɛ ∈ (0, Δ) and the following property holds: (i) for each .x ∈ B(0, 2M1 ) satisfying

.

d(x, Ci ) ≤ Δ, i = 1, . . . , m

.

the inequality d(x, Ci ) ≤ L max{d(xi , Cs ) : s = 1, . . . , m}

.

is true. Assume that .{xk }∞ k=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m, wk,i ≥ Δ,

.

||x0 || ≤ M,

.

and for each integer .k ≥ 0, m Σ .

i=1

wk,i = 1

104

3 The Cimmino Algorithm in a Hilbert Space

and B(xk+1 , Δk ) ∩

m Σ

.

wk,i Pi,Δk (xk ) = / ∅.

i=1

Then ˆ −1 . Card({k ∈ {0, 1, . . . } : d(x, C) > ɛ }) ≤ M2 ɛ −2 L2 Δ

.

Proof Theorem 3.9 implies that Card({k ∈ {0, 1, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} ≥ ɛ L−1 })

.

ˆ −1 M2 L2 , ≤ ɛ −2 Δ

(3.58)

||xi || ≤ 2M1 , i = 0, 1, . . . ,

(3.59)

.

.

Property (i) implies that if an integer .k ≥ 0 and d(xk , C) > ɛ ,

.

then .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ L−1 .

Together with (3.58) his implies that ˆ −1 . Card({k ∈ {0, 1, . . . } : d(x, C) > ɛ }) ≤ M2 ɛ −2 L2 Δ

.

U ∩

Theorem 3.13 is proved.

Theorem 3.14 Assume that the bounded regularity property and the following property hold: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let M > max{1, ||z∗ ||},

.

{Δi }∞ i=0 ⊂ (0, 1],

.

∞ Σ .

i=0

1/2

Δi

< ∞,

(3.60)

Inexact Cimmino Iterates with Summable Errors

105

M1 = 8M + 162 (

∞ Σ

.

1/2

Δ i )2 ,

(3.61)

i=0

ɛ ∈ (0, 1). Then there exists a natural number Q such that for each each {xi }∞ i=0 ⊂ X, each

. .

ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

which satisfy ||x0 || ≤ M,

.

(3.62)

for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.63)

i=1

and B(xk+1 , Δk ) ∩

m Σ

.

wk,i Pi,Δk (xk ) /= ∅

(3.64)

i=1

the inequality ||xi || ≤ 2M1 , i = 0, 1, . . .

.

(3.65)

holds, there exists .z ∈ C such that ||z − xi || ≤ ɛ for each integer i ≥ Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Theorem 3.12 implies that there exists a natural number Q such that ∞ Σ .

(Δi + (Δi (8M1 + 8))1/2 ) < ɛ /2

(3.66)

i=Q

and that the following property holds: ˆ (iii) for each .{xi }∞ i=0 ⊂ X, each .wk,i ≥ Δ, k = 0, 1, . . . , i = 1, . . . , m satisfying (3.62)–(3.64) for each integer .k ≥ 0, the inequality (3.65) hold and there is

106

3 The Cimmino Algorithm in a Hilbert Space

j ∈ {Q, . . . , 2Q} for which

.

d(xj , C) < ɛ /2.

.

ˆ Assume that .{xi }∞ i=0 ⊂ X, .wk,i ≥ Δ, k = 0, 1, . . . , i = 1, . . . , m and (3.62)–(3.64) hold for each integer .k ≥ 0. Property (ii) implies that ||xi || ≤ 2M1 , i = 0, 1, . . .

(3.67)

k ∈ {Q, . . . , 2Q}

(3.68)

.

and there is .

for which d(xk , C) < ɛ /2.

.

Clearly, there exists z∈C

(3.69)

||xk − z|| < ɛ /2.

(3.70)

||z|| ≤ 2M1 + 1.

(3.71)

.

such that .

In view of (3.67) and (3.70), .

Set γq = 0

.

and for each integer .q > k, γq =

q−1 Σ

.

(Δi + (2Δi (8M1 + 2))1/2 ).

(3.72)

i=k

We show that for each integer .q ≥ k, ||xq − z|| ≤ ɛ /3 + γq .

.

(3.73)

(In view of (3.70), Eq. (3.73) holds for .q = k.) Assume that .q ≥ k is an integer and (3.73) holds. By (3.64), for each integer .i ∈ {1, . . . , m}, there exists yi ∈ Pi,Δq (xq )

.

(3.74)

Inexact Cimmino Iterates with Summable Errors

107

such that ||xq+1 −

m Σ

.

wq,i yi || ≤ Δq .

(3.75)

i=1

Let .i ∈ {1, . . . , m}. In view of (3.74), ||xq − yi || ≤ d(xq , Ci ) + Δq .

.

(3.76)

Property (i), (3.67), (3.69), (3.71), (3.73), (3.76) and Lemma 2.3 applied with x = xq , x˜ = ξ = yi , δ = Δq , z∗ = z, D = Ci

.

imply that ||z − yi || ≤ ||z − xq || + 2(2Δq (2||xq − z|| + 1))1/2

.

.

≤ ||z − xq || + 2(2Δq )1/2 (8M1 + 2)1/2 .

(3.77)

It follows from (3.63), (3.72), (3.73), (3.75). (3.77) and the convexity of the norm that ||z − xq+1 ||

.

.

≤ ||z −

m Σ

wk,i yk,i || + ||

i=1

.

≤

m Σ

wk,i yk,i − xq+1 ||

i=1 m Σ

wk,i ||z − yk,i || + Δq

i=1 .

.

≤ Δq + (2Δq (8M1 + 2))1/2 + ||z − xq ||

< Δq + (2Δq (8M1 + 2))1/2 + γq + ɛ /2 ≤ ɛ /2 + γq+1 .

Thus by induction we showed that (3.73) holds for each integer .q ≥ k. By (3.66), (3.68) and (3.73), for each integer .q ≥ k, ||z − xq || < ɛ /2 +

.

∞ Σ (Δi + (2Δi (8M1 + 2))1/2 ) < ɛ . i=k

Theorem 3.14 is proved. U ∩

108

3 The Cimmino Algorithm in a Hilbert Space

The First Result with Nonsummable Errors for the Cimmino Algorithm Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

(3.78)

We continue to assume that assumption (A1) introduced in section “Exact Cimmino Iterates” holds. (A1) There exists z∗ ∈ C

.

(3.79)

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

(3.80)

M > max{1, ||z∗ ||},

(3.81)

ˆ −1 (8M + 3)−1 ), δ ∈ (0, 4−1 Δ

(3.82)

.

ˆ −1 )1/2 , 1 ≥ ɛ 0 ≥ (4δ(8M + 3)Δ

(3.83)

ˆ −1 ⎦. n0 = 1 + L8(4M + 1)2 ɛ 0−2 Δ

(3.84)

.

Theorem 3.15 Let .

.

.

Assume that .{xi }∞ i=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m, wk,i ≥ Δ,

(3.85)

||x0 || ≤ M

(3.86)

.

.

and that for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.87)

i=1

and B(xk+1 , δ) ∩

m Σ

.

i=1

wk,i Pi,0 (xk ) /= ∅.

(3.88)

The First Result with Nonsummable Errors for the Cimmino Algorithm

109

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Note that in Theorem 3.15 we assume that for each integer .k ≥ 0 and each i ∈ {1, . . . , m},

.

Pi,0 (xk ) /= ∅.

.

This assumption holds if the space X is finite-dimensional or all the sets .Ci , i = 1, . . . , m are convex. It should be mentioned that in Theorem 3.15 .δ is the computational error made by our computer system, we obtain a point z such that .d(z, Cs ) ≤ ɛ 0 and in order to obtain this point we need .n0 iterations. It is not difficult to see that .ɛ 0 = c1 δ 1/2 and .n0 = Lc2 δ −1 ⎦ + 1, where .c1 and .c2 are positive constants depending on M. Applying by induction Theorem 3.15 we obtain the following result. .

Theorem 3.16 Suppose that .ɛ ¯ ∈ (0, 1), M > max{1, ||z∗ ||},

.

{x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

Let .ɛ 0 ∈ (0, ɛ ), ¯ ˆ −1 (8M + 3)−1 ), δ ∈ (0, 4−1 Δ

.

ˆ −1 )1/2 , 1 ≥ ɛ 0 ≥ (2δ(8M + 3)Δ

.

ˆ −1 ⎦. n0 = 1 + L2(4M + 1)2 ɛ 0−2 Δ

.

Assume that .{xi }∞ i=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m, wk,i ≥ Δ,

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, m Σ .

i=1

wk,i = 1

110

3 The Cimmino Algorithm in a Hilbert Space

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,0 (xk ) = / ∅.

i=1

Then xi ∈ B(0, 5M + 1) for all integers i ≥ 0

.

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 0 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

and that for each integer .p ≥ 0, d(xqp , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Proof of Theorem 3.15 Let .k ≥ 0 be an integer. By (3.88), for each .i ∈ {1, . . . , m}, there exists yk,i ∈ Pi,0 (xk )

(3.89)

||yk,i − xk || = d(xk , Ci ),

(3.90)

.

such that .

||xk+1 −

m Σ

.

wk,i yk,i || ≤ δ.

(3.91)

i=1

Let .i ∈ {1, . . . , m}. In view of (3.79), (3.86), (3.90) and (3.91), ||y0,i − x0 || ≤ ||x0 − z∗ || ≤ 2M,

.

||y0,i || ≤ 3M,

.

||x1 || ≤ max{||yk,i || : i = 1, . . . , m} + 1 ≤ 3M + 1.

.

(3.92)

Assume that s is a natural number and for each integer .k ∈ [1, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(3.93)

The First Result with Nonsummable Errors for the Cimmino Algorithm

111

By (3.92), ||x1 − z∗ || ≤ 4M + 1.

.

(3.94) U ∩

We prove the following auxiliary result. Lemma 3.17 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 1.

.

(3.95)

Then ˆ ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 2−1 ɛ 02 Δ.

.

Proof Let .i ∈ {1, . . . , m}. Proposition 2.4, (3.79), (3.89) and (3.90) imply that ||z∗ − yk,i ||2 + ||xk − yk,i ||2 ≤ ||z∗ − xk ||2 .

.

(3.96)

In view of (3.90) and (3.93), there exists .p ∈ {1, . . . , m} such that ||yk,p − xk || = d(xk , Cp ) > ɛ 0 .

.

By (3.96) and the equation above, ||z∗ − yk,p ||2 ≤ ||z∗ − xk ||2 − ɛ 02 .

.

It follows from (3.73), (3.85), (3.87), (3.96), the equation above and the convexity of the function .|| · ||2 that ||z∗ −

m Σ

.

wk,i yk,i ||2 ≤

i=1 .

.

≤ wk,p ||z∗ − yk,p ||2 +

wk,i ||z∗ − yk,i ||2

i=1

Σ {wk,i ||z∗ − yk,i ||2 : i ∈ {1, . . . , m} \ {p}}

≤ wk,p (||z∗ − xk ||2 − ɛ 02 ) + .

m Σ

Σ {wk,i ||z∗ − xk ||2 : i ∈ {1, . . . , m} \ {p}}

ˆ ≤ ||z∗ − xk ||2 − wk,p ɛ 02 ≤ ||z∗ − xk ||2 − ɛ 02 Δ

and ||z∗ −

m Σ

.

i=1

wk,i yk,i || ≤ ||z∗ − xk ||.

112

3 The Cimmino Algorithm in a Hilbert Space

By (3.91), (3.95) and the equation above, |||z∗ − xk+1 ||2 − ||z∗ −

m Σ

.

wk,i yk,i ||2 |

i=1

.

m Σ

≤ |||z∗ − xk+1 || − ||z∗ −

wk,i yk,i |||(||z∗ − xk+1 || + ||z∗ −

i=1

.

≤ ||xk+1 −

m Σ

m Σ

wk,i yk,i ||)

i=1

wk,i yk,i ||(2||z∗ −

i=1

m Σ

wk,i yk,i || + ||

i=1 .

m Σ

wk,i yk,i − xk+1 ||)

i=1

≤ δ(8M + 3).

Equation (3.80) and the equation above imply that ||z∗ − xk+1 ||2 ≤ ||z∗ −

m Σ

.

wk,i yk,i ||2 + δ(8M + 3)

i=1 .

ˆ 02 + δ(8M + 3) ≤ ||z∗ − xk ||2 − Δɛ .

ˆ 02 /2. ≤ ||z∗ − xk ||2 − Δɛ

This completes the proof of Lemma 3.17. It follows from (3.94) and Lemma 3.17 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 1,

.

and for all .k = 1, . . . , s, ˆ 02 /2. ||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − Δɛ

.

By (3.95) and the equation above, 4(M + 1)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ ˆ 02 s (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ≥ 2−1 Δɛ k=1

and ˆ −1 . s ≤ 2(4M + 1)2 ɛ 0−2 Δ

.

The Second Result with Nonsummable Errors for the Cimmino Algorithm

113

Thus we have shown that the following property holds: (i) if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (3.93) is true, then (3.95) holds for .k = 1, . . . , s + 1 and ˆ −1 . s ≤ 2(4M + 1)2 ɛ 0−2 Δ

.

Together with (3.84) this implies that there exists a natural number ˆ −1 ⎦ + 1 = n0 q ≤ L2(4M + 1)2 ɛ 0−2 Δ

.

such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

U ∩

Theorem 3.15 is proved.

The Second Result with Nonsummable Errors for the Cimmino Algorithm Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in section “Exact Cimmino Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

Theorem 3.18 Let M > max{1, ||z∗ ||}, 0 < δ ≤ 1,

.

ˆ −1/2 , n0 = 1+L12(4M +1)2 ɛ −2 Δ ˆ −1 ⎦+1. 1 ≥ ɛ 0 = 16(δ(8M +3))3/4 Δ 0

.

(3.97)

114

3 The Cimmino Algorithm in a Hilbert Space

Assume that .{xi }∞ i=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m, wk,i ≥ Δ,

(3.98)

||x0 || ≤ M,

(3.99)

.

.

for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.100)

i=1

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,δ (xk ) /= ∅.

(3.101)

i=1

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

It should be mentioned that in Theorem 3.18 .δ is the computational error made by our computer system, we obtain a point z such that .d(z, Cs ) ≤ ɛ 0 and in order to obtain this point we need .n0 iterations. It is not difficult to see that .ɛ 0 = c1 δ 3/4 and −3/2 ⎦ + 1, where .c and .c are positive constants depending on M. .n0 = Lc2 δ 1 2 Applying by induction Theorem 3.18 we obtain the following result. Theorem 3.19 Suppose that .ɛ ¯ ∈ (0, 1], M > max{1, ||z∗ ||},

.

{x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

Let .ɛ 0 ∈ (0, ɛ ), ¯ .δ ∈ (0, 1], ˆ −1/2 , 1 ≥ ɛ 0 ≥ 16(δ(8M + 3))3/4 Δ

.

ˆ −1 ⎦. n0 = 1 + L2(4M + 1)2 ɛ 0−2 Δ

.

Assume that .{xi }∞ i=0 ⊂ X, (3.98)–(3.101) hold for each integer .k ≥ 0. Then xi ∈ B(0, 5M + 1) for all integers i ≥ 0

.

The Second Result with Nonsummable Errors for the Cimmino Algorithm

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

and that for each integer .p ≥ 0, d(xqp , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 3.19 implies the following result. Theorem 3.20 Suppose that .ɛ ¯ ∈ (0, 1], M > max{1, ||z∗ ||},

.

{x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

Let .δ ∈ (0, 2−1 ], .ɛ 0 ∈ (0, ɛ ), ¯ ˆ −1/2 . ɛ 0 = 16(δ(8M + 3))3/4 Δ

.

Assume that .{δi }∞ i=0 ⊂ (0, δ], lim δi = 0,

.

i→∞

{xi }∞ i=0 ⊂ X,

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, ˆ i = 1, . . . , m, wk,i ≥ Δ,

m Σ

.

wk,i = 1

i=1

and B(xk+1 , δk ) ∩

m Σ

.

wk,i Pi,δk (xk ) = / ∅.

i=1

Then xk ∈ B(0, 5M + 1) for all integers k ≥ 0

.

115

116

3 The Cimmino Algorithm in a Hilbert Space

and .

lim inf max{d(xk , Cs ) : s = 1, . . . , m} = 0. k→∞

Theorem 3.19 implies the following result. Theorem 3.21 Assume that the assumptions of Theorem 3.19 holds and the bounded regularity property holds. Then .

lim inf d(xk , C) = 0. k→∞

Proof of Theorem 3.18 Let .k ≥ 0 be an integer. By (3.101), for each .i ∈ {1, . . . , m}, there exists yk,i ∈ Pi,δ (xk )

.

(3.102)

such that ||xk+1 −

m Σ

.

wk,i yk,i || ≤ δ.

(3.103)

i=1

In view of (3.102), ||yk,i − xk || ≤ d(xk , Ci ) + δ, i = 1, . . . , m.

.

(3.104)

In view of (A1), (3.97), (3.99) and (3.104), ||y0,i − x0 || ≤ ||x0 − z∗ || + δ ≤ 2M + δ ≤ 2M + 1/2,

.

||y0,i || ≤ 3M + 1/2.

.

Together with (3.100) and (3.103) this implies that ||x1 || ≤ max{||y0,i || : i = 1, . . . , m} + δ ≤ 3M + 1.

.

(3.105)

Assume that s is a natural number and for each integer .k ∈ [1, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(3.106)

||x1 − z∗ || ≤ 4M + 1.

(3.107)

By (3.105), .

We prove the following auxiliary result.

U ∩

The Second Result with Nonsummable Errors for the Cimmino Algorithm

117

Lemma 3.22 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 1.

.

(3.108)

Then ˆ ||xk+1 − z∗ ||2 ≤ ||xk − z∗ ||2 − −2−1 ɛ 02 Δ.

.

Proof Let .i ∈ {1, . . . , m}. Assumption (A1), (3.102), (3.103) and Lemma 2.3 applied with x = xk , x˜ = ξ = yk,i , D = Ci

.

imply that ||z∗ − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xi − yk,i ||2

.

.

− 32(2δ(2||xk − z∗ || + 1))3/2 .

(3.109)

In view of (3.108), ||z∗ − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xk − yk,i ||2

.

− 32(2δ(8M + 3))3/2 .

(3.110)

||xk − yk,i || ≥ d(xk , Ci ).

(3.111)

.

By (3.102), .

Equations (3.106) and (3.111) imply that there exists .p ∈ {1, . . . , m} for which ||xk − yk,p || ≥ d(xk , Cp ) > ɛ 0 .

.

(3.112)

It follows from (3.110) and (3.112) imply that ||z∗ − yk,p ||2 ≤ ||z∗ − xk ||2 − ɛ 02 + 32(2δ(8M + 3))3/2 .

.

(3.113)

It follows from (3.98), (3.100), (3.110), (3.113) and the convexity of the function || · ||2 that

.

||z∗ −

m Σ

.

i=1

wk,i yk,i || ≤ 2

m Σ

wk,i ||z∗ − yk,i ||2

i=1

Σ 2 . ≤ wk,p ||z∗ − yk,p || + {wk,i (||z∗ − yk,i ||2 : i ∈ {1, . . . , m} \ {p}}

118

3 The Cimmino Algorithm in a Hilbert Space

≤ wk,p (||z∗ − xk ||2 − ɛ 02 + 32(δ(4M + 3))3/2 )

.

.

+

Σ {wk,i (||z∗ − xk ||2 + 32(δ(4M + 3))3/2 ) : i ∈ {1, . . . , m} \ {p}} .

≤ ||z∗ − xk ||2 + 32(δ(8M + 3))3/2 − wk,p ɛ 02 .

ˆ ≤ ||z∗ − xk ||2 + 32(δ(8M + 3))3/2 − ɛ 02 Δ.

(3.114)

By (3.97), (3.103) and (3.114), ||z∗ −

m Σ

.

wk,i yk,i || ≤ ||z∗ − xk ||.

(3.115)

i=1

It follows from (3.103), (3.108) and (3.115) that |||z∗ − xk+1 || − ||z∗ − 2

.

m Σ

wk,i yk,i ||2 |

i=1

.

≤ |||z∗ − xk+1 || − ||z∗ −

m Σ

wk,i yk,i |||(|||z∗ − xk+1 || + ||z∗ −

i=1

.

≤ ||xk+1 −

m Σ

wk,i yk,i ||(2||z∗ −

i=1

wk,i yk,i ||)

i=1 m Σ

wk,i yk,i || + ||xk+1 −

i=1 .

m Σ

m Σ

wk,i yk,i ||)

i=1

≤ δ(2||z∗ − xk || + δ) ≤ δ(8M + 3).

(3.116)

Equations (3.97), (3.114) and (3.116) imply that ||z∗ − xk+1 ||2 ≤ ||z∗ −

m Σ

.

wk,i yk,i ||2 + δ(8M + 3)

i=1 .

ˆ 02 + 32(2δ(8M + 3))3/2 + δ(8M + 3) ≤ ||z∗ − xk+1 ||2 − Δɛ .

ˆ 02 /2 + 64(2δ(8M + 3))3/2 ≤ ||z∗ − xk ||2 − Δɛ .

ˆ 02 /2. ≤ ||z∗ − xk ||2 − Δɛ

This completes the proof of Lemma 3.22.

U ∩

The Second Result with Nonsummable Errors for the Cimmino Algorithm

119

It follows from (3.107) and Lemma 3.22 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 1,

(3.117)

ˆ 02 /2. ||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − Δɛ

(3.118)

.

and for all .k = 1, . . . , s, .

By (3.107) and (3.118), (4M + 1)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ

ˆ 02 s (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 2−1 Δɛ

k=1

and ˆ −1 . s ≤ 2(4M + 1)2 ɛ 0−2 Δ

.

Thus we have shown that the following property holds: (i) if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (3.106) is true, then (3.117) holds for .k = 1, . . . , s + 1 and ˆ −1 . s ≤ 2(4M + 1)2 ɛ 0−2 Δ

.

Together with (3.97) this implies that there exists a natural number ˆ −1 ⎦ + 1 = n0 q ≤ L2(4M + 1)2 ɛ 0−2 Δ

.

such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 3.18 is proved. Theorem 3.18 implies the following result. Theorem 3.23 Assume that the bounded regularity property holds, M > max{1, ||z∗ ||}, 0 < ɛ < 1.

.

120

3 The Cimmino Algorithm in a Hilbert Space

Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each t .{xi }∞ i=0 ⊂ X, each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

satisfying ||x0 || ≤ M,

.

for each integer .k ≥ 0, m Σ .

wk,i = 1

i=1

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,δ (xk ) = / ∅

i=1

there exists a natural number .q ≤ n0 such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , C) ≤ ɛ .

.

Theorem 3.23 implies the following result. Theorem 3.24 Assume that the bounded regularity property holds, .ɛ ∈ (0, 1), ¯ M > M¯ + 1. M¯ > 0, C ⊂ B(0, M),

.

Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each .{xi }∞ i=0 ⊂ X, each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

satisfying ||x0 || ≤ M,

.

for each integer .k ≥ 0, m Σ .

i=1

wk,i = 1

The Second Result with Nonsummable Errors for the Cimmino Algorithm

121

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,δ (xk ) /= ∅

i=1

the equation ||xk || ≤ 5M + 1, k = 0, 1, . . .

.

holds and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

and that for each integer .p ≥ 0, d(xqp , C) ≤ ɛ .

.

Theorem 3.25 Assume that the bounded regularity property holds, .Δ0 ∈ (0, 1), ¯ M > M¯ + 1 M¯ > 0, C ⊂ B(0, M),

.

and that the following property holds: (i) for each .ξ ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .η ∈ Cs ∩ B(ξ, Δ0 ), αξ + (1 − α)η ∈ Cs .

.

Let .ɛ ∈ (0, Δ0 /2). Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each .{xi }∞ i=0 ⊂ X and each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

satisfying ||x0 || ≤ M,

.

(3.119)

for each integer .k ≥ 0, m Σ .

i=1

wk,i = 1

(3.120)

122

3 The Cimmino Algorithm in a Hilbert Space

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,δ (xk ) = / ∅

(3.121)

i=1

the inequality ||xi || ≤ 5M + 1 for all integers i ≥ 0

.

(3.122)

is true and d(xi , C) ≤ ɛ for all integers i ≥ n0 .

.

Proof By Theorem 3.24, there exist .δ0 ∈ (0, ɛ ) and a natural number .n0 such that the following property holds: (ii) for each .{xi }∞ i=0 ⊂ X and each ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

.

which satisfy ||x0 || ≤ M,

.

for each integer .k ≥ 0, m Σ .

wk,i = 1

i=1

and B(xk+1 , δ0 ) ∩

m Σ

.

wk,i Pi,δ0 (xk ) /= ∅

i=1

the inequality ||xi || ≤ 5M + 1 for all integers i ≥ 0

.

holds and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

The Second Result with Nonsummable Errors for the Cimmino Algorithm

123

and that for each integer .p ≥ 0, d(xqp , C) ≤ ɛ /4.

.

Choose δ ∈ (0, δ0 )

(3.123)

2n0 (2δ)1/2 (10M + 3)1/2 < ɛ /3.

(3.124)

.

such that .

Assume that .{xi }∞ i=0 ⊂ X, (3.119)–(3.121) hold for each integer .k ≥ 0. Property (ii), (3.119)–(3.121) and (3.123) imply that ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

is true and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0

.

and that for all integers .p ≥ 0, 1 ≤ qp+1 − qp ≤ n0 ,

(3.125)

d(xqp , C) ≤ ɛ /4.

(3.126)

.

.

In order to complete the proof it is sufficient to show that d(xk , C) ≤ ɛ

.

for all integers .k ≥ q0 . Assume the contrary. Then there exists an integer k ≥ q0

.

such that d(xk , C) > ɛ .

.

There exists an integer .p ≥ 0 such that qp ≤ k < (qp + 1),

(3.127)

1 ≤ qp+1 − qp ≤ n0 ,

(3.128)

.

.

124

3 The Cimmino Algorithm in a Hilbert Space

d(xqp , C) ≤ ɛ /4.

.

(3.129)

In view of (3.129), there exists z ∈ C such that ||xqp − z|| < ɛ /3.

.

Assume that .j ∈ {qp , . . . , qp+1 ) − 1} satisfies ||xj − z0 || ≤ ɛ /3 + 2(j − qp )(2δ)1/2 (10M + 3)1/2 .

.

(3.130)

It follows from (3.121) that for each .i ∈ {1, . . . , m} there exists yj,i ∈ Pi,δ (xj )

.

(3.131)

such that ||xj +1 −

m Σ

.

wj,i yj,i || ≤ δ.

(3.132)

i=1

Equations (3.121), (3.124), (3.126), (3.130), property (i) and Lemma 2.3 applied with D = Ci , z∗ = z0 , x = xj , x˜ = ξ = yj,i

.

imply that ||z − yj,i || ≤ ||z − xj || + 2(2δ)1/2 (||xj − z|| + 1)1/2

.

.

≤ ||z − xj || + 2(6δ)1/2 .

Together with (3.132) and the convexity of the norm this implies that ||z − xj +1 || ≤ ||z − xj || + 2(6δ)1/2 + δ ≤ ||xj − z|| + 2(2δ)1/2 (10M + 3)1/2

.

.

≤ ɛ /3 + 2(j + 1 − qp )(2δ)1/2 (10M + 3)1/2 .

(3.133)

It follows from (3.124), (3.127), (3.128), (3.130) and (3.133) that ||xk − z|| ≤ ||xqp − z|| + 2(k − qp )(2δ)1/2 (10M + 3)1/2

.

.

< ɛ /3 + 2n0 (2δ)1/2 (10M + 3)1/2 < ɛ .

This contradicts our assumption. The contradiction we have reached completes the proof of Theorem 3.25. U ∩

The Third Result with Nonsummable Errors for the Cimmino Algorithm

125

The Third Result with Nonsummable Errors for the Cimmino Algorithm Suppose that .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets, .δC ∈ (0, 1]. Set C = ∩m i=1 Ci .

.

Assume that the following assumption holds. (A2) There exists z∗ ∈ X

.

(3.134)

and for each .s ∈ {1, . . . , m} there exists z∗s ∈ B(z∗ , δC ) ∩ Cs

.

(3.135)

such that for each .x ∈ Cs and each .α ∈ [0, 1], αz∗s + (1 − α)x ∈ Cs .

.

(3.136)

Clearly, assumption (A2) is weaker than its prototype (A1). In (A1) we assume that all the sets .Ci , .i = 1, . . . , m are star-shaped at the same point. In (A2) we suppose that for each .i ∈ {1, . . . , m} the set .Ci is star-shaped at .z∗i belonging to a .δC -neighborhood of a fixed point .z∗ ∈ X. Theorem 3.26 Let M > max{1, ||z∗ ||}, δ ∈ (0, 1], ɛ 0 ∈ (0, 1],

.

ˆ 02 /4, 33δ 1/2 (8M + 9)3/2 ≤ Δɛ

(3.137)

ˆ δC (16M + 8) ≤ 4−1 ɛ 02 Δ

(3.138)

ˆ −1 ⎦. n0 = 1 + L2(4M + 3)2 ɛ 0−2 Δ

(3.139)

.

.

and .

Assume that .{xi }∞ i=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

(3.140)

||x0 || ≤ M,

(3.141)

.

satisfy .

126

3 The Cimmino Algorithm in a Hilbert Space

for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.142)

i=1

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,δ (xk ) /= ∅.

(3.143)

i=1

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Proof Let .k ≥ 0 be an integer. By (3.143), for each .i ∈ {1, . . . , m}, there exists yk,i ∈ Pi,δ (xk )

.

(3.144)

such that ||xk+1 −

m Σ

.

wk,i yk,i || ≤ δ.

(3.145)

i=1

In view of (3.144), ||yk,i − xk || ≤ d(xk , Ci ) + δ., i = 1, . . . , m.

.

(3.146)

In view of (3.135), (3.141) and (3.146), for each .i ∈ {1, . . . , m}, ||y0,i − x0 || ≤ ||x0 − z∗i || + δ

.

.

≤ ||x0 − z∗ || + ||z∗ − z∗i || + δ ≤ 2M + δ + δC , ||y0,i − x0 || ≤ 2M + 2, ||y0,i || ≤ 3M + 2.

.

Together with (3.140), (3.142) and (3.145) this implies that ||x1 || ≤ max{||y0,i || : i = 1, . . . , m} + δ ≤ 3M + 3.

.

(3.147)

In view of (3.147), ||x1 − z∗ || ≤ 4M + 3.

.

(3.148)

The Third Result with Nonsummable Errors for the Cimmino Algorithm

127

Assume that s is a natural number and for each integer .k ∈ [1, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(3.149) U ∩

We prove the following auxiliary result. Lemma 3.27 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 3.

.

Then ˆ ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 2−1 ɛ 02 Δ.

.

Proof Let .i ∈ {1, . . . , m}. Assumption (A2), (3.144), (3.146) and Lemma 2.3 applied with x = xk , x˜ = ξ = yk,i , D = Ci , z∗ = z∗i

.

imply that ||z∗i − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xi − yk,i ||2

.

− 32(2(2||xk − z∗ || + 1))3/2 δ 1/2 .

(3.150)

||z∗i − xk || ≤ ||z∗ − xk || + 1 ≤ 4M + 4.

(3.151)

.

By (3.135) and (3.150), .

In view of (3.150) and (3.151), ||z∗i − xk ||2 ≥ ||z∗i − yk,i ||2 + ||xk − yk,i ||2

.

.

− 32(2(8M + 9))3/2 δ 1/2 .

(3.152)

Equations (3.135) and (3.150) imply that |||z∗i − xk ||2 − ||z∗ − xk ||2 |

.

.

.

≤ |||z∗i − xk || − ||z∗ − xk |||(||z∗i − xk || + ||z∗ − xk ||)

≤ ||z∗i − z∗ ||(2||z∗ − xk || + ||z∗i − z∗ ||) ≤ δC (8M + 7).

(3.153)

128

3 The Cimmino Algorithm in a Hilbert Space

By (3.144), ||xk − yk,i || ≥ d(xk , Ci ).

.

Equations (3.131), (3.135), (3.150) and (3.152) imply that ||z∗i − yk,i ||2 ≤ ||z∗i − xk ||2 + 1 ≤ (4M + 4)2 + 1,

.

||z∗i − yk,i || ≤ 4M + 5, ||z∗ − yk,i || ≤ 4M + 6.

.

(3.154)

Equations (3.135) and (3.154) imply that |||z∗i − yk,i ||2 − ||z∗ − yk,i ||2 |

.

.

≤ |||z∗i − yk,i || − ||z∗ − yk,i |||(||z∗i − yk,i || + ||z∗ − yk,i ||) .

≤ ||z∗i − z∗ ||(8M + 11) ≤ δC (8M + 11).

(3.155)

It follows from (3.152), (3.153) and (3.155) that |||z∗ − xk ||2 ≥ |||z∗i − xk ||2 − δC (8M + 7)

.

.

.

≥ ||z∗i − yk,i ||2 + ||xk − yk,i ||2 − 32(2(8M + 9))3/2 δ 1/2 − δC (8M + 7)

≥ ||z∗ −yk,i ||2 +||xk −yk,i ||2 −δC (16M +18)−32(2(8M +9))3/2 δ 1/2 .

(3.156)

By (3.149) and (3.153), there exists .p ∈ {1, . . . , m} for which ||xk − yk,p || ≥ d(xk , Cp ) > ɛ 0 .

.

It follows from (3.156) and (3.157) imply that ||z∗ − yk,p ||2 ≤ ||z∗ − xk ||2 − ||xk − yk,p ||2

.

.

+ δC (16M + 18) + 32(2(8M + 9))3/2 δ 1/2 .

.

≤ ||z∗ − xk ||2 − ɛ 02

+ δC (16M + 18) + 32(2(8M + 9))3/2 δ 1/2 .

(3.157)

The Third Result with Nonsummable Errors for the Cimmino Algorithm

129

It follows from (3.137), (3.138), (3.140), (3.142), (3.150), (3.156), the inequality above and the convexity of the function .|| · ||2 that ||z∗ −

m Σ

.

wk,i yk,i ||2 ≤

i=1 .

.

+

.

Σ {wk,i ||z∗ − yk,i ||2 : i ∈ {1, . . . , m} \ {p}}

≤ wk,p ||z∗ − xk ||2 − wk,p ɛ 02

+ wk,p (δC (16M + 18) + 32(2(8M + 9))3/2 δ 1/2 )

Σ {wk,i ||z∗ − xk ||2 + wk,i (δC (16M + 18) + 32(2(8M + 9))3/2 δ 1/2 ) : .

.

wk,i ||z∗ − yk,i ||2

i=1

≤ wk,p ||z∗ − yk,p ||2 + .

m Σ

i ∈ {1, . . . , m} \ {p}}

ˆ 02 + δC (16M + 18) + 32(2(8M + 9))3/2 δ 1/2 ≤ ||z∗ − xk ||2 − Δɛ

(3.158)

and ||z∗ −

m Σ

.

wk,i yk,i || ≤ ||z∗ − xk || ≤ 4M + 3.

(3.159)

i=1

It follows from (3.145) and (3.159) that |||z∗ − xk+1 ||2 − ||z∗ −

m Σ

.

wk,i yk,i ||2 |

i=1

.

≤ |||z∗ − xk+1 || − ||z∗ −

m Σ

wk,i yk,i |||(|||z∗ − xk+1 || + ||z∗ −

i=1

.

≤ ||xk+1 −

m Σ

m Σ

wk,i yk,i ||)

i=1

wk,i yk,i ||(2||z∗ −

i=1

m Σ

wk,i yk,i || + ||xk+1 −

i=1 .

m Σ

≤ δ(8M + 7).

Equations (3.137), (3.138), (3.159) and (3.160) imply that ||z∗ − xk+1 ||2 ≤ ||z∗ −

m Σ

.

i=1

wk,i yk,i ||)

i=1

wk,i yk,i ||2 + δ(8M + 7)

(3.160)

130

.

3 The Cimmino Algorithm in a Hilbert Space

ˆ 02 + δC (16M + 18) + 32(2(8M + 4))3/2 δ 1/2 + δ(8M + 7) ≤ ||z∗ − xk+1 ||2 − Δɛ .

ˆ 02 + δC (16M + 18) + 33(2(8M + 9))3/2 δ 1/2 ≤ ||z∗ − xk ||2 − Δɛ .

ˆ 02 /2. ≤ ||z∗ − xk ||2 − Δɛ

This completes the proof of Lemma 3.27.

U ∩

It follows from (3.148) and Lemma 3.27 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 3,

(3.161)

ˆ 02 /2. ||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − Δɛ

(3.162)

.

and for all .k = 1, . . . , s, .

By (3.147) and (3.162), 4(M + 3)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ

ˆ 02 s (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 2−1 Δɛ

k=1

and ˆ −1 . s ≤ 2(4M + 3)2 ɛ 0−2 Δ

.

Thus we have shown that the following property holds: If an integer .s ≥ 1 and for each integer .k ∈ [1, s] (3.149) is true, then (3.161) holds for .k = 1, . . . , s + 1 and ˆ −1 . s ≤ 2(4M + 3)2 ɛ 0−2 Δ

.

Together with (3.139) this implies that there exists a natural number ˆ −1 ⎦ + 1 = n0 q ≤ L2(4M + 3)2 ɛ 0−2 Δ

.

such that xi ∈ B(0, 5M + 3), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 3.26 is proved.

The Cimmino Algorithm for Almost Star-Shaped Feasibility Problems

131

The Cimmino Algorithm for Almost Star-Shaped Feasibility Problems Suppose that .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

Assume that .M > 1, .δM ∈ (0, 1]. z∗ ∈ B(0, M),

(3.163)

B(0, M) ∩ Cs /= ∅, s = 1, . . . , m

(3.164)

.

.

and that the following assumption holds. (A3) For each .s ∈ {1, . . . , m}, each .x ∈ B(0, 5M + 9) ∩ Cs and each .α ∈ [0, 1], d(αz∗ + (1 − α)x, Cs ) < δM .

.

(3.165)

Assumption (A3), (3.163) and (3.165) imply that for each .s ∈ {1, . . . , m} there exists z∗s ∈ Cs ∩ B(z∗ , δM ).

.

(3.166)

In view of (3.163) and (3.166), ||z∗s || ≤ M + δM ≤ M + 1, s = 1, . . . , m.

.

(3.167)

We study the following feasibility problem Find z ∈ C

.

which is called, in view of (A3), as an almost star-shaped feasibility problem. Theorem 3.28 Let δ ∈ (0, 1), ɛ 0 ∈ (0, 1],

(3.168)

ˆ −1 , ɛ 02 ≥ 66(δ + δM )1/2 (8M + 8)3/2 Δ

(3.169)

.

.

a natural number .n0 satisfy ˆ −1 ⎦. n0 ≥ 1 + L2(4M + 3)2 ɛ 0−2 Δ

.

(3.170)

132

3 The Cimmino Algorithm in a Hilbert Space

Assume that .{xi }∞ i=0 ⊂ X, ˆ k = 0, 1, . . . , i = 1, . . . , m wk,i ≥ Δ,

(3.171)

||x0 || ≤ M,

(3.172)

.

satisfy .

for each integer .k ≥ 0, m Σ .

wk,i = 1

(3.173)

i=1

and B(xk+1 , δ) ∩

m Σ

.

wk,i Pi,δ (xk ) /= ∅.

(3.174)

i=1

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Proof Let .k ≥ 0 be an integer. By (3.174), for each .i ∈ {1, . . . , m}, there exists yk,i ∈ Pi,δ (xk )

.

(3.175)

such that ||xk+1 −

m Σ

.

wk,i yk,i || ≤ δ.

(3.176)

i=1

In view of (3.175), ||yk,i − xk || ≤ d(xk , Ci ) + δ, i = 1, . . . , m.

.

In view of (3.166), (3.172), (3.175) and (3.177), for each .i ∈ {1, . . . , m}, ||y0,i − x0 || ≤ ||x0 − z∗i || + δ,

.

.

≤ ||x0 − z∗ || + ||z∗ − z∗i || + δ ≤ 2M + δ + δM ,

||y0,i − x0 || ≤ 2M + 2 ||y0,i || ≤ 3M + 2, ||y0,i − z∗ || ≤ 4M + 2.

.

(3.177)

The Cimmino Algorithm for Almost Star-Shaped Feasibility Problems

133

Together with (3.173), (3.176) and the convexity of the norm this implies that ||x1 − z∗ || ≤ 4M + 3.

.

(3.178)

Assume that s is a natural number and for each integer .k ∈ [0, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(3.179) U ∩

We prove the following auxiliary result. Lemma 3.29 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 3.

.

(3.180)

Then ˆ ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 2−1 ɛ 02 Δ.

.

Proof Let .i ∈ {1, . . . , m}. Set E0 = {αz∗ + (1 − α)yk,i : α ∈ [0, 1]}.

.

(3.181)

By (3.166), (3.177) and (3.180), ||xk − yk,i || ≤ ||xk − z∗i || + 1 ≤ 4M + 5, ||yk,i || ≤ 9M + 9.

.

(3.182)

Let y ∈ E0 .

.

(3.183)

Assumption (A3) and Eqs. (3.175), (3.181) and (3.183) imply that there exists ˆ y ∈ Ci ∩ {v ∈ X : ||y − v|| < δM }.

.

(3.184)

By (3.177) and (3.184), ||xk − y|| ≥ ||xk − ˆ y|| − ||ˆ y − y|| > ||xk − ˆ y|| − δM

.

.

≥ d(xk , Ci ) − δM ≥ ||xk − yk,i || − δ − δM

and ||xk − yk,i || ≤ ||xk − y|| + δ + δM for each y ∈ E0 .

.

(3.185)

134

3 The Cimmino Algorithm in a Hilbert Space

Equations (3.181), (3.185) and Lemma 2.3 applied with D = E0 , x = xk , x˜ = ξ = yk,i

.

imply that ||z∗ − xk ||2 ≥ ||z∗ − yk,i ||2 + ||xk − yk,i ||2

.

.

− 32(δ + δM )1/2 (2||xk − z∗ || + 1)3/2 .

(3.186)

By (3.180) and (3.186), ||z∗ − yk,i ||2 ≤ ||z∗ − xk ||2 − ||xk − yk,i ||2

.

.

+ 32(δ + δM )1/2 (8M + 8)3/2 .

(3.187)

By (3.175) and (3.179), there exists .p ∈ {1, . . . , m} for which ||xk − yk,p || ≥ d(xk , Cp ) > ɛ 0 .

.

(3.188)

It follows from (3.187) and (3.188) that ||z∗ − yk,p ||2 ≤ ||z∗ − xk ||2 − ɛ 02 + 32(δ + δM )1/2 (8M + 8)3/2 .

.

(3.189)

It follows from (3.171), (3.173), (3.187), (3.189) and the convexity of the function || · ||2 that

.

||z∗ −

m Σ

.

wk,i yk,i ||2 ≤

i=1 .

≤ wk,p ||z∗ − yk,p ||2 + .

.

.

+

Σ

m Σ

wk,i ||z∗ − yk,i ||2

i=1

Σ {wk,i ||z∗ − yk,i ||2 : i ∈ {1, . . . , m} \ {p}}

≤ wk,p ||z∗ − xk ||2 − wk,p ɛ 02

+ 32wk,p (δ + δM )1/2 (8M + 8)3/2

{wk,i ||z∗ − xk ||2 + 32wk,i (δ + δM )1/2 (8M + 8)3/2 : i ∈ {1, . . . , m} \ {p}} .

ˆ 02 + 32(δ + δM )1/2 (8M + 8)3/2 . ≤ ||z∗ − xk ||2 − Δɛ

It follows from (3.169), (3.176), (3.180) and (3.190) that |||z∗ − xk+1 || − ||z∗ −

.

2

m Σ i=1

wk,i yk,i ||2 |

(3.190)

The Cimmino Algorithm for Almost Star-Shaped Feasibility Problems

.

m Σ

≤ |||z∗ − xk+1 || − ||z∗ −

135

wk,i yk,i |||(||z∗ − xk+1 || + ||z∗ −

i=1

.

m Σ

≤ ||xk+1 −

m Σ

wk,i yk,i ||)

i=1

wk,i yk,i ||(2||z∗ −

i=1

m Σ

wk,i yk,i || + ||xk+1 −

i=1 .

m Σ

wk,i yk,i ||)

i=1

≤ δ(8M + 8).

(3.191)

Equations (3.169), (3.190)) and (3.191) imply that ||z∗ − xk+1 ||2 ≤ ||z∗ −

m Σ

.

wk,i yk,i ||2 + δ(8M + 8)

i=1 .

ˆ 02 + 32(δ + δM )1/2 (8M + 8)3/2 + δ(8M + 8) ≤ ||z∗ − xk ||2 − Δɛ .

ˆ 02 + 33(δ + δM )1/2 (8M + 8)3/2 ≤ ||z∗ − xk ||2 − Δɛ .

ˆ 02 /2. ≤ ||z∗ − xk ||2 − Δɛ

This completes the proof of Lemma 3.29. U ∩ It follows from (3.178) and Lemma 3.29 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 3

(3.192)

ˆ 02 /2. ||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − Δɛ

(3.193)

.

and for all .k = 1, . . . , s, .

By (3.178) and (3.193), (4M + 3)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ

ˆ 02 s (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 2−1 Δɛ

k=1

and ˆ −1 . s ≤ 2(4M + 3)2 ɛ 0−2 Δ

.

136

3 The Cimmino Algorithm in a Hilbert Space

Thus we have shown that the following property holds: if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (3.179) is true, then (3.192) holds for .k = 1, . . . , s + 1 and ˆ −1 . s ≤ 2(4M + 3)2 ɛ 0−2 Δ

.

This implies that there exists a natural number ˆ −1 ⎦ + 1 = n0 q ≤ L2(4M + 3)2 ɛ 0−2 Δ

.

such that xi ∈ B(0, 5M + 3), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 3.28 is proved.

Chapter 4

Dynamic String-Averaging Methods in Hilbert Spaces

In this chapter we study the convergence of dynamic string-averaging methods for solving star-shaped feasibility problems in a Hilbert space. Our main goal is to obtain an approximate solution of the problem in the presence of computational errors. We show that the dynamic string-averaging methods generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

Preliminaries Let .(X, ) be a Hilbert space with an inner product . which induces a complete norm .||x|| = 1/2 , x ∈ X For each .x ∈ X and each nonempty set .E ⊂ X put d(x, E) = inf{||x − y|| : y ∈ E}.

.

For every point .x ∈ X and every positive number .r > 0 set B(x, r) = {y ∈ X : ||x − y|| ≤ r}.

.

Suppose that m is a natural number, .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. We study the following feasibility problem Find x ∈ C := ∩ni=1 Ci .

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_4

137

138

4 Dynamic String-Averaging Methods in Hilbert Spaces

If all the sets .Ci , .i = 1, . . . , m are convex this problem is called as the convex feasibility problem. In our study we do not assume the convexity of the sets .Ci , .i = 1, . . . , m. For each .i ∈ {1, . . . , m}, each .x ∈ X and each .ɛ ≥ 0 set Pi,ɛ (x) := {v ∈ Ci : ||x − v|| ≤ d(x, Ci ) + ɛ }

.

(4.1)

which is a set-valued maping. We apply a dynamic string-averaging method with variable strings and weights in order to obtain a good approximative solution of our feasibility problem. It should be mentioned that first this class of methods was introduced by Censor et al. in [53] for solving a convex feasibility problem. Next we describe the dynamic string-averaging method with variable strings and weights. By an index vector, we a mean a vector .t = (t1 , . . . , tp ) such that .ti ∈ {1, . . . , m} for all .i = 1, . . . , p. For an index vector .t = (t1 , . . . , tq ) and each .ɛ ≥ 0 set p(t) = q, P [t, ɛ ] = Ptq ,ɛ · · · Pt1 ,ɛ .

.

(4.2)

Denote by .M the collection of all pairs .(Ω, w), where .Ω is a finite set of index vectors and Σ .w : Ω → (0, ∞) satisfies w(t) = 1. (4.3) t∈Ω

Let .(Ω, w) ∈ M and .ɛ ≥ 0. Define PΩ,w,ɛ (x) =

Σ

.

w(t)P [t, ɛ ](x), x ∈ X.

(4.4)

t∈Ω

The dynamic string-averaging method with variable strings and variable weights can now be described by the following algorithm. Initialization: select an arbitrary point .x0 ∈ X. Iterative step: given a current iteration vector .xk pick a pair (Ωk+1 , wk+1 ) ∈ M

.

and .ɛ k+1 > 0 and calculate the next iteration vector .xk+1 by xk+1 ∈ PΩk+1 ,wk+1 ,ɛ k+1 (xk ).

.

Fix a number Δ ∈ (0, m−1 ]

.

(4.5)

Preliminaries

139

and an integer q¯ ≥ m.

(4.6)

.

Denote by .M∗ the set of all .(Ω, w) ∈ M such that .

p(t) ≤ q¯ for all t ∈ Ω,

(4.7)

w(t) ≥ Δ for all t ∈ Ω.

(4.8)

.

¯ Fix a natural number .N. We apply an algorithm generated by {(Ωi , wi )}∞ i=1 ⊂ M∗

.

and a sequence of nonnegative numbers .{ɛ k }∞ k=1 such that for each natural number j, j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }).

(4.9)

This algorithm generates, for any starting point .x0 ∈ X, a sequence .{xk }∞ k=0 .⊂ X, where xk+1 = PΩk+1 ,wk+1 ,ɛ k+1 (xk ).

.

In the sequel we need the following definitions. Let .δ ≥ 0, .x ∈ X and let .t = (t1 , . . . , tp(t) ) be an index vector. Define p(t)

A0 (x, t, δ) = {(y, λ) ∈ X × R 1 : there is a sequence {yi }i=0 ⊂ X such that

.

y0 = x and for all i = 1, . . . , p(t),

.

B(yi , δ) ∩ Pti ,δ (yi−1 ) /= ∅,

.

y = yp(t) ,

.

λ = max{||yi − yi−1 || : i = 1, . . . , p(t)}}.

.

(4.10)

Let .δ ≥ 0, .x ∈ X and let .(Ω, w) ∈ M. Define A(x, (Ω, w), δ) = {(y, λ) ∈ X × R 1 : there exist

.

(yt , λt ) ∈ A0 (x, t, δ), t ∈ Ω such that

.

||y −

Σ

.

t∈Ω

w(t)yt || ≤ δ, λ = max{λt : t ∈ Ω}}.

(4.11)

140

4 Dynamic String-Averaging Methods in Hilbert Spaces

As usual we denote by Card.(A) the cardinality of a set A and suppose that the sum over empty set is zero.

Exact Iterates Assume that z∗ ∈ ∩m i=1 Ci

.

(4.12)

and the following assumption holds: (A1) for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

Proposition 2.4 implies the following result. Proposition 4.1 Assume that Pi,0 (x) /= ∅, i ∈ {1, . . . , m}, x ∈ X.

.

Then for each .i ∈ {1, . . . , m}, each .x ∈ X and each .y ∈ Pi,0 (x), ||z∗ − x||2 ≥ ||z∗ − y||2 + ||x − y||2 .

.

Theorem 4.2 Assume that Pi,0 (x) /= ∅, i ∈ {1, . . . , m}, x ∈ X.

.

Let .M > max{||z∗ ||, 1}, .ɛ ∈ (0, 1]. Assume that {(Ωi , wi )}∞ i=1 ⊂ M∗

.

(4.13)

satisfies for every natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

x0 ∈ B(0, M)

.

(4.14) (4.15)

∞ and .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfy for every natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), 0).

.

(4.16)

Exact Iterates

141

Then ||xi || ≤ 3M, i = 0, 1, . . .

.

and Card({j ∈ {1, 2, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ })

.

.

≤ 4N¯ M 2 Δ−1 ɛ −2 (N¯ + 1)2 q¯ 2 .

Proof Set γ0 = ɛ (N¯ + 1)−1 q¯ −1 .

.

(4.17)

Let .i ≥ 0 be an integer. The inclusion (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), 0)

.

(4.18)

is true. By (4.11) and (4.18) there exist vectors (yt , αt ) ∈ A0 (xi , t, 0), t ∈ Ωi+1

.

(4.19)

such that xi+1 =

Σ

.

wi+1 (t)yt ,

(4.20)

t∈Ωi+1

λi+1 = max{αt : t ∈ Ωi+1 }.

.

(4.21)

It follows from (4.19) that for every .t = (t1 , . . . , tp(t) ) ∈ Ωi+1 there exists a finite (t) p(t) sequence .{yj }j =0 ⊂ X such that (t)

(t)

y0 = xi , yp(t) = yt ,

.

(t)

(t)

yj ∈ Ptj ,0 (yj −1 ) for each integer j = 1, . . . , p(t),

.

(t)

(t)

αt = max{||yj +1 − yj || : j = 0, . . . , p(t) − 1}.

.

Let t = (t1 , . . . , tp(t) ) ∈ Ωi+1 .

.

(4.22) (4.23) (4.24)

142

4 Dynamic String-Averaging Methods in Hilbert Spaces

By (4.23) and Proposition 4.1, for every integer j satisfying .0 ≤ j < p(t), we have (t)

(t)

(t)

(t)

||z∗ − yj ||2 − ||z∗ − yj +1 ||2 ≥ ||yj − yj +1 ||2 .

.

(4.25)

In view of (4.22), (4.24) and (4.25), (t)

(t)

||z∗ − xi ||2 − ||z∗ − yt ||2 = ||z∗ − y0 ||2 − ||z∗ − yp(t) ||2

.

.

=

p(t)−1 Σ j =0

.

≥

(t)

(t)

(||z∗ − yj ||2 − ||z∗ − yj +1 ||2 )

p(t)−1 Σ j =0

(t)

(t)

||yj − yj +1 ||2 ≥ αt2 .

(4.26)

It follows from (4.3), (4.8), (4.13), (4.20), (4.21) and (4.26) that Σ

||z∗ − xi+1 ||2 = ||z∗ −

.

wi+1 (t)yt ||2 ≤

t∈Ωi+1

.

≤

Σ

Σ

||z∗ − yt ||2 wi+1 (t)

t∈Ωi+1

wi+1 (t)(||z∗ − xi ||2 − αt2 )

t∈Ωi+1 .

≤ ||z∗ − xi ||2 − Δλ2i+1 .

Thus ||z∗ − xi+1 ||2 ≤ ||z∗ − xi ||2 − Δλ2i+1 for all integers i ≥ 0.

.

(4.27)

In view of (4.15) and (4.27), for each integer .i ≥ 0, ||z∗ − xi || ≤ ||z∗ − x0 || ≤ 2M, ||xi || ≤ 3M.

.

By (4.15) and (4.25), for each natural number n, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xn ||2

.

n−1 Σ . = (||z∗ − xi ||2 − ||z∗ − xi+1 ||2 ) i=0

.

≥

n−1 Σ i=0

Δλ2i+1 ≥ Δγ02 Card({i ∈ {1, . . . , n} : λi ≥ γ0 }).

(4.28)

Exact Iterates

143

Since the relation above holds for every natural number n we conclude that Card({i ∈ {1, 2, . . . , } : λi ≥ γ0 })

.

.

≤ 4M 2 Δ−1 γ0−2 = 4M 2 Δ−1 ɛ −2 q −2 (N¯ + 1)2 .

(4.29)

Assume that an integer .i ≥ 1 and λ i < γ0 .

(4.30)

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), 0)

(4.31)

.

The inclusion .

is true. By (4.11) and (4.31) there exist vectors (yt , αt ) ∈ A0 (xi−1 , t, 0), t ∈ Ωi

.

(4.32)

such that xi =

Σ

.

wi (t)yt ,

(4.33)

t∈Ωi

λi = max{αt : t ∈ Ωi }.

.

(4.34)

It follows from (4.10) and (4.32) that for every .t = (t1 , . . . , tp(t) ) ∈ Ωi there exists (t) p(t) a finite sequence .{yj }j =0 ⊂ X such that (t)

(t)

y0 = xi−1 , yp(t) = yt ,

.

(t)

(t)

yj ∈ Ptj ,0 (yj −1 ) for each integer j = 1, . . . , p(t),

.

(t)

(t)

αt = max{||yj +1 − yj || : j = 0, . . . , p(t) − 1}.

.

(4.35) (4.36) (4.37)

Let t = (t1 , . . . , tp(t) ) ∈ Ωi

.

be an index vector. By (4.1), (4.30), (4.34), (4.36), (4.37), for every .j = 0, . . . , p(t) − 1, αt < γ0 , ||yj(t)+1 − yj(t) || < γ0

.

(4.38)

144

4 Dynamic String-Averaging Methods in Hilbert Spaces

and d(yj(t) , Ctj +1 ) = ||yj(t) − yj(t)+1 || < γ0 .

.

(4.39)

It follows from (4.7), (4.34), (4.35), (4.37)–(4.39) that for every integer .j = 0, . . . , p(t) we have ||xi−1 − yj(t) || ≤ j λi ≤ j γ0 ≤ qγ ¯ 0

.

and if .j < p(t), then (t)

(t)

d(xi−1 , Ctj +1 ) ≤ ||xi−1 − yj || + d(yj , Ctj +1 )

.

≤ j γ0 + γ0 ≤ qγ ¯ 0.

.

Therefore d(xi−1 , Ctj ) ≤ qγ ¯ 0 , j = 1, . . . , p(t),

(4.40)

||xi−1 − yt || ≤ qγ ¯ 0

(4.41)

.

.

for all .t ∈ Ωi . In view of (4.40), d(xi−1 , Cs ) ≤ qγ ¯ 0 , s ∈ ∪t∈Ωi {t1 , . . . , tp(t) }.

.

(4.42)

It follows from (4.3), (4.33) and (4.41) and the convexity of the norm that ||xi−1 − xi || = ||xi−1 −

Σ

.

wi (t)yt ||

t∈Ωi

Σ

wi (t)||xi−1 − yt || ≤ γ0 q. ¯

(4.43)

E0 = {i ∈ {1, 2, . . . } : λi ≥ γ0 }.

(4.44)

.

≤

t∈Ωi

Set .

By (4.29), (4.43) and (4.44), Card(E0 ) ≤ 4M 2 Δ−1 γ0−2

.

(4.45)

and the following property holds: (P1) if an integer .i ≥ 1 satisfies the inequality .λi < γ0 , then (4.42) holds and ||xi−1 − xi || ≤ γ0 q. ¯

.

(4.46)

Exact Iterates

145

Set E1 = {i ∈ {1, 2, . . . } : [i, i + N¯ − 1] ∩ E0 /= ∅}.

.

(4.47)

By (4.11), (4.45) and (4.47) we have Card(E1 ) ≤ N¯ Card(E0 ) ≤ 4N¯ M 2 Δ−1 γ0−2

.

¯ 2 Δ−1 q¯ 2 (N¯ + 1)2 ɛ −2 . ≤ 4NM

.

(4.48)

Assume that a natural number .j /∈ E1 . In view of (4.47) and (4.50), {j, . . . , j + N¯ − 1} ∩ E0 = ∅.

.

(4.49)

Equations (4.1) and (4.49) imply that for every integer .i ∈ {j, . . . , j + N¯ − 1} (4.42) and (4.46) hold. In view of (4.46) which holds for every integer .i ∈ {j, . . . , j + N¯ − 1} we obtain that for every pair of integers .i1 , i2 ∈ {j − 1, . . . , j + N¯ − 1}, ||xi1 − xi2 || ≤ γ0 N¯ q. ¯

.

(4.50)

By (4.17), (4.42) and (4.57) which holds for each .i ∈ {j, . . . , j + N¯ − 1}, we have that for each .i ∈ {j, . . . , j + N¯ − 1} and each .s ∈ ∪t∈Ωi {t1 , . . . , tp(t) }, d(xj , Cs ) ≤ ||xj − xi−1 || + d(xi−1 , Cs )

.

.

¯ 0 q¯ + γ0 q¯ ≤ ɛ ≤ Nγ

and for each .s ∈ {1, . . . , m}, d(xj , Cs ) ≤ ɛ .

.

Thus for each .j ∈ {1, 2, . . . , } \ E1 , d(xj , Cs ) ≤ ɛ , s = 1, . . . , m.

.

Theorem 4.2 is proved. U ∩ Theorem 4.2 implies the following result. Theorem 4.3 Assume that M > max{1, ||z∗ ||},

.

146

4 Dynamic String-Averaging Methods in Hilbert Spaces

ɛ ∈ (0, 1) and that the bounded regularity property holds. Then there exists a natural number Q such that for each

.

{(Ωi , wi )}∞ i=1 ⊂ M∗

.

(4.51)

satisfying for every natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

(4.52)

∞ each .x0 ∈ B(0, M) and each pair .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfying for every natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), 0)

.

(4.53)

the inequality Card({k ∈ {1, 2, . . . } : d(xk , C) > ɛ }) ≤ Q

.

holds. Corollary 4.4 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1), the bounded regularity property holds and that a natural number Q is as guaranteed by Theorem 4.3. Assume that

.

{(Ωi , wi )}∞ i=1 ⊂ M∗ ,

.

for every natural number j (4.52) holds, x0 ∈ B(0, M)

.

∞ and .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfy (4.53) for every natural number i.. Then there exists a natural number .j ≤ Q such that

d(xj , C) ≤ ɛ

.

and .

lim d(xk , C) = 0.

k→∞

Theorem 4.2 implies the following result.

Exact Iterates

147

Theorem 4.5 Let .M > max{1, ||z∗ ||}, .Δ0 ∈ (0, 1), .L > 1 and let the following property hold: (i) for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, C) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Assume that .ɛ ∈ (0, Δ0 ], {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfy for every natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

x0 ∈ B(0, M)

.

∞ and .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfy for every natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), 0).

.

Then ||xi || ≤ 3M, i = 0, 1, . . .

.

¯ N¯ + 1)2 ɛ −2 L2 . Card({i ∈ {1, 2, . . . } : d(xi , C) > ɛ }) < 4M 2 Δ−1 N(

.

Theorem 4.6 Assume that the bounded regularity property holds, and the following property holds: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .ɛ 0 ∈ (0, 1). Then there exists a natural number Q such that for each {(Ωi , wi )}∞ i=1 ⊂ M∗

.

148

4 Dynamic String-Averaging Methods in Hilbert Spaces

satisfying for every natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

(4.54)

each x0 ∈ B(0, M)

.

∞ and each pait .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfying for every natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), 0)

.

(4.55)

there exists .z ∈ C such that ||z − xk || < ɛ for each integer k ≥ Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Let a natural number Q be as guaranteed by Theorem 4.3 with .ɛ = ɛ 0 /4. Assume that {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfying for every natural number j Eq. (4.54), x0 ∈ B(0, M)

.

∞ and .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfy (4.55) for every natural number i. By (4.54), (4.55), Theorem 4.3 and the choice of Q, that there exists .p ∈ {0, . . . , Q} such that

d(xp , C) ≤ ɛ 0 /4.

.

There exists z0 ∈ C

.

such that ||xp − z0 || < ɛ 0 /2.

.

Property (i), Proposition 4.1 and the relation above imply that for each integer .i ≥ p, ||xi − z0 || ≤ ||xp − z0 || < ɛ 0 .

.

Inexact Iterates with Summable Errors

149

U ∩

Theorem 4.6 is proved. Theorem 4.5 implies the following result. Theorem 4.7 Assume that the following property holds: for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs

.

and that all the assumptions of Theorem 4.5 holds. Then for each integer ¯ N¯ + 1)2 ɛ −2 L2 , j ≥ 4M 2 Δ−1 N(

.

d(xi , C) ≤ ɛ

.

and the sequence .{xi }∞ i=0 converges to a point of C.

Inexact Iterates with Summable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in section “Inexact Cimmino Iterates with Summable Errors” holds. (A1) There exists z∗ ∈ C

.

(4.56)

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

The following theorem describes the behavior of inexact iterates with computational errors. Theorem 4.8 Let M > max{||z||∗ , 1}, ɛ ∈ (0, 1],

.

150

4 Dynamic String-Averaging Methods in Hilbert Spaces

let a sequence .{Δi }∞ i=1 ⊂ (0, ∞) satisfy ∞ Σ .

1/2

Δi

< ∞,

(4.57)

i=1

M1 = 8M + 162 (q¯ + 1)2 (

∞ Σ

.

1/2

Δi )2 + 4,

(4.58)

i=1

M2 = 4M 2 + 34(2M1 + 1)3/2 (

∞ Σ

.

1/2

Δi )q¯

(4.59)

i=1

and let .p¯ be a natural number such that Δi < 2−1 ɛ (N¯ + 1)−1 for each integer i ≥ p. ¯

(4.60)

{(Ωi , wi )}∞ i=1 ⊂ M∗

(4.61)

.

Assume that .

satisfies for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

x0 ∈ B(0, M)

.

(4.62) (4.63)

∞ and that sequences .{xi }∞ i=1 ⊂ X, .{λi }i=1 ⊂ [0, ∞) satisfies for each natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), Δi ).

.

(4.64)

Then ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({i ∈ {1, 2, . . . } : max{d(xk , Cs ) : s = 1, . . . , m} > ɛ })

.

.

¯ + N) ¯ 2 M2 q¯ 2 Δ−1 ɛ −2 . ≤ p¯ + 4N(1

Proof Set γ0 = 2−1 ɛ (N¯ + 1)q¯ −1 .

.

(4.65)

Inexact Iterates with Summable Errors

151

Let a nonnegative integer i be given. The inclusion (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), Δi+1 )

.

(4.66)

is true. It follows from (4.11) and (4.66) that there exist (yt(i) , αt(i) ) ∈ A0 (xi , t, Δi+1 ), t ∈ Ωi+1

.

(4.67)

such that ||xi+1 −

Σ

.

(i)

wi+1 (t)yt || ≤ Δi+1 ,

(4.68)

t∈Ωi+1

λi+1 = max{αt(i) : t ∈ Ωi+1 }.

(4.69)

.

By (4.10) and (4.67), for each index vector .t = (t1 , . . . , tp(t) ) ∈ Ωi+1 there exists a p(t) finite sequence .{yj(i,t) }j =0 ⊂ X such that (i,t) y0(i,t) = xi , yp(t) = yt(i) ,

(4.70)

.

for every integer .j = 1, . . . , p(t), we have B(yj(i,t) , Δi+1 ) ∩ Ptj ,Δi+1 (yj(i,t) −1 ) /= ∅,

(4.71)

.

(i,t) αt((i) = max{||yj(i,t) +1 − yj || : j = 0, . . . , p(t) − 1}.

.

(4.72)

Let t = (t1 , . . . , tp(t) ) ∈ Ωi+1 .

.

For each integer .j ∈ {1, . . . , p(t)}, Assumption (A1), (4.71) and Lemma 2.3 applied (i,t) with .δ = Δi+1 , .x = yj(i,t) −1 , .x˜ = yj +1 , .D = Ctj imply that (i,t) 1/2 ||z∗ − yj(i,t) || ≤ ||z∗ − yj(i,t) , −1 || + 2(2Δi+1 (2||z∗ − yj −1 || + 1))

.

(i,t)

(i,t) 2

||z∗ − yj −1 ||2 ≥ ||z∗ − yj

.

.

1/2

(i,t)

|| + ||yj (i,t)

(4.73)

(i,t)

− yj −1 ||2

− 32Δi+1 (2||z∗ − yj −1 || + 1)3/2 .

(4.74)

152

4 Dynamic String-Averaging Methods in Hilbert Spaces

We show that for each integer .i ≥ 0 and each .t ∈ Ωi+1 , ||z∗ − xi || ≤ M1 ,

(4.75)

(i,t)

(4.76)

.

||z∗ − yj

.

|| ≤ M1 .

Set Δ0 = 0.

.

By (4.58), ∞ Σ 1/2 2M + 4( Δi )(2M1 + 1)1/2 (q¯ + 1)

.

i=0

.

1/2

≤ M1 /4 + 8(q¯ + 1)M1

∞ Σ

1/2

≤ M1 /4 + M1 /2.

(4.77)

Δi (2M1 + 1)1/2 (q¯ + 1).

(4.78)

Δi

i=0

Assume that .k ≥ 0 is an integer, ||z∗ − xk || ≤ 2M + 4

k Σ

.

1/2

i=0

(Clearly, for .k = 0 our assumption holds.) Let t = (t1 , . . . , tp(t) ) ∈ Ωk+1 .

.

We show that for each .j ∈ {0, . . . , p(t)}, (k,t)

||z∗ − yj

.

1/2

|| ≤ ||z∗ − xk || + 4j Δk+1 (2M1 + 1)1/2 .

(4.79)

(In view of (4.70), our assumption holds for .j = 0.) Assume that .j ∈ {0, . . . , p(t)− 1} and (4.79) holds. Equations (4.77)–(4.79) imply that ||z∗ − yj(k,t) || ≤ M1 .

(4.80)

.

It follows from (4.73), (4.79) and (4.80) that (k,t)

(k,t)

|| + 2(2Δk+1 (2||z∗ − yj

(k,t)

|| + 4Δk+1 (2M1 + 1)1/2

||z∗ − yj +1 || ≤ ||z∗ − yj

.

.

.

≤ ||z∗ − yj

(k,t)

|| + 1))1/2

1/2

1/2

≤ ||z∗ − xk || + 4(j + 1)Δk+1 (2M1 + 1)1/2 .

Inexact Iterates with Summable Errors

153

Thus by induction we showed that for all .j = 0, . . . , p(t) (4.79) holds. By (4.79), for all .t = (t1 , . . . , tp(t) ) ∈ Ωk+1 and each .j ∈ {0, . . . , p(t)}, (k,t)

||z∗ − yj

.

≤ 2M + 4

.

k Σ

||

1/2

1/2

Δi (2M1 + 1)1/2 (q¯ + 1) + 4qΔ ¯ k+1 (2M1 + 1)1/2 .

(4.81)

k=0

It follows from (4.3), (4.61), (4.68), (4.70), (4.81) and the convexity of the norm that ||z∗ − xk+1 ||

.

.

Σ

≤ ||z∗ −

t∈Ωk+1

.

≤

Σ

(k)

wk+1 (t)yt || + ||

(k)

wk+1 (t)yt

− xk+1 ||

t∈Ωk+1

Σ

wk+1 (t)||z∗ − yt(k) || + Δk+1

t∈Ωk+1

.

≤ 2M + 4

k Σ

1/2

1/2

Δi (2M1 + 1)1/2 (q¯ + 1) + 4qΔ ¯ k+1 (2M1 + 1)1/2 + Δk+1

i=0

.

≤ 2M + 4

k+1 Σ

1/2

Δi (2M1 + 1)1/2 (q¯ + 1)

i=0

and the assumption made for k also holds for .k+1. Thus we have shown by induction that (4.78) holds for each integer .k ≥ 0. Equations (4.77), (4,78) and (4.81) imply that (4.75) and (4.76) hold for each integer .i ≥ 0, each .t ∈ Ωi+1 and each .j ∈ {0, . . . , p(t)}. It follows from (4.73) and (4.76) that for each integer .i ≥ 0, .t = (t1 , . . . , tp(t) ) ∈ Ωi+1 and each .j ∈ {1, . . . , p(t)}, (i,t)

||z∗ − yj −1 ||2

.

.

(i,t) 2

≥ ||z∗ − yj

(i,t)

|| + ||yj

(i,t)

1/2

− yj −1 ||2 − 32Δi+1 (2M1 + 1)3/2 .

(4.82)

Let .i ≥ 0 be an integer and .t = (t1 , . . . , tp(t) ) ∈ Ωi+1 . By (4.7), (4.61), (4.70), (4.72) and (4.80), (i)

||z∗ − xi ||2 − ||z∗ − yt ||2

.

.

(i,t) 2

= ||z∗ − y0

(i,t)

|| − ||z∗ − yp(t) ||2

154

4 Dynamic String-Averaging Methods in Hilbert Spaces

.

=

p(t)−1 Σ

(i,t) 2

(||z∗ − yj

j =0

.

≥

p(t)−1 Σ

(i,t)

||yj

j =0

.

(i,t)

|| − ||z∗ − yj +1 ||2 )

1/2

(i,t)

− yj +1 ||2 − 32Δi+1 (2M1 + 1)3/2 p(t) 1/2

≥ αt2 − 32Δi+1 (2M1 + 1)3/2 q. ¯

(4.83)

It follows from (4.3), (4.8), (4.61), (4.83) and the convexity of the function .|| · ||2 that Σ Σ (i) (i) .||z∗ − wi+1 (t)yt ||2 ≤ wi+1 (t)||z∗ − yt ||2 t∈Ωi+1

.

≤ ||z∗ − xi ||2 −

t∈Ωi+1

Σ

1/2

(i)

wi+1 (t)(αt )2 + 32Δi+1 (2M1 + 1)3/2 q¯

t∈Ωi+1

.

1/2

≤ ||z∗ − xi ||2 − Δλ2i+1 + 32Δi+1 (2M1 + 1)3/2 q. ¯

(4.84)

It follows from (4.3), (4.61), (4.76) and the convexity of the norm that Σ

||z∗ −

Σ

(i)

wi+1 (t)yt || ≤

.

t∈Ωi+1

(i)

wi+1 (t)||z∗ − yt || ≤ M1 .

(4.85)

t∈Ωi+1

By (4.68), (4.75) and (4.85), |||z∗ − xi+1 ||2 − ||z∗ −

Σ

.

(i)

wi+1 (t)yt ||2 |

t∈Ωi+1

.

≤ ||xi+1 −

Σ

(i)

wi+1 (t)yt ||(||z∗ − xi+1 ||

t∈Ωi+1

.

+ ||z∗ −

Σ

(i)

wi+1 (t)yt || ≤ 2M1 Δi+1 .

t∈Ωi+1

In view of (4.84) and (4.86), ||z∗ − xi+1 ||2 ≤ 2M1 Δi+1 + ||z∗ −

Σ

wi+1 (t)yt(i) ||2

.

t∈Ωi+1

.

1/2

≤ 2M1 Δi+1 + ||z∗ − xi ||2 − Δλ2i+1 + 32Δi+1 (2M1 + 1)3/2 q¯

(4.86)

Inexact Iterates with Summable Errors

155 1/2

≤ ||z∗ − xi ||2 − Δλ2i+1 + 34Δi+1 (2M1 + 1)3/2 q. ¯

.

(4.87)

Equations (4.58) and (4.75) imply that ||xi || ≤ 2M1 , i = 0, 1, . . . .

(4.88)

.

By (4.63) and (4.87), for each natural number n, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xn ||2

.

.

=

n−1 Σ (||z∗ − xi ||2 − ||z∗ − xi+1 ||2 ) i=0

.

=

n−1 Σ

Δλ2i+1 − 34

i=0

n−1 Σ

1/2

Δi+1 (2M1 + 1)3/2 q. ¯

i=0

Together with (4.59) this implies that M2 = 4M 2 + 34(2M1 + 1)3/2

∞ Σ

.

1/2

Δi

i=1 .

≥

n−1 Σ

Δλ2i+1

i=0

≥ Δγ02 Card({i ∈ {1, . . . , n} : λi ≥ γ0 }).

Since the relation above holds for any natural number n we conclude that Card({i ∈ {1, 2, . . . } : λi ≥ γ0 }) ≤ Δ−1 γ0−2 M2 .

.

(4.89)

Assume that an integer .i ≥ 1 and (4.90)

λ i < γ0 .

.

By (4.6), (4.71), (4.72) and (4.90), for every .t = (t1 , . . . , tp(t) ) ∈ Ωi and every j ∈ {0, . . . , p(t) − 1},

.

(i−1)

αt

.

< γ0 ,

||yj(i−1,t) − yj(i−1,t) +1 || < γ0 ,

.

(i−1,t)

d(yj

.

(i−1,t)

, Ctj +1 ) ≤ ||yj

(i−1,t)

− yj +1

|| + Δi < γ0 + Δi .

(4.91) (4.92) (4.93)

156

4 Dynamic String-Averaging Methods in Hilbert Spaces

It follows from (4.7), (4.61), (4.70), (4.92) and (4.93) that for every .t = (t1 , . . . , tp(t) ) ∈ Ωi and every .j ∈ {0, . . . , p(t)}, ||xi−1 − yj(i−1,t) || ≤ qγ ¯ 0

(4.94)

.

and if .j < p(t), then (i−1,t)

d(xi−1 , Ctj +1 ) ≤ ||xi−1 − yj

.

.

(i−1,t)

|| + d(yj

, Ctj +1 )

≤ j γ0 + γ0 + Δi ≤ qγ ¯ 0 + Δi

and d(xi−1 , Ctj ) ≤ qγ ¯ 0 + Δi , j = 1, . . . , p(t).

.

(4.95)

In view of (4.70) and (4.94), (i−1)

||xi−1 − yt

.

|| ≤ qγ ¯ 0 , t ∈ Ωi .

(4.96)

By (4.95), d(xi−1 , Cs ) ≤ qγ ¯ 0 + Δi , s ∈ ∪t∈Ωi {t1 , . . . , tp(t) }.

.

(4.97)

It follows from (4.3), (4.61) and (4.96) and the convexity of the norm that ||xi−1 − xi ||

.

.

Σ

≤ ||xi−1 −

wi (t)yt(i−1) || + ||

t∈Ωi

.

≤

Σ

Σ

wi (t)yt(i−1) − xi ||

t∈Ωi (i−1)

wi (t)||xi−1 − yt

|| + Δi ≤ qγ ¯ 0 + Δi .

t∈Ωi

Set E0 = {i ∈ {1, 2, . . . } : λi ≥ γ0 }.

(4.98)

Card(E0 ) ≤ M2 Δ−1 γ0−2 .

(4.99)

.

By (4.89) and (4.98), .

We have shown that the following property holds: (P1) If an integer .i ≥ 1 satisfies the inequality .λi < γ0 , then (4.97) holds and ||xi−1 − xi || ≤ γ0 q¯ + Δi .

.

(4.100)

Inexact Iterates with Summable Errors

157

Set E1 = {i ∈ {1, 2, . . . } : [i, i + N¯ − 1] ∩ E0 /= ∅}.

.

(4.101)

By (4.65), (4.99) and (4.101) we have ¯ 2 Δ−1 γ −2 Card(E1 ) ≤ N¯ Card(E0 ) ≤ NM 0

.

−1 2 ¯ ¯ ≤ 4NM + 1)2 ɛ −2 . 2 Δ q¯ (N

.

(4.102)

Assume that a natural number j /∈ E1 .

(4.103)

{j, . . . , j + N¯ − 1} ∩ E0 = ∅.

(4.104)

.

In view of (4.101)–(4.103), .

Together with (4.98) and property (P1) this implies that for every integer .i ∈ {j, . . . , j + N¯ − 1} (4.97) holds and for every pair of integers .i1 , i2 ∈ {j − 1, . . . , j + N¯ − 1}, ¯ i. ||xi1 − xi2 || ≤ γ0 N¯ q¯ + NΔ

.

(4.105)

By (4.65), (4.97) and (4.105) which holds for each .i ∈ {j, . . . , j + N¯ − 1}, we have that for each .i ∈ {j, . . . , j + N¯ − 1} and each .s ∈ ∪t∈Ωi {t1 , . . . , tp(t) }, d(xj , Cs ) ≤ ||xj − xi−1 || + d(xi−1 , Cs )

.

.

.

¯ 0 q¯ + NΔ ¯ i + γ0 q¯ + Δi ≤ Nγ

= qγ ¯ 0 (N¯ + 1) + (N¯ + 1)Δi ≤ ɛ /2 + (N¯ + 1)Δi .

(4.106)

By (4.62) and (4.106), d(xj , Cs ) ≤ ɛ /2 + (N¯ + 1)Δi , s = 1, . . . , m.

.

Together with (4.100) this implies that for each .j ∈ {p, ¯ p¯ + 1, . . . , } \ E1 , .

max{d(xj , Cs ) : s = 1, . . . , m} ≤ ɛ .

Combined with (4.102) this implies that Card({j ∈ {1, 2, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} > ɛ })

.

158

4 Dynamic String-Averaging Methods in Hilbert Spaces .

.

≤ p¯ + Card(E1 )

≤ p¯ + 4N¯ M2 Δ−1 ɛ −2 (N¯ + 1)2 q¯ 2 .

Theorem 4.18 is proved. U ∩ Corollary 4.9 Assume that the assumptions of Theorem 4.8 holds. Then there exist a natural number ¯ N¯ + 1)2 q¯ 2 j ≤ p¯ + 4M2 ɛ −2 Δ−1 N(

.

such that d(xj , Cs ) ≤ ɛ , s = 1, . . . , m.

.

Theorem 4.8 implies the following result. Theorem 4.10 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1], the bounded regularity property holds and that .{Δi }∞ i=1 ⊂ (0, 1] satisfies

.

∞ Σ .

1/2

Δi

< ∞.

i=1

Then there exist a natural number Q and .M0 > M such that for each {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfying for every natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

each x0 ∈ B(0, M)

.

∞ and each pair .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfying for every natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), Δi )

.

Inexact Iterates with Summable Errors

159

the inequalities ||xi || ≤ M0 , i = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) ≤ Q

.

hold. Theorem 4.8 implies the following result. Theorem 4.11 Let .M > max{1, ||z∗ ||}, .Δ0 ∈ (0, 1), .ɛ ∈ (0, Δ0 ), .L > 1, {Δi }∞ i=1 ⊂ (0, 1] satisfy

.

∞ Σ .

1/2

Δi

< ∞,

i=1

M1 = 8M + 162 (q¯ + 1)2 (

∞ Σ

.

1/2

Δi )2 + 4,

i=1

M2 = 4M 2 + 34(2M1 + 1)3/2

∞ Σ

.

1/2

Δi q, ¯

i=1

p¯ be a natural number such that

.

Δi < 2−1 ɛ L−1 (N¯ + 1)−1 for each integer i ≥ p¯

.

and let the following property hold: (i) for each .x ∈ B(0, 2M1 ) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, C) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Assume that {(Ωi , wi )}∞ i=1 ⊂ M∗

.

160

4 Dynamic String-Averaging Methods in Hilbert Spaces

satisfies for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

x0 ∈ B(0, M)

.

∞ and that sequences .{xi }∞ i=1 ⊂ X, .{λi }i=1 ⊂ [0, ∞) satisfies for each natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), Δi )

.

then ||xi || ≤ 2M1 , i = 0, 1, . . . ,

.

Card({i ∈ {1, 2, . . . } : d(xk , C) > ɛ })

.

.

¯ 2 q¯ 2 Δ−1 ɛ −2 L2 M2 . ≤ p¯ + 4N¯ (1 + N)

Theorem 4.12 Assume that the bounded regularity property holds, and the following property holds: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

(4.107)

Let .M > max{1, ||z∗ ||}, .ɛ 0 ∈ (0, 1), .{Δi }∞ i=1 ⊂ (0, 1] satisfy ∞ Σ .

1/2

Δi

< ∞.

i=1

Then there exist a natural number .Q0 and .M0 > M such that for each {(Ωi , wi )}∞ i=1 ⊂ M∗

.

(4.108)

satisfying for every natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

each x0 ∈ B(0, M)

.

(4.109)

Inexact Iterates with Summable Errors

161

∞ and .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfy for every natural number i,

(xi , λi ) ∈ A(xi−1 , (Ωi , wi ), Δi )

(4.110)

||xi || ≤ M0 , i = 0, 1, . . .

(4.111)

.

the inequality .

there exists .z ∈ C such that ||z − xk || < ɛ 0 for each integer k ≥ Q0

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Let a natural number Q and .M0 > M be as guaranteed by Theorem 4.10 with .ɛ = ɛ 0 /4. Choose M1 ≥ 4M0 + 2 + 16(q¯ + 1)(4M1 + 1)1/2 (

∞ Σ

.

1/2

Δi

+ 1).

i=1

There exists a natural number .p¯ such that for each integer .i ≥ p, ¯ 4(q¯ + 1)(2M1 + 1)1/2

∞ Σ

.

1/2

Δi

< ɛ 0 /4.

(4.112)

i=p¯

Set Q0 = Q + p. ¯

.

Assume that {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfying for every natural number j (4.109) holds, x0 ∈ B(0, M)

.

(4.113)

∞ and .{xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞) satisfy for every natural number i, (4.110) holds. It follows from Theorem 4.10, the choice of .M0 , (4.109), (4.110) and (4.113) that (4.111) holds,

Card({i ∈ {1, 2, . . . } : d(xi , C) > ɛ 0 /4}) ≤ Q.

.

(4.114)

162

4 Dynamic String-Averaging Methods in Hilbert Spaces

By (4.114), that there exists p ∈ {p, ¯ . . . , p¯ + Q}

.

(4.115)

such that d(xp , C) ≤ ɛ 0 /4.

.

By the equation above there exists z∈C

(4.116)

||xp − z|| < ɛ 0 /3.

(4.117)

||z|| ≤ 2M0 + 1.

(4.118)

.

such that .

In view of (4.111), .

Assume that .i ≥ p is an integer and ||z−xi || ≤ ||z−xp ||+

Σ

.

1/2

{Δj

: j is an integer satisfying p < j ≤ i}.

(4.119)

We have (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), Δi+1 ).

.

By (4.11), there exist (i)

(i)

(yt , αt ) ∈ A0 (xi , t, Δi+1 ), t ∈ Ωi+1

.

(4.120)

such that ||xi+1 −

Σ

.

wi+1 (t)yt(i) || ≤ Δi+1 ,

(4.121)

t∈Ωi+1 (i)

λi+1 = max{αt

.

: t ∈ Ωi+1 }.

(4.122)

By (4.10) and (4.120), for each index vector .t = (t1 , . . . , tp(t) ) ∈ Ωi+1 there exists (i,t) p(t) a finite sequence .{yj }j =0 ⊂ X such that (i,t)

y0

.

(i,t)

(i)

= xi , yp(t) = yt ,

(4.123)

Inexact Iterates with Summable Errors

163

for every integer .j = 1, . . . , p(t), we have (i,t)

B(yj

.

(i,t)

, Δi+1 ) ∩ Ptj ,Δi+1 (yj −1 ) /= ∅,

(i,t)

(i,t)

αt (i) = max{||yj +1 − yj

.

(4.124)

|| : j = 0, . . . , p(t) − 1}.

(4.125)

Let t = (t1 , . . . , tp(t) ) ∈ Ωi+1 , j ∈ {1, . . . , p(t)}.

.

Property (i), (4.116), (4.124) and Lemma 2.3 applied with .δ = Δi+1 , .x = yj(i,t) −1 , (i,t)

x˜ = yj

.

, .D = Ctj , .z∗ = z imply that (i,t)

||z∗ − yj

.

(i,t)

(i,t)

|| ≤ ||z∗ − yj −1 || + 2(2Δi+1 (2||z∗ − yj −1 || + 1))1/2 .

(4.126)

We show that (i,t)

||z − yj

.

|| ≤ M1 , j = 0, . . . , p(t).

By the choice of .M1 , 2M0 + 1 + 4(q¯ + 1)(4M1 + 1)1/2 (

∞ Σ

.

1/2

Δi

+ 1)

i=0 .

≤ M1 /2 + M1 /2 = M1 .

(4.127)

We show that for .j = 0, . . . , p(t), (i,t)

||z − yj

.

1/2

|| ≤ ||z − xi || + 4j Δi+1 (2M1 + 1)1/2 .

(4.128)

In view of (4.119) and (4.123), our assumption holds for .j = 0. Assume that .j ∈ {0, . . . , p(t) − 1} and (4.128) holds. Equations (4.117), (4.119), (4.127) and (4.128) imply that (i,t)

||z − yj

.

|| ≤ M1 .

By (4.126) and (4.128), (i,t)

(i,t)

||z − yj +1 || ≤ ||z − yj

.

.

|| + 2(2Δi+1 (2M1 + 1))1/2 1/2

≤ ||z − xi || + 4(j + 1)Δi+1 (2M1 + 1)1/2 .

(4.129)

164

4 Dynamic String-Averaging Methods in Hilbert Spaces

Thus by induction we showed that (4.128) holds for .j = 0, . . . , p(t). By (4.123) and (4.128), for every .j ∈ {0, . . . , p(t)}, (i,t)

||z − yj

.

1/2

|| ≤ ||z − xi || + 4qΔ ¯ i+1 (2M1 + 1)1/2 , 1/2

(i)

||z − yt || ≤ ||z − xi || + 4qΔ ¯ i+1 (2M1 + 1)1/2 .

.

It follows from (4.110), (4.121), the equation above and the convexity of the norm imply that ||z − xi+1 ||

.

.

≤ ||z −

Σ

(i)

wi+1 (t)yt || + ||

t∈Ωi+1

≤

.

Σ

(i)

wi+1 (t)yt − xi+1 ||

t∈Ωi+1

Σ

(i)

wi+1 (t)||z − yt || + Δi+1

t∈Ωi+1

.

1/2

≤ ||z − xi || + 4qΔ ¯ i+1 (2M1 + 1)1/2 + Δi+1 .

1/2

≤ ||z − xi || + 4(q¯ + 1)Δi+1 (2M1 + 1)1/2 .

Thus we showed by induction that (4.119) holds for each integer .i ≥ p. By (4.112), (4.117) and (4.119), for each integer .i > p, ||z − xi || ≤ ||z − xp || + 4(q¯ + 1)

∞ Σ

.

1/2

Δi (2M1 + 1)1/2 < ɛ 0 .

i=p¯

U ∩

Theorem 4.12 is proved.

The First Result with Nonsummable Errors In this section we use the following assumption: .

For each i ∈ {1, . . . , m} and each x ∈ X, Pi,0 (x) = / ∅.

(4.130)

Let .δ ≥ 0, .x ∈ X and let .t = (t1 , . . . , tp(t) ) be an index vector. Define p(t)

B0 (x, t, δ) = {(y, λ) ∈ X × R 1 : there is a sequence {yi }i=0 ⊂ X such that

.

y0 = x and for all i = 1, . . . , p(t),

.

The First Result with Nonsummable Errors

165

B(yi , δ) ∩ Pti ,0 (yi−1 ) /= ∅,

.

y = yp(t) ,

.

λ = max{||yi − yi−1 || : i = 1, . . . , p(t)}}.

.

(4.131)

Let .δ ≥ 0, .x ∈ X and let .(Ω, w) ∈ M. Define B(x, (Ω, w), δ) = {(y, λ) ∈ X × R 1 : there exist

.

(yt , λt ) ∈ B0 (x, t, δ), t ∈ Ω such that

.

||y −

Σ

.

w(t)yt || ≤ δ, λ = max{λt : t ∈ Ω}}.

(4.132)

z∗ ∈ C := ∩m i=1 Ci

(4.133)

t∈Ω

Assume that .

and the following assumption holds: (A1) For each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

(4.134)

Theorem 4.13 Assume that (4.130) holds., .M > max{||z∗ ||, 1}, .δ > 0 satisfies ¯ −1 , δ ≤ ((q¯ + 1)N)

(4.135)

¯ ɛ 0 = (12(q¯ + 1)N(4M + 2)Δ−1 δ)1/2

(4.136)

.

.

and a natural number .n0 satisfies n0 > 16M 2 Δ−1 ɛ 02 + 1.

(4.137)

{(Ωi , wi )}∞ i=1 ⊂ M∗

(4.138)

.

Let .

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

(4.139) (4.140)

166

4 Dynamic String-Averaging Methods in Hilbert Spaces

satisfy for each natural number i, (xi , λi ) ∈ B(xi−1 , (Ωi , wi ), δ).

.

(4.141)

Then there exists an integer .q ∈ [0, n0 − 1] such that ||xi || ≤ 3M + 1, i = 0, . . . , q N¯ ,

(4.142)

¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

(4.143)

.

.

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (4.143), then for each .i = q N¯ , . . . , (q + 1)N¯ and each .s ∈ {1, . . . , m}, d(xi , Cs ) ≤ ɛ 0 (q¯ + 1)(N¯ + 1)

.

and ||xi − xj || ≤ ɛ 0 (q¯ + 1)N¯

.

for each .i, j ∈ {q N¯ , . . . , (q + 1)N¯ }. Note that in Theorem 4.13 we assume that for each natural number i and each x ∈ X,

.

{ξ ∈ Ci : ||x − ξ || = d(x, Ci )} /= ∅.

.

This assumption holds if the space X is finite-dimensional or all the sets .Ci , i = 1, . . . , m are convex. It should be mentioned that in Theorem 4.13 .δ is the computational error made by our computer system, we obtain a point z such that ¯ + 1)(q¯ + 1)ɛ 0 and in order to obtain this point we need .(n0 − 1)N¯ .d(z, Cs ) ≤ (N iterations. It is not difficult to see that .ɛ 0 = c1 δ 1/2 and .n0 = Lc2 δ −1 ⎦ + 1, where .c1 and .c2 are positive constants depending on M. .

Proof Let .i ≥ 0 be an integer. We have (xi+1 , λi+1 ) ∈ B(xi , (Ωi+1 , wi+1 ), δ).

.

(4.144)

By (4.132) and (4.144) there exists vectors (i)

(i)

(yt , αt ) ∈ B0 (xi , t, δ), t ∈ Ωt+1

.

(4.145)

such that ||xi+1 −

Σ

.

(i)

(4.146)

: t ∈ Ωi+1 }.

(4.147)

wi+1 (t)yt || ≤ δ,

t∈Ωi+1 (i)

λi+1 = max{αt

.

The First Result with Nonsummable Errors

167

It follows from (4.131) and (4.145) that for each index vector .t p(t) (t1 , . . . , tp(t) ) ∈ Ωi+1 there exists a finite sequence .{yj(i,t) }j =0 ⊂ X such that (i,t) y0(i,t) = xi , yp(t) = yt(i) ,

.

B(yj(i,t) , δ) ∩ Ptj ,0 (yj(i,t) −1 ) /= ∅ for each integer j = 1, . . . , p(t),

.

(i)

αt

.

(i,t)

(i,t)

= max{||yj +1 − yj

|| : j = 0, . . . , p(t) − 1}.

=

(4.148) (4.149) (4.150)

Assume that s is a nonnegative integer and that for each integer .k ∈ [0, s], .

max{λi : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 .

(4.151)

By (4.140), ||x0 − z∗ || ≤ 2M.

.

(4.152)

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

.

(4.153) U ∩

We prove the following auxiliary result. Lemma 4.14 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ ||xk N¯ − z∗ || + iδ(q¯ + 1).

.

(4.154)

Then ||xk N¯ +i+1 − z∗ || ≤ δ(q¯ + 1) + ||xk N¯ +i − z∗ ||

.

and ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i − z∗ ||2 + 3δ(q¯ + 1)(4M + 2) − Δλ2k N¯ +i+1 .

.

If .λk N¯ +i+1 > ɛ 0 , then ||xk N¯ +i+1 − z||2 ≤ ||xk N¯ +i − z||2 ≤ −2−1 Δɛ 02 .

.

168

4 Dynamic String-Averaging Methods in Hilbert Spaces

Proof Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and .j ∈ {1, . . . , p(t)}. By (4.149), there exists ¯

¯

N +i,t) ξt,j ∈ Ptj ,0 (yj(k−1 ) ∩ B(yj(k N +i,t) , δ)

.

(4.155)

such that (k N¯ +i,t)

||ξt,j − yj −1

.

(k N¯ +i,t)

|| = d(yj −1

(4.156)

, Ctj ).

Proposition 2.4, assumption (A1) and (4.156) imply that (k N¯ +i,t) 2

||z∗ − yj −1

.

(k N¯ +i,t) 2

|| ≥ ||z∗ − ξt,j ||2 + ||ξt,j − yj −1

|| .

(4.157)

In view of (4.157), (k N¯ +i,t)

||z∗ − ξt,j || ≤ ||z∗ − yj −1

.

||.

(4.158)

Equations (4.155) and (4.158) imply that (k N¯ +i,t)

||z∗ − yj

.

(k N¯ +i,t)

|| ≤ ||z∗ − ξt,j || + ||ξt,j − yj (k N¯ +i,t)

≤ ||z∗ − yj −1

.

||

|| + δ.

(4.159)

It follows from (4.7), (4.148) and (4.159) that for each .s ∈ {1, . . . , p(t)}, ¯

¯

||z∗ − ys(k N +i,t) || ≤ ||z∗ − y0(k N +i,t) || + qδ ¯

.

≤ ||z∗ − xk N¯ +i || + qδ. ¯

.

(4.160)

By (4.148) and (4.160), (k N¯ +i)

||z∗ − yt

.

|| ≤ ||z∗ − xk N¯ +i || + qδ. ¯

(4.161)

By (4.3), (4.146), (4.161) and the convexity of the norm, ||z∗ − xk N¯ +i+1 ||

.

.

Σ

≤ ||z∗ −

¯

wk N¯ +i+1 (t)yt(k N+i) ||

t∈Ωk N+i+1 ¯

.

+ ||xk N¯ +i+1 −

Σ t∈Ωk N+i+1 ¯

(k N¯ +i)

wk N¯ +i+1 (t)yt

||

The First Result with Nonsummable Errors

.

Σ

≤δ+

169 (k N¯ +i)

wk N+i+1 (t)||yt ¯

− z∗ ||

t∈Ωk N+i+1 ¯ .

≤ ||xk N¯ +i − z∗ || + δ(q¯ + 1).

(4.162)

Equations (4.135), (4.153), (4.154) and (4.160) imply that for each .t (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and each .s ∈ {1, . . . , p(t)}. ¯

||z∗ − xk N¯ +i ||, ||z∗ − xk N¯ +i+1 ||, ||z∗ − ys(k N +i,t) || ≤ 2M + 1.

.

=

(4.163)

Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and .j ∈ {1, . . . , p(t)}. By (4.155), (4.158) and (4.165), (k N¯ +i,t) 2

|||z∗ − ξt,j ||2 − ||z∗ − yj

.

.

(k N¯ +i,t)

≤ ||ξt,j − yj

|| |

(k N¯ +i,t)

||(|||z∗ − ξt,j || + ||z∗ − yj

||) ≤ δ(4M + 2).

(4.164)

By (4.155), (4.158) and (4.163), (k N¯ +i,t) 2

|||ξt,j − yj −1

.

.

(k N¯ +i,t)

≤ ||ξt,j − yj

(k N¯ +i,t)

|| − ||yj

(k N¯ +i,t)

||(|||ξt,j − yj −1 .

(k N¯ +i,t) 2

− yj −1

|| |

(k N¯ +i,t)

|| + ||yj

(k N¯ +i,t)

− yj −1

≤ δ(8M + 4).

||) (4.165)

In view of (4.157), (4.164) and (4.165), (k N¯ +i,t) 2

||z∗ − yj −1

.

.

(k N¯ +i,t) 2

|| ≥ ||z∗ − ξt,j ||2 + ||ξt,j − yj −1

(k N¯ +i,t) 2

≥ ||z∗ − yj

(k N¯ +i,t)

|| − δ(4M + 8) + ||yj

||

(k N¯ +i,t) 2

− yj −1

|| − δ(8M + 4).

Thus we have shown that the following property holds: (P1) for each index vector .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and each integer .j ∈ {1, . . . , p(t)}, (k N¯ +i,t) 2

||z∗ − yj −1

.

(k N¯ +i,t) 2

|| ≥ ||z∗ − yj

(k N¯ +i,t)

|| + ||yj

(k N¯ +i,t) 2

− yj −1

|| − 3δ(4M + 2).

Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 . Property (P1), (4.7), (4.148) and (4.150) imply that ¯

||z∗ − xk N¯ +i ||2 − ||z∗ − yt(k N +i) ||2

.

170

4 Dynamic String-Averaging Methods in Hilbert Spaces (k N¯ +i,t) 2

= ||z∗ − y0

.

.

(k N¯ +i,t) 2

|| − ||z∗ − yp(t)

||

p(t) Σ (k N¯ +i,t) 2 (k N¯ +i,t) 2 (||z∗ − yj −1 || − ||z∗ − yj || )

=

j =1

.

≥

p(t) Σ j =1

¯

¯

N +i,t) 2 (||yj(k N +i,t) − yj(k−1 || − 3qδ(4M ¯ + 2)

and (k N¯ +i) 2

||z∗ − yt

.

(k N¯ +i) 2

|| ≤ ||z∗ − x(k N¯ +i) ||2 + 3qδ(4M ¯ + 2) − (αt

(4.166)

) .

It follows from (4.3), (4.8), (4.117), (4.147), (4.166) and the convexity of the function .|| · ||2 that Σ

||z∗ −

.

(k N¯ +i) 2

||

wk N¯ +i+1 (t)yt

t∈Ωk N+i+1 ¯

.

Σ

≤

¯

wk N+i+1 (t)||z∗ − yt(k N +i) ||2 ¯

t∈Ωk N+i+1 ¯

.

Σ

≤ ||xk N¯ +i − z∗ ||2 + 3qδ(4M ¯ + 2) −

¯ (k N+i) 2

wk N¯ +i+1 (t)(αt

)

t∈Ωk N+i+1 ¯ .

≤ ||xk N¯ +i − z∗ ||2 + 3qδ(4M ¯ + 2) − Δλ2k N+i+1 . ¯

(4.167)

Equations (4.3), (4.146), (4.163) and the convexity of the norm imply that |||z∗ −

Σ

(k N¯ +i) 2

|| − ||z∗ − xk N¯ +i+1 ||2 |

wk N+i+1 (t)yt ¯

.

t∈Ωk N+i+1 ¯

.

≤ ||

Σ

(k N¯ +i)

wk N+i+1 (t)yt ¯

− xk N¯ +i+1 ||

t∈Ωk N+i+1 ¯

.

× (||

Σ

(k N¯ +i)

wk N+i+1 (t)yt ¯

− z∗ || + ||z∗ − xk N¯ +i+1 ||) ≤ δ(4M + 2).

t∈Ωk N+i+1 ¯

(4.168)

The First Result with Nonsummable Errors

171

In view of (4.167) and (4.168), Σ

||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ −

.

¯ (k N+i) 2

wk N¯ +i+1 (t)yt

|| + δ(4M + 2)

t∈Ωk N+i+1 ¯ .

≤ ||xk N¯ +i − z∗ ||2 + 3(q¯ + 1)δ(4M + 2) − Δλ2k N+i+1 . ¯

(4.169)

Assume that λk N¯ +i+1 > ɛ 0 .

.

(4.170)

It follows from (4.136), (4.269) and (4.170) that ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i − z∗ ||2 − 3(q¯ + 1)δ(4M + 2) − Δɛ 02

.

.

≤ ||xk N¯ +i+1 − z∗ ||2 − Δɛ 02 /2. U ∩

Lemma 4.14 is proved.

It follows from (4.135), (4.136), (4.151), (4.153) and Lemma 4.14 that for all integers .i = 0, . . . , N¯ , ||xk N¯ +i − z∗ || ≤ 2M + 1,

.

||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2

.

=

.

¯ N−1 Σ

[||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ]

i=0 .

≤ −2−1 Δɛ 02 + 3N¯ δ(q¯ + 1)(4M + 2) ≤ −4−1 Δɛ 02 .

Thus we have shown that the following property holds: (P2) if an integer .k ∈ [0, s] satisfies .||xk N¯ − z∗ || ≤ 2M, then ¯ ||xj − z∗ || ≤ 2M + 1, j = k N¯ , . . . , (k + 1)N,

.

||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2 ≤ −4−1 Δɛ 02 .

.

In view of (4.152) and property (P2) we have ||xj − z|| ≤ 2M + 1, j = 0, . . . , (s + 1)N¯

.

and (4.171) is true for every integer .k = 0, . . . , s.

(4.171)

172

4 Dynamic String-Averaging Methods in Hilbert Spaces

By (4.152) and (4.171), 4−1 Δɛ 02 (s + 1) ≤

.

s Σ [||xk N¯ − z∗ ||2 − ||x(k+1)N¯ − z∗ ||2 ] k=0

.

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 Δ−1 ɛ 0−2 .

.

Thus we have shown that the following property holds: (P3) If an integer .s ≥ 0 and for every integer .k ∈ [0, s] relation (4.151) holds, then s ≤ 16M 2 Δ−1 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , s + 1.

.

By property (P3) and (4.137), there exists an integer .q ∈ [0, n0 − 1] such that for every integer k satisfying .0 ≤ k < q, .

¯ > ɛ 0 , max{λi : i = k N¯ + 1, . . . , (k + 1)N}

.

max{λi : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 , ||xq N¯ − z|| ≤ 2M,

.

||xj − z|| ≤ 2M + 1, j = 0, . . . , q N¯ ,

.

||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(4.172)

Let j ∈ {q N¯ , . . . , (q + 1)N¯ − 1}.

.

It follows from (4.147) and (4.172) that (j )

αt

.

≤ λj +1 ≤ ɛ 0 .

(4.173)

The First Result with Nonsummable Errors

173

By (4.148), (4.150), (4.173), for every index vector .t ∈ Ωj +1 and every integer i = 1, . . . , p(t) we have

.

(j,t)

(j,t)

= xj , ||yi

y0

.

(j,t)

− yi−1 || ≤ ɛ 0 .

(4.174)

In view of (4.7), (4.148) and (4.174), for every index vector .t ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (j,t)

||xj − yi

.

(j )

|| ≤ iɛ 0 ≤ qɛ ¯ 0 , ||xj − yt || ≤ ɛ 0 q. ¯

(4.175)

In view of (4.149) and (4.175) for every index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (t,j )

d(xj , Cti ) ≤ ||xj − yi

.

(t,j )

|| + d(yi

, Cti ) ≤ ɛ 0 q¯ + δ.

(4.176)

It follows from (4.3), (4.136), (4.146), (4.175) and the convexity of the norm that Σ

||xj +1 − xj || ≤ ||xj +1 −

(j )

wj +1 (t)yt ||

.

t∈Ωj +1

.

Σ

+ ||

(j )

wj +1 (t)yt

− xj ||

t∈Ωj +1

.

≤δ+

Σ

(j )

wj +1 (t)||yt

− xj || ≤ δ + ɛ 0 q¯

t∈Ωj +1

and ||xj +1 − xj || ≤ ɛ 0 (q¯ + 1).

.

(4.177)

¯ . . . , (q + 1)N¯ }, In view of (4.177), for every pair of integers .j1 , j2 ∈ {q N, ||xj1 − xj2 || ≤ ɛ 0 (q¯ + 1)N¯ .

.

(4.178)

Let .s ∈ {1, . . . , m}. By (4.139), there exist an integer .j ∈ {q N¯ , . . . , (q + 1)N¯ − 1} and an index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 such that s ∈ {t1 , . . . , tp(t) }.

.

By (4.136) and (4.176), d(xj , Cs ) ≤ ɛ 0 (q¯ + 1).

.

(4.179)

174

4 Dynamic String-Averaging Methods in Hilbert Spaces

¯ . . . , (q + 1)N}, ¯ By (4.178) and (4.179), for each .i ∈ {q N, d(xi , Cs ) ≤ ||xi − xj || + d(xj , Cs ) ≤ ɛ 0 (q¯ + 1)N¯ + ɛ 0 (q¯ + 1).

.

Thus for every .s ∈ {1, . . . , m} and every .i ∈ {q N¯ , . . . , (q + 1)N¯ }, d(xi , Cs ) ≤ ɛ 0 (q¯ + 1)(N¯ + 1).

.

This completes the proof of Theorem 4.13. Applying by induction Theorem 4.13 we obtain the following result. Theorem 4.15 Assume that (4.130) holds, .M¯ > max{||z∗ ||, 1}, ¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

¯ .δ > 0 satisfies Let .M > M, ¯ −1 , δ ≤ ((q¯ + 1)N)

.

ɛ 0 = (12(q¯ + 1)N¯ (4M + 2)Δ−1 δ)1/2 ≤ ɛ ¯ (q¯ + 1)(N¯ + 1)−1

.

and a natural number .n0 satisfy n0 > 16M 2 Δ−1 ɛ 02 + 1.

.

Let {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

satisfy for each natural number i, (xi , λi ) ∈ B(xi−1 , (Ωi , wi ), δ).

.

Then xi ∈ B(0, 3M + 1) for all integers i ≥ 0

.

The Second Result with Nonsummable Errors

175

and there exists a strictly increasing sequence of integers .{qp }∞ p=1 such that ¯ 0 ≤ q1 ≤ n0 N,

.

N¯ ≤ qp+1 − qp ≤ n0 N¯ for all integers p ≥ 1

.

and that for each integer .p ≥ 1, ¯ d(xi , Cs ) ≤ ɛ 0 (q¯ + 1)(N¯ + 1), s = 1, . . . , m, i = qp , . . . , qp + N,

.

¯ i, j ∈ {qp , . . . , qp + N}. ¯ ||xi − xj || ≤ ɛ 0 (q¯ + 1)N,

.

The Second Result with Nonsummable Errors Assume that z∗ ∈ ∩m i=1 Ci = C

.

(4.180)

and the following assumption holds: (A1) for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

(4.181)

Theorem 4.16 Assume that .M > max{||z∗ ||, 1}, .δ > 0 satisfies δ ≤ (4q¯ + 4)−2 N¯ −2 (4M + 4)−1 ,

(4.182)

¯ ɛ 0 = (132q¯ N(4M + 9)3/2 Δ−1 δ 1/2 )1/2

(4.183)

.

.

and a natural number .n0 satisfy n0 > 16M 2 Δ−1 ɛ 0−2 + 1.

(4.184)

{(Ωi , wi )}∞ i=1 ⊂ M∗

(4.185)

.

Let .

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

(4.186) (4.187)

176

4 Dynamic String-Averaging Methods in Hilbert Spaces

satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ).

.

(4.188)

Then there exists an integer .q ∈ [0, n0 − 1] such that ||xi || ≤ 3M + 1, i = 0, . . . , q N¯ ,

(4.189)

¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

(4.190)

.

.

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (4.190), then for each .i = q N¯ , . . . , (q + 1)N¯ and each .s ∈ {1, . . . , m}, d(xi , Cs ) ≤ ɛ 0 (q + 1)(N¯ + 1)

.

and ||xi − xj || ≤ ɛ 0 (q¯ + 1)N¯

.

for each .i, j ∈ {q N¯ , . . . , (q + 1)N¯ }. It should be mentioned that in Theorem 4.16 .δ is the computational error made by our computer system, we obtain a point z such that .d(z, Cs ) ≤ (N¯ + 1)(q¯ + 1)ɛ 0 and in order to obtain this point we need .(n0 − 1)N¯ iterations. It is not difficult to see that .ɛ 0 = c1 δ 1/4 and .n0 = Lc2 δ −1/2 ⎦ + 1, where .c1 and .c2 are positive constants depending on M. Proof Let .i ≥ 0 be an integer. We have (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), δ).

.

(4.191)

By (4.11) and (4.191) there exists vectors (i)

(i)

(yt , αt ) ∈ A0 (xi , t, δ), t ∈ Ωt+1

.

(4.192)

such that ||xi+1 −

Σ

.

wi+1 (t)yt(i) || ≤ δ,

(4.193)

t∈Ωi+1

λi+1 = max{αt : t ∈ Ωi+1 }.

.

(4.194)

It follows from (4.10) and (4.192) that for each index vector .t = (t1 , . . . , tp(t) ) ∈ (i,t) p(t) Ωi+1 there exists a finite sequence .{yj }j =0 ⊂ X such that (i,t) y0(i,t) = xi , yp(t) = yt(i) , B(yj(i,t) , δ) ∩ Ptj ,δ (yj(i,t) / ∅ −1 ) =

.

The Second Result with Nonsummable Errors .

(i)

αt

.

177

for all integers j = 1, . . . , p(t),

(4.195)

(i,t)

(4.196)

(i,t)

= max{||yj +1 − yj

|| : j = 0, . . . , p(t) − 1}.

Assume that s is a nonnegative integer and that for each integer .k ∈ [0, s], .

max{λi : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 .

(4.197)

By (4.187), ||x0 − z∗ || ≤ 2M.

(4.198)

.

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

(4.199)

.

U ∩ We prove the following auxiliary result. Lemma 4.17 Assume that an integer i ∈ [0, N¯ − 1]

(4.200)

.

satisfies ||xk N¯ +i − z∗ || ≤ 2M + 4iδ 1/2 (q¯ + 1)(4M + 4)1/2 .

.

(4.201)

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + 2(q¯ + 1)(2δ(4M + 4))1/2

.

and ||xk N¯ +i+1 −z∗ ||2 ≤ ||xk N¯ +i −z∗ ||2 +33δ 1/2 q(4M ¯ +4)3/2 −Δλ2k N¯ +i+1 .

.

(4.202)

If .λk N¯ +i+1 > ɛ 0 , then ||xk N¯ +i+1 − z||2 − ||xk N¯ +i − z||2 ≤ −2−1 Δɛ 02 .

.

(4.203)

Proof Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and .j ∈ {1, . . . , p(t)}. Assumption (A1) (k N¯ +i,t)

and Lemma 2.3 applied to .x = yj −1 ¯

¯

(k N¯ +i,t)

, .x˜ = yj

, .D = Ctj imply that ¯

N +i,t) N +i,t) ||z∗ − yj(k N +i,t) || ≤ ||z∗ − yj(k−1 || + 2(2δ(2||z∗ − yj(k−1 ||))1/2 ,

.

(4.204)

178

4 Dynamic String-Averaging Methods in Hilbert Spaces (k N¯ +i,t) 2

||z∗ − yj −1

.

(k N¯ +i,t) 2

|| ≥ ||z∗ − yj

.

(k N¯ +i,t)

|| + ||yj

(k N¯ +i,t)

− 32δ 1/2 (2||z∗ − yj −1

(k N¯ +i,t) 2

− yj −1

|| + 1)3/2 .

||

(4.205)

We show that for each integer .j ∈ {0, . . . , p(t)}, (k N¯ +i,t)

||z∗ − yj

.

|| ≤ ||z∗ − xk N¯ +i || + j [2(2δ(4M + 4))1/2 ].

(4.206)

Assume that .j ∈ {0, . . . , p(t) − 1} and (4.206) holds. (In view of (4.195), (4.206) holds for .j = 0.) By (4.7), (4.182), (4.200), (4.201) and (4.206), (k N¯ +i,t)

||z∗ − yj

.

.

|| ≤ 2M + 4iδ 1/2 (q¯ + 1)(4M + 4)1/2 + 4δ 1/2 q(4M ¯ + 4)1/2

≤ 2M + 4N¯ δ 1/2 (q¯ + 1)(4M + 4)1/2 ≤ 2M + 1.

(4.207)

Equations (4.204), (4.206) and (4.207) imply that (k N¯ +i,t)

||z∗ − yj +1

.

.

(k N¯ +i,t)

|| ≤ ||z∗ − yj

|| + 2(2δ(4M + 3))1/2

≤ ||z∗ − xk N¯ +i || + 2(j + 1)(2δ(4M + 4))1/2 .

Thus we have showed by induction that for all integers .j = 0, . . . , p(t) (4.206) holds and (k N¯ +i,t)

||z∗ − yj

.

|| ≤ 2M + 1.

(4.208)

By (4.205) and (4.208), for .j = 1, . . . , p(t), (k N¯ +i,t) 2

||z∗ − yj −1

.

.

(k N¯ +i,t) 2

≥ ||z∗ − yj

(k N¯ +i,t)

|| + ||yj

||

(k N¯ +i,t) 2

− yj −1

|| − 32δ 1/2 (4M + 4)3/2 .

(4.209)

Thus we have shown that the following property holds: (P1) for each .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and each .j ∈ {1, . . . , p(t)} (4.209) holds, for each .j = 0, . . . , p(t), (4.208) holds and for each .j = 0, . . . , p(t) (4.206) is true. Property (P1) and (4.195), (4.206) imply that for each .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 , (k N¯ +i)

||z∗ −yt

.

(k N¯ +i,t)

|| = ||z∗ −yp(t)

1/2 || ≤ ||z∗ −xk N¯ +i ||+2q(2δ(4M+4)) ¯ .

(4.210)

The Second Result with Nonsummable Errors

179

By (4.3), (4.185), (4.193), (4.210) and the convexity of the norm, ||z∗ − xk N¯ +i+1 ||

.

.

Σ

≤ ||z∗ −

¯ (k N+i)

wk N¯ +i+1 (t)yt

||

t∈Ωk N+i+1 ¯

.

Σ

+ ||xk N¯ +i+1 −

¯

wk N¯ +i+1 (t)yt(k N +i) ||

t∈Ωk N+i+1 ¯

.

Σ

≤δ+

(k N¯ +i)

wk N+i+1 (t)||yt ¯

− z∗ ||

t∈Ωk N+i+1 ¯

≤ ||xk N¯ +i − z∗ || + 2(2δ(4M + 4))1/2 (q¯ + 1).

.

(4.211)

In view of (4.182), (4.201) and (4.211), ||z∗ − xk N¯ +i || ≤ 2M + 1.

(4.212)

.

Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 . By (4.7), (4.195), (4.196), (4.209), (k N¯ +i)

||z∗ − xk N¯ +i ||2 − ||yt

.

.

.

=

(k N¯ +i,t)

= ||y0

− z∗ ||2

(k N¯ +i,t)

− z∗ ||2 − ||yp(t)

− z∗ ||2

p(t) Σ ¯ N¯ +i,t) (||yj(k−1 − z∗ ||2 − ||yj(k N +i,t) − z∗ ||2 ) j =1

.

≥

p(t) Σ

(k N¯ +i,t)

||yj

j =1

.

(k N¯ +i,t) 2

− yj −1

|| − 32qδ ¯ 1/2 (4M + 4)3/2

(k N¯ +i) 2

≤ (αt

) − 32qδ ¯ 1/2 (4M + 4)3/2

and (k N¯ +i)

||yt

.

(k N¯ +i) 2

− z∗ ||2 ≤ ||z∗ − xk N¯ +i ||2 − (αt

) + 32qδ ¯ 1/2 (4M + 4)3/2 .

(4.213)

180

4 Dynamic String-Averaging Methods in Hilbert Spaces

It follows from (4.3), (4.8), (4.185), (4.194), (4.213) and the convexity of the function .|| · ||2 that Σ

||z∗ −

.

(k N¯ +i) 2

||

wk N¯ +i+1 (t)yt

t∈Ωk N+i+1 ¯

.

Σ

≤

(k N¯ +i) 2

wk N+i+1 (t)||z∗ − yt ¯

||

t∈Ωk N+i+1 ¯

.

Σ

≤ ||xk N¯ +i − z∗ ||2 + 32qδ ¯ 1/2 (4M + 4)3/2 −

¯ (k N+i) 2

wk N¯ +i+1 (t)(αt

)

t∈Ωk N+i+1 ¯ .

≤ ||xk N¯ +i − z∗ ||2 + 32qδ ¯ 1/2 (4M + 4)3/2 − Δλ2k N+i+1 . ¯

(4.214)

Equations (4.182), (4.193), (4.201), (4.210) and (4.212) imply that Σ

|||z∗ − xk N¯ +i+1 ||2 − ||z∗ −

¯ (k N+i) 2

||

wk N¯ +i+1 (t)yt

.

t∈Ωk N+i+1 ¯

.

Σ

≤ |||z∗ − xk N¯ +i+1 || − ||z∗ −

(k N¯ +i)

wk N+i+1 (t)yt ¯

|||

t∈Ωk N+i+1 ¯

.

Σ

× (||z∗ − xk N¯ +i+1 || + ||z∗ −

¯

wk N+i+1 (t)yt(k N +i) ||) ¯

t∈Ωk N+i+1 ¯

.

≤ ||xk N¯ +i+1 −

Σ

(k N¯ +i)

wk N+i+1 (t)yt ¯

||(4M + 2) ≤ δ(4M + 2).

(4.215)

t∈Ωk N+i+1 ¯

In view of (4.214) and (4.215), Σ

||z∗ − xk N¯ +i+1 ||2 ≤ ||z∗ −

.

¯ (k N+i) 2

wk N¯ +i+1 (t)yt

|| + δ(4M + 2)

t∈Ωk N+i+1 ¯ .

≤ ||xk N¯ +i − z∗ ||2 + 32qδ ¯ 1/2 (4M + 4)3/2 − Δλ2k N+i+1 + δ(4M + 2) ¯ .

≤ ||xk N¯ +i − z∗ ||2 + 33qδ ¯ 1/2 (4M + 4)3/2 − Δλ2k N+i+1 . ¯

(4.216)

Assume that λk N¯ +i+1 > ɛ 0 .

.

(4.217)

The Second Result with Nonsummable Errors

181

It follows from (4.183), (4.216) and (4.217) that ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i − z||2 − 33qδ ¯ 1/2 (4M + 4)3/2 − Δɛ 02

.

.

≤ ||xk N¯ +i+1 − z∗ ||2 − Δɛ 02 /2. U ∩

Lemma 4.17 is proved.

It follows from (4.182), (4.199) and Lemma 4.17 that for all integers .i = 0, . . . , N¯ − 1, equation (4.202) holds and for all .i = 0, . . . , N¯ , ¯ q¯ + 1)δ 1/2 (4M + 4)1/2 ≤ 2M + 1. ||xk N¯ +i − z∗ || ≤ 2M + 4N(

.

(4.218)

Lemma 4.17 and Eqs. (4.183), (4.202) and (4.207) imply that ||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2

.

.

=

¯ N−1 Σ

[||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ]

i=0 .

≤ −2−1 Δɛ 02 + 33N¯ δ 1/2 q(4M ¯ + 4)3/2 ≤ −4−1 Δɛ 02 .

Thus we have shown that the following property holds: (P2) if an integer .k ∈ [0, s] satisfies .||xk N¯ − z∗ || ≤ 2M, then ¯ ||xj − z∗ || ≤ 2M + 1, j = k N¯ , . . . , (k + 1)N,

.

||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2 ≤ −4−1 Δɛ 02 .

.

In view of (4.198) and property (P2) we have ||xj − z|| ≤ 2M + 1, j = 0, . . . , (s + 1)N¯

.

and (4.219) is true for every integer .k = 0, . . . , s. By (4.198) and (4.219), 4−1 Δɛ 02 (s + 1) ≤

.

s Σ [||xk N¯ − z∗ ||2 − ||x(k+1)N¯ − z∗ ||2 ] k=0

.

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 Δ−1 ɛ 0−2 .

.

(4.219)

182

4 Dynamic String-Averaging Methods in Hilbert Spaces

Thus we have shown that the following property holds: (P3) If an integer .s ≥ 0 and for every integer .k ∈ [0, s] relation (4.197) holds, then s ≤ 16M 2 Δ−1 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , s + 1.

.

By property (P3), there exists an integer .q ∈ [0, n0 − 1] such that for every integer k satisfying .0 ≤ k < q, .

¯ > ɛ 0 , max{λi : i = k N¯ + 1, . . . , (k + 1)N} .

(4.220)

max{λi : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 .

Property (P3) and (4.220) imply that ||xq N¯ − z|| ≤ 2M,

.

||xj − z|| ≤ 2M + 1, j = 0, . . . , q N¯ ,

.

||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(4.221)

Let j ∈ {q N¯ , . . . , (q + 1)N¯ − 1}.

.

It follows from (4.194)–(4.196) and (4.221) that (j )

αt

.

≤ λj +1 ≤ ɛ 0 , t ∈ Ωj +1

and that for every index vector .t ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (j,t)

y0

.

(j,t)

||yi

.

= xj ,

(4.222)

(j,t)

(4.223)

− yi−1 || ≤ ɛ 0 .

The Second Result with Nonsummable Errors

183

In view of (4.195), (4.222) and (4.223), for every index vector .t ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (j,t)

||xj − yi

.

|| ≤ iɛ 0 ≤ ɛ 0 q, ¯

(4.224)

(j )

(4.225)

||xj − yt || ≤ ɛ 0 q. ¯

.

In view of (4.195) and (4.225) for every index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (t,j )

d(xj , Cti ) ≤ ||xj − yi

.

(t,j )

|| + d(yi

, Cti ) ≤ ɛ 0 q¯ + δ.

(4.226)

It follows from (4.3), (4.193), (4.225) and the convexity of the norm that Σ

||xj +1 − xj || ≤ ||xj +1 −

.

(j )

wj +1 (t)yt ||

t∈Ωj +1

.

Σ

+ ||

(j )

wj +1 (t)yt

− xj ||

t∈Ωj +1

.

≤δ+

Σ

(j )

wj +1 (t)||yt

− xj || ≤ δ + ɛ 0 q. ¯

t∈Ωj +1

By (4.193) and the relation above, ||xj +1 − xj || ≤ ɛ 0 (q¯ + 1).

.

(4.227)

¯ . . . , (q + 1)N¯ }, In view of (4.227), for every pair of integers .j1 , j2 ∈ {q N, ||xj1 − xj2 || ≤ ɛ 0 (q¯ + 1)N¯ .

.

(4.228)

Let .s ∈ {1, . . . , m}. By (4.183), (4.186) and (4.226), there exist an integer .j ∈ {q N¯ , . . . , (q + 1)N¯ − 1} and an index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 such that s ∈ {t1 , . . . , tp(t) }

(4.229)

d(xj , Cs ) ≤ ɛ 0 (q¯ + 1).

(4.230)

.

and .

¯ . . . , (q + 1)N}, ¯ By (4.228) and (4.230), for each .i ∈ {q N, d(xi , Cs ) ≤ ||xi − xj || + d(xj , Cs ) ≤ ɛ 0 (q¯ + 1)N¯ + ɛ 0 (q¯ + 1).

.

184

4 Dynamic String-Averaging Methods in Hilbert Spaces

Thus for every .s ∈ {1, . . . , m} and every .i ∈ {q N¯ , . . . , (q + 1)N¯ )}, d(xi , Cs ) ≤ ɛ 0 (q¯ + 1)(N¯ + 1).

.

This completes the proof of Theorem 4.16. Applying by induction Theorem 4.16 we obtain the following result. Theorem 4.18 Assume that .ɛ ¯ ∈ (0, 1), .M¯ > max{||z∗ ||, 1}, ¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

¯ .δ > 0 satisfies Let .M > M, δ ≤ (4q¯ + 4)−2 N¯ −2 (4M + 4)−1 ,

.

¯ ɛ 0 = ((264qδ) ¯ 1/2 N(4M + 4)3/2 Δ−1 )1/2 ≤ N¯ −1 (q¯ + 1)−1 ɛ ¯

.

and a natural number .n0 satisfy n0 > 164M 2 Δ−1 ɛ 0−2 + 1.

.

Let {(Ωi , wi )}∞ i=1 ⊂ M∗ ,

.

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ).

.

Then xi ∈ B(0, 3M + 1) for all integers i ≥ 0

.

and there exists a strictly increasing sequence of integers .{qp }∞ p=1 such that ¯ 0 ≤ q1 ≤ n0 N,

.

N¯ ≤ qp+1 − qp ≤ n0 N¯ for all integers p ≥ 1

.

The Second Result with Nonsummable Errors

185

and that for each integer .p ≥ 1, ¯ d(xi , Cs ) ≤ ɛ 0 (q¯ + 1)(N¯ + 1), s = 1, . . . , m, i = qp , . . . , qp + N,

.

¯ i, j ∈ {qp , . . . , qp + N. ¯ ||xi − xj || ≤ ɛ 0 (q¯ + 1)N,

.

Theorem 4.18 implies the following result. Theorem 4.19 Assume that .ɛ ¯ ∈ (0, 1), .M¯ > max{||z∗ ||, 1}, ¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

¯ .δ¯ > 0 satisfy Let .M > M, δ¯ ≤ (4q¯ + 4)−2 N¯ −2 (4M + 4)−1 ,

.

¯ ((264qδ) ¯ 1/2 N(4M + 4)3/2 Δ−1 )1/2 ≤ N¯ −1 (q¯ + 1)−1 ɛ ¯

.

¯ and .{δi }∞ i=0 ⊂ (0, δ) satisfies .

lim δi = 0.

i→∞

Let {(Ωi , wi )}∞ i=1 ⊂ M∗ ,

.

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δi ).

.

Then xi ∈ B(0, 3M + 1) for all integers i ≥ 0

.

and .

lim inf max{d(xi , Cs ) : s = 1, . . . , m} = 0. i→∞

Theorem 4.19 implies the following result.

186

4 Dynamic String-Averaging Methods in Hilbert Spaces

Theorem 4.20 Assume that all the condition of Theorem 4. 19 are true and the bounded regularity property holds. Then .

lim inf d(xi , C) = 0. i→∞

Theorem 4.16 implies the following result. Theorem 4.21 Assume that the bounded regularity property holds, .M > max{||z∗ ||, 1}, .ɛ ∈ (0, 1). Then there exists .δ ∈ (0, ɛ ) and a natural number .n0 such that for each {(Ωi , wi )}∞ i=1 ⊂ M∗

.

which satisfies for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

each ∞ x0 ∈ B(0, M) and each {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

which satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ).

.

there exists an integer .q ∈ [0, n0 − 1] such that ||xi || ≤ 3M + 1, i = 0, . . . , q N¯ ,

.

¯ and for each .i = q N¯ , . . . , (q + 1)N, d(xi , C) ≤ ɛ .

.

Theorem 4.21 implies the following result. Theorem 4.22 Assume that the bounded regularity property holds, .M¯ max{||z∗ ||, 1}, .M > M¯ + 1,

>

¯ C ⊂ B(0, M),

.

ɛ ∈ (0, 1). Then there exists .δ ∈ (0, ɛ ) and a natural number .n0 such that for each

.

{(Ωi , wi )}∞ i=1 ⊂ M∗

.

which satisfies for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

The Second Result with Nonsummable Errors

187

each ∞ x0 ∈ B(0, M) and each {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

which satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ)

.

the inequality ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

holds and there exists a strictly increasing sequence of integers .{qp }∞ p=1 such that 0 ≤ q1 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

and that for each integer .p ≥ 0, ¯ ¯ . . . , (qp + 1)N. d(xi , C) ≤ ɛ , i = qp N,

.

Theorem 4.23 Assume that the bounded regularity property holds, .M¯ max{||z∗ ||, 1}, .M > M¯ + 1,

>

¯ C ⊂ B(0, M),

.

Δ ∈ (0, 1), .ɛ 0 ∈ (0, Δ) and that the following property holds:

.

(i) for each .ξ ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .η ∈ Cs ∩B(ξ, Δ), αξ + (1 − α)η ∈ Cs .

.

(4.231)

Then there exists .δ0 ∈ (0, ɛ ) and a natural number .n1 such that for each {(Ωi , wi )}∞ i=1 ⊂ M∗

.

which satisfies for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

each ∞ x0 ∈ B(0, M) and each {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

(4.232)

188

4 Dynamic String-Averaging Methods in Hilbert Spaces

which satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ0 )

.

(4.233)

the inequality ||xi || ≤ 3M + 1 for all integers i ≥ 0

.

holds and , d(xi , C) ≤ ɛ 0

.

for each integer .i ≥ n1 . Proof We may assume that ɛ 0 < Δ/4.

(4.234)

ɛ = ɛ 0 /4.

(4.235)

.

Set .

Let .δ ∈ (0, ɛ ) and a natural number .n0 be as guaranteed by Theorem 4.22. Set δ0 = min{δ, 36−2 ɛ 2 (q¯ + 1)−2 (N¯ + 1)−2 (n0 + 1)−2 }.

.

(4.236)

Assume that {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfies(4.232) for each natural number j , x0 ∈ B(0, M)

.

(4.237)

∞ {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞), for each natural number i, (4.233) holds. By Theorem 4.22 and the choice of .δ, n0 ,

.

||xi || ≤ 3M + 1 for all integers i ≥ 0

.

(4.238)

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

(4.239)

1 ≤ qp+1 − qp ≤ n0 ,

(4.240)

.

and for each integer .p ≥ 0, .

The Second Result with Nonsummable Errors

189

and, d(xi , C) ≤ ɛ = ɛ 0 /4, i = qp N¯ , . . . , (qp + 1)N¯ .

.

(4.241)

Let .i ≥ 0 be an integer. We have (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), δ0 ).

.

(4.242)

By (4.11) and (4.242) there exists vectors (i)

(i)

(yt , αt ) ∈ A0 (xi , t, δ0 ), t ∈ Ωt+1

.

(4.243)

such that Σ

||xi+1 −

(i)

wi+1 (t)yt || ≤ δ0 ,

.

(4.244)

t∈Ωi+1

λi+1 = max{αt : t ∈ Ωi+1 }.

.

(4.245)

It follows from (4.11) and (4.243) that for each index vector .t = (t1 , . . . , tp(t) ) ∈ (i,t) p(t) Ωi+1 there exists a finite sequence .{yj }j =0 ⊂ X such that (i,t)

y0

.

(i,t)

B(yj

.

(i,t)

(i)

= xi , yp(t) = yt ,

(4.246)

, δ0 ) ∩ Ptj ,δ0 (yj −1 ) /= ∅

(i,t)

(4.247)

(i,t)

|| : j = 0, . . . , p(t) − 1}.

(4.248)

for each integer .j = 1, . . . , p(t), (i)

αt

.

(i,t)

= max{||yj +1 − yj

Set 1/2

γ0 = 12δ0 .

.

(4.249)

In view (4.236) and (4.249), γ0 (q¯ + 1)(N¯ + 1)(n0 + 1) ≤ ɛ /3.

.

Assume that .p ≥ 0 is an integer. In view of (4.241), d(xi , C) ≤ ɛ = ɛ 0 /4, i = qp N¯ , . . . , (qp + 1)N¯ .

.

(4.250)

By (4.250). there exists z∈C

.

(4.251)

190

4 Dynamic String-Averaging Methods in Hilbert Spaces

such that ||x(qp +1)N¯ − z|| < ɛ 0 /3.

(4.252)

¯ qp+1 N¯ ≤ n0 N¯ + qp N.

(4.253)

.

By (4.240), .

Assume that .i ∈ {0, . . . , qp+1 N¯ − qp N¯ } and ||z − x(qp +1)N¯ +i || ≤ γ0 (q¯ + 1)i + ɛ 0 /3.

(4.254)

.

(In view of (4.252) Eq. (4.254) hold for .i = 0.) Assume that .t ∈ Ω(qp +1)N¯ +i+1 , .j ∈ {1, . . . , p(t)} and ¯ (q +1)N+i,t

p ||z − yj −1

.

|| ≤ ɛ 0 /3 + γ0 (q¯ + 1)i + (j − 1)γ0 .

(4.255)

(In view of (4.246) and (4.254), Eq. (4.255) holds for .j = 1.) By (4.247), there exists ((q +1)N¯ +i,t)

ξ ∈ Ptj ,δ0 (yj −1p

.

¯ ((qp +1)N+i,t)

) ∩ B(yj

(4.256)

), δ0 ).

Equations (4.251) and (4.256) imply that ¯ ((q +1)N+i,t)

||yj −1p

.

((q +1)N¯ +i,t)

− ξ || ≤ δ0 + ||z − yj −1p

||.

(4.257)

It follows from (4.234), (4.249), (4.250), (4.255) and (4.257) that ¯ ((q +1)N+i,t)

||z − ξ || ≤ ||z − yj −1p

.

.

((q +1)N¯ +i,t)

|| + ||yj −1p

− ξ ||

≤ 2ɛ 0 /3 + 2[γ0 (q¯ + 1)i + j γ0 ] ≤ Δ.

(4.258)

Assumption (A1), (4.247), (4.249), (4.250), (4.255), (4.256) and (4.258) and ((q +1)N¯ +i,t)

Lemma 2.3 applied with .z∗ = z, x = yj −1p .δ = δ0 imply that ¯ ((qp +1)N+i,t)

||z − yj

.

.

((q +1)N¯ +i,t)

|| ≤ ||z − yj −1p ¯ ((q +1)N+i,t)

≤ ||z − yj −1p

, .D = Ctj ,

¯ ((q +1)N+i,t)

|| + 2(2δ0 (2||z − yj −1p

1/2

|| + 12δ0

((qp +1)N¯ +i,t)

, .x˜ = yj

≤ ɛ 0 /3 + γ0 (q¯ + 1)i + j γ0 .

|| + 1))1/2

The Third Result with Nonsummable Errors

191

Thus we have shown by induction that (4.255) holds for all .j = 1, . . . , p(t) + 1. Together with (4.246) this implies that ((qp +1)N¯ +i)

||z − yt

.

|| ≤ ɛ 0 /3 + γ0 (q¯ + 1)i + qγ ¯ 0.

(4.259)

It follows from (4.3), (4.244), (4.249) and (4.259) and the convexity of the norm that ||z − x(qp +1)N+i+1 || ¯

.

.

Σ

≤ ||z −

¯ ((qp +1)N+i)

w(qp +1)N¯ +i+1 (t)yt

||

t∈Ω(qp +1)N+i+1 ¯

.

Σ

+ ||x(qp +1)N+i+1 − ¯

((qp +1)N¯ +i)

w(qp +1)N¯ +i+1 yt

||

t∈Ω(qp +1)N+i+1 ¯ .

≤ ɛ 0 /3 + γ0 (q¯ + 1)i + qγ ¯ 0 + δ0 ≤ ɛ 0 /3 + γ0 (q¯ + 1)(i + 1).

¯ Thus by induction we showed that (4.254) holds for all .i = 0, . . . , qp+1 N¯ − qp N. ¯ ¯ Together with (4.250) this implies that for all .i = 0, . . . , qp+1 N − qp N, ¯ + 1) ≤ ɛ . ||z − x(qp +1)N+i ¯ || ≤ ɛ 0 /3 + γ0 (q¯ + 1)(n0 + 1)(N

.

U ∩

Theorem 4.23 is proved.

The Third Result with Nonsummable Errors Suppose that .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets, .δC ∈ (0, 1]. Set C = ∩m i=1 Ci .

.

Assume that the following assumption holds. (A2) There exists z∗ ∈ X

.

(4.260)

and for each .s ∈ {1, . . . , m} there exists z∗s ∈ B(z∗ , δC ) ∩ Cs

.

(4.261)

192

4 Dynamic String-Averaging Methods in Hilbert Spaces

such that for for each .x ∈ Cs and each .α ∈ [0, 1], αz∗s + (1 − α)x ∈ Cs .

.

(4.262)

Clearly, assumption (A2) is weaker than its prototype (A1). In (A1) we assume that all the sets .Ci , .i = 1, . . . , m are star-shaped at the same point. In (A2) we suppose that for each .i ∈ {1, . . . , m} the set .Ci is star-shaped at .z∗i belonging to a .δC -neighborhood of a fixed point .z∗ ∈ X. Theorem 4.24 Assume that .M > max{||z∗ ||, 1}, ¯ q¯ + 1))−1 , 0 < δC ≤ (4N(

(4.263)

δ ≤ (4q¯ + 4)−2 N¯ −2 (8M + 32)−1 ,

(4.264)

.

δ ∈ (0, 1) satisfies

.

.

¯ ¯ ɛ 0 = (32q¯ N(4M + 4)δC + 136q¯ N(4M + 4)1/2 δ 1/2 Δ−1 )1/2

.

(4.265)

and a natural number .n0 satisfy n0 > 16M 2 Δ−1 ɛ −2 + 1.

.

(4.266)

Let {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

(4.267)

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ).

.

(4.268)

Then there exists an integer .q ∈ [0, n0 − 1] such that ||xi || ≤ 3M + 1, i = 0, . . . , q N¯ ,

(4.269)

¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

(4.270)

.

.

The Third Result with Nonsummable Errors

193

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (4.270), then for each .i = q N¯ , . . . , (q + 1)N¯ and each .s ∈ {1, . . . , m}, d(xi , Cs ) ≤ ɛ 0 (q + 1)(N¯ + 1)

.

and ||xi − xj || ≤ ɛ 0 (q¯ + 1)N¯

.

for each .i, j ∈ {q N¯ , . . . , (q + 1)N¯ }. Proof Let .i ≥ 0 be an integer. In view of (4.268), (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), δ).

.

(4.271)

By (4.11) and (4.271) there exists vectors (i)

(i)

(yt , αt ) ∈ A0 (xi , t, δ), t ∈ Ωt+1

.

(4.272)

such that Σ

||xi+1 −

.

(i)

(4.273)

: t ∈ Ωi+1 }.

(4.274)

wi+1 (t)yt || ≤ δ,

t∈Ωi+1 (i)

λi+1 = max{αt

.

In view of (4.100) and (4.272), for each index vector .t = (t1 , . . . , tp(t) ) ∈ Ωi+1 (i,t) p(t) there exists a finite sequence .{yj }j =0 ⊂ X such that (i,t)

y0

.

(i,t)

(i,t)

(4.275) (4.276)

(i)

(4.277)

αt

.

(i)

, δ) ∩ Ptj ,δ (yj −1 ) /= ∅ for each integer j = 1, . . . , p(t),

B(yj

.

(i,t)

= xi , yp(t) = yt ,

(i,t)

(i,t)

= max{||yj +1 − yj

|| : j = 0, . . . , p(t) − 1}.

Assume that s is a nonnegative integer and for each integer .k ∈ [0, s], .

max{λi : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 .

(4.278)

By (4.268), ||x0 − z∗ || ≤ 2M.

.

(4.279)

194

4 Dynamic String-Averaging Methods in Hilbert Spaces

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

(4.280)

γ0 = 2δC q¯ + 2(q¯ + 1)δ 1/2 (8M + 32)1/2 .

(4.281)

.

Set .

U ∩ We prove the following auxiliary result. Lemma 4.25 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ 2M + iγ0 .

(4.282)

.

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + γ0

.

and ||xk N¯ +i+1 − z∗ ||2

.

.

≤ ||xk N¯ +i −z∗ ||2 −Δλ2k N+i+1 +8qδ ¯ C (4M +4)+33δ 1/2 q(4M ¯ +4)3/2 . ¯

(4.283)

If .λk N¯ +i+1 > ɛ 0 , then ||xk N¯ +i+1 − z||2 ≤ ||xk N¯ +i − z||2 − 2−1 Δɛ 02 .

.

Proof Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and .j ∈ {1, . . . , p(t)}. Assumption (A2), (k N¯ +i,t)

(4.276) and Lemma 2.3 applied with .x = yj −1 .D = Ctj imply that (k N¯ +i,t)

||z∗tj − yj

.

(k N¯ +i,t)

|| ≤ ||z∗tj − yj −1

(k N¯ +i,t) 2

||z∗tj − yj −1

.

.

(k N¯ +i,t)

|| + 2(2δ(2||z∗tj − yj −1

(k N¯ +i,t) 2

|| ≥ ||z∗tj − yj

(k N¯ +i,t)

, .x˜ = yj

(k N¯ +i,t)

|| + ||yj

(k N¯ +i,t)

− 32δ 1/2 ((2||z∗tj − yj −1

|| + 1)3/2 .

, .z∗ = z∗tj ,

|| + 1))1/2 , (4.284)

(k N¯ +i,t) 2

− yj −1

||

(4.285)

The Third Result with Nonsummable Errors

195

By (4.261) and (4.284), (k N¯ +i,t)

||z∗ − yj

.

(k N¯ +i,t)

≤ δC + ||z∗tj − yj −1

.

.

(k N¯ +i,t)

|| ≤ ||z∗ − z∗tj || + ||z∗tj − yj

(k N¯ +i,t)

≤ 2δC + ||z∗ − yj −1

(k N¯ +i,t)

|| + 2(2δ(2δC + 2||z∗ − yj −1 (k N¯ +i,t)

|| + 2(2δ(2||z∗ − yj −1

||

|| + 1))1/2

|| + 3))1/2 .

(4.286)

We show that for each integer .j ∈ {0, . . . , p(t)}, (k N¯ +i,t)

||z∗ − yj

.

|| ≤ ||z∗ − xk N¯ +i || + 2j δC + 2j δ 1/2 (8M + 32)1/2 .

(4.287)

(In view of (4.275), (4.287) holds for .j = 0.) Assume that .j ∈ {0, . . . , p(t) − 1} and (4.287) holds. By (4.7), (4.263), (4.264), (4.281), (4.282) and (4.287), (k N¯ +i,t)

||z∗ − yj

.

.

|| ≤ 2M + iγ0 + 2qδ ¯ C + 2qδ ¯ 1/2 (8M + 32)1/2

≤ 2M + (i + 1)γ0 ≤ 2M + N¯ γ0 ≤ 2M + 1.

(4.288)

Assume that .j ∈ {0, . . . , p(t) − 1} and (4.287) holds. By (4.286)–(4.288), (k N¯ +i,t)

||z∗ − yj +1

.

.

(k N¯ +i,t)

|| ≤ ||z∗ − yj

|| + 2δC + 2δ 1/2 (8M + 10)1/2

≤ ||z∗ − xk N¯ +i || + 2(j + 1)δC + 2j δ 1/2 (8M + 32)1/2 + 2δ 1/2 (8M + 10)1/2 .

≤ ||z∗ − xk N¯ +i || + 2(j + 1)δC + 2(j + 1)δ 1/2 (8M + 32)1/2 .

Thus we have showed by induction that (4.287) and (4.288) hold for all integers j = 0, . . . , p(t). By (4.261), (4.285) and (4.288), for all .j = 1, . . . , p(t),

.

¯

¯

¯

¯

N +i,t) 2 N +i,t) 2 ||z∗tj − yj(k−1 || ≥ ||z∗tj − yj(k N +i,t) ||2 + ||yj(k N +i,t) − yj(k−1 ||

.

.

− 32δ 1/2 (4M + 4)3/2 .

(4.289)

Equations (4.261) and (4.288) imply that for .s = j, j − 1, ¯

¯

|||z∗ − ys(k N +i,t) ||2 − ||z∗tj − ys(k N +i,t) ||2 |

.

.

¯

≤ ||z∗ − z∗tj ||(2||z∗ − ys(k N +i,t) || + ||z∗tj − z∗ ||) .

≤ δC (4M + 4).

(4.290)

196

4 Dynamic String-Averaging Methods in Hilbert Spaces

It follows from (4.289) and (4.290) that (k N¯ +i,t) 2

||z∗ − yj −1

.

.

(k N¯ +i,t) 2

|| ≥ ||z∗tj − yj −1 (k N¯ +i,t) 2

≥ ||z∗tj − yj .

.

|| − (4M + 4)δC

(k N¯ +i,t)

|| + ||yj

(k N¯ +i,t) 2

− yj −1

||

− 32δ 1/2 (4M + 4)3/2 − (4M + 4)δC ¯

¯

¯

N +i,t) 2 ≥ ||z∗ − yj(k N +i,t) ||2 + ||yj(k N +i,t) − yj(k−1 || .

− 32δ 1/2 (4M + 4)3/2 − 8(4M + 4)δC .

(4.291)

By (4.281), (4.282) and (4.288), the following property holds: (P1) for each .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and each .j ∈ {0, . . . , p(t)}, ¯

||z∗ − yj(k N +i,t) || ≤ ||z∗ − xk N¯ +i || + 2j δC + 2j δ 1/2 (8M + 32)1/2 ≤ 2M + 1

.

and for each .j = {1, . . . , p(t)}, ¯

¯

¯

¯

N +i,t) 2 N +i,t) 2 ||z∗ − yj(k−1 || ≥ ||z∗ − yj(k N +i,t) ||2 + ||yj(k N +i,t) − yj(k−1 ||

.

.

− 32δ 1/2 (4M + 4)3/2 − 8(4M + 4)δC .

Property (P1) and (4.275) imply that for each .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 , (k N¯ +i)

||z∗ − yt

.

(k N¯ +i,t)

|| = ||z∗ − yp(t)

||

≤ ||z∗ − xk N¯ +i || + 2qδ ¯ C + 2qδ ¯ 1/2 (8M + 32)1/2 . By (4.3), (4.273), (4.281), (4.292) and the convexity of the norm, ||z∗ − xk N¯ +i+1 ||

.

.

Σ

≤ ||z∗ −

¯

wk N¯ +i+1 (t)yt(k N+i) ||

t∈Ωk N+i+1 ¯

.

Σ

+ ||xk N¯ +i+1 −

(k N¯ +i)

wk N¯ +i+1 (t)yt

||

t∈Ωk N+i+1 ¯

.

≤δ+

Σ t∈Ωk N+i+1 ¯

¯

wk N+i+1 (t)||yt(k N +i) − z∗ || ¯

(4.292)

The Third Result with Nonsummable Errors

197

≤ ||xk N¯ +i − z∗ || + 2qδ ¯ C + 2(q¯ + 1)δ 1/2 (8M + 32)1/2

.

.

≤ ||xk N¯ +i − z∗ || + γ0 .

(4.293)

In view of (4.263), (4.264), (4.281), (4.282) and (4.293), ||z∗ − xk N¯ +i || ≤ 2M + 1.

(4.294)

.

Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 . Property (P1) and (4.217) imply that (k N¯ +i,t)

||y0

.

(k N¯ +i,t)

− z∗ ||2 − ||yp(t)

− z∗ ||2

p(t) Σ (k N¯ +i,t) (k N¯ +i,t) . = (||yj −1 − z∗ ||2 − ||yj − z∗ ||2 ) j =1

.

≥

p(t) Σ

(k N¯ +i,t)

||yj

j =1

||

− 32qδ ¯ 1/2 (4M + 4)3/2 − 8qδ ¯ C (4M + 4)

.

.

(k N¯ +i,t) 2

− yj −1

¯

≥ (αt(k N +i) )2 − 32qδ ¯ 1/2 (4M + 4)3/2 − 8qδ ¯ C (4M + 4).

(4.295)

By (4.275) and (4.295), ¯

(k N +i,t) ||yp(t) − z∗ ||2 ≤ ||xk N¯ +i − z∗ ||2

.

.

(k N¯ +i) 2

− (αt

) + 32qδ ¯ 1/2 (4M + 4)3/2 + 8qδ ¯ C (4M + 4).

(4.296)

It follows from (4.8), (4.274), (4.296) and the convexity of the function .|| · ||2 that Σ

||z∗ −

(k N¯ +i) 2

||

wk N¯ +i+1 (t)yt

.

t∈Ωk N+i+1 ¯

.

≤

Σ

(k N¯ +i) 2

wk N+i+1 (t)||z∗ − yt ¯

||

t∈Ωk N+i+1 ¯

.

≤ ||xk N¯ +i − z∗ ||2 −

Σ

¯ (k N+i) 2

wk N¯ +i+1 (t)(αt

t∈Ωk N+i+1 ¯ .

+ 32qδ ¯ 1/2 (4M + 4)3/2 + 8qδ ¯ C (4M + 4)

)

198

4 Dynamic String-Averaging Methods in Hilbert Spaces

.

.

≤ ||xk N¯ +i − z∗ ||2 − Δλ2k N¯ +i+1

+ 32qδ ¯ 1/2 (4M + 4)3/2 + 8qδ ¯ C (4M + 4).

(4.297)

Equations (4.273), (4.294) imply that Σ

|||z∗ − xk N¯ +i+1 ||2 − ||z∗ −

.

(k N¯ +i) 2

|| |

wk N+i+1 (t)yt ¯

t∈Ωk N+i+1 ¯

.

Σ

≤ ||xk N¯ +i+1 −

(k N¯ +i)

wk N¯ +i+1 (t)yt

||

Σ

(k N¯ +i)

t∈Ωk N+i+1 ¯

.

× (||z∗ − xk N¯ +i+1 || + ||z∗ −

wk N+i+1 (t)yt ¯

||)

t∈Ωk N+i+1 ¯

≤ δ(4M + 4).

.

(4.298)

In view of (4.297) and (4.298), ||z∗ − xk N¯ +i+1 ||2

.

.

≤ ||xk N¯ +i −z∗ ||2 −Δλ2k N¯ +i+1 +32qδ ¯ 1/2 (4M +4)3/2 +8qδ ¯ C (4M +4)+δ(4M +4) .

≤ ||xk N¯ +i −z∗ ||2 −Δλ2k N+i+1 +33qδ ¯ 1/2 (4M +4)3/2 +8qδ ¯ C (4M +4). ¯

(4.299)

Assume that λk N¯ +i+1 > ɛ 0 .

.

It follows from (4.265), (4.299) and the equation above that ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i − z||2 − Δɛ 02 /2.

.

(4.300) U ∩

Lemma 4.25 is proved.

It follows from (4.280) and Lemma 4.25 that for all integers .i = 0, . . . , N¯ − 1, Eq. (4.283) holds and for all .i = 0, . . . , N¯ , (4.282) holds and if λk N¯ +i+1 > ɛ 0 ,

.

then (4.300) is true. By (4.250) and (4.282), for all .i = 0, . . . , N¯ , ¯ 0 ≤ 2M + 1. ||xk N¯ +i − z∗ || ≤ 2M + Nγ

.

(4.301)

The Third Result with Nonsummable Errors

199

Lemma 4.25 and Eqs. (4.265), (4.278) and (4.283) imply that ||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2

.

.

=

¯ N−1 Σ

[||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ]

i=0 .

≤ −2−1 Δɛ 02 + 8N¯ qδ ¯ C (4M + 4) + 33N¯ δ 1/2 q(4M ¯ + 4)3/2 ≤ −4−1 Δɛ 02 .

Thus we have shown that the following property holds: (P2) if an integer .k ∈ [0, s] satisfies .||xk N¯ − z∗ || ≤ 2M, then ¯ ||xj − z∗ || ≤ 2M + 1, j = k N¯ , . . . , (k + 1)N,

.

||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2 ≤ −4−1 Δɛ 02 .

.

(4.302)

In view of (4.279) and property (P2) we have ||xj − z|| ≤ 2M + 1, j = 0, . . . , (s + 1)N¯

.

and (4.302) is true for every integer .k = 0, . . . , s. By (4.280) and (4.302), 4−1 Δɛ 02 (s + 1) ≤

.

s Σ [||xk N¯ − z∗ ||2 − ||x(k+1)N¯ − z∗ ||2 ] k=0

.

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 Δ−1 ɛ 0−2 .

.

(4.303)

Thus we have shown that the following property holds: (P3) If an integer .s ≥ 0 and for every integer .k ∈ [0, s] relation (4.278) holds, then s ≤ 16M 2 Δ−1 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , s + 1.

.

200

4 Dynamic String-Averaging Methods in Hilbert Spaces

By property (P3) and (4.266), there exists an integer .q ∈ [0, n0 − 1] such that for every integer k satisfying .0 ≤ k < q, .

¯ > ɛ 0 , max{λi : i = k N¯ + 1, . . . , (k + 1)N}

.

max{λi : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 ,

||xj − z∗ || ≤ 2M + 1, ||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(4.304)

Let j ∈ {q N¯ , . . . , (q + 1)N¯ − 1}.

.

It follows from (4.274), (4.277) and (4.304) that (j )

≤ λj +1 ≤ ɛ 0 , t ∈ Ωj +1

αt

.

(4.305)

and that for every index vector .t ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (j,t)

y0

.

(j,t)

||yi

.

= xj ,

(4.306)

(j,t)

(4.307)

− yi−1 || ≤ ɛ 0 .

In view of (4.306) and (4.307), for every index vector .t ∈ Ωj +1 and every integer i = 1, . . . , p(t),

.

(j,t)

||xj − yi

.

|| ≤ iɛ 0 ≤ ɛ 0 q, ¯

In view of (4.276) and the relation above for every index vector t = (t1 , . . . , tp(t) ) ∈ Ωj +1

.

and every integer .i = 1, . . . , p(t), (t,j )

d(xj , Cti ) ≤ ||xj − yi

.

(t,j )

|| + d(yi

, Cti ) ≤ ɛ 0 q¯ + δ.

(4.308)

Almost Star-Shaped Feasibility Problems

201

It follows from (4.265), (4.273), (4.275), (4.308) and the convexity of the norm that ||xj +1 − xj || ≤ ||xj +1 −

Σ

.

(j )

wj +1 (t)yt ||

t∈Ωj +1

.

Σ

+ ||

(j )

wj +1 (t)yt

− xj ||

t∈Ωj +1

.

Σ

≤δ+

(j )

wj +1 (t)||yt

− xj || ≤ δ + ɛ 0 q, ¯

t∈Ωj +1

||xj +1 − xj || ≤ ɛ 0 (q¯ + 1).

.

(4.309)

¯ . . . , (q + 1)N¯ }, In view of (4.309), for every pair of integers .j1 , j2 ∈ {q N, ||xj1 − xj2 || ≤ ɛ 0 (q¯ + 1)N¯ .

.

(4.310)

Let .s ∈ {1, . . . , m}. By (4.267), , there exist an integer .j ∈ {q N¯ , . . . , (q + 1)N¯ − 1} and an index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 such that s ∈ {t1 , . . . , tp(t) }

.

and by (4.308), d(xj , Cs ) ≤ ɛ 0 (q¯ + 1).

.

¯ . . . , (q + 1)N}, ¯ By (4.310) and (4.311), for each .i ∈ {q N, d(xi , Cs ) ≤ ||xi − xj || + d(xj , Cs ) ≤ ɛ 0 (q¯ + 1)N¯ + ɛ 0 (q¯ + 1).

.

Thus for every .s ∈ {1, . . . , m} and every .i ∈ {q N¯ , . . . , (q + 1)(N¯ + 1)}, d(xi , Cs ) ≤ ɛ 0 (q¯ + 1)(N¯ + 1).

.

This completes the proof of Theorem 4.24.

Almost Star-Shaped Feasibility Problems Suppose that .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

(4.311)

202

4 Dynamic String-Averaging Methods in Hilbert Spaces

Assume that .M > 1, .δM ∈ (0, 1]. z∗ ∈ B(0, M),

(4.312)

B(0, M) ∩ Cs /= ∅, s = 1, . . . , m

(4.313)

.

.

and that the following assumption holds. (A3) For each .s ∈ {1, . . . , m}, .x ∈ B(0, 5M + 8) ∩ Cs and each .α ∈ [0, 1], d(αz∗ + (1 − α)x, Cs ) < δM .

.

(4.314)

Assumption (A3) implies that for each .s ∈ {1, . . . , m} there exists z∗s ∈ Cs

(4.315)

||z∗s − z∗ || ≤ δM .

(4.316)

||z∗s || ≤ M + δM ≤ M + 1, s = 1, . . . , m.

(4.317)

.

such that .

In view of (4.312) and (4.316), .

We study the following feasibility problem Find z ∈ C

.

which is called, in view of (A3), as an almost star-shaped feasibility problem. Theorem 4.26 Assume that ¯ q¯ + 1))−1 0 < δM ≤ (8N(

.

(4.318)

δ ∈ (0, 1) satisfies

.

δ ≤ (8q¯ + 8)−2 N¯ −2 (8M + 32)−1 , δ + δM ≤ (16q) ¯ −2 (4M + 4)−1 ,

(4.319)

¯ ɛ 0 = (136q¯ N(4M + 4)3/2 (δM + δ)Δ−1 )1/2

(4.320)

.

.

and a natural number .n0 satisfy n0 > 16M 2 Δ−1 ɛ 0−2 + 1.

.

(4.321)

Almost Star-Shaped Feasibility Problems

203

Let {(Ωi , wi )}∞ i=1 ⊂ M∗

.

satisfy for each natural number j j +N¯ −1

{1, . . . , m} ⊂ ∪i=j

.

(∪t∈Ωi {t1 , . . . , tp(t) }),

∞ x0 ∈ B(0, M) and {xi }∞ i=1 ⊂ X, {λi }i=1 ⊂ [0, ∞)

.

(4.322) (4.323)

satisfy for each natural number i, (xi , λi ) ∈ A(xi−1 , (Ωi , wi ), δ).

.

(4.324)

Then there exists an integer .q ∈ [0, n0 − 1] such that ||xi || ≤ 3M + 1, i = 0, . . . , q N¯ ,

(4.325)

¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

(4.326)

.

.

Moreover, if an integer .q ∈ [0, n0 − 1] satisfies (4.326), then for each .i = q N¯ , . . . , (q + 1)N¯ and each .s ∈ {1, . . . , m}, d(xi , Cs ) ≤ ɛ 0 (q + 1)(N¯ + 1)

.

and ||xi − xj || ≤ ɛ 0 (q¯ + 1)N¯

.

for each .i, j ∈ {q N¯ , . . . , (q + 1)N¯ }. Proof Let .i ≥ 0 be an integer. In view of (4.324), (xi+1 , λi+1 ) ∈ A(xi , (Ωi+1 , wi+1 ), δ).

.

(4.327)

By (4.11) and (4.327) there exists vectors (i)

(i)

(yt , αt ) ∈ A0 (xi , t, δ), t ∈ Ωt+1

.

(4.328)

such that ||xi+1 −

Σ

.

wi+1 (t)yt(i) || ≤ δ,

(4.329)

t∈Ωi+1 (i)

λi+1 = max{αt

.

: t ∈ Ωi+1 }.

(4.330)

204

4 Dynamic String-Averaging Methods in Hilbert Spaces

It follows from (4.10) and (4.328) that for each index vector .t = (t1 , . . . , tp(t) ) ∈ (i,t) p(t) Ωi+1 there exists a finite sequence .{yj }j =0 ⊂ X such that (i,t)

(i,t)

(i,t)

(i,t)

(4.332)

(i)

(4.333)

αt

.

(4.331)

, δ) ∩ Ptj ,δ (yj −1 ) /= ∅ for each integer j = 1, . . . , p(t),

B(yj

.

(i)

= xi , yp(t) = yt ,

y0

.

(i,t)

(i,t)

= max{||yj +1 − yj

|| : j = 0, . . . , p(t) − 1}.

Assume that a nonnegative integer s satisfies for each integer .k ∈ [0, s], .

max{λi : i = k N¯ + 1, . . . , (k + 1)N¯ } > ɛ 0 .

(4.334)

By (4.312) and (4.323), ||x0 − z∗ || ≤ 2M.

(4.335)

γ1 = 4(δM + δ)1/2 (4M + 4)1/2 ,

(4.336)

γ0 = (q¯ + 1)γ1 .

(4.337)

.

Set .

.

Assume that an integer .k ∈ [0, s] satisfies ||xk N¯ − z∗ || ≤ 2M.

.

(4.338) U ∩

We prove the following auxiliary result. Lemma 4.27 Assume that an integer i ∈ [0, N¯ − 1]

.

satisfies ||xk N¯ +i − z∗ || ≤ 2M + iγ0 .

.

(4.339)

Then ||xk N¯ +i+1 − z∗ || ≤ ||xk N¯ +i − z∗ || + (q¯ + 1)γ1

.

(4.340)

Almost Star-Shaped Feasibility Problems

205

and ||xk N¯ +i+1 − z∗ ||2

.

.

≤ ||xk N¯ +i − z∗ ||2 − Δλ2k N+i+1 + δ(4M + 2) + 32(δM + δC )1/2 q(4M ¯ + 4)3/2 . ¯ (4.341)

If .λk N¯ +i+1 > ɛ 0 , then ||xk N¯ +i+1 − z||2 ≤ ||xk N¯ +i − z||2 − 2−1 Δɛ 02 .

.

(4.342)

Proof Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and .j ∈ {1, . . . , p(t)}. Assume that (k N¯ +i,t)

||z∗ − yj −1

.

|| ≤ ||z∗ − xk N¯ +i || + (j − 1)γ1 .

(4.343)

(In view of (4.331), Eq. (4.343) holds for .j = 1.) By (4.332), there exists (k N¯ +i,t)

ξt,j ∈ B(yj

.

, δ) ∩ Ctj

(4.344)

such that (k N¯ +i,t)

||ξt,j − yj −1

.

(k N¯ +i,t)

|| ≤ d(yj −1

, Ctj ) + δ.

(4.345)

Equations (4.316) and (4.345) imply that (k N¯ +i,t)

||ξt,j − yj −1

.

.

(k N¯ +i,t)

≤ ||yj −1

(k N¯ +i,t)

|| ≤ ||yj −1

− z∗tj || + δ

− z∗ || + ||z∗ − z∗tj || + δ

¯

N +i,t) ≤ ||yj(k−1 − z∗ || + δ + δM .

(4.346)

E0 = {αz∗ + (1 − α)ξt,j : α ∈ [0, 1]}.

(4.347)

.

Set .

By (4.313), (4.319), (4.336), (4.337) and (4.339), (k N¯ +i,t)

||yj −1

.

− z∗ || ≤ 2M + γ0 i + (j − 1)γ1 ≤ 2M + 1.

(4.348)

In view of (4.312) and (4.348), (k N¯ +i,t)

||yj −1

.

|| ≤ 3M + 1.

(4.349)

206

4 Dynamic String-Averaging Methods in Hilbert Spaces

Let y ∈ E0 .

(4.350)

.

Assumption (A3) and Eqs. (4.344), (4.346), (4.347), (4.349), (4.350) imply that there exists ˆ y ∈ Ctj ∩ {v ∈ X : ||y − v|| < δM }.

(4.351)

.

By (4.345), (4.350) and (4.351), (k N¯ +i,t)

||yj −1

.

.

(k N¯ +i,t)

− y|| ≥ ||yj −1

¯

−ˆ y || − ||ˆ y − y|| ¯

N +i,t) N +i,t) > ||yj(k−1 −ˆ y || − δM ≥ d(yj(k−1 , Ctj ) − δM (k N¯ +i,t)

≥ ||ξt,j − yj −1

.

|| − δ − δM .

(4.352)

, E0 ) + δ + δM .

(4.353)

In view of (4.352), (k N¯ +i,t)

||ξt,j − yj −1

.

(k N¯ +i,t)

|| ≤ d(yj −1

Equations (4.344), (4.347), (4.348), (4.353) and Lemma 2.3 applied with (k N¯ +i,t)

D = E0 , x = yj −1

.

(k N¯ +i,t)

, x˜ = yj

imply that ¯

¯

N +i,t) ||z∗ − yj(k N +i,t) || ≤ ||z∗ − yj(k−1 || + 2(2(δ + δM )(4M + 4))1/2 ,

.

(k N¯ +i,t) 2

||z∗ − yj −1

.

(k N¯ +i,t) 2

|| ≥ ||z∗ − yj .

(k N¯ +i,t)

|| + ||yj −1

(k N¯ +i,t) 2

− yj

− 32(δM + δ)1/2 (4M + 4)3/2 .

(4.354) ||

(4.355)

By (4.336), (4.343) and (4.354), ¯

||z∗ − yj(k N +i,t) || ≤ ||z∗ − xk N¯ +i || + j γ1 .

.

(4.356)

Therefore by induction we showed that for .j = 1, . . . , p(t) Eq. (4.336) holds and that by (4.318), (4.319), (4.336), (4.337), (4.339) and (4.356), for .j = 1, . . . , p(t), (k N¯ +i,t)

||z∗ − yj

.

|| ≤ 2M + 1

(4.357)

Almost Star-Shaped Feasibility Problems

207

and (4.355) is true. Thus we have shown that the following property holds: (P1) for each .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 and each .j ∈ {0, . . . , p(t)}, (k N¯ +i,t)

||z∗ − yj

.

|| ≤ ||z∗ − xk N¯ +i || + j γ1

and for .j = 1, . . . , p(t), (4.357) is true and (k N¯ +i,t) 2

||z∗ − yj −1

.

.

¯

||

¯

¯

N +i,t) ≥ ||z∗ − yj(k N +i,t) ||2 + ||yj(k−1 − yj(k N +i,t) ||2 .

− 32(δM + δ)1/2 (4M + 4)3/2 .

(4.358)

Property (P1) and (4.331), (4.357) imply that for each .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 , (k N¯ +i)

||z∗ − yt

.

(k N¯ +i,t)

|| = ||z∗ − yp(t)

|| ≤ ||z∗ − xk N¯ +i || + qγ ¯ 1,

¯

||z∗ − yt(k N +i) || ≤ 2M + 1.

.

By (4.329), (4.336), the equation above and the convexity of the norm, ||z∗ − xk N¯ +i+1 ||

.

.

Σ

≤ ||z∗ −

¯ (k N+i)

wk N¯ +i+1 (t)yt

||

t∈Ωk N+i+1 ¯

.

Σ

+ ||xk N¯ +i+1 −

(k N¯ +i)

wk N¯ +i+1 (t)yt

||

t∈Ωk N+i+1 ¯

.

Σ

≤δ+

(k N¯ +i)

wk N+i+1 (t)||yt ¯

− z∗ ||

t∈Ωk N+i+1 ¯ .

.

≤ ||xk N¯ +i − z∗ || + qγ ¯ 1+δ

≤ ||xk N¯ +i − z∗ || + (q¯ + 1)γ1 .

(4.359)

In view of (4.336)–(4.339) and (4.359), ||z∗ − xk N¯ +i+1 || ≤ 2M + 1.

.

(4.360)

208

4 Dynamic String-Averaging Methods in Hilbert Spaces

Let .t = (t1 , . . . , tp(t) ) ∈ Ωk N¯ +i+1 . By (4.7), (4.331), (4.358) and (P1), ¯

¯

(k N +i,t) ||y0(k N +i,t) − z∗ ||2 − ||yp(t) − z∗ ||2

.

.

p(t) Σ (k N¯ +i,t) (k N¯ +i,t) (||yj −1 − z∗ ||2 − ||yj − z∗ ||2 )

=

j =1

≥

.

p(t) Σ

(k N¯ +i,t)

(k N¯ +i,t) 2

||yj

− yj −1

j =1

|| − 32q(δ ¯ M + δ)1/2 (4M + 4)3/2 .

Together with (4.331) and (4.333) this implies that (k N¯ +i)

||yt

.

.

−

p(t) Σ j =1

.

¯

− z∗ ||2 ≤ ||z∗ − xk N¯ +i ||2 ¯

N +i,t) 2 ||yj(k N +i,t) − yj(k−1 || + 32q(δ ¯ M + δ)1/2 (4M + 4)3/2

¯

≤ ||z∗ − xk N¯ +i ||2 + 32q(δ ¯ M + δ)1/2 (4M + 4)3/2 − (αt(k N +i) )2 .

(4.361)

It follows from (4.3), (4.8), (4.330), (4.361) and the convexity of the function .|| · ||2 that Σ ¯ .||z∗ − wk N¯ +i+1 (t)yt(k N +i) ||2 t∈Ωk N+i+1 ¯

.

≤

Σ

(k N¯ +i) 2

wk N+i+1 (t)||z∗ − yt ¯

||

t∈Ωk N+i+1 ¯

.

≤ ||xk N¯ +i − z∗ ||2 + 32q(δ ¯ M + δ)1/2 (4M + 4)3/2 −

Σ

¯

wk N¯ +i+1 (t)(αt(k N+i) )2

t∈Ωk N+i+1 ¯ .

≤ ||xk N¯ +i − z∗ ||2 + 32q(δ ¯ M + δ)1/2 (4M + 4)3/2 − Δλ2k N¯ +i+1 .

(4.362)

Equations (4.329), (4.359) and (4.360) imply that Σ

|||z∗ − xk N¯ +i+1 ||2 − ||z∗ −

.

(k N¯ +i) 2

|| |

wk N+i+1 (t)yt ¯

t∈Ωk N+i+1 ¯

.

≤ |||z∗ − xk N¯ +i+1 || − ||z∗ −

Σ t∈Ωk N+i+1 ¯

(k N¯ +i)

wk N+i+1 (t)yt ¯

|||

Almost Star-Shaped Feasibility Problems

.

209

× (||z∗ − xk N¯ +i+1 || + ||z∗ −

Σ

(k N¯ +i)

wk N+i+1 (t)yt ¯

||)

t∈Ωk N+i+1 ¯

.

Σ

≤ ||xk N¯ +i+1 −

(k N¯ +i)

wk N+i+1 (t)yt ¯

||(4M + 2) ≤ δ(4M + 2).

(4.363)

t∈Ωk N+i+1 ¯

In view of (4.362) and (4.363), ||z∗ − xk N¯ +i+1 ||2 ≤ ||xk N¯ +i − z∗ ||2 + δ(4M + 2)

.

+ 32q(δ ¯ M + δ)1/2 (4M + 2)3/2 − Δλ2k N+i+1 ¯

.

.

≤ ||xk N¯ +i − z∗ ||2 + 33q(δ ¯ + δM )1/2 (4M + 4)3/2 − Δλ2k N¯ +i+1 .

(4.364)

Assume that λk N¯ +i+1 > ɛ 0 .

.

(4.365)

It follows from (4.320), (4.364) and (4.365) that ||xk N¯ +i+1 − z∗ ||2 ≤ ||xk N¯ +i+1 − z∗ ||2 − Δɛ 02 /2.

.

U ∩

Lemma 4.27 is proved.

It follows from (4.337), (4.338) and Lemma 4.27 applied by induction that the following property holds: (P2) that for all integers .i = 0, . . . , N¯ , Eq. (4.339) holds, for all .i = 0, . . . , N¯ − 1, Eq. (4.341) holds and if λk N+i+1 > ɛ 0 , ¯

.

then (4.342) is true. Property (P2) and (4.319), (4.336), (4.337) and (4.339) imply that ¯ 0 ≤ 2M + 1. ||xk N¯ +i − z∗ || ≤ 2M + Nγ

.

Property (P2) and (4.320), (4.334) (4.341) imply that ||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2

.

.

=

¯ N−1 Σ

[||xk N¯ +i+1 − z∗ ||2 − ||xk N¯ +i − z∗ ||2 ]

i=0 .

≤ −2−1 Δɛ 02 + 33N¯ (δ + δM )1/2 q(4M ¯ + 4)3/2 ≤ −4−1 Δɛ 02 .

210

4 Dynamic String-Averaging Methods in Hilbert Spaces

Thus we have shown that the following property holds: (P3) if an integer .k ∈ [0, s] satisfies .||xk N¯ − z∗ || ≤ 2M, then ¯ ||xj − z∗ || ≤ 2M + 1, j = k N¯ , . . . , (k + 1)N,

.

||x(k+1)N¯ − z∗ ||2 − ||xk N¯ − z∗ ||2 ≤ −4−1 Δɛ 02 .

.

(4.366)

In view of (4.335) and property (P3) we have ||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯

.

and (4.366) is true for every integer .k = 0, . . . , s. By (4.335) and (4.366), 4−1 Δɛ 02 (s + 1) ≤

.

s Σ [||xk N¯ − z∗ ||2 − ||x(k+1)N¯ − z∗ ||2 ] k=0

.

= ||x0 − z∗ ||2 − ||x(s+1)N¯ − z∗ ||2 ≤ ||x0 − z∗ ||2 ≤ 4M 2 , s + 1 ≤ 16M 2 Δ−1 ɛ 0−2 .

.

Thus we have shown that the following property holds: (P4) If an integer .s ≥ 0 and for every integer .k ∈ [0, s] relation (4.334) holds, then s ≤ 16M 2 Δ−1 ɛ 0−2 − 1,

.

||xj − z∗ || ≤ 2M + 1, j = 0, . . . , (s + 1)N¯ ,

.

||xk N¯ − z∗ || ≤ 2M, k = 0, . . . , s + 1.

.

By property (P4), (4.321) and (4.335), there exists an integer .q ∈ [0, n0 − 1] such that for every integer k satisfying .0 ≤ k < q, .

¯ > ɛ 0 , max{λi : i = k N¯ + 1, . . . , (k + 1)N}

.

max{λi : i = q N¯ + 1, . . . , (q + 1)N¯ } ≤ ɛ 0 ,

||xj − z∗ || ≤ 2M + 1, ||xj || ≤ 3M + 1, j = 0, . . . , q N¯ .

.

Assume that an integer .q ∈ [0, n0 − 1] satisfies ¯ λi ≤ ɛ 0 , i = q N¯ + 1, . . . , (q + 1)N.

.

(4.367)

Almost Star-Shaped Feasibility Problems

211

Let j ∈ {q N¯ , . . . , (q + 1)N¯ − 1}.

.

It follows from (4.330), (4.333) and (4.367) that (j )

≤ λj +1 ≤ ɛ 0 , t ∈ Ωj +1

αt

.

(4.368)

and that for every index vector .t ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (j,t)

y0

.

(j,t)

||yi

.

= xj , (j,t)

− yi−1 || ≤ ɛ 0 .

(4.369)

In view of (4.331) and (4.369), for every index vector .t ∈ Ωj +1 and every integer i = 1, . . . , p(t),

.

(j,t)

||xj − yi

.

|| ≤ iɛ 0 ≤ ɛ 0 q, ¯

(4.370)

In view of (4.332) and (4.370) for every index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 and every integer .i = 1, . . . , p(t), (t,j )

d(xj , Cti ) ≤ ||xj − yi

.

(t,j )

|| + d(yi

, Cti ) ≤ ɛ 0 q¯ + δ.

(4.371)

It follows from (4.3), (4.320), (4.329), (4.370) and the convexity of the norm that Σ

||xj +1 − xj || ≤ ||xj +1 −

.

(j )

wj +1 (t)yt ||

t∈Ωj +1

.

Σ

+ ||

(j )

wj +1 (t)yt

− xj ||

t∈Ωj +1

.

≤δ+

Σ

(j )

wj +1 (t)||yt

− xj || ≤ δ + ɛ 0 q¯

t∈Ωj +1

and ||xj +1 − xj || ≤ ɛ 0 (q¯ + 1).

.

(4.372)

¯ . . . , (q + 1)N¯ }, In view of (4.372), for every pair of integers .j1 , j2 ∈ {q N, ||xj1 − xj2 || ≤ ɛ 0 (q¯ + 1)N¯ .

.

(4.373)

212

4 Dynamic String-Averaging Methods in Hilbert Spaces

Let .s ∈ {1, . . . , m}. By (4.322), there exist an integer .j ∈ {q N¯ , . . . , (q + 1)N¯ − 1} and an index vector .t = (t1 , . . . , tp(t) ) ∈ Ωj +1 such that s ∈ {t1 , . . . , tp(t) }

(4.374)

d(xj , Cs ) ≤ ɛ 0 (q¯ + 1).

(4.375)

.

and .

¯ By (4.371), (4.373)–(4.375), for each .i ∈ {q N¯ , . . . , (q + 1)N}, d(xi , Cs ) ≤ ||xi − xj || + d(xj , Cs ) ≤ ɛ 0 (q¯ + 1)N¯ + ɛ 0 (q¯ + 1).

.

This completes the proof of Theorem 4.26.

Chapter 5

Methods with Remotest Set Control in a Hilbert Space

In this chapter we study the convergence of methods with remotest set control for solving star-shaped feasibility problems in a Hilbert space. Our main goal is to obtain an approximate solution of the problem in the presence of computational errors. We show that the iterative method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

Exact Iterates Let .(X, ) be a Hilbert space equipped with an inner product . which induces the norm ||x|| = 1/2 , x ∈ X.

.

Suppose that m is a natural number, .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets, for each .i ∈ {1, . . . , m} and each .x ∈ X Pi (x) := {v ∈ Ci : ||x − v|| ≤ ||x − y|| for every y ∈ Ci }.

.

(5.1)

is nonempty. Set C = ∩m i=1 Ci .

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_5

213

214

5 Methods with Remotest Set Control in a Hilbert Space

We study the following feasibility problem Find x ∈ C.

.

Assume that the following assumption holds: (A1) z∗ ∈ ∩m i=1 Ci

.

for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

(5.2)

In other words all the sets .Ci , .i = 1, . . . , m are star-shaped at .z∗ . We consider the following algorithm with remotest set control [66]. Initialization: select an arbitrary .x0 ∈ X. Iterative step: given a current iteration point .xk choose mk ∈ {1, . . . , m}

.

such that d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m}

.

and choose xk+1 ∈ Pmk (xk ).

.

Theorem 5.1 Assume that M > max{||z∗ ||, 1},

.

(5.3)

ɛ ∈ (0, 1], .{xk }∞ k=0 ⊂ X,

.

{mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M

(5.4)

d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m},

(5.5)

xk+1 ∈ Pmk (xk ).

(5.6)

.

and that for each integer .k ≥ 0, .

and .

Exact Iterates

215

Then ||xk || ≤ 3M, k = 0, 1, . . . ,

.

Card({j ∈ {0, 1, . . . } : max{d(xj , Cs ) : s = 1, . . . , m} ≥ ɛ ) ≤ 4M 2 ɛ −2

.

and .

lim max{d(xk , Cs ) : s = 1, . . . , m} = 0.

k→∞

Proof By (A1), Proposition 2.4, (5.1), (5.4)–(5.6), for each integer .k ≥ 0, ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ||xk − xk+1 ||2

.

.

= ||z∗ − xk+1 ||2 + max{d(xk , Ci ) : i = 1, . . . , m}2 ,

(5.7)

||z∗ − xk+1 || ≤ ||z∗ − xk ||,

.

||z∗ − xk || ≤ ||z∗ − x0 || ≤ 2M, ||xk || ≤ 3M.

.

It follows from (5.3), (5.4) and (5.7) that for each natural number Q, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xQ ||2

.

.

=

Q−1 Σ

(||z∗ − xk ||2 − ||z∗ − xk+1 ||2 )

k=0

.

≥

Q−1 Σ

max{d(xk , Ci )2 : i = 1, . . . , m}

k=0 .

≥ ɛ 2 Card({k ∈ {0, . . . , Q − 1} : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ })

and Card({k ∈ {0, . . . , Q − 1} : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ }) ≤ 4M 2 ɛ −2 .

.

Since the relation above holds for every natural number Q we conclude that Card({k ∈ {0, 1, . . . , } : max{d(xk , Ci ) : i = 1, . . . , m} ≥ ɛ }) ≤ 4M 2 ɛ −2 .

.

Theorem 5.1 is proved. Theorem 5.1 implies the following result.

216

5 Methods with Remotest Set Control in a Hilbert Space

Theorem 5.2 Assume that M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1] and that the bounded regularity property holds. Then there exists a natural number Q such that for each .{xk }∞ k=0 ⊂ X, each

.

{mk }∞ k=0 ⊂ {1, . . . , m}

.

such that ||x0 || ≤ M

.

and that for each integer .k ≥ 0, d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m},

.

and xk+1 ∈ Pmk (xk )

.

the inequalities ||xk || ≤ 3M, k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : d(xk , C) ≥ ɛ }) < Q

.

are true and .

lim d(xk , C) = 0.

k→∞

Theorem 5.1 implies the following result. Theorem 5.3 Let .M > max{1, ||z∗ ||}, .Δ0 ∈ (0, 1), .L > 1 and let the following property hold: for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, C) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds.

Exact Iterates

217

Assume that . ɛ ∈ (0, Δ0 ), .{xk }∞ k=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

and that for each integer .k ≥ 0, (5.5) and (5.6) hold. Then ||xk || ≤ 3M, k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) ≤ 4M 2 ɛ −2 L2 .

.

Theorem 5.1 and Proposition 2.4 imply the following result. Theorem 5.4 Assume that the bounded regularity property the following property hold. (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, . ɛ ∈ (0, 1). Then there exists a natural number Q such that for each .{xk }∞ k=0 ⊂ X, each {mk }∞ k=0 ⊂ {1, . . . , m}

.

such that ||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m},

.

and xk+1 ∈ Pmk (xk )

.

there exists .z¯ ∈ C such that ||z − xk || for each integer k ≥ Q

.

and the sequence .{xi }∞ i=0 converges to a point of C.

218

5 Methods with Remotest Set Control in a Hilbert Space

Theorem 5.5 Assume that the following property holds: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .L > 1, .Δ0 ∈ (0, 1), . ɛ ∈ (0, Δ0 ) and the following property holds: (ii) for each .x ∈ B(0, 3M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, Ci ) ≤ L max{d(x, Ci ) : i = 1, . . . , m}

.

holds. Assume that .{xk }∞ k=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

and that for each integer .k ≥ 0, (5.5) and (5.6) hold. Then d(xk , C) < ɛ for each integer k > 16M 2 ɛ −2 L2 .

.

Inexact Iterates with Summable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in Sect. “Exact Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

Inexact Iterates with Summable Errors

219

For each .i ∈ {1, . . . , m}, each .x ∈ X and each . ɛ ≥ 0 set Pi, ɛ (x) = {y ∈ Ci : ||x − y|| ≤ d(x, Ci ) + ɛ }.

.

(5.8)

The following theorem describes the behavior of inexact iterates with computational errors. Theorem 5.6 Assume that M > max{1, ||z∗ ||},

(5.9)

.

ɛ ∈ (0, 1], .{Δi }∞ i=0 ⊂ (0, 1],

.

∞ Σ .

1/2

Δi

< ∞,

(5.10)

i=0

M1 = 8M + 162 (

∞ Σ

.

1/2

Δ j )2 ,

(5.11)

j =0

M2 = 4M + 32(2M1 + 1) 2

.

3/2

∞ Σ

1/2

Δi ,

(5.12)

i=0

p¯ is a natural number such that

.

Δi < ɛ /4 for each i ≥ p, ¯

.

(5.13)

{xk }∞ k=0 ⊂ X,

.

{mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

(5.14)

for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − Δk

.

(5.15)

and B(xk+1 , Δk ) ∩ Pmk ,Δk (xk ) /= ∅.

.

Then ||xk || ≤ M1 + M, k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : ||xk − xk+1 || ≥ ɛ /4}) ≤ 16M 2 ɛ −2ˆ .

.

(5.16)

220

5 Methods with Remotest Set Control in a Hilbert Space

Moreover if .k ≥ p¯ is an integer and .||xk − xk+1 || ≤ ɛ /4, then d(xk , Cs ) ≤ ɛ , s = 1, . . . , m.

.

and .

lim max{d(xk , Cs ) : s = 1, . . . , m} = 0.

k→∞

Proof Set ɛ 0 = ɛ /4.

.

Let .k ≥ 0 be an integer. By (5.8) and (5.16), there exists yk ∈ Cmk ∩ B(xk+1 , Δk )

(5.17)

||xk − yk || ≤ d(xk , Cmk ) + Δk .

(5.18)

.

such that .

Equations (5.11) and (5.17) and Lemma 2.3 applied with .δ = Δk , .x = xk , .x˜ = xk+1 , ξ = yk , .D = Cmk imply that

.

||z∗ − xk+1 || ≤ ||z∗ − xk || + 2(2Δk (2||x − z∗ || + 1))1/2 ,

(5.19)

.

||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ||xi − xk+1 ||2

.

.

− 8(2Δk (2||xk − z∗ || + 1))1/2 ||z∗ − xk || − 16Δk (2||xk − z∗ || + 1).

(5.20)

Equation (5.11) implies that ∞ Σ

2M +

4(2Δj (2M1 + 1))1/2 ≤ 2M + 12

.

j =0

∞ Σ

1/2

1/2

Δ j M1

j =0

.

≤ M1 /4 + 12M1 M1 16−1 ≤ M1 . 1/2

1/2

(5.21)

We show that for each integer .k ≥ 1, ||z∗ − xk || ≤ 2M + 4

.

k−1 Σ (2Δi (2M1 + 1))1/2 . i=0

(5.22)

Inexact Iterates with Summable Errors

221

(In view of (5.9), (5.11), (5.14) and (5.19), equation (5.22) holds for .k = 1.) Assume that .k ≥ 1 is an integer and (5.22) holds. By (5.19), (5.21) and (5.22), ||z∗ − xk+1 || ≤ ||z∗ − xk || + 4(2Δk (2M1 + 1))1/2

.

.

≤ 2M + 4

k−1 Σ (2Δi (2M1 + 1))1/2 + 2(2Δk (2M1 + 1))1/2 . i=0

Thus our assumption holds for .k + 1 too. Therefore by induction we showed that (5.22) holds for all natural numbers k. Equations (5.9), (5.21) and (5.22) imply that for all natural numbers k ||z∗ − xk || ≤ M1 , ||xk || ≤ M1 + M.

.

(5.23)

By (A1) and (5.9), (5.14), (5.20) and (5.23), for each integer .k ≥ 0, ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ||xk − xk+1 ||2

.

.

1/2

− 32Δk (2M1 + 1)3/2 .

It follows from (5.9), (5.14) and (5.24) that for each natural number Q, 4M 2 ≥ ||z∗ − x0 ||2 ≥ ||z∗ − x0 ||2 − ||z∗ − xQ ||2

.

.

=

Q−1 Σ

[||z∗ − xi || − ||z∗ − xi+1 ||2 ]

i=0

.

≥

Q−1 Σ

1/2

(||xk − xk+1 ||2 − 32Δk (2M1 + 1)3/2 )

i=0

and in view of (5.12), M2 ≥ 4M 2 +

∞ Σ

.

1/2

32Δi (2M1 + 1)3/2

i=0

.

≥

Q−1 Σ

||xk − xk+1 ||2

i=0 .

≥ ɛ 2 Card({i ∈ {0, . . . , Q − 1}) : ||xk − xk+1 || ≥ ɛ 0 })

(5.24)

222

5 Methods with Remotest Set Control in a Hilbert Space

and Card({i ∈ {0, . . . , Q − 1} : ||xk − xk+1 || ≥ ɛ 0 }) ≤ ɛ 0−2 M2 .

.

Since the relation above holds for every natural number Q we conclude that Card({k ∈ {0, 1, . . . } : ||xk − xk+1 || ≥ ɛ 0 }) ≤ ɛ 0−2 M2 .

.

Assume that k ≥ p¯

(5.25)

||xk − xk+1 || < ɛ 0 .

(5.26)

.

is an integer and .

By (5.13), (5.15), (5.17), (5.25), (5.26) and the relation . ɛ 0 = ɛ /4, d(xk , Cmk ) ≤ ||xk − yk || ≤ ||xk − xk+1 || + ||xk+1 − yk || < ɛ 0 + Δk

.

and for each .i ∈ {1, . . . , m} d(xk , Ci ) ≤ d(xk , Cmk ) + Δk < ɛ 0 + 2Δk < ɛ .

.

Theorem 5.6 is proved. Theorem 5.6 implies the following result. Theorem 5.7 Assume that the bounded regularity property holds, M > max{1, ||z∗ ||},

.

ɛ ∈ (0, 1], .{Δi }∞ i=0 ⊂ (0, 1],

.

∞ Σ .

1/2

Δi

< ∞.

i=0

Then there exists a natural number Q and .M1 > M such that for each .{xk }∞ k=0 ⊂ X, each {mk }∞ k=0 ⊂ {1, . . . , m}

.

Inexact Iterates with Summable Errors

223

for which (5.14) holds and (5.15) and (5.16) are true for each integer .k ≥ 0 the following relations hold: ||xk || ≤ 2M1 , k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : d(x, C) ≥ ɛ }) < Q

.

.

lim d(xk , C) = 0.

k→∞

Theorem 5.6 implies the following result. Theorem 5.8 Let .M > max{1, ||z∗ ||}, .Δ0 ∈ (0, 1), .L > 1, .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

i=0

M1 , M2 be defined by (5.11) and (5.12), . ɛ ∈ (0, Δ0 ) and let the following property hold:

.

(i) for each .x ∈ B(0, M1 + M) satisfying d(x, Ci ) ≤ Δ0 , i = 1, . . . , m

.

the inequality d(x, Ci ) ≤ L max{d(xi , Cs ) : s = 1, . . . , m}

.

is true. Let .p¯ be a natural number such that Δi < L−1 ɛ /4 for each i ≥ p. ¯

.

Assume that .{xk }∞ k=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − Δk

.

and B(xk+1 , Δk ) ∩ Pmk ,Δk (xk ) = / ∅.

.

224

5 Methods with Remotest Set Control in a Hilbert Space

Then ||xk || ≤ M1 + M, k = 0, 1, . . . ,

.

Card({k ∈ {0, 1, . . . } : d(xk , C) > ɛ }) ≤ p¯ + 16M2 ɛ −2 L2 .

.

Theorem 5.9 Assume that the bounded regularity property and the following property hold: (i) for each .x ∈ C, each .s ∈ {1, . . . , m}, each .α ∈ [0, 1] and each .y ∈ Cs , αx + (1 − α)y ∈ Cs .

.

Let .M > max{1, ||z∗ ||}, .{Δi }∞ i=0 ⊂ (0, 1], ∞ Σ .

1/2

Δi

< ∞,

i=0

M1 = 8M + 162 (

∞ Σ

.

1/2

Δ i )2 ,

i=0

ɛ ∈ (0, 1). Then there exists a natural number Q such that for each each .{xi }∞ i=0 ⊂ X, each

.

{mk }∞ k=0 ⊂ {1, . . . , m}

.

such that ||x0 || ≤ M,

.

(5.27)

and for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − Δk ,

.

(5.28)

and B(xk+1 , Δk ) ∩ Pmk ,Δk (xk ) /= ∅.

.

the inequality ||xk || ≤ 2M1 , k = 0, 1, . . . ,

.

(5.29)

Inexact Iterates with Summable Errors

225

holds, there exists .z ∈ C such that ||z − xi || ≤ ɛ for each integer i ≥ Q

.

and the sequence .{xi }∞ i=0 converges to a point of C. Proof Theorem 5.7 implies that there exists a natural number Q such that ∞ Σ .

(Δi + (Δi (8M1 + 8))1/2 ) < ɛ /8

i=Q

and the following property holds: (iii) for each .{xi }∞ i=0 ⊂ X and each {mk }∞ k=0 ⊂ {1, . . . , m},

.

satisfying (5.27)–(5.29) for each integer .i ≥ 0, the inequality ||xk || ≤ 2M1 , k = 0, 1, . . . ,

.

holds and there is .j ∈ {Q, . . . , 2Q} for which d(xj , C) < ɛ /2.

.

Assume that .{xi }∞ i=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m}

.

and (5.27)–(5.29) hold for each integer .i ≥ 0. Property (ii) implies that ||xi || ≤ 2M1 , i = 0, 1, . . .

(5.30)

k ∈ {Q, . . . , 2Q}

(5.31)

d(xk , C) < ɛ /2.

(5.32)

z∈C

(5.33)

.

and there is .

for which .

By (5.32), there exists .

226

5 Methods with Remotest Set Control in a Hilbert Space

such that ||xk − z|| < ɛ /2.

(5.34)

||z|| ≤ 2M1 + 1.

(5.35)

.

In view of (5.30) and (5.34), .

Set γk = 0

.

and for each integer .q > k, q−1 Σ .γq = 2 (2Δi (8M1 + 2))1/2 .

(5.36)

i=k

We show that for each integer .q ≥ k, ||xq − z|| ≤ ɛ /3 + γq .

.

(5.37)

(In view of (5.34), Eq. (5.37) holds for .q = k.) Assume that .q ≥ k is an integer and (5.37) holds. By (5.29), there exists yq ∈ Pmq ,Δq (xq ) ∩ B(xq+1 , Δq ).

(5.38)

||xq − yq || ≤ d(xq , Cmq ) + Δq .

(5.39)

.

In view of (5.38), .

Property (i), (5.31), (5.36)–(5.39) and Lemma 2.3 applied with x = xq , x˜ = xq+1 , ξ = yq , δ = Δq , z∗ = z, D = Cmq

.

imply that ||z − xq+1 || ≤ ||z − xq || + 2(2Δq (2||xq − z|| + 1))1/2

.

.

≤ ||z − xq || + 2(8Δq )1/2 ≤ ɛ /2 + γq+1 .

Thus by induction we showed that (5.37) holds for each integer .q ≥ k. By (5.36) and the choice of Q, for each integer .q ≥ k, ||z − xq || < ɛ .

.

Theorem 5.9 is proved.

The First Result with Nonsummable Errors

227

The First Result with Nonsummable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in Sect. “Exact Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

Theorem 5.10 Let M ≥ max{1, ||z∗ ||},

(5.40)

δ ∈ (0, 4−1 (2M + 3)−1 ],

(5.41)

1 ≥ ɛ 0 = (2δ(8M + 3))1/2 ,

(5.42)

n0 = 1 + L2(4M + 1)2 ɛ 0−2 ⎦ + 1.

(5.43)

.

.

.

.

Assume that .{xi }∞ i=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m}

(5.44)

B(xk+1 , δ) ∩ Pmk ,0 (xk ) /= ∅.

(5.45)

.

and .

228

5 Methods with Remotest Set Control in a Hilbert Space

Then there exists an integer .q ∈ [0, n0 − 1] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Applying by induction Theorem 5.10 we obtain the following result. Theorem 5.11 Suppose that . ɛ ¯ ∈ (0, 1), M¯ > max{1, ||z∗ ||},

.

¯ {x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

¯ . ɛ 0 ∈ (0, ɛ ), Let .M > M, ¯ δ ∈ (0, 4−1 (2M + 3)−1 ),

.

1 ≥ ɛ 0 = (2δ(8M + 3))1/2 ,

.

n0 = 1 + L2(4M + 1)2 ɛ 0−2 ⎦ + 1.

.

Assume that .{xk }∞ k=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m}

.

and B(xk+1 , δ) ∩ Pmk ,0 (xk ) = / ∅.

.

Then xi ∈ B(0, 5M + 1) for all integers i ≥ 0

.

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

The First Result with Nonsummable Errors

229

and that for each integer .p ≥ 0, d(xqp , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Proof of Theorem 5.10 By (5.45) for each integer .k ≥ 0 there exists yk ∈ Pmk ,0 (xk ) ∩ B(xk+1 , δ).

(5.46)

||yk − xk || = d(xk , Cmk ).

(5.47)

.

By (5.46), .

Assumption (A1) and (5.40), (5.44) and (5.47) imply that ||y0 − x0 || ≤ ||x0 − z∗ || ≤ 2M, ||y0 || ≤ 3M, ||x1 || ≤ 3M + 1.

.

Assume that s is a natural number and that for each integer .k ∈ [1, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(5.48)

By (5.40) and the equations above, ||x1 − z∗ || ≤ 4M + 1.

.

(5.49)

Equations (5.44), (5.46) and (5.48) imply that for each .k ∈ {1, . . . , s}, ||xk − yk || = d(xk , Cmk ) = max{d(xk , Ci ) : i = 1, . . . , m} > ɛ 0 .

.

(5.50)

We prove the following auxiliary result. Lemma 5.12 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 1.

.

Then ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 2−1 ɛ 02 .

.

Proof Proposition 2.4, (A1), (5.46) and (5.47) imply that ||z∗ − yk ||2 + ||xk − yk ||2 ≤ ||z∗ − xk ||2 .

.

(5.51)

230

5 Methods with Remotest Set Control in a Hilbert Space

In view of (5.50), (5.51) and the relation above, ||z∗ − yk ||2 ≤ ||z∗ − xk ||2 − ɛ 02 ,

.

(5.52)

||z∗ − yk || ≤ 4M + 1.

.

By (5.46) and the equation above, |||z∗ − xk+1 ||2 − ||z∗ − yk ||2 |

.

.

≤ |||z∗ − xk+1 || − ||z∗ − yk |||(|||z∗ − xk+1 || + ||z∗ − yk ||) .

≤ ||xk+1 − yk ||(2|||z∗ − yk || + ||xk+1 − yk ||) .

≤ δ(8M + 3).

Equations (5.42), (5.52) and the equation above imply that ||z∗ − xk+1 ||2 ≤ ||z∗ − yk ||2 + δ(8M + 3)

.

.

≤ ||z∗ − xk ||2 − ɛ 02 + δ(8M + 3) ≤ ||z∗ − xk ||2 − ɛ 02 /2.

.

This completes the proof of Lemma 5.12. It follows from (5.49) and Lemma 5.12 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 1,

(5.53)

||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − ɛ 02 /2.

(5.54)

.

and for all .k = 1, . . . , s, .

By (5.49) and (5.54), 4(M + 1)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 2−1 ɛ 02 s k=1

and s ≤ 2(4M + 1)2 ɛ 0−2 .

.

The Second Result with Nonsummable Errors

231

Thus we have shown that the following property holds: if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (5,48) is true, then (5.53) holds for .k = 1, . . . , s + 1 and s ≤ 2(4M + 1)2 ɛ 0−2 .

.

Together with (5.43) this property implies that there exists a natural number q ≤ L2(4M + 1)2 ɛ 0−2 ⎦ + 1 = n0

.

such that xi ∈ B(0, 5M + 1), i = 1, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

U ∩

Theorem 5.10 is proved.

The Second Result with Nonsummable Errors Suppose that m is a natural number and .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

We continue to assume that assumption (A1) introduced in Sect. “Exact Iterates” holds. (A1) There exists z∗ ∈ C

.

such that for each .i ∈ {1, . . . , m}, each .x ∈ Ci and each .α ∈ [0, 1], αz∗ + (1 − α)x ∈ Ci .

.

Theorem 5.13 Let M > max{1, ||z∗ ||}, 0 < δ ≤ 1/2,

(5.55)

ɛ 0 = 16(2δ(8M + 3))3/4 ,

(5.56)

n0 = 1 + L8(4M + 1)2 ɛ 0−2 ⎦ + 1.

(5.57)

.

.

.

232

5 Methods with Remotest Set Control in a Hilbert Space

Assume that .{xi }∞ i=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

(5.58)

for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − δ

.

(5.59)

and B(xk+1 , δ) ∩ Pmk ,δ (xk ) = / ∅.

.

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Applying by induction Theorem 5.13 we obtain the following result. Theorem 5.14 Suppose that . ɛ ¯ ∈ (0, 1), M > max{1, ||z∗ ||},

.

{x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

Let . ɛ 0 ∈ (0, ɛ ), ¯ δ ∈ (0, 2−1 ],

.

1 ≥ ɛ 0 = 16(2δ(8M + 3))3/4 ,

.

n0 = 1 + L8(4M + 1)2 ɛ 0−2 ⎦.

.

Assume that .{xk }∞ k=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, (5.59) and (5.60) hold. Then xi ∈ B(0, 5M + 1) for all integers i ≥ 0

.

(5.60)

The Second Result with Nonsummable Errors

and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

and that for each integer .p ≥ 0, d(xqp , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 5.14 implies the following result. Theorem 5.15 Suppose that . ɛ ¯ ∈ (0, 1], M > max{1, ||z∗ ||},

.

{x ∈ X : d(x, Cs ) ≤ ɛ ¯ , s = 1, . . . , m} ⊂ B(0, M).

.

Let .δ ∈ (0, 2−1 ], . ɛ 0 ∈ (0, ɛ ), ¯ ɛ 0 = 16(2δ(8M + 3))3/4 .

.

Assume that .{δi }∞ i=0 ⊂ (0, δ], lim δi = 0,

.

i→∞

{xk }∞ k=0 ⊂ X,

.

{mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − δ,

.

and B(xk+1 , δk ) ∩ Pmk ,δk (xk ) = / ∅.

.

Then xk ∈ B(0, 5M + 1) for all integers k ≥ 0

.

233

234

5 Methods with Remotest Set Control in a Hilbert Space

and .

lim inf max{d(xk , Cs ) : s = 1, . . . , m} = 0. k→∞

Theorem 5.15 implies the following result. Theorem 5.16 Assume that the assumptions of Theorem 5.15 holds and the bounded regularity property holds. Then lim inf d(xk , C) = 0.

.

k→∞

Theorem 5.13 implies the following result. Theorem 5.17 Let M > max{1, ||z∗ ||}, 0 < ɛ < 1

.

and let the bounded regularity property hold. Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each .{xi }∞ i=0 ⊂ X and each {mk }∞ k=0 ⊂ {1, . . . , m}

.

such that ||x0 || ≤ M,

.

and for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − δ,

.

(5.61)

and B(xk+1 , δ) ∩ Pmk ,δ (xk ) = / ∅

.

(5.62)

there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , C) ≤ ɛ .

.

Theorem 5.17 implies the following result. Theorem 5.18 Suppose that the bounded regularity property holds, . ɛ ∈ (0, 1), M¯ > max{1, ||z∗ ||},

.

The Second Result with Nonsummable Errors

235

M > M¯ + 1 and

.

¯ C ⊂ B(0, M).

.

Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each .{xk }∞ k=0 ⊂ X and each {mk }∞ k=0 ⊂ {1, . . . , m}

.

satisfying ||x0 || ≤ M

.

and for each integer .k ≥ 0, (5.61) and (5.62) are true the inequality xi ∈ B(0, 5M + 1) for all integers i ≥ 0

.

holds and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

and that for each integer .p ≥ 0, d(xqp , C) ≤ ɛ .

.

Proof of Theorem 5.13 By (5.60), for each integer .k ≥ 0 there exists yk ∈ B(xk+1 , δ) ∩ PmK ,δ (xk ).

.

(5.63)

In view of (5.59) and (5.63), for each integer .k ≥ 0, ||xk+1 − yk || ≤ δ,

(5.64)

||xk − yk || ≤ d(xk , Cmk ) + δ,

(5.65)

.

.

d(xk , Cmk ) ≥ max{d(xk , Cs ) : s = 1, . . . , m} − δ.

.

(5.66)

Assumption (A1) and (5.55), (5.58), (5.64) and (5.65) imply that ||y0 − x0 || ≤ ||x0 − z∗ || + δ ≤ ||x0 − z∗ || + 1 ≤ 2M + 1/2,

.

||y0 || ≤ 3M + 1/2, ||x1 || ≤ 3M + 1,

(5.67)

||x1 − z∗ || ≤ 4M + 1.

(5.68)

.

.

236

5 Methods with Remotest Set Control in a Hilbert Space

Assume that s is a natural number and for each integer .k ∈ [1, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(5.69)

We prove the following auxiliary result. Lemma 5.19 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 1.

.

(5.70)

Then ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 8−1 ɛ 02 .

.

Proof Assumption (A1), (5.63), (5.65) and Lemma 2.3 applied with .x = xk , x˜ = xk+1 , .ξ = yk , D = Cmk imply that ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ||xk − xk+1 ||2

.

.

− 32(δ(2||xk − z∗ || + 1))3/2 .

(5.71)

In view of (5.56), (5.70) and (5.71), ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ||xk − xk+1 ||2

.

.

− 32(δ(8M + 3))3/2 .

(5.72)

By (5.63), (5.64) and (5.66), ||xk − xk+1 || ≥ ||xk − yk || − ||yk − xk+1 || ≥ ||xk − yk || − δ

.

.

≥ d(xk , Cmk ) − δ ≥ max{d(xk , Ci ) : i = 1, . . . , m} − 2δ > ɛ 0 − 2δ ≥ ɛ 0 /2. (5.73)

Equations (5.56), (5.72) and (5.73) imply that ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 + ɛ 02 /4 − 32(δ(8M + 3))3/2

.

.

≥ ||z∗ − xk+1 ||2 + ɛ 02 /8.

This completes the proof of Lemma 5.19. It follows from Lemma 5.19 applied by induction that for all .k = 1, . . . , s + 1,

The Second Result with Nonsummable Errors

||z∗ − xk || ≤ 4M + 1,

.

237

(5.74)

and for all .k = 1, . . . , s, ||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − ɛ 02 /8.

.

By the equation above and (5.68), (4M + 1)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 8−1 ɛ 02 s k=1

and s ≤ 8(4M + 1)2 ɛ 0−2 .

.

Thus we have shown that the following property holds: (i) if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (5.69) is true, then (5.74) holds for .k = 1, . . . , s + 1 and s ≤ 8(4M + 1)2 ɛ 0−2 .

.

Property (i) implies that there exists a natural number q ≤ L8(4M + 1)2 ɛ 0−2 ⎦ + 1 = n0

.

such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 5.13 is proved. Theorem 5.20 Assume that the bounded regularity property holds, .Δ0 ∈ (0, 1), ¯ M > M¯ + 1 M¯ > 1, C ⊂ B(0, M),

.

238

5 Methods with Remotest Set Control in a Hilbert Space

and that the following property holds: (i) for each .ξ ∈ C, each .s ∈ {1, . . . , m} and each .α ∈ [0, 1] and each .η ∈ Cs ∩ B(ξ, Δ0 ), αξ + (1 − α)η ∈ Cs .

.

Let . ɛ ∈ (0, Δ0 /2). Then there exist .δ ∈ (0, ɛ ) and a natural number .n0 such that for each .{xi }∞ i=0 ⊂ X and each each {mk }∞ k=0 ⊂ {1, . . . , m}

.

satisfying ||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) ≤ max{d(xk , Ci ) : i = 1, . . . , m} − δ

.

(5.75)

and B(xk+1 , δ) ∩ Pmk ,δ (xk ) = / ∅.

.

(5.76)

there is .z ∈ C such that ||xi − z|| ≤ ɛ for all integers i ≥ n0 .

.

Proof By Theorem 5.18, there exist .δ0 ∈ (0, ɛ ) and a natural number .n0 such that the following property holds: (ii) for each .{xi }∞ i=0 ⊂ X and each {mk }∞ k=0 ⊂ {1, . . . , m}

.

satisfying ||x0 || ≤ M,

.

for each integer .k ≥ 0, d(xk , Cmk ) ≤ max{d(xk , Ci ) : i = 1, . . . , m} − δ0

.

and B(xk+1 , δ0 ) ∩ Pmk ,δ0 (xk ) /= ∅

.

The Second Result with Nonsummable Errors

239

the inequality ||xi || ≤ 5M + 1 for all integers i ≥ 0

.

holds and there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0 ,

.

1 ≤ qp+1 − qp ≤ n0 for all integers p ≥ 0

.

(5.77)

and that for each integer .p ≥ 0, d(xqp , C) ≤ ɛ /4.

.

(5.78)

Choose δ ∈ (0, δ0 )

.

such that 2n0 (2δ)1/2 (10M + 3)1/2 < ɛ /3.

.

(5.79)

Assume that .{xi }∞ i=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

(5.80)

for each integer .k ≥ 0, (5.75) and (5.76). Property (ii), (5.75), (5.76), (5.79) and (5.80) imply that there exists a strictly increasing sequence of integers .{qp }∞ p=0 such that 1 ≤ q0 ≤ n0

.

and that for all integers .p ≥ 0, (5.77) and (5.78) hold. We show that d(xk , C) ≤ ɛ

.

for all integers .k ≥ q0 . Assume the contrary. Then there exists an integer k ≥ q0

.

240

5 Methods with Remotest Set Control in a Hilbert Space

such that d(xk , C) > ɛ .

.

(5.81)

There exists an integer .p ≥ 0 such that qp ≤ k < qp+1 .

.

(5.82)

In view of (5.78), 1 ≤ qp+1 − qp ≤ n0 , d(xqp , C) ≤ ɛ /4.

.

There exists z ∈ C such that ||xqp − z|| < ɛ /3.

.

(5.83)

Assume that .j ∈ {qp , . . . , qp+1 ) − 1} satisfies ||xj − z|| ≤ ɛ /3 + 2(j − qp )(2δ)1/2 (10M + 3)1/2 .

.

(5.84)

It follows from (5.76) that there exists .

yj ∈ Pmj ,δ (xj ) ∩ B(xj +1 , δ).

(5.85)

||xj − yj || ≤ d(xj , Cmj ) + δ.

(5.86)

By (5.85), .

Equations (5.77), (5.79), (5.84)–(5.86), properties (i) and (ii) and Lemma 2.3 applied with D = Cmj , z∗ = z, x = xj , x˜ = xj +1 , ξ = yj

.

imply that ||z − xj +1 || ≤ ||z − xj || + 2(2δ)1/2 (2||xj − z|| + 1)1/2

.

.

.

≤ ||z − xj || + 2(2δ)1/2 (10M + 3)1/2

≤ ɛ /3 + 2(j − qp + 1)(2δ)1/2 (10M + 3)1/2 .

The Third Result with Nonsummable Errors

241

Thus we have shown by induction that (5.84) holds for .j = qp , . . . , qp+1 . In particular, in view of (5.79), ||z − xk || ≤ ɛ /3 + 2(k − qp )(2δ)1/2 (10M + 3)1/2

.

.

< ɛ /3 + 2n0 (2δ)1/2 (10M + 3)1/2 < ɛ .

This contradicts (5.81). The contradiction we have reached completes the proof of Theorem 5.20.

The Third Result with Nonsummable Errors Suppose that .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets, .δC ∈ (0, 1]. Set C = ∩m i=1 Ci .

.

Assume that the following assumption holds. (A2) There exists z∗ ∈ X

.

and for each .s ∈ {1, . . . , m} there exists z∗s ∈ B(z∗ , δC ) ∩ Cs

.

such that for each .x ∈ Cs and each .α ∈ [0, 1], αz∗s + (1 − α)x ∈ Cs .

.

Clearly, assumption (A2) is weaker than its prototype (A1). In (A1) we assume that all the sets .Ci , .i = 1, . . . , m are star-shaped at the same point. In (A2) we suppose that for each .i ∈ {1, . . . , m} the set .Ci is star-shaped at .z∗i belonging to a .δC -neighborhood of a fixed point .z∗ ∈ X. Theorem 5.21 Let M > max{1, ||z∗ ||}, δ ∈ (0, 1], ɛ 0 ∈ (0, 1],

(5.87)

32δ 1/2 (8M + 9)3/2 ≤ ɛ 02 /16,

(5.88)

δC (16M + 64) ≤ 16−1 ɛ 02

(5.89)

.

.

.

242

5 Methods with Remotest Set Control in a Hilbert Space

and n0 = 1 + L8(4M + 3)2 ɛ 0−2 ⎦.

.

(5.90)

Assume that .{xk }∞ k=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m}

.

satisfy ||x0 || ≤ M,

.

(5.91)

for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − δ,

.

(5.92)

and B(xk+1 , δ) ∩ Pmk ,δ (xk ) = / ∅.

.

(5.93)

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Proof Let .k ≥ 0 be an integer. By (5.93), there exists .

yk ∈ B(xk+1 , δ) ∩ Pmk ,δ (xk ).

(5.94)

||xk − yk || ≤ d(xk , Cmk ) + δ.

(5.95)

In view of (5.94), .

Assumption (A2) and (5.87), (5.91), (5.94) and (5.95) imply that ||y0 − x0 || ≤ ||x0 − z∗m0 || + δ

.

.

≤ ||x0 − z∗ || + ||z∗ − z∗m0 || + δ ≤ 2M + δ + δC ,

||y0 − x0 || ≤ 2M + 2, ||y0 || ≤ 3M + 2, ||x1 || ≤ 3M + 3, ||x1 − z∗ || ≤ 4M + 3. (5.96)

.

The Third Result with Nonsummable Errors

243

Assume that s is a natural number and for each integer .k ∈ [1, s], .

max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

(5.97)

We prove the following auxiliary result. Lemma 5.22 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 3.

.

(5.98)

Then ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 8−1 ɛ 02 .

.

Proof Assumption (A2), (5.94), (5.95) and Lemma 2.3 applied with x = xk , x˜ = xk+1 , ξ = yk , D = Cmk , z∗ = z∗i

.

imply that ||z∗mk − xk ||2 ≥ ||z∗mk − xk+1 ||2 + ||xk − xk+1 ||2

.

− 32(2(2||xk − z∗ || + 1))3/2 δ 1/2 .

(5.99)

||z∗mk − xk || ≤ ||z∗ − xk || + 1 ≤ 4M + 4.

(5.100)

.

By (A2) and (5.99), .

In view of (5.99) and (5.100), ||z∗mk − xk ||2 ≥ ||z∗mk − xk+1 ||2 + ||xk − xk+1 ||2

.

.

− 32(2(8M + 9))3/2 δ 1/2 .

(5.101)

It follows from (5.88), (5.92), (5.94) and (5.97) that ||xk − xk+1 || ≥ ||xk − yk || − ||yk − xk+1 ||

.

.

≥ d(xk , Cmk ) − δ ≥ max{d(xk , Ci ) : i = 1, . . . , m} − 2δ .

> ɛ 0 − 2δ > ɛ 0 /2.

(5.102)

244

5 Methods with Remotest Set Control in a Hilbert Space

Equations (5.98) and (5.100) and (A2) imply that |||z∗mk − xk ||2 − ||z∗ − xk ||2 |

.

≤ ||z∗mk − z∗ ||(||z∗ − xk || + ||z∗mk − xk ||)

.

.

≤ ||z∗mk − z∗ ||(8M + 7) ≤ δC (8M + 7).

(5.103)

Equations (5.101)–(5.103) imply that ||z∗ − xk ||2 ≤ ||z∗mk − xk ||2 − δC (8M + 7)

.

.

≥ ||z∗mk − xk+1 ||2 + ɛ 02 /4 − 32(2(8M + 9))3/2 δ 1/2 − δC (8M + 7).

(5.104)

In view of (5.88), (5.89) and (5.104), ||z∗mk − xk+1 || ≤ ||z∗ − xk || ≤ 4M + 3.

.

(5.105)

Assumption (A2) and (5.105) imply that |||z∗mk − xk+1 ||2 − ||z∗ − xk+1 ||2 |

.

.

≤ ||z∗mk − z∗ ||(2||z∗,mk − xk+1 || + ||z∗mk − z∗ ||) ≤ δC (8M + 6 + δC ) ≤ δC (8M + 7).

.

(5.106)

It follows from (5.88), (5.89), (5.104) and (5.106) that ||z∗ − xk ||2 ≥ |||z∗ − xk+1 ||2 − δC (8M + 7)

.

.

+ 4−1 ɛ 02 − 32(2(8M + 9))3/2 δ 1/2 − δC (8M + 7) .

≥ |||z∗ − xk+1 ||2 + ɛ 0 /8.

This completes the proof of Lemma 5.22. It follows from (5.96) and Lemma 5.22 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 3,

(5.107)

||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − ɛ 02 /8.

(5.108)

.

and for all .k = 1, . . . , s, .

Almost Star-Shaped Feasibility Problems

245

By (5.107) and (5.108), 4(M + 3)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 8−1 ɛ 02 s k=1

and s ≤ 8(4M + 3)2 ɛ 0−2 .

.

Thus we have shown that the following property holds: (a) if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (5.97) is true, then (5.107) holds for .k = 1, . . . , s + 1 and s ≤ 8(4M + 3)2 ɛ 0−2 .

.

Property (a) and (5.90) imply that there exists a natural number q ≤ L8(4M + 3)2 ɛ 0−2 ⎦ + 1 = n0

.

such that xi ∈ B(0, 5M + 3), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 5.21 is proved.

Almost Star-Shaped Feasibility Problems Suppose that .Ci ⊂ X, .i = 1, . . . , m are nonempty, closed sets. Set C = ∩m i=1 Ci .

.

Assume that .M > 1, .δM ∈ (0, 1]. z∗ ∈ B(0, M),

(5.109)

B(0, M) ∩ Cs /= ∅, s = 1, . . . , m

(5.110)

.

.

and that the following assumption holds.

246

5 Methods with Remotest Set Control in a Hilbert Space

(A3) For each .s ∈ {1, . . . , m}, .x ∈ B(0, 9M + 9) ∩ Cs and each .α ∈ [0, 1], d(αz∗ + (1 − α)x, Cs ) < δM .

.

We study approximate solutions the following feasibility problem Find z ∈ C

.

which is called, in view of (A3), as an almost star-shaped feasibility problem. Assumption (A3) and (5.109) imply that for each .s ∈ {1, . . . , m} there exists z∗s ∈ Cs ∩ B(z∗ , δM ).

(5.111)

||z∗s || ≤ M + δM ≤ M + 1, s = 1, . . . , m.

(5.112)

.

In view of (5.111), .

Theorem 5.23 Let .δ ∈ (0, 1), ɛ 0 ∈ (0, 1], ɛ 0 ≥ (8 · 33(δ + δM )1/2 (8M + 8)3/2 )1/2 ,

.

(5.113)

a natural number .n0 satisfy n0 ≥ 1 + L8(4M + 3)2 ɛ 0−2 ⎦.

.

(5.114)

Assume that .{xi }∞ i=0 ⊂ X, {mk }∞ k=0 ⊂ {1, . . . , m},

.

||x0 || ≤ M,

.

(5.115)

and that for each integer .k ≥ 0, d(xk , Cmk ) ≥ max{d(xk , Ci ) : i = 1, . . . , m} − δ,

.

(5.116)

and B(xk+1 , δ) ∩ Pmk ,δ (xk ) = / ∅.

.

Then there exists an integer .q ∈ [1, n0 ] such that xi ∈ B(0, 5M + 1), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

(5.117)

Almost Star-Shaped Feasibility Problems

247

Proof Let .k ≥ 0 be an integer. By (5.117), there exists .

yk ∈ B(xk+1 , δ) ∩ Pmk ,δ (xk ).

(5.118)

||xk − yk || ≤ d(xk , Cmk ) + δ.

(5.119)

In view of (5.118), .

By (5.109), (5.111), (5.115), (5.116) and (5.119), ||y0 − x0 || ≤ ||x0 − z∗mk || + δ

.

.

≤ ||x0 − z∗ || + ||z∗ − z∗mk || + δ ≤ 2M + δ + δM ,

||y0 − x0 || ≤ 2M + 2, ||y0 || ≤ 3M + 2, ||x1 || ≤ 3M + 3, ||x1 − z∗ || ≤ 4M + 3. (5.120)

.

Assume that a natural number s satisfies for each integer .k ∈ [1, s], max{d(xk , Cs ) : s = 1, . . . , m} > ɛ 0 .

.

(5.121)

We prove the following auxiliary result. Lemma 5.24 Assume that an integer .k ∈ [1, s] satisfies ||xk − z∗ || ≤ 4M + 3.

.

(5.122)

Then ||xk+i − z∗ ||2 ≤ ||xk − z∗ ||2 − 8−1 ɛ 02 .

.

Proof Set E0 = {αz∗ + (1 − α)yk : α ∈ [0, 1]}.

.

(5.123)

By (5.111), (5.119) and (5.123), ||xk − yk || ≤ ||xk − z∗mk || + 1

.

.

≤ ||xk − z∗ || + ||z∗ − z∗mk || + 1 ≤ 4M + 5, ||yk || ≤ 4M + 5 + ||xk || ≤ 9M + 9.

.

(5.124)

248

5 Methods with Remotest Set Control in a Hilbert Space

Let y ∈ E0 .

.

(5.125)

Assumption (A3) and Eqs. (5.117), (5.118), (5.123)–(5.125) imply that there exists ˆ y ∈ Cmk

(5.126)

||y − ˆ y || < δM .

(5.127)

.

and .

By (5.119), (5.124), (5.126) and (5.127), ||xk − y|| ≥ ||xk − ˆ y || − ||ˆ y − y|| > ||xk − ˆ y || − δM

.

≥ d(xk , Cmk ) − δM ≥ ||xk − yk || − δ − δM .

(5.128)

||xk − yk || ≤ ||xk − y|| + δ + δM for each y ∈ E0 .

(5.129)

.

By (5.128), .

Equations (5.129) and Lemma 2.3 applied with D = E0 , x = xk , x˜ = yk ,

.

imply that ||z∗ − xk ||2 ≥ ||z∗ − yk ||2 + ||xk − yk ||2

.

.

− 32(δ + δM )1/2 (2||xk − z∗ || + 1)3/2 .

(5.130)

By (5.113), (5.116), (5.118) and (5.121), ||xk − yk || ≥ d(xk , Cmk )

.

.

≥ max{d(xk , Ci ) : i = 1, . . . , m} − δ .

≥ ɛ 0 − δ > ɛ 0 /2.

(5.131)

Equations (5.113), (5.119), (5.130) and (5.131) imply that ||z∗ − xk ||2 ≥ ||z∗ − yk ||2 + ɛ 02 /4 − 32(δ + δM )1/2 (8M + 7)3/2

.

.

≥ ||z∗ − yk ||2 + ɛ 02 /8.

(5.132)

Almost Star-Shaped Feasibility Problems

249

In view of (5.132), ||z∗ − yk || ≤ ||z∗ − xk ||.

.

(5.133)

Equations (5.118), (5.122) and (5.132) imply that |||z∗ − yk ||2 − ||z∗ − xk+1 ||2 |

.

.

≤ ||yk − xk+1 ||(2||z∗ − yk || + ||yk − xk+1 ||) .

≤ δ(8M + 6).

(5.134)

It follows from (5.111), (5.132) and (5.134) that ||z∗ − xk ||2 ≥ ||z∗ − xk+1 ||2 − δ(8M + 6)

.

.

+ 4−1 ɛ 02 − 32(8M + 7)3/2 (δ + δM )1/2 .

≥ ||z∗ − xk+1 ||2 + ɛ 02 /8.

This completes the proof of Lemma 5.24. It follows from Lemma 5.24 applied by induction that for all .k = 1, . . . , s + 1, ||z∗ − xk || ≤ 4M + 3,

(5.135)

||z∗ − xk+1 ||2 ≤ ||z∗ − xk ||2 − ɛ 02 /8.

(5.136)

.

and for all .k = 1, . . . , s, .

By (5.135) and (5.136), 4(M + 3)2 ≥ ||x1 − z∗ ||2 ≥ ||x1 − z∗ ||2 − ||xs+1 − z∗ ||2

.

.

=

s Σ (||z∗ − xk ||2 − ||z∗ − xk+1 ||2 ) ≥ 8−1 ɛ 02 s k=1

and s ≤ 8(4M + 3)2 ɛ 0−2 .

.

250

5 Methods with Remotest Set Control in a Hilbert Space

Thus we have shown that the following property holds: (a) if an integer .s ≥ 1 and for each integer .k ∈ [1, s] (5.118) is true, then (5.135) holds for .k = 1, . . . , s + 1 and s ≤ 8(4M + 3)2 ɛ 0−2 .

.

Property (a) implies that there exists a natural number q ≤ L8(4M + 3)2 ɛ 0−2 ⎦ + 1 = n0

.

such that xi ∈ B(0, 5M + 3), i = 0, . . . , q,

.

d(xq , Cs ) ≤ ɛ 0 , s = 1, . . . , m.

.

Theorem 5.23 is proved.

Chapter 6

Algorithms Based on Unions of Nonexpansive Maps

In this chapter we analyze iterative algorithms, which can be described in terms of a structured set-valued operator. Namely, at every point in the ambient space, it is assumed that the value of the operator can be expressed as a finite union of values of single-valued quasi-nonexpansive operators. For such algorithms it is shown their global convergence for an arbitrary starting point. An analogous result is also proved for the Krasnosel’ski-Mann iterations. Our main goal is also to obtain an approximate solution of the fixed point problem in the presence of computational errors. We show that the iterative method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

Global Convergence of Iterates Suppose that .(X, ρ) is a metric space and that .C ⊂ X is its nonempty, closed set. For every point .x ∈ X and every positive number r define B(x, r) = {y ∈ X : ρ(x, y) ≤ r}.

.

For each .x ∈ X and each nonempty set .D ⊂ X set ρ(x, D) = inf{ρ(x, y) : y ∈ D}.

.

For every operator .S : C → C set Fix(S) = {x ∈ C : S(x) = x}.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_6

251

252

6 Algorithms Based on Unions of Nonexpansive Maps

Fix θ ∈ C.

.

Suppose that the following assumption holds: (A1) For each .M > 0 the set .B(θ, M) ∩ C is compact. Assume that m is a natural number, .Ti : C → C, .i = 1, . . . , m are continuous operators and that the following assumption holds: (A2) For every natural number .i ∈ {1, . . . , m}, every point .z ∈ Fix(Ti ), every point .x ∈ C and every .y ∈ C \ Fix(Ti ), we have ρ(z, Ti (x)) ≤ ρ(z, x)

.

and ρ(z, Ti (y)) < ρ(z, y).

.

Note that operators satisfying (A2) are called paracontractions [85]. Assume that for every point .x ∈ X, a nonempty set φ(x) ⊂ {1, . . . , m}

.

(6.1)

is given. In other words, φ : X → 2{1,...,m} \ {∅}.

.

Suppose that the following assumption holds: (A3) For each .x ∈ C there exists .δ > 0 such that for each .y ∈ B(x, δ) ∩ C, φ(y) ⊂ φ(x).

.

Define T (x) = {Ti (x) : i ∈ φ(x)}

(6.2)

F¯ (T ) = {z ∈ C : Ti (z) = z, i = 1, . . . , m}

(6.3)

F (T ) = {z ∈ C : z ∈ T (z)}.

(6.4)

.

for each .x ∈ C, .

and .

Global Convergence of Iterates

253

Assume that F¯ (T ) /= ∅.

.

Denote by Card.(D) the cardinality of a set D. For each .z ∈ R 1 set Lz⎦ = inf{i : i is an integer and i ≤ z}.

.

In the sequel we suppose that the sum over empty set is zero. We study the asymptotic behavior of sequences of iterates .xt+1 ∈ F (xt ), .t = 0, 1, . . . . In particular we are interested in their convergence to a fixed point of T . This iterative algorithm was introduced in [144] which also contains its application to sparsity constrained minimisation. The following result, which is proved in Sect. “Proof of Theorem 6.1”, shows that almost all iterates of our set-valued mappings are approximated solutions of the corresponding fixed point problem. It was obtained in [169]. Theorem 6.1 Assume that .M > 0, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) = / ∅.

.

(6.5)

Then there exists an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfy ρ(x0 , θ ) ≤ M

.

and xt+1 ∈ F (xt ) for each integer t ≥ 0

.

the inequality ρ(xt , θ ) ≤ 3M

.

holds for all integers .t ≥ 0, Card({t ∈ {0, 1, . . . , } : ρ(xt , xt+1 ) > ɛ }) ≤ Q

.

and .limt→∞ ρ(xt , xt+1 ) = 0. The following global convergence result is proved in Sect. “Proof of Theorem 6.2”. It was also obtained in [169]. Theorem 6.2 Assume that a sequence .{xt }∞ t=0 ⊂ C and that for each integer .t ≥ 0, xt+1 ∈ F (xt ).

.

254

6 Algorithms Based on Unions of Nonexpansive Maps

Then there exist x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 φ(xt ) ⊂ φ(x∗ )

.

and if an integer .i ∈ φ(xt ) satisfies .xt+1 = Ti (xt ), then Ti (x∗ ) = x∗ .

.

Theorem 6.2 generalizes the main result of [144] which establishes a local convergence of the iterative algorithm for iterates starting from a point belonging to a neighborhood of a strong fixed point belonging to the set .F¯ (T ).

An Auxiliary Result Lemma 6.3 Assume that M, ɛ > 0 and that z∗ ∈ C satisfies Ti (z∗ ) = z∗ , i = 1, . . . , m.

.

(6.6)

Then there exists δ > 0 such that for each s ∈ {1, . . . , m} and each x ∈ C ∩B(θ, M) satisfying ρ(x, Ts (x)) > ɛ

(6.7)

ρ(z∗ , Ts (x)) ≤ ρ(z∗ , x) − δ

(6.8)

.

the inequality .

is true. Proof Let s ∈ {1, . . . , m}. It is sufficient to show that there exists δ > 0 such that for each x ∈ C ∩ B(θ, M) satisfying (6.7) inequality (6.8) is true. Assume the contrary. Then for each integer k ≥ 1, there exists xk ∈ C ∩ B(θ, M)

(6.9)

ρ(xk , Ts (xk )) > ɛ

(6.10)

.

such that .

Proof of Theorem 6.1

255

and ρ(z∗ , Ts (xk )) > ρ(z∗ , xk ) − k −1 .

.

(6.11)

In view of (A1) and (6.9), extracting a subsequence and re-indexing, we may assume without loss of generality that there exists x∗ = lim xk .

.

k→∞

(6.12)

By (6.9)–(6.12) and the continuity of Ts , ρ(x∗ , θ ) ≤ M,

.

ρ(x∗ , Ts (x∗ )) = lim ρ(xk , Ts (xk )) ≥ ɛ

.

k→∞

and ρ(z∗ , Ts (x∗ )) ≥ ρ(z∗ , x∗ ).

.

This contradicts (6.6) and (A2). The contradiction we have reached proves Lemma 6.3.

Proof of Theorem 6.1 By (6.5), there exists z∗ ∈ B(θ, M) ∩ F¯ (T ).

.

(6.13)

Lemma 6.3 implies that there exists .δ ∈ (0, ɛ ) such that the following property holds: (a) for each .s ∈ {1, . . . , m} and each .x ∈ C ∩ B(z∗ , 2M) satisfying ρ(x, Ts (x)) > ɛ

.

we have ρ(z∗ , Ts (x)) ≤ ρ(z∗ , x) − δ.

.

Choose a natural number Q ≥ 2Mδ −1 .

.

(6.14)

256

6 Algorithms Based on Unions of Nonexpansive Maps

Assume that .{xi }∞ i=0 ⊂ C, ρ(x0 , θ ) ≤ M

(6.15)

xt+1 ∈ F (xt ).

(6.16)

.

and that for each integer .t ≥ 0, .

Let .t ≥ 0 be an integer. By (6.2) and (6.16), there exists .s ∈ {1, . . . , m} such that xt+1 = Ts (xt ).

.

(6.17)

Assumption (A2) and Eqs. (6.3), (6.13) and (6.17) imply that ρ(z∗ , xt+1 ) = ρ(z∗ , Ts (xt )) ≤ ρ(z∗ , xt ).

.

(6.18)

Since t is an arbitrary nonnegative integer equations (6.13), (6.15) and (6.18) imply that for each integer .i ≥ 0, ρ(z∗ , xi ) ≤ ρ(z∗ , x0 ) ≤ 2M

.

(6.19)

and ρ(xi , θ ) ≤ 3M.

.

Assume that ρ(xt+1 , xt ) > ɛ .

.

(6.20)

Property (a) and Eqs. (6.17), (6.19) and (6.20) imply that ρ(z∗ , xt+1 ) = ρ(z∗ , Ts (xt )) ≤ ρ(z∗ , xt ) − δ.

.

Thus we have shown that the following property holds: (b) if an integer .t ≥ 0 satisfies (6.20), then ρ(z∗ , xt+1 ) ≤ ρ(z∗ , xt ) − δ.

.

Assume that .n ≥ 1 is an integer. Property (b) and Eqs. (6.18)–(6.20) imply that 2M ≥ ρ(z∗ , x0 ) ≥ ρ(z∗ , x0 ) − ρ(z∗ , xn+1 )

.

.

=

n Σ (ρ(z∗ , xt ) − ρ(z∗ , xt+1 )) t=0

Proof of Theorem 6.2

.

≥

Σ

257

{ρ(z∗ , xt ) − ρ(z∗ , xt+1 ) : t ∈ {0, . . . , n}, ρ(xt , xt+1 ) > ɛ } .

≥ δCard({t ∈ {0, . . . , n} : ρ(xt , xt+1 ) > ɛ })

and in view of (6.14), Card({t ∈ {0, . . . , n} : ρ(xt , xt+1 ) > ɛ }) ≤ 2Mδ −1 ≤ Q.

.

Since n is an arbitrary natural number we conclude that Card({t ∈ {0, 1, . . . } : ρ(xt , xt+1 ) > ɛ }) ≤ Q.

.

Since . ɛ is any element of .(0, 1) Theorem 6.1 is proved.

Proof of Theorem 6.2 In view of Theorem 6.1, the sequence .{xt }∞ t=0 is bounded. In view of (A1), it has a limit point .x∗ ∈ C and a subsequence .{xtk }∞ k=0 such that x∗ = lim xtk .

(6.21)

.

k→∞

In view of (A3) and (6.21), we may assume without loss of generality that φ(xtk ) ⊂ φ(x∗ ), k = 1, 2, . . .

.

(6.22)

and that there exists .

p ˆ ∈ φ(x∗ )

such that xtk +1 = Tpˆ(xtk ), k = 1, 2, . . . .

.

(6.23)

It follows from Theorem 6.1, the continuity of .Tpˆ and Eqs. (6.21) and (6.23) that Tpˆ(x∗ ) = lim Tpˆ(xtk ) = lim xtk +1 = lim xtk = x∗ .

(6.24)

I1 = {i ∈ φ(x∗ ) : Ti (x∗ ) = x∗ }, I2 = φ(x∗ ) \ I1 .

(6.25)

.

k→∞

k→∞

k→∞

Set .

258

6 Algorithms Based on Unions of Nonexpansive Maps

In view of (6.24) and (6.25), .

p ˆ ∈ I1 .

Fix .δ0 ∈ (0, 1) such that ρ(x∗ , Ti (x∗ )) > 2δ0 , i ∈ I2 .

.

(6.26)

Assumption (A3), the continuity of .Ti , i = 1, . . . , m and (6.26) imply that there exists .δ1 ∈ (0, δ0 ) such that for each .x ∈ B(x∗ , δ1 ) ∩ C, φ(x) ⊂ φ(x∗ ),

(6.27)

ρ(x, Ti (x)) > δ0 , i ∈ I2 .

(6.28)

.

.

Theorem 6.1 implies that there exists an integer .q1 ≥ 1 such that for each integer t ≥ q1 ,

.

ρ(xt , xt+1 ) ≤ δ0 /2.

(6.29)

ɛ ∈ (0, δ1 ),

(6.30)

t ≥ q1

(6.31)

ρ(xt , x∗ ) ≤ ɛ .

(6.32)

.

Assume that .

.

is an integer and that .

It follows from (6.27), (6.28), (6.30) and (6.32) that φ(xt ) ⊂ φ(x∗ )

(6.33)

ρ(xt , Ti (xt )) > δ0 , i ∈ I2 .

(6.34)

.

and .

In view of (6.33), there exists s ∈ φ(x∗ )

.

such that xt+1 = Ts (xt ).

.

(6.35)

Krasnosel’ski-Mann Iterations

259

By (6.29), (6.31) and (6.35), ρ(xt , Ts (xt )) = ρ(xt , xt+1 ) ≤ δ0 /2.

.

(6.36)

It follows from (6.25), (6.34) and (6.36) that s ∈ I1 , Ts (x∗ ) = x∗ .

.

Combined with assumption (A2) and Eqs. (6.32) and (6.35) this implies that ρ(xt+1 , x∗ ) = ρ(Ts (xt ), x∗ ) ≤ ρ(xt , x∗ ) ≤ ɛ .

.

Thus we have shown that if .t ≥ q1 is an integer and (6.32) holds, then (6.33) is true and if .s ∈ φ(x∗ ) and (6.35) holds, then .s ∈ I1 and .ρ(xt+1 , x∗ ) ≤ ɛ . By induction and (6.21), we obtain that ρ(xi , x∗ ) ≤ ɛ

.

for all sufficiently large natural numbers i. Since . ɛ i an arbitrary element of .(0, δ1 ) we conclude that lim xt = x∗

.

t→∞

and Theorem 6.2 is proved.

Krasnosel’ski-Mann Iterations The results of this section were obtained in [169]. Assume that .(X, || · ||) is a normed space and that .ρ(x, y) = ||x − y||, x, y ∈ X. We use the notation, definitions and assumptions introduced in Sect. “Global Convergence of Iterates”. In particular, we assume that assumptions (A1)–(A3) hold. Suppose that the set C is convex and denote by .I d : X → X the identity operator: .I d(x) = x, .x ∈ X. Let κ ∈ (0, 2−1 ).

.

We consider Kraasnosel’ski-Mann iteration associated with our set-valued mapping T and obtain the global convergence result (see Theorem 6.4 below) which generalizes the local convergence result of [76] for iterates starting from a point belonging to a neighborhood of a strong fixed point belonging to the set .F¯ (T ). The following result is proved in Sect. “Proof of Theorem 6.4”.

260

6 Algorithms Based on Unions of Nonexpansive Maps

Theorem 6.4 Assume that .M > 0, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) /= ∅

.

(6.37)

Then there exists an integer .Q ≥ 1 such that for each {λt }∞ t=0 ⊂ (κ, 1 − κ)

.

(6.38)

and each sequence .{xi }∞ i=0 ⊂ C which satisfies ||x0 − θ || ≤ M

.

and xt+1 ∈ (1 − λt )xt + λt T (xt ) for each integer t ≥ 0

.

the inequality ||xt − θ || ≤ 3M

.

holds for all integers .t ≥ 0, Card({t ∈ {0, 1, . . . , } : ||xt − xt+1 || > ɛ }) ≤ Q

.

and .limt→∞ ||xt − xt+1 || = 0. The following result is proved in Sect. “Proof of Theorem 6.5”. Theorem 6.5 Assume that {λt }∞ t=0 ⊂ (κ, 1 − κ)

.

and that a sequence .{xt }∞ t=0 ⊂ C satisfies (6.39). Then there exist x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 φ(xt ) ⊂ φ(x∗ )

.

and if an integer .i ∈ φ(xt ) satisfies xt+1 = λt Ti (xt ) + (1 − λ)xt ,

.

then Ti (x∗ ) = x∗ .

.

(6.39)

Proof of Theorem 6.4

261

Proof of Theorem 6.4 By (6.37), there exists z∗ ∈ B(θ, M) ∩ F¯ (T ).

.

(6.40)

Lemma 6.3 implies that there exists .δ ∈ (0, ɛ ) such that the following property holds: (c) for each .s ∈ {1, . . . , m} and each .x ∈ C ∩ B(z∗ , 2M) satisfying ρ(x, Ts (x)) > ɛ

.

we have ρ(z∗ , Ts (x)) ≤ ρ(z∗ , x) − δ.

.

Choose a natural number Q ≥ 2Mδ −1 κ −1 .

.

(6.41)

Assume that (6.38) holds and that a sequence .{xi }∞ i=0 ⊂ C satisfies (6.39) and ||x0 − θ || ≤ M.

.

(6.42)

Let .t ≥ 0 be an integer. By (6.2) and (6.39), there exists .s ∈ {1, . . . , m} such that xt+1 = λt Ts (xt ) + (1 − λt )xt .

.

(6.43)

Assumption (A2) and Eqs. (6.3), (6.40) and (6.43) imply that ||xt+1 − z∗ || = ||λt Ts (xt ) + (1 − λt )xt − z∗ ||

.

.

≤ λt ||Ts (xt ) − z∗ || + (1 − λt )||xt − z∗ || ≤ ||z∗ − xt ||.

(6.44)

Since t is an arbitrary nonnegative integer equations (6.40), (6.42) and (6.44) imply that for each integer .i ≥ 0, ||z∗ − xi || ≤ ||z∗ − x0 || ≤ 2M

.

and ||xi − θ || ≤ 3M.

.

262

6 Algorithms Based on Unions of Nonexpansive Maps

Assume that ||xt+1 − xt || > ɛ .

.

(6.45)

It follows from (6.38), (6.43) and (6.45) that ɛ < ||xt+1 − xt || = ||λt Ts (xt ) + (1 − λt )xt − xt || = λt ||Ts (xt ) − xt ||

.

and −1 ||Ts (xt ) − xt || ≥ ɛ λ−1 t ≥ ɛ (1 − κ) .

.

(6.46)

Property (c) and Eq. (6.46) imply that ||z∗ − Ts (xt )|| ≤ ||z∗ − xt || − δ.

.

(6.47)

By (6.38), (6.43) and (6.47), ||xt+1 − z∗ || = ||λt Ts (xt ) + (1 − λt )xt − z∗ ||

.

.

.

≤ λt ||Ts (xt ) − z∗ || + (1 − λt )||xt − z∗ ||

≤ λt (||xt − z∗ || − δ) + (1 − λt )||xt − z∗ || .

≤ ||xt − z∗ || − λt δ ≤ ||xt − z∗ || − δκ.

(6.48)

Thus we have shown that the following property holds: (d) if an integer .t ≥ 0 satisfies (6.45), then ||z∗ − xt+1 || ≤ ||z∗ − xt || − δκ.

.

Assume that .n ≥ 1 is an integer. Property (d) and Eqs. (6.40), (6.42) and (6.44) imply that 2M ≥ ||z∗ − x0 || ≥ ||z∗ − x0 || − ||z∗ − xn+1 ||

.

.

=

n Σ (||z∗ − xt || − ||z∗ − xt+1 ||) t=0

.

≥

Σ {||z∗ − xt || − ||z∗ − xt+1 || : t ∈ {0, . . . , n}, ||xt − xt+1 || > ɛ } .

≥ δκCard({t ∈ {0, . . . , n} : ||xt − xt+1 || > ɛ })

Proof of Theorem 6.5

263

and in view of (6.41), Card({t ∈ {0, . . . , n} : ||xt − xt+1 || > ɛ }) ≤ 2M(δκ)−1 ≤ Q.

.

Since n is an arbitrary natural number we conclude that Card({t ∈ {0, 1, . . . } : ||xt − xt+1 || > ɛ }) ≤ Q.

.

Since . ɛ is any element of .(0, 1) we obtain that .

lim ||xt − xt+1 || = 0.

t→∞

Theorem 6.4 is proved.

Proof of Theorem 6.5 In view of Theorem 6.4, the sequence .{xt }∞ t=0 is bounded. In view of (A1), it has a limit point .x∗ ∈ C and a subsequence .{xtk }∞ k=0 such that x∗ = lim xtk .

.

k→∞

(6.49)

In view of (A3) and Eqs. (6.38), (6.39) and (6.49), extracting a subsequence and re-indexing, we may assume without loss of generality that φ(xtk ) ⊂ φ(x∗ ), k = 1, 2, . . .

.

(6.50)

and that there exists .

p ˆ ∈ φ(x∗ )

such that xtk +1 = λtk Tpˆ(xtk ) + (1 − λtk )xtk , k = 1, 2, . . .

.

(6.51)

and that there exists λ∗ = lim λtk ∈ [κ, 1 − κ].

.

k→∞

(6.52)

It follows from Theorem 6.4, the continuity of .Tpˆ and Eqs. (6.49), (6.51) and (6.52) that λ∗ Tpˆ(x∗ ) + (1 − λ∗ )x∗

.

264

6 Algorithms Based on Unions of Nonexpansive Maps .

= lim (λtk Tpˆ(xtk ) + (1 − λtk )xtk ) k→∞

.

= lim xtk +1 = lim xtk = x∗ . k→∞

k→∞

(6.53)

Set I1 = {i ∈ φ(x∗ ) : Ti (x∗ ) = x∗ }, I2 = φ(x∗ ) \ I1 .

.

(6.54)

In view of (6.53) and (6.54), .

p ˆ ∈ I1 .

Fix .δ0 ∈ (0, 1) such that ||x∗ − Ti (x∗ )|| > 2δ0 , i ∈ I2 .

.

(6.55)

Assumption (A3), the continuity of .Ti , i = 1, . . . , m and (6.55) imply that there exists .δ1 ∈ (0, δ0 ) such that for each .x ∈ B(x∗ , δ1 ) ∩ C, φ(x) ⊂ φ(x∗ ),

(6.56)

||x − Ti (x)|| > δ0 , i ∈ I2 .

(6.57)

.

.

Theorem 6.4 implies that there exists an integer .q1 ≥ 1 such that for each integer t ≥ q1 ,

.

||xt − xt+1 || ≤ κδ0 /2.

(6.58)

ɛ ∈ (0, δ1 ),

(6.59)

t ≥ q1

(6.60)

||xt − x∗ || ≤ ɛ .

(6.61)

.

Assume that .

.

is an integer and that .

It follows from (6.56), (6.57), (6.59) and (6.61) that φ(xt ) ⊂ φ(x∗ )

(6.62)

||xt − Ti (xt )|| > δ0 , i ∈ I2 .

(6.63)

.

and .

Proof of Theorem 6.5

265

In view of (6.39), there exists s ∈ φ(xt ) ⊂ φ(x∗ )

.

such that xt+1 = λt Ts (xt ) + (1 − λt )xt .

.

(6.64)

By (6.38), (6.58) and (6.64), κδ0 /2 ≥ ||xt+1 − xt || = λt ||Ts (xt ) − xt ||

.

and ||xt − Ts (xt )|| ≤ κδ0 (2λt )−1 ≤ δ0 /2.

.

(6.65)

It follows from (6.54), (6.56), (6.57), (6.59), (6.61) and (6.65) that s ∈ I1 , Ts (x∗ ) = x∗ .

.

Combined with assumption (A2) and Eqs. (6.39), (6.61) and (6.64) this implies that ||xt+1 − x∗ || = ||λt Ts (xt ) + (1 − λt )xt − x∗ ||

.

.

≤ λt ||Ts (xt ) − x∗ || + (1 − λt )||xt − x∗ || .

≤ ||xt − x∗ || ≤ ɛ .

Thus we have shown that if .t ≥ q1 is an integer and (6.61) holds, then .||xt+1 −x∗ || ≤ ɛ . By induction and (6.49), we obtain that ||xi − x∗ || ≤ ɛ

.

for all sufficiently large natural numbers i. Since . ɛ i an arbitrary element of .(0, δ1 ) we conclude that .

and Theorem 6.5 is proved.

lim xt = x∗

t→∞

266

6 Algorithms Based on Unions of Nonexpansive Maps

The Cimmino Type Algorithm Suppose that .(X, || · ||) is a normed space and that .C ⊂ X is its nonempty, closed, convex set. For each .x, y ∈ X, set ρ(x, y) = ||x − y||, x, y ∈ X.

.

We use the notation, definitions and assumptions introduced in Sect. “Global Convergence of Iterates”. In particular we assume that assumptions (A1)–(A3) hold. We prove the following two results. Theorem 6.6 Assume that .M > 0, .c¯ ∈ (0, m−1 ], . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(0, M) /= ∅

.

(6.66)

Then there exists an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C and each .γt,i ≥ c, ¯ .t = 0, 1, . . . , i = 1, . . . , m which satisfy ||x0 || ≤ M

.

(6.67)

and for each integer .t ≥ 0, Σ .

γt,i = 1

(6.68)

i∈φ(xt )

and xt+1 =

Σ

.

γt,i Ti (xt )

i∈φ(xt )

the inequality ||xt || ≤ 3M

.

holds for all integers .t ≥ 0 and that Card({t ∈ {0, 1, . . . , } : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ }) ≤ Q.

.

Theorem 6.7 Assume that .c¯ ∈ (0, m−1 ] and that a sequence .{xi }∞ i=0 ⊂ C and .γt,i ≥ c, ¯ .t = 0, 1, . . . , i = 1, . . . , m satisfy for each integer .t ≥ 0, Σ .

i∈φ(xt )

γt,i = 1

(6.69)

Proof of Theorem 6.6

267

and xt+1 =

Σ

.

γt,i Ti (xt ).

(6.70)

i∈φ(xt )

Then there exists x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 and each .i ∈ φ(xt ), φ(xt ) ⊂ φ(x∗ )

.

and Ti (x∗ ) = x∗ .

.

Proof of Theorem 6.6 In view of (6.2), there exists z∗ ∈ B(0, M) ∩ F¯ (T ).

.

(6.71)

Lemma 6.3 implies that there exists .δ ∈ (0, ɛ ) such that the following property holds: (a) for each .s ∈ {1, . . . , m} and each .x ∈ C ∩ B(z∗ , 2M) satisfying ||x − Ts (x)|| > ɛ

.

we have ||z∗ − Ts (x)|| ≤ ||z∗ − x|| − δ.

.

Choose a natural number Q ≥ 2Mδ −1 c¯−1 .

(6.72)

γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m,

(6.73)

||x0 || ≤ M

(6.74)

.

Assume that .{xi }∞ i=0 ⊂ C, .

.

268

6 Algorithms Based on Unions of Nonexpansive Maps

and that for each integer .t ≥ 0, Σ .

γt,i = 1

(6.75)

i∈φ(xt )

and Σ

xt+1 =

γt,i Ti (xt ).

.

(6.76)

i∈φ(xt )

Let .t ≥ 0 be an integer. Assumption (A2), the convexity of the norm and Eqs. (6.3), (6.71), (6.73), (6.75) and (6.76) imply that ||z∗ − xt+1 || = ||z∗ −

Σ

.

γt,i Ti (xt )||

i∈φ(xt )

.

Σ

≤

Σ

γt,i ||z∗ − Ti (xt )|| ≤

i∈φ(xt )

γt,i ||z∗ − xt || ≤ ||z∗ − xt ||.

(6.77)

i∈φ(xt )

Since t is an arbitrary nonnegative integer equations (6.71), (6.74) and (6.77) imply that ||z∗ − xj || ≤ ||z∗ − x0 || ≤ 2M, j = 0, 1, . . . .

(6.78)

max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ .

(6.79)

.

Assume that .

In view of (6.79), there exists s ∈ φ(xt )

.

such that ||xt − Ts (xt )|| > ɛ .

.

(6.80)

By (6.73), (6.75) and (6.76), ||z∗ − xt+1 || = ||z∗ −

Σ

.

γt,i Ti (xt )||

i∈φ(xt )

.

≤

Σ i∈φ(xt )

γt,i ||z∗ − Ti (xt )||.

(6.81)

Proof of Theorem 6.6

269

Property (a), assumption (A1) and Eqs. (6.3), (6.71), (6.78) and (6.80) imply that for each .i ∈ φ(xt ), ||z∗ − Ti (xt )|| ≤ ||z∗ − xt ||,

(6.82)

||z∗ − Ts (xt )|| ≤ ||z∗ − xt || − δ.

(6.83)

.

.

It follows from (6.73), (6.75) and (6.81)–(6.83) that ||z∗ − xt+1 || ≤

Σ

.

.

.

≤

{γt,i ||z∗ − Ti (xt )|| : i ∈ φ(xt ) \ {s}}

+ γt,s ||z∗ − Ts (xt )||

Σ {γt,i ||z∗ − xt || : i ∈ φ(xt ) \ {s}} + γt,s ||z∗ − xt || − δγt,s .

≤ ||z∗ − xt || − δ c. ¯

Thus we have shown that the following property holds: (b) if an integer .t ≥ 0 satisfies (6.79), then ||z∗ − xt+1 || ≤ ||z∗ − xt || − δ c. ¯

.

Assume that .n ≥ 1 is an integer. Property (b) and Eqs. (6.77) and (6.78) imply that 2M ≥ ||z∗ − x0 || ≥ ||z∗ − x0 || − ||z∗ − xn+1 ||

.

.

=

n Σ (||z∗ − xt || − ||z∗ − xt+1 ||) t=0

.

≥

Σ

.

.

{||z∗ − xt || − ||z∗ − xt+1 || : t ∈ {0, . . . , n}, max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ }

≥ δ cCard({t ¯ ∈ {0, . . . , n} : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ })

and Card({t ∈ {0, . . . , n} : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ }) ≤ 2Mδ −1 c¯−1 .

.

270

6 Algorithms Based on Unions of Nonexpansive Maps

Since n is an arbitrary natural number we conclude that in view of (6.72), Card({t ∈ {0, 1, . . . } :

.

max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ }) ≤ 2Mδ −1 c¯−1 ≤ Q.

.

Theorem 6.6 is proved.

Proof of Theorem 6.7 In view of Theorem 6.6, the sequence .{xt }∞ t=0 is bounded. Assumption (A1 implies that it has a limit point .x∗ ∈ C and a subsequence .{xtk }∞ k=0 such that x∗ = lim xtk .

.

k→∞

(6.84)

By (A2), we may assume without loss of generality that φ(xtk ) ⊂ φ(x∗ ), k = 1, 2, . . . .

.

(6.85)

Set I1 = {i ∈ φ(x∗ ) : Ti (x∗ ) = x∗ }, I2 = φ(x∗ ) \ I1 .

.

(6.86)

Fix .δ0 ∈ (0, 1) such that ||x∗ − Ti (x∗ )|| > 2δ0 , i ∈ I2 .

.

(6.87)

Assumption (A2), the continuity of .Ti , i = 1, . . . , m and (6.87) imply that there exists .δ1 ∈ (0, δ0 ) such that for each .x ∈ B(x∗ , δ1 ) ∩ C, φ(x) ⊂ φ(x∗ ),

(6.88)

||x − Ti (x)|| > δ0 , i ∈ I2 .

(6.89)

.

.

Theorem 6.6 implies that there exists an integer .Q0 ≥ 1 such that Card({t ∈ {0, 1, . . . } : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > δ0 /2}) ≤ Q0 .

.

This implies that there exists an integer .Q1 ≥ 1 such that for each integer .t ≥ Q1 , .

max{||xt − Ti (xt )|| : i ∈ φ(xt )} ≤ δ0 /2.

(6.90)

Proof of Theorem 6.7

271

Assume that ɛ ∈ (0, δ1 ),

(6.91)

.

t ≥ Q1

.

is an integer and that ||xt − x∗ || ≤ ɛ .

(6.92)

.

It follows from (6.88), (6.89), (6.91) and (6.92) that φ(xt ) ⊂ φ(x∗ )

(6.93)

||xt − Ti (xt )|| > δ0 , i ∈ I2 .

(6.94)

.

and .

In view of (6.90) and the inequality .t ≥ Q1 , ||xt − Ti (xt )|| ≤ δ0 /2, i ∈ φ(xt ).

.

(6.95)

It follows from (6.83), (6.86) and (6.92)–(6.95) that φ(xt ) ⊂ I1 .

(6.96)

.

Assumption (A2) and Eq. (6.86) imply that for each .i ∈ I1 , ||Ti (xt ) − x∗ || ≤ ||xt − x∗ || ≤ ɛ .

.

By (6.69), (6.76) and (6.97), ||x∗ − xt+1 || = ||x∗ −

Σ

.

γt,i Ti (xt )||

i∈φ(xt )

.

≤

Σ

γt,i ||x∗ − Ti (xt )|| ≤ ||x∗ − xt || ≤ ɛ .

i∈φ(xt )

This implies that for all sufficiently large natural numbers t, ||xt − x∗ || ≤ ɛ .

.

Since . ɛ is an arbitrary element of .(0, δ1 ) we conclude that .

lim xt = x∗ .

t→∞

(6.97)

272

6 Algorithms Based on Unions of Nonexpansive Maps

By (6.86) and (6.96), for each sufficiently large natural number t, φ(xt ) ⊂ I1 = {i ∈ φ(x∗ ) : x∗ = Ti (x∗ )}.

.

Theorem 6.7 is proved.

Global Convergence of Iterates with Summable Errors Suppose that .(X, ρ) is a metric space and that .C ⊂ X is its nonempty, closed set. We use the notation, notation and definitions introduced in Sect. “Global Convergence of Iterates”. In particular we assume that assumptions (A1)–(A3) hold. The following result shows that almost all inexact iterates of our set-valued mappings with summable computational errors are approximated solutions of the corresponding fixed point problem. Many results of this type are collected in [163, 165]. Theorem 6.8 Assume that .M > 0, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) /= ∅,

.

(6.98)

{rt }∞ t=0 ⊂ (0, ∞),

.

∞ Σ .

rt < ∞.

(6.99)

t=0

Then there exists an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfy ρ(x0 , θ ) ≤ M

.

and B(xt+1 , rt ) ∩ F (xt ) /= ∅

.

for each integer .t ≥ 0 the inequality ρ(xt , θ ) ≤ 3M +

∞ Σ

.

rj

j =0

holds for all integers .t ≥ 0, Card({t ∈ {0, 1, . . . , } : ρ(xt , xt+1 ) > ɛ }) ≤ Q

.

and .limt→∞ ρ(xt , xt+1 ) = 0.

Proof of Theorem 6.8

273

The following result shows the global convergence of inexact iterates of our setvalued mappings with summable computational errors. Theorem 6.9 Assume that .{rt }∞ t=0 ⊂ (0, ∞), ∞ Σ .

rt < ∞.

t=0

a sequence .{xt }∞ t=0 ⊂ C and that for each integer .t ≥ 0, B(xt+1 , rt ) ∩ F (xt ) /= ∅.

.

Then there exist x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 φ(xt ) ⊂ φ(x∗ )

.

and if an integer .i ∈ φ(xt ) satisfies .ρ(xt+1 , Ti (xt )) ≤ rt , then Ti (x∗ ) = x∗ .

.

Proof of Theorem 6.8 By (6.98), there exists z∗ ∈ B(θ, M) ∩ F¯ (T ).

.

(6.100)

Lemma 6.3 and (6.100) imply that there exists .δ ∈ (0, ɛ ) such that the following property holds: Σ (a) for each .s ∈ {1, . . . , m} and each .x ∈ C ∩ B(z∗ , 2M + ∞ j =0 rj ) satisfying ρ(x, Ts (x)) > ɛ /2

.

we have ρ(z∗ , Ts (x)) ≤ ρ(z∗ , x) − δ.

.

By (6.99), there exists a natural number .m0 such that rt < δ/4 for each integer t ≥ m0 .

.

(6.101)

274

6 Algorithms Based on Unions of Nonexpansive Maps

Choose a natural number Q ≥ m0 + 2δ −1 (M +

∞ Σ

.

(6.102)

rt ).

t=0

Assume that .{xi }∞ i=0 ⊂ C, ρ(x0 , θ ) ≤ M

(6.103)

B(xt+1 , rt ) ∩ F (xt ) /= ∅.

(6.104)

.

and that for each integer .t ≥ 0, .

Let .t ≥ 0 be an integer. By (6.2) and (6.104), there exists .s ∈ {1, . . . , m} such that ρ(xt+1 , Ts (xt )) ≤ rt .

(6.105)

.

Assumption (A2) and Eqs. (6.4), (6.100) and (6.105) imply that ρ(z∗ , xt+1 ) = ρ(z∗ , Ts (xt )) + ρ(Ts (xt ), xt+1 ) ≤ ρ(z∗ , xt ) + rt .

.

(6.106)

Since t is an arbitrary nonnegative integer equations (6.100) and (6.103) imply that for each integer .k ≥ 1, k−1 Σ

ρ(z∗ , xk ) ≤ ρ(z∗ , x0 ) +

.

ri ≤ 2M +

i=0

∞ Σ

ri

(6.107)

i=0

and ρ(xk , θ ) ≤ 3M +

∞ Σ

.

ri .

(6.108)

i=0

Assume that t ≥ m0

(6.109)

ρ(xt+1 , xt ) > ɛ .

(6.110)

.

is an integer, .

By (6.101), (6.105), (6.109) and (6.110), ρ(xt , Ts (xt )) ≥ ρ(xt , xt+1 ) − ρ(xt+1 , Ts (xt ))

.

.

> ɛ − rt > ɛ − δ/4 > ɛ /2.

(6.111)

Proof of Theorem 6.8

275

Property (a) and Eqs. (6.107) and (6.111) imply that ρ(z∗ , Ts (xt )) ≤ ρ(z∗ , xt ) − δ.

(6.112)

.

By (6.101), (6.105), (6.109) and (6.112), ρ(z∗ , xt+1 ) ≤ ρ(z∗ , Ts (xt )) + ρ(Ts (xt ), xt+1 )

.

.

≤ ρ(z∗ , xt )) − δ + rt ≤ ρ(z∗ , xt ) − δ/2.

Thus we have shown that the following property holds: (b) if an integer .t ≥ m0 satisfies (6.110), then ρ(z∗ , xt+1 ) ≤ ρ(z∗ , xt ) − δ/2.

.

Assume that .n ≥ m0 + 1 is an integer. Property (b) and Eqs. (6.100) and (6.103) imply that 2M ≥ ρ(z∗ , x0 ) ≥ ρ(z∗ , x0 ) − ρ(z∗ , xn+1 )

.

.

=

n Σ (ρ(z∗ , xt ) − ρ(z∗ , xt+1 )).

(6.113)

t=0

It follows from (6.106), (6.1106) and (6.113) that 2M +

∞ Σ

.

i=0 .

≥

ri ≥

n Σ

(ρ(z∗ , xk ) − ρ(z∗ , xk+1 ) + rk )

k=0

Σ {ρ(z∗ , xk ) − ρ(z∗ , xk+1 ) + rk : k ∈ {m0 , . . . , n}, ρ(xt , xt+1 ) > ɛ } .

≥ (δ/2)Card({k ∈ {m0 , . . . , n} : ρ(xk , xk+1 ) > ɛ })

and Card({k ∈ {m0 , . . . , n} : ρ(xt , xt+1 ) > ɛ }) ≤ 2δ −1 (2M +

∞ Σ

.

ri ).

i=0

Since n is an arbitrary natural number satisfying .n ≥ m0 + 1 using (6.102) we conclude that Card({k ∈ {0, 1, . . . } : ρ(xk , xk+1 ) > ɛ })

.

276

6 Algorithms Based on Unions of Nonexpansive Maps

.

≤ m0 + 2δ −1 (2M +

∞ Σ

ri ) ≤ Q.

i=0

Since . ɛ is any element of .(0, 1) Theorem 6.8 is proved.

Proof of Theorem 6.9 In view of Theorem 6.8, the sequence .{xt }∞ t=0 is bounded. In view of (A1), it has a limit point .x∗ ∈ C and a subsequence .{xtk }∞ k=0 such that x∗ = lim xtk .

(6.114)

.

k→∞

By our assumptions, for each integer .t ≥ 0, B(xt+1 , rt ) ∩ F (xt ) /= ∅.

.

(6.115)

In view of (A3), we may assume without loss of generality that φ(xtk ) ⊂ φ(x∗ ), k = 1, 2, . . .

(6.116)

p ˆ ∈ φ(x∗ )

(6.117)

ρ(xtk +1 , Tpˆ(xtk )) ≤ rtk , k = 1, 2, . . . .

(6.118)

.

and that there exists .

such that .

It follows from Theorem 6.8, the continuity of .Tpˆ and Eqs. (6.114) and (6.118) that Tpˆ(x∗ ) = lim Tpˆ(xtk ) = lim xtk +1 = lim xtk = x∗ .

(6.119)

I1 = {i ∈ φ(x∗ ) : Ti (x∗ ) = x∗ }, I2 = φ(x∗ ) \ I1 .

(6.120)

.

k→∞

k→∞

k→∞

Set .

In view of (6.119) and (6.120), p ˆ ∈ I1 .

(6.121)

ρ(x∗ , Ti (x∗ )) > 2δ0 , i ∈ I2 .

(6.122)

.

Fix .δ0 ∈ (0, 1) such that .

Proof of Theorem 6.9

277

Assumption (A3), the continuity of .Ti , i = 1, . . . , m imply that there exists .δ1 ∈ (0, δ0 ) such that for each .x ∈ B(x∗ , δ1 ) ∩ C, φ(x) ⊂ φ(x∗ ),

(6.123)

ρ(x, Ti (x)) > 3δ0 /2, i ∈ I2 .

(6.124)

.

.

Theorem 6.8 implies that there exists an integer .q1 ≥ 1 such that for each integer t ≥ q1 ,

.

ρ(xt , xt+1 ) ≤ δ0 /4,

(6.125)

.

rt ≤ δ0 /4.

(6.126)

ɛ ∈ (0, δ1 ),

(6.127)

t ≥ q1

(6.128)

ρ(xt , x∗ ) < ɛ .

(6.129)

.

Assume that .

.

is an integer and that .

It follows from (6.123), (6.127) and (6.129) that φ(xt ) ⊂ φ(x∗ )

(6.130)

ρ(xt , Ti (xt )) > 3δ0 /2, i ∈ I2 .

(6.131)

.

and .

In view of (6.115), there exists s ∈ φ(xt )

(6.132)

ρ(xt+1 , Ts (xt )) ≤ rt .

(6.133)

.

such that .

Equations (6.125) and (6.128) imply that ρ(xt , xt+1 ) ≤ δ0 /2.

.

(6.134)

278

6 Algorithms Based on Unions of Nonexpansive Maps

By (6.126), (6.128), (6.133) and (6.134), ρ(xt , Ts (xt )) ≤ ρ(xt , xt+1 ) + ρ(xt+1 , Ts (xt ))

.

.

≤ rt + δ0 /2 ≤ 3δ0 /4.

(6.135)

It follows from (6.120), (6.130)–(6.132) and (6.135) that s ∈ I1 , Ts (x∗ ) = x∗ .

(6.136)

.

Assumption (A2) and Eqs. (6.133) and (6.136) imply that ρ(Ts (xt ), x∗ ) ≤ ρ(xt , x∗ ),

.

ρ(xt+1 , x∗ ) ≤ ρ(xt+1 , Ts (xt )) + ρ(Ts (xt ), x∗ )

.

≤ ρ(xt , x∗ ) + rt .

.

Thus we have shown that the following property holds: (c) if .t ≥ q1 is an integer and .ρ(xt , x∗ ) ≤ ɛ , then (6.130) and (6.131) are true and if and integer s satisfies (6.132), (6.133), then .s ∈ I1 and ρ(xt+1 , x∗ ) ≤ ρ(xt , x∗ ) + rt .

.

Fix an integer .τ ≥ q1 such that ∞ Σ .

rt < ɛ /4, ρ(xτ , x∗ ) < ɛ /2.

(6.137)

t=τ

Property (c) and Eqs. (6.136) and (6.137) imply that ρ(xτ +1 , x∗ ) ≤ ρ(xτ , x∗ ) + rτ < 3 ɛ /4.

.

(6.138)

Assume that .j ≥ 1 is an integer and ρ(xτ +j , x∗ ) ≤ ρ(xτ , x∗ ) +

j −1 Σ

.

rτ +i .

(6.139)

i=0

(By (6.138) Eq. (6.139) holds for .j = 1.) Equations (6.136), (6.137) and (6.139) imply that ρ(xτ +j , x∗ ) < ɛ .

.

(6.140)

The Cimmino Algorithm with Summable Errors

279

Property (c) and Eqs. (6.139) and (6.140) imply that ρ(xτ +j +1 , x∗ ) ≤ ρ(xτ +j , x∗ ) + rτ +j

.

.

< ρ(xτ , x∗ ) +

j −1 Σ

rτ +i + rτ +j = ρ(xτ , x∗ ) +

i=0

j Σ

rτ +i .

i=0

Thus by induction we showed that (6.139) holds for each integer .j ≥ 1. By (6.136), (6.137) and (6.139), for each integer .j ≥ 1, ρ(xτ +j , x∗ ) < ρ(xτ , x∗ ) +

j −1 Σ

.

rτ +i < ɛ .

i=0

Since . ɛ i an arbitrary element of .(0, δ1 ) we conclude that .

lim xt = x∗ .

t→∞

Together with property (c) this implies Theorem 6.9.

The Cimmino Algorithm with Summable Errors Suppose that .(X, || · ||) is a normed space and that .C ⊂ X is its nonempty, closed, convex set. For each .x, y ∈ X, set ρ(x, y) = ||x − y||, x, y ∈ X.

.

We use the notation, definitions and assumptions introduced in Sect. “Global Convergence of Iterates”. In particular we assume that assumptions (A1)–(A3) hold. Theorem 6.10 Assume that .M > 0, .c¯ ∈ (0, m−1 ], . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(0, M) = / ∅

.

(6.141)

and .{rt }∞ t=0 ⊂ (0, ∞) satisfies ∞ Σ .

rt < ∞.

(6.142)

t=0

Then there exists an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C and each γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m

.

(6.143)

280

6 Algorithms Based on Unions of Nonexpansive Maps

which satisfy Σ

γt,i = 1, t = 0, 1, . . . ,

.

(6.144)

i∈φ(xt )

||x0 || ≤ M

(6.145)

.

and for each integer .t ≥ 0, Σ

||xt+1 −

.

γt,i Ti (xt )|| ≤ rt

(6.146)

i∈φ(xt )

the inequality ||xt || ≤ 3M +

∞ Σ

.

rj

j =0

holds for all integers .t ≥ 0 and that Card({t ∈ {0, 1, . . . , } : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ }) ≤ Q.

.

Proof In view of (6.141), there exists z∗ ∈ B(0, M) ∩ F¯ (T ).

.

(6.147)

Lemma 6.3 and (6.3) (6.147) imply that there exists .δ ∈ (0, ɛ ) such that the following property holds: Σ (a) for each .s ∈ {1, . . . , m} and each .x ∈ C ∩ B(z∗ , 2M + ∞ j =0 rj ) satisfying ||x − Ts (x)|| > ɛ

.

we have ||z∗ − Ts (x)|| ≤ ||z∗ − x|| − δ.

.

In view of (6.142), there exists a natural number .m0 such that rt < cδ/4 ¯ for each integer t ≥ m0 .

.

(6.148)

Choose a natural number Q ≥ 4Mδ −1 c¯−1 + m0 .

.

(6.149)

The Cimmino Algorithm with Summable Errors

281

Assume that .{xi }∞ i=0 ⊂ C and (6.143)–(6.146) hold. Let .t ≥ 0 be an integer. Assumption (A2), the convexity of the norm and Eqs. (6.144), (6.146) and (6.147) imply that Σ

||z∗ − xt+1 || = ||z∗ −

γt,i Ti (xt )||

.

i∈φ(xt )

.

Σ

+ ||

γt,i Ti (xt ) − xt+1 ||

i∈φ(xt )

Σ

≤

.

γt,i ||z∗ − Ti (xt )|| + rt ≤ ||z∗ − xt || + rt .

(6.150)

i∈φ(xt )

Equations (6.145), (6.147) and (6.150) imply that for each integer .i ≥ 0, ||z∗ − xi || ≤ ||z∗ − x0 || +

∞ Σ

.

rj ≤ 2M +

j =0

||xi || ≤ 3M +

∞ Σ

rj ,

(6.151)

j =0 ∞ Σ

.

rj .

j =0

Assume that .

max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ .

(6.152)

In view of (6.152), there exists s ∈ φ(xt )

(6.153)

||xt − Ts (xt )|| > ɛ .

(6.154)

.

such that .

By (6.150), ||z∗ − xt+1 || ≤

Σ

.

γt,i ||z∗ − Ti (xt )|| + rt .

(6.155)

i∈φ(xt )

Assumption (A2) and Eq. (6.147) imply that for each .i ∈ φ(xt ), ||z∗ − Ti (xt )|| ≤ ||z∗ − xt ||.

.

(6.156)

282

6 Algorithms Based on Unions of Nonexpansive Maps

Property (a) and Eqs. (6.151), (6.153) and (6.154) imply that ||z∗ − Ts (xt )|| ≤ ||z∗ − xt || − δ.

.

(6.157)

It follows from (6.143), (6.144) and (6.155)–(6.157) that ||z∗ − xt+1 || ≤ rt +

.

.

.

Σ {γt,i ||z∗ − Ti (xt )|| : i ∈ φ(xt ) \ {s}} + γt,s ||z∗ − Ts (xt )||

≤ rt + ||z∗ − xt || − δ c. ¯

(6.158)

If .t ≥ m0 , then in view of (5.148), (5.158), ||z∗ − xt+1 || ≤ ||z∗ − xt || − δ c/2. ¯

.

Thus we have shown that the following property holds: (b) if an integer .t ≥ m0 satisfies (6.152), then ||z∗ − xt+1 || ≤ ||z∗ − xt || − δ c/2. ¯

.

Assume that .n ≥ m0 + 1 is an integer. Property (b) and Eqs. (6.143), (6.145), (6.150) and (6.152) imply that 2M ≥ ||z∗ − x0 || ≥ ||z∗ − x0 || − ||z∗ − xn+1 ||

.

.

=

n Σ

(||z∗ − xt || − ||z∗ − xt+1 ||),

t=0

2M +

∞ Σ

.

i=0 .

≥

ri ≥

n Σ (||z∗ − xt || − ||z∗ − xt+1 || + rt ) t=0

Σ {||z∗ − xt || − ||z∗ − xt+1 || + rt :

t ∈ {m0 , . . . , n}, max{||xt − Ti (xt )|| : i ∈ φ(xt )} ≥ ɛ }

.

.

≥ 2−1 δ cCard({t ¯ ∈ {m0 , . . . , n} : max{||xt − Ti (xt )|| : i ∈ φ(xt )} ≥ ɛ })

and Card({t ∈ {m0 , . . . , n} : max{||xt − Ti (xt )|| : i ∈ φ(xt )} ≥ ɛ }) ≤ 4Mδ −1 c¯−1 .

.

The Cimmino Algorithm with Summable Errors

283

Together with (6.149) this implies that Card({t ∈ {0, . . . , n} : max{||xt − Ti (xt )|| : i ∈ φ(xt )} ≥ ɛ })

.

.

≤ m0 + 4Mδ −1 c¯−1 ≤ Q.

Since n is an arbitrary natural number satisfying .n ≥ m0 + 1 we conclude that Card({t ∈ {0, 1, . . . } : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > ɛ }) ≤ Q.

.

Theorem 6.10 is proved. Theorem 6.11 Assume that .c¯ ∈ (0, m−1 ], .{rt }∞ t=0 ⊂ (0, ∞), ∞ Σ .

rt < ∞,

t=0

and that a sequence .{xi }∞ i=0 ⊂ C and for each .t = 0, 1, . . . , i = 1, . . . , m γt,i ≥ c, ¯

.

for each integer .t ≥ 0, Σ .

γt,i = 1

i∈φ(xt )

and Σ

||xt+1 −

.

γt,i Ti (xt )|| ≤ rt .

i∈φ(xt )

Then there exist x∗ = lim xt

.

t→∞

and a natural number .t0 such that for each integer .t ≥ t0 and each .i ∈ φ(xt ), φ(xt ) ⊂ φ(x∗ )

.

and Ti (x∗ ) = x∗ .

.

284

6 Algorithms Based on Unions of Nonexpansive Maps

Proof In view of Theorem 6.10, the sequence .{xt }∞ t=0 is bounded. Assumption (A1) implies that it has a limit point .x∗ ∈ C and a subsequence .{xtk }∞ k=0 such that x∗ = lim xtk .

.

k→∞

By (A3) and by our assumptions, we may assume without loss of generality that φ(xtk ) ⊂ φ(x∗ ), k = 1, 2, . . . .

.

Set I1 = {i ∈ φ(x∗ ) : Ti (x∗ ) = x∗ }, I2 = φ(x∗ ) \ I1 .

.

Fix .δ0 ∈ (0, 1) such that ||x∗ − Ti (x∗ )|| > 2δ0 , i ∈ I2 .

.

Assumption (A3), the continuity of .Ti , i = 1, . . . , m and by the relation above imply that there exists .δ1 ∈ (0, δ0 ) such that for each .x ∈ B(x∗ , δ1 ) ∩ C, φ(x) ⊂ φ(x∗ ),

.

||x − Ti (x)|| > 3δ0 /2, i ∈ I2 .

.

(6.159)

Theorem 6.10 implies that there exists an integer .Q0 ≥ 1 such that Card({t ∈ {0, 1, . . . } : max{||xt − Ti (xt )|| : i ∈ φ(xt )} > δ0 /2}) ≤ Q0 .

.

Clearly, there exists an integer .Q1 ≥ 1 such that for each integer .t ≥ Q1 , .

max{||xt − Ti (xt )|| : i ∈ φ(xt )} ≤ δ0 /2.

(6.160)

ɛ ∈ (0, δ1 ),

(6.161)

t ≥ Q1

(6.162)

||xt − x∗ || < ɛ .

(6.163)

Assume that .

.

is an integer and that .

It follows from (6.158), (6.159), (6.161)–(6.163) that φ(xt ) ⊂ φ(x∗ )

.

(6.164)

The Cimmino Algorithm with Summable Errors

285

and ||xt − Ti (xt )|| > 3δ0 /2, i ∈ I2 .

.

(6.165)

On the other hand, in view of (6.160) and (6.162), ||xt − Ti (xt )|| ≤ δ0 /2, i ∈ φ(xt ).

.

(6.166)

It follows from the definition of .I1 and .I2 (6.165) and (6.166) that φ(xt ) ⊂ I1 .

(6.167)

.

Assumption (A2) and the definition of .I1 and .I2 imply that for each .i ∈ I1 , ||Ti (xt ) − x∗ || ≤ ||xt − x∗ ||.

.

by our assumptions (6.168) and the convexity of the norm, ||x∗ − xt+1 || ≤ ||x∗ −

Σ

.

γt,i Ti (xt )||

i∈φ(xt )

Σ

+ ||

.

γt,i Ti (xt ) − x∗ ||

i∈φ(xt )

.

≤ rt +

Σ

γt,i ||x∗ − Ti (xt )||

i∈φ(xt )

≤

.

Σ

γt,i ||x∗ − xt || + rt

i∈φ(xt ) .

≤ rt + ||xt − x∗ ||.

Thus we have shown that the following property holds: (c) If an integer .t ≥ Q1 and ||xt − x∗ || < ɛ ,

.

then φ(xt ) ⊂ φ(x∗ ), φ0 (xt ) ⊂ I1 ,

.

||xt+1 − x∗ || ≤ ||xt − x∗ || + rt .

.

(6.168)

286

6 Algorithms Based on Unions of Nonexpansive Maps

Assume that τ ≥ Q1

(6.169)

.

is an integer, ∞ Σ .

(6.170)

rt < ɛ /4,

t=τ

||xτ − z∗ || < ɛ /2.

(6.171)

.

Property (c) and Eqs. (6.169)–(6.171) imply that ||xτ +1 − x∗ || ≤ ||xτ − x∗ || + rτ < 3 ɛ /4.

.

(6.172)

Assume that .j ≥ 1 is an integer and that ||xτ +j − x∗ || ≤ ||xτ − x∗ || +

j −1 Σ

.

rτ +i .

(6.173)

i=0

(In view of (6.172) Eq. (6.173) is true for .j = 1.) By (6.170), (6.171) and (6.173), ||xτ +j − x∗ || < ɛ .

.

Property (c) and Eqs. (6.169) and (6.173) imply that ||xτ +j +1 − x∗ || ≤ ||xτ +j − x∗ || + rτ +j

.

.

≤ ||xτ − x∗ || +

j −1 Σ

rτ +i + rτ +j

i=0

.

= ||xτ − x∗ || +

j Σ

rτ +i .

i=0

Thus (6.173) is true for each integer .j ≥ 1 and together (6.170), (6.171) this implies that for each integer .j ≥ 1, ||xτ +j − x∗ || ≤ ||xτ − x∗ || +

j −1 Σ

.

i=0

rτ +i < ɛ .

Iterates with Nonsummable Errors

287

Since . ɛ is an arbitrary element of .(0, δ1 ) we conclude that .

lim xt = x∗ .

t→∞

Combined with property (c) this completes the proof of Theorem 6.11.

Iterates with Nonsummable Errors Suppose that .(X, ρ) is a metric space and that .C ⊂ X is its nonempty, closed set. We use the notation, notation and definitions introduced in Sect. “Global Convergence of Iterates”. In particular we assume that assumptions (A1)–(A3) hold. Theorem 6.12 Assume that .M > 1, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) /= ∅,

.

(6.174)

Then there exist .δ ∈ (0, ɛ ) and an integer .Q > 2 such that for each sequence {xi }∞ i=Q ⊂ C which satisfy

.

ρ(x0 , θ ) ≤ M

(6.175)

ρ(xt+1 , T (xt )) ≤ δ

(6.176)

.

and for each .t = 0, . . . , Q − 1, .

there exists an integer .p ∈ [2, Q) such that ρ(xt , θ ) ≤ 3M + 1, t = 1, . . . , p

.

and ρ(xp , xp+1 ) ≤ ɛ .

.

Proof In view of (6.174), there exists z∗ ∈ F¯ (T ) ∩ B(θ, M).

.

(6.177)

Lemma 6.3 implies that there exists .δ ∈ (0, ɛ /2) such that the following property holds: (i) for each .s ∈ {1, . . . , m} and each .x ∈ C ∩ B(θ, 3M + 1) satisfying ρ(x, Ts (x)) > ɛ /4

.

288

6 Algorithms Based on Unions of Nonexpansive Maps

we have ρ(z∗ , Ts (x)) ≤ ρ(z∗ , x) − 2δ.

.

Choose a natural number Q ≥ 2 + δ −1 (2M + 1).

.

(6.178)

Assume that .{xi }Q i=0 ⊂ C, (6.175) holds and (6.176) holds for each .t ∈ 0, . . . , Q − 1. By (6.176), for each integer .t ∈ {0, . . . , Q − 1}, there exists Tt ∈ {Ts : s ∈ φ(xt )}

(6.179)

ρ(xt+1 , Tt (xt )) ≤ δ.

(6.180)

.

such that .

Assumption (A2) and Eqs. (6.3), (6.175), (6.177), and (6.180) imply that ρ(z∗ , x1 ) ≤ ρ(z∗ , T0 (x0 )) + ρ(T0 (x0 ), x1 ) ≤ ρ(z∗ , x0 ) + δ ≤ 2M + 1,

.

(6.181)

ρ(θ, x0 ) ≤ 3M + 1.

.

Assume that .s ∈ {1, . . . , Q − 1} and that for each .t ∈ {1, . . . , s}, ρ(xt , xt+1 ) > ɛ .

.

(6.182)

Assume that .t ∈ {1, . . . , s} and that ρ(xt , z∗ ) ≤ 2M + 1.

.

(6.183)

(In view of (6.181), Eq. (6.183) holds for .t = 1.) By (6.179), (6.180) and (6.182), ρ(xt , Tt (xt )) ≥ ρ(xt , xt+1 ) − ρ(xt+1 , Tt (xt )) > ɛ − δ ≥ ɛ /2.

.

(6.184)

Property (i) and Eqs. (6.177), (6.179), (6.183) and (6.184) imply that ρ(z∗ , Tt (xt )) ≤ ρ(z∗ , xt ) − 2δ.

.

By (6.180) and (6.185), ρ(z∗ , xt+1 ) ≤ ρ(z∗ , Tt (xt )) + ρ(Tt (xt )), xt+1 )

.

.

≤ ρ(z∗ , xt ) − 2δ + δ = ρ(z∗ , xt ) − δ.

(6.185)

Iterates with Nonsummable Errors

289

Thus we have shown that the following property holds: (ii) if an integer .t ∈ {1, . . . , s} satisfies (6.183), then ρ(z∗ , xt+1 ) ≤ ρ(z∗ , xt ) − δ.

.

By induction, property (ii) and (6.181) imply that for all .t = 1, . . . , s + 1, ρ(z∗ , xt ) ≤ 2M + 1

(6.186)

ρ(z∗ , xt+1 ) ≤ ρ(z∗ , xt ) − δ.

(6.187)

.

and for all .t = 1, . . . , s, .

It follows from (6.178), (6.181) and (6.187) that 2M + 1 ≥ ρ(z∗ , x1 ) ≥ ρ(z∗ , x1 ) − ρ(z∗ , xs+1 )

.

.

=

s Σ

(ρ(z∗ , xt ) − ρ(z∗ , xt+1 )) ≥ δs

t=1

and s ≤ (2M + 1)δ −1 ≤ Q − 2.

.

This implies that there exists a natural number p such that 2 ≤ p ≤ (2M + 1)δ −1 + 1,

.

for each .t ∈ {1, . . . , p}, ρ(xt , z∗ ) ≤ 2M + 1

.

and ρ(xp , xp+1 ) ≤ ɛ .

.

Theorem 6.12 is proved. Theorem 6.13 Assume that .M > 1, .r0 ∈ (0, 1), F¯ (T ) ∩ B(θ, M) /= ∅,

.

{x ∈ C : ρ(x, T (x)) ≤ r0 } ⊂ B(θ, M),

.

ɛ ∈ (0, r0 , 2)

.

(6.188)

290

6 Algorithms Based on Unions of Nonexpansive Maps

Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence {xi }∞ i=0 ⊂ C which satisfies

.

ρ(x0 , θ ) ≤ M

(6.189)

ρ(xt+1 , T (xt )) ≤ δ

(6.190)

.

.

for every integer .t ≥ 0, the inequality ρ(xt , θ ) ≤ 3M + 1, t = 1, 2, . . .

.

is true and there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

.

and ρ(xqp , xqp +1 ) ≤ ɛ .

.

Proof Let .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 be as guaranteed by Theorem 6.12. Assume that .{xi }∞ i=0 ⊂ C satisfies (6.189) and (6.190) for each integer .t ≥ 0. Theorem 6.12, the choice of .δ, Q and Eqs. (6.189) and (6.190) imply that there exists a natural number .q1 ≤ Q such that ρ(xt , θ ) ≤ 3M + 1, t = 1, . . . , q1 + 1,

.

ρ(xq1 , xq1 +1 ) ≤ ɛ .

.

Assume that .p ≥ 1 is an integer and that we defined natural numbers .qi , .i = 1, . . . , p such that 1 ≤ q1 ≤ Q

.

for every integer .i ∈ {1, . . . , p} \ {p}, 1 ≤ qp+1 − qp ≤ Q,

.

ρ(xt , θ ) ≤ 3M + 1, t = 1, . . . , qp + 1

.

(6.191)

Iterates with Nonsummable Errors

291

and ρ(xqi , xqi +1 ) ≤ ɛ , i = 1, . . . , p.

.

(6.192)

(Clearly, for .p = 1, our assumption holds.) Consider the sequence .{xt+qp }∞ t=0 . In view of (6.190), there exists s ∈ φ(xqp )

.

such that ρ(xqp +1 , Ts (xqp )) ≤ δ < ɛ .

.

(6.193)

By (6.192) and (6.193), ρ(xqp , Ts (xqp )) ≤ ρ(xqp , xqp +1 ) + ρ(xqp +1 , Ts (xqp ))

.

.

≤ δ + ɛ < 2 ɛ ≤ r0 .

Together with (6.188) this implies that ρ(xqp , θ ) ≤ M.

.

(6.194)

By the choice of .δ, Q, Eqs. (6.190), (6.194) and Theorem 6.12 applied to the sequence .{xt+qp }∞ t=0 we obtain that there exists qp+1 ∈ {qp + 1, . . . , qp + Q}

.

such that ρ(xt , θ ) ≤ 3M + 1, t = qp + 1, . . . , qp+1 + 1,

.

ρ(xqp+1 , xqp+1 +1 ) ≤ ɛ

.

and our assumption holds for .p + 1 too. Thus by induction Theorem 6.13 is proved. Applying Theorem 6.12 by induction we easily obtain the the following result is true. Theorem 6.14 Assume that .M > 1, F¯ (T ) ∩ B(θ, M) /= ∅,

.

292

6 Algorithms Based on Unions of Nonexpansive Maps

ɛ ∈ (0, 1). Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfies

.

ρ(xt , θ ) ≤ M, t = 0, 1, 2, . . .

.

ρ(xt+1 , T (xt )) ≤ δ

.

for every integer .t ≥ 0, there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

.

and ρ(xqp , xqp +1 ) ≤ ɛ .

.

In this chapter we prove the following result. Theorem 6.15 Assume that the following property holds: (A4) If .x ∈ C satisfies .x ∈ T (x), then for each .i ∈ φ(x), Ti (x) = x.

.

Let .M ≥ 1 be an integer, F¯ (T ) ∩ B(θ, M) /= ∅,

.

(6.195)

ɛ ∈ (0, 1). Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfies

.

ρ(xt , θ ) ≤ M, t = 1, 2, . . .

.

ρ(xt+1 , T (xt )) ≤ δ

.

for every integer .t ≥ 0, the inequality ρ(xt , xt+1 ) ≤ ɛ

.

holds for all integer .t ≥ Q. Theorems 6.13 and 6.15 imply the following result.

Auxiliary Results for Theorem 6.15

293

Theorem 6.16 Assume that (A4) holds, .M > 1, F¯ (T ) ∩ B(θ, M) /= ∅,

.

r0 ∈ (0, 1),

.

{x ∈ C : ρ(x, T (x)) ≤ r0 } ⊂ B(θ, M),

.

ɛ ∈ (0, 1). Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfies

.

ρ(xt , θ ) ≤ M, t = 0, 1, . . .

.

ρ(xt+1 , T (xt )) ≤ δ

.

for every integer .t ≥ 0, the inequality ρ(xt , xt+1 ) ≤ ɛ

.

holds for all integer .t ≥ Q.

Auxiliary Results for Theorem 6.15 For each .x ∈ X and each .r > 0 set B 0 (x, r) = {y ∈ X ρ(x, y) < r}.

.

In this section we continue to assume that (A4) holds. The next result follows easily from the continuity of the mappings .Ti , .i = 1, . . . , m. Proposition 6.17 The set .{x ∈ C : x ∈ T (x)} is closed. Let .x ∈ C satisfy x ∈ T (x).

.

Assumption (A3) implies that there exists .rx ∈ (0, 1) such that for each .y ∈ B(x, rx ), φ(y) ⊂ φ(x).

.

(6.196)

294

6 Algorithms Based on Unions of Nonexpansive Maps

Assumptions (A2) and (A4) imply the following property holds: (A5) For each .x ∈ C satisfy .x ∈ T (x), each .y ∈ B(x, rx ), each .i ∈ φ(y), the inclusion .i ∈ φ(x) holds and ρ(Ti (y), x) ≤ ρ(y, x).

.

Lemma 6.18 Let .M > 1 be an integer and . ɛ ∈ (0, 1). Then there exists .δ ∈ (0, ɛ ) such that for each .x ∈ B(θ, M) ∩ C and each .i ∈ φ(x) satisfying ρ(x, Ti (x)) ≤ δ

.

there exists z ∈ B(θ, M + 1) ∩ C

.

such that Ti (z) = z

.

and ρ(z, x) ≤ ɛ .

.

Proof Assume the contrary. Then for each natural number k there exist xk ∈ B(θ, M) ∩ C, ik ∈ φ(xk )

.

such that ρ(xk , Tik (xk )) ≤ k −1 ɛ ,

.

{z ∈ B(xk , ɛ ) ∩ C : Tik (z) = z} = ∅.

.

(6.197)

In view of (A1), extracting a subsequence and re-indexing we may assume without loss of generality that ik = i1 , k = 1, 2, . . .

.

and that there exists x∗ = lim xk .

.

k→∞

By the continuity of .Ti1 and the equation above, x∗ = Ti1 (x∗ ).

.

Auxiliary Results for Theorem 6.15

295

This contradicts (6.197). The contradiction we have reached proves Lemma 6.18. Let .M, q be natural numbers and ΩM = {x ∈ C ∩ B(θ, M) : x ∈ T (x)}

.

(6.198)

which is compact by Proposition 6.17. Clearly, ΩM ⊂ ∪{B 0 (x, (4q)−1 rx ) : x ∈ ΩM }.

.

Since the set .ΩM is compact there exist a natural number .p(M, q) and (M,q)

zi

.

∈ ΩM , i = 1, . . . , p(M, q)

(6.199)

such that , (4q)−1 rz(M,q) )}.

(6.200)

r (M,q) = min{(4q)−1 rz(M,q) : i = 1, . . . , p(M, q)}.

(6.201)

p(M,q)

ΩM ⊂ ∪i=1

.

(M,q)

{B 0 (zi

i

Set .

i

Proposition 6.19 Let .M > 1, q ≥ 1 be integers, . ɛ ∈ (0, 1), q −1 < ɛ /8,

(6.202)

δ0 ∈ (0, r (M,q)/4 )

(6.203)

.

let (A4) hold and .

be such the following property holds: (a) for each .x ∈ B(θ, M) ∩ C satisfying .ρ(x, T (x)) ≤ δ0 there exists z ∈ B(θ, M + 1) ∩ C

.

such that .z ∈ T (z) and ρ(z, x) ≤ r (M,q) /4

.

(it exists by Lemma 6.18).

296

6 Algorithms Based on Unions of Nonexpansive Maps

Assume that Q is a natural number. Then there exists .δ ∈ (0, δ0 ) such that for Q each finite sequence .{xt }t=0 ⊂ C satisfying ρ(x0 , θ ) ≤ M,

(6.204)

ρ(x0 , T (x0 )) ≤ δ0 ,

(6.205)

B(xt+1 , δ) ∩ T (xi ) /= ∅

(6.206)

.

.

for each .t ∈ {0, . . . , Q − 1}, .

the inequality .ρ(xi+1 , xi ) ≤ ɛ holds for all .t = 0, . . . , Q − 1. Proof Choose a positive number .δ such that δ < min{δ0 , (8Q)−1 r (M,q) }.

.

(6.207)

Q

Assume that a finite sequence .{xt }t=0 ⊂ C satisfies (6.204), (6.205) and (6.206) for each .t ∈ {0, . . . , Q − 1}. By (6.205), there exists s˜0 ∈ φ(x0 )

(6.208)

ρ(x0 , Ts˜0 (x0 ))) ≤ δ0 .

(6.209)

.

such that .

In view of (6.206), for each .t ∈ {0, . . . , Q − 1}, there exists s(t) ∈ φ(xt ))

(6.210)

ρ(xt+1 , Ts(t) (xt )) ≤ δ.

(6.211)

.

such that .

Property (a) and Eqs. (6.204) and (6.205) imply that there exists z ∈ ΩM

(6.212)

ρ(z, x0 ) ≤ r (M,q) .

(6.213)

.

such that .

By (6.200) and (6.212), there exists j ∈ {1, . . . , p(M, q)}

.

Auxiliary Results for Theorem 6.15

297

such that (M,q)

ρ(z, zj

.

) < (4q)−1 rz(M,q) .

(6.214)

j

In view of (6.201), (6.213) and (6.214), (M,q)

ρ(x0 , zj

.

.

(M,q)

) ≤ ρ(x0 , z) + ρ(z, zj

)

< r (M,q) /4 + (4q)−1 rz(M,q) < (3q)−1 rz(M,q) . j

(6.215)

j

It follows from (6.196), (6.198), (6.199), (6.208) and (6.215), (M,q)

s(0) ∈ φ(zj

.

).

Assumptions (A2) and (A4) and Eqs. (6.199) and (6.215) and the equation above imply that (M,q)

ρ(Ts(0) (x0 ), zj

.

(M,q)

) ≤ ρ(x0 , zj

) ≤ (3q)−1 rz(M,q) .

(6.216)

j

By (6.211) and (6.216), (M,q)

ρ(x1 , zj

.

(M,q)

) ≤ ρ(x1 , Ts(0) (x0 )) + ρ(Ts(0) (x0 ), zj ≤ δ + (3q)−1 rz(M,q) .

.

) (6.217)

j

We show that for all .k = 0, . . . , Q, (M,q)

ρ(xk , zj

.

) ≤ kδ + (3q)−1 rz(M,q) .

(6.218)

j

(By (6.215) and (6.217) Eq. (6.218) holds for .k = 0, 1.) Assume that .k < Q is a natural number and (6.218). By (6.201), (6.207) and (6.218), (M,q)

ρ(xk , zj

.

.

) ≤ (3q)−1 rz(M,q) + Qδ j

≤ (3q)−1 rz(M,q) + 8−1 r (M,q) ≤ 2−1 rz(M,q) . j

(6.219)

j

By (6.196) and (6.219), (M,q)

φ(xk ) ⊂ φ(zj

.

).

(6.220)

298

6 Algorithms Based on Unions of Nonexpansive Maps

Assumptions (A2), (A4) and (6.220) imply that (M,q)

ρ(Ts(k) (xk ), zj

.

(M,q)

) ≤ ρ(xk , zj

(6.221)

).

In view of (6.211), (6.218) and (6.221), (M,q)

ρ(xk+1 , zj

.

.

(M,q)

) ≤ ρ(xk+1 , Ts(k) (xk )) + ρ(Ts(k) (xk ), zj (M,q)

≤ ρ(xk , zj

)

) + δ ≤ (3q)−1 rz(M,q) + (k + 1)δ. j

Thus by induction we showed that (6.218) holds for all .k = 1, . . . , Q. By (6.201), (6.202), (6.207) and (6.218), for all integers .k = 1, . . . , Q, (M,q)

ρ(xk , zj

.

) ≤ (3q)−1 rz(M,q) + Qδ < (2q)−1 rz(M,q) . j

(6.222)

j

By (6.202) and (6.222), for all .k = 0, . . . , Q − 1, (M,q)

ρ(xk+1 , xk ) ≤ ρ(xk , zj

.

(M,q)

) + ρ(xk+1 , zj

) ≤ q −1 rz(M,q) < ɛ . j

Proposition 6.19 is proved. Proof of Theorem 6.15 Choose an integer .q ≥ 1 such that q −1 < ɛ /8.

.

(6.223)

Lemma 6.18 implies that there exists δ0 ∈ (0, r (M,q) /4)

.

(6.224)

such that property (a) of Proposition 6.19 holds. Theorem 6.14 implies that there exist .δ1 ∈ (0, δ0 /2) and an integer .Q ≥ 1 such that the following property holds: (b) for each sequence .{xi }∞ i=0 ⊂ C which satisfies ρ(xt , θ ) ≤ M, t = 1, 2, . . .

.

ρ(xt+1 , T (xt )) ≤ δ1

.

for every integer .t ≥ 0, there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q

.

Auxiliary Results for Theorem 6.15

299

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

.

and ρ(xqp , xqp +1 ) ≤ δ0 /2.

.

Proposition 6.19 implies that there exists .δ ∈ (0, δ1 ) such that the following property holds: Q (c) for each finite sequence .{xt }t=0 ⊂ C satisfying ρ(x0 , θ ) ≤ M, ρ(x0 , T (x0 )) ≤ δ0 ,

.

for each .t ∈ {0, . . . , Q − 1}, B(xt+1 , δ) ∩ T (xi ) /= ∅

.

the inequality .ρ(xt+1 , xt ) ≤ ɛ holds for all .t = 0, . . . , Q − 1. Assume that .{xi }∞ i=0 ⊂ C satisfies ρ(xt , θ ) ≤ M, t = 0, 1, . . . ,

(6.225)

ρ(xt+1 , T (xt )) ≤ δ, t = 0, 1, . . .

(6.226)

.

.

Property (b) and Eqs. (6.225) and (6.226) imply that there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

(6.227)

ρ(xqp , xqp +1 ) ≤ δ0 /2.

(6.228)

.

and .

Q

Let .p ≥ 1 be an integer. Property (c) applied to the sequence .{xi+qp }i=0 and (6.225), (6.226) and (6.228) imply that ρ(xqp , T (xqp )) ≤ ρ(xqp , xqp +1 ) + ρ(xqp +1 , T (xqp )) ≤ δ0 /2 + δ ≤ δ,

.

ρ(xt , xt+1 ) ≤ ɛ , t = qp , . . . , qp + Q.

.

300

6 Algorithms Based on Unions of Nonexpansive Maps

In view of (6.227), ρ(xt , xt+1 ) ≤ ɛ , t = qp , . . . , qp+1 .

.

Since p is any natural number we conclude that ρ(xt , xt+1 ) ≤ ɛ

.

for every integer .t ≥ q1 . Theorem 6.15 is proved.

The Cimmino Algorithm with Nonsummable Errors Let .(X, || · ||) be a normed space equipped with a norm .|| · ||. For each .x, y ∈ X set ρ(x, y) = ||x − y||.

.

We use the notation, notation and definitions introduced in Sect. “Global Convergence of Iterates”. In particular we assume that assumptions (A1)–(A3) hold. Let c¯ ∈ (0, m−1 ).

.

Theorem 6.20 Assume that .M > 1, . ɛ ∈ (0, 1) and that F¯ (T ) ∩ B(θ, M) = / ∅.

.

Then there exist .δ ∈ (0, ɛ ) and an integer .Q > 2 such that for each sequence Q {xi }i=0 ⊂ C and each

.

γt,i ≥ c, ¯ t = 0, . . . , Q − 1, i = 1, . . . , m

.

satisfying ||x0 || ≤ M,

.

(6.229)

for each integer .t = 0, . . . , Q − 1, Σ .

γt,i = 1

(6.230)

γt,i Ti (xt )|| ≤ δ

(6.231)

i∈φ(xt )

and ||xt+1 −

Σ

.

i∈φ(xt )

The Cimmino Algorithm with Nonsummable Errors

301

there exists a nonnegative integer .s < Q such that ||xt || ≤ 3M + 1, t = 1, . . . , s + 1,

.

for each .t ∈ {0, . . . , s}, .

max{||Ti (xs ) − xs || : i ∈ φ(xs )} > ɛ

and .

max{||Ti (xs+1 ) − xs+1 || : i ∈ φ(xs+1 )} ≤ ɛ ,

Proof Fix z∗ ∈ F¯ (T ) ∩ B(0, M).

(6.232)

Ti (z∗ ) = z∗ , i = 1, . . . , m.

(6.233)

.

In view of (6.3) and (6.232), .

By (6.233), there exists .δ0 ∈ (0, ɛ /2) such that the following property holds: (a) for each .i ∈ {1, . . . , m} and each .x ∈ C ∩ B(0, 3M + 1) satisfying ||x − Ti (x)) > ɛ

.

we have ||z∗ − Ti (x)|| ≤ ||z∗ − x|| − δ0 .

.

Choose a natural number Q ≥ 1 + 2δ0−1 (2M + 1)c¯−1

(6.234)

δ < δ0 c/2. ¯

(6.235)

.

and a positive number .

Q

Assume that .{xi }i=0 ⊂ C, (6.229) holds, γt,i ≥ c, ¯ t = 0, . . . , Q − 1, i = 1, . . . , m,

.

302

6 Algorithms Based on Unions of Nonexpansive Maps

for each .t ∈ 0, . . . , Q − 1, (6.230), (6.231) hold. Assumption (A2), the convexity of the norm and Eqs. (6.229)–(6.233) imply that ||z∗ − x1 || ≤ ||z∗ −

Σ

.

i∈φ(x0 )

.

≤

Σ

γ0,i Ti (x0 )|| + ||x1 −

Σ

γ0,i Ti (x0 )||

i∈φ(x0 )

γ0,i ||z∗ − Ti (x0 )|| + δ

i∈φ(x0 ) .

= ||z∗ − x0 || + 1 ≤ 2M + 1

(6.236)

and ||x1 || ≤ 3M + 1.

.

Assume that s ∈ {1, . . . , Q − 1}

.

and for each .t ∈ {1, . . . , s}, .

max{||Ti (xt ) − xt || : i ∈ φ(xt )} > ɛ .

(6.237)

Assume that .t ∈ {1, . . . , s} and that ||xt − z∗ || ≤ 2M + 1.

(6.238)

.

(In view of (6.236), Eq. (6.238) holds for .t = 1.) In view of (6.232) and (6.238), ||xt || ≤ 3M + 1.

(6.239)

.

Equation (6.237) implies that there exists a number s such that s ∈ φ(xt ), ||Ts (xt ) − xt || > ɛ .

(6.240)

.

It follows from Assumption (A2), property (a), the convexity of the norm, (6.230), (6.231), (6.233), (6.235), (6.239), (6.240) and the relation γt,i ≥ c, ¯ t = 0, . . . , Q − 1, i = 1, . . . , m

.

that ||z∗ − xt+1 || ≤ ||xt+1 −

Σ

.

i∈φ(xt )

γt,i Ti (xt )|| + ||

Σ i∈φ(xt )

γt,i Ti (xt ) − z∗ ||

The Cimmino Algorithm with Nonsummable Errors

Σ

≤δ+

.

303

γt,i ||z∗ − Ti (xt )|| + δ

i∈φ(xt )

=δ+

.

.

Σ {γt,i ||z∗ − Ti (xt )|| : i ∈ φ(xt ) \ {s}} + γt,s ||z∗ − Ts (xt )||

≤ δ + ||z∗ − xt || .

Σ

{γt,i : i ∈ φ(xt ) \ {s}} + γt,s (||z∗ − xt || − δ0 )

≤ δ + ||z∗ − xt || − δ0 c¯ ≤ ||z∗ − xt || − δ0 c/2. ¯

Thus we have shown that the following property holds: if (6.238) holds, then ||z∗ − xt+1 || ≤ ||z∗ − xt || − δ0 c/2. ¯

.

By induction, we obtain that for all .t = 1, . . . , s + 1, ||z∗ − xt || ≤ 2M + 1

.

(6.241)

and ||z∗ − xt+1 || ≤ ||z∗ − xt || − δ¯0 /2, t = 1, . . . , s.

.

(6.242)

It follows from (6.236) and (6.242) that 2M + 1 ≥ ||z∗ − x1 || ≥ ||z∗ − x1 || − ||z∗ − xs+1 ||

.

.

=

s Σ (||z∗ − xt || − ||z∗ − xt+1 ||) ≥ cδ ¯ 0 s/2 t=1

and s ≤ 2(2M + 1)δ0−1 c¯−1 < Q.

.

Together with (6.241) and (6.242) this implies that there exists a nonnegative integer s ≤ 2(2M + 1)δ0−1 c¯−1

.

such that for each .t ∈ {0, . . . , s} \ {0}, (6.237) holds, ||xt − z∗ || ≤ 2M + 1, t = 1, . . . , s + 1

.

304

6 Algorithms Based on Unions of Nonexpansive Maps

and .

max{||Ti (xs+1 ) − xs+1 || : i ∈ φ(xs+1 )} ≤ ɛ ,

Theorem 6.20 is proved. Theorem 6.21 Assume that .M > 1, .r0 ∈ (0, 1), .c¯ ∈ (0, m−1 ), F¯ (T ) ∩ B(0, M) /= ∅,

.

{x ∈ C : max{ρ(x, Ti (x)) : i ∈ φ(x)} ≤ r0 } ⊂ B(0, M),

.

(6.243)

ɛ ∈ (0, r0 , 2) Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C which satisfies

.

||x0 || ≤ M

.

and each γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m

.

satisfying for each integer .t = 0, 1, . . . , Σ .

γt,i = 1

(6.244)

γt,i Ti (xt )|| ≤ δ

(6.245)

i∈φ(xt )

and Σ

||xt+1 −

.

i∈φ(xt )

the inequality .||xt || ≤ 3M + 1 holds for all integers .t ≥ 0 and there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

.

and .

max{||Ti (xqp ) − xqp || : i ∈ φ(xqp )} ≤ ɛ .

The Cimmino Algorithm with Nonsummable Errors

305

Proof Let .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 be as guaranteed by Theorem 6.20. Assume that .{xi }∞ i=0 ⊂ C, γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m

.

and (6.244), (6.245) hold each integer .t ≥ 0. Theorem 6.20, the choice of .δ, Q and Eqs. (6.244) and (6.245) imply that there exists a natural number .q1 ≤ Q such that ||xt || ≤ 3M + 1, t = 0, . . . , q1 ,

.

max{||Ti (xq1 ) − xq1 || : i ∈ φ(xq1 )} ≤ ɛ .

.

Assume that .p ≥ 1 is an integer and that we defined natural numbers .qi , .i = 1, . . . , p such that ||xt || ≤ 3M + 1, t = 0, . . . , qp ,

.

1 ≤ q1 ≤ Q,

.

for every integer .i ∈ {1, . . . , p} \ {p}, 1 ≤ qp+1 − qp ≤ Q

.

and .

max{||Ti (xqj ) − xqj || : i ∈ φ(xqj )} ≤ ɛ , j = 1, . . . , p.

(6.246)

(Clearly, for .p = 1, our assumption holds.) Consider the sequence .{xt+qp }∞ t=0 . In view of (6.243), (6.246) and the inequality . ɛ < r0 /2, ||xqp || ≤ M.

.

By the choice of .δ, Q, Eqs. (6.244), (6.245) and Theorem 6.20 applied to the sequence .{xt+qp }∞ t=0 we obtain that there exists qp+1 ∈ {qp + 1, . . . , qp + Q}

.

such that ||xt || ≤ 3M + 1, t = qp + 1, . . . , qp+1 ,

.

.

max{||Ti (xqp+1 ) − xqp+1 || : i ∈ φ(xqp+1 )} ≤ ɛ

and our assumption holds for .p + 1 too. Thus by induction Theorem 6.21 is proved.

306

6 Algorithms Based on Unions of Nonexpansive Maps

Applying by induction Theorem 6.20 we obtain the following result. Theorem 6.22 Assume that .M > 1, .c¯ ∈ (0, m−1 ), F¯ (T ) ∩ B(0, M) /= ∅.

.

Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence {xi }∞ i=0 ⊂ C and each

.

γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m

.

satisfying for each integer .t = 0, 1, . . . , ||xt || ≤ M,

.

Σ .

γt,i = 1

i∈φ(xt )

and Σ

||xt+1 −

.

γt,i Ti (xt )|| ≤ δ

i∈φ(xt )

there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q,

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

.

and .

max{||Ti (xqp ) − xqp || : i ∈ φ(xqp )} ≤ ɛ .

In this chapter we prove the following result. Theorem 6.23 Assume that the following property holds (see Theorem 6.15): (A4) If .x ∈ C satisfies .x ∈ T (x), then for each .i ∈ φ(x), Ti (x) = x.

.

Let .M > 1, .c¯ ∈ (0, m−1 ), F¯ (T ) ∩ B(θ, M) /= ∅,

.

The Cimmino Algorithm with Nonsummable Errors

307

ɛ ∈ (0, 1). Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C and each

.

γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m

.

satisfying for each integer .t = 0, 1, . . . ||xt || ≤ M,

.

Σ .

γt,i = 1

i∈φ(xt )

and ||xt+1 −

Σ

.

γt,i Ti (xt )|| ≤ δ

i∈φ(xt )

the inequality .

max{||Ti (xt ) − xt || : i ∈ φ(xt )} ≤ ɛ

holds for all integers .t ≥ Q. Theorems 6.21 and 6.23 imply the following result. Theorem 6.24 Assume that (A4) holds, .M > 1, .c¯ ∈ (0, m−1 ), F¯ (T ) ∩ B(0, M) /= ∅,

.

r0 ∈ (0, 1),

.

{x ∈ C : max{ρ(x, Ti (x)) : i ∈ φ(x)} ≤ r0 } ⊂ B(0, M),

.

ɛ ∈ (0, r0 , 2). Then there exist .δ ∈ (0, ɛ ) and an integer .Q ≥ 1 such that for each sequence .{xi }∞ i=0 ⊂ C and each

.

γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m

.

satisfying for each integer .t = 0, 1, . . . , ||x0 || ≤ M,

.

Σ .

i∈φ(xt )

γt,i = 1

308

6 Algorithms Based on Unions of Nonexpansive Maps

and Σ

||xt+1 −

.

γt,i Ti (xt )|| ≤ δ

i∈φ(xt )

the inequalities ||xt || ≤ 3M + 1

.

and .

max{||Ti (xt ) − xt || : i ∈ φ(xt )} ≤ ɛ

hold for all integers .t ≥ Q.

Auxiliary Results for Theorem 6.23 Lemma 6.25 Let M, ɛ > 0 and (A4) hold. Then there exists δ > 0 such that for each x ∈ B(0, M) ∩ C satisfying ||x − Ti (x)|| ≤ δ, i ∈ φ(x)

.

there exists z ∈ C ∩ B(x, ɛ )

.

such that φ(x) ⊂ φ(z),

.

Ti (z) = z, i ∈ φ(x).

.

Proof Assume the contrary. Then for each natural number k there exist xk ∈ B(0, M) ∩ C

(6.247)

||xk − Ti (xk )|| ≤ k −1 , i ∈ φ(xk ),

(6.248)

.

such that .

{z ∈ B(xk , ɛ ) ∩ C : Ti (z) = z, i ∈ φ(z),

.

φ(xk ) ⊂ φ(z)} = ∅.

.

(6.249)

Auxiliary Results for Theorem 6.23

309

In view of (A1), (A3) and (6.247), we may assume without loss of generality that there exists x∗ = lim xk

.

k→∞

(6.250)

and for all sufficiently large natural numbers k, φ(xk ) ⊂ φ(x∗ ).

.

(6.251)

We may assume without loss of generality that (6.251) holds for each integer k ≥ 1. By (6.250), there exists a natural number k0 such that for each integer k ≥ k0 , x∗ ∈ B(xk , ɛ ).

.

Assumption (A4) and (6.249) imply that ||Ti (x∗ ) − x∗ || > 0, i ∈ φ(x∗ ).

.

In view of (6.250) and the continuity of Ts , s = 1, . . . , m for each i ∈ φ(x∗ ), .

lim ||Ti (xk ) − xk || = ||Ti (x∗ ) − x∗ || > 0.

k→∞

This contradicts (6.248) and (6.251). The contradiction we have reached proves Lemma 6.25. For each x ∈ C set φ0 (x) = {i ∈ φ(x) : Ti (x) = x}.

.

(6.252)

Assumption (A4), if x ∈ C and φ0 (x) /= ∅, then φ0 (x) = φ(x).

.

Lemma 6.26 Let x ∈ C and (A4) hold. Then there exists r > 0 such that for each y ∈ B(x, r) ∩ C, the inclusions φ(y) ⊂ φ(x), φ0 (y) ⊂ φ0 (x)

.

hold. Proof Assumption (A3) implies that there exists r0 > 0 such that φ(y) ⊂ φ(x) for all y ∈ B(x, r0 ) ∩ C.

.

310

6 Algorithms Based on Unions of Nonexpansive Maps

We may assume without loss of generality that φ0 (x) /= φ(x).

.

By (A4), φ0 (x) = ∅.

.

For each i ∈ φ(x), ||x − Ti (x)|| > 0

.

and since the operator Ti is continuous there exists ri > 0 such that ||y − Ti (y)|| > 2−1 ||x − Ti (x)|| > 0, y ∈ B(x, ri ) ∩ C.

.

(6.253)

Set r = min{ri : i ∈ {0} ∪ φ(x)}.

.

It is clear that for each y ∈ B(x, r) ∩ C, φ(y) ⊂ φ(x),

.

for each i ∈ φ(x), (6.253) holds and φ0 (y) = ∅.

.

Lemma 6.26 is proved.

Proof of Theorem 6.23 Lemma 6.26 implies that for each .x ∈ C there exists .rx ∈ (0, 1) such that for each y ∈ B(x, rx ) ∩ C, the inclusions

.

φ(y) ⊂ φ(x), φ0 (y) ⊂ φ0 (x)

.

hold. Recall that .M, q are natural numbers and ΩM = {x ∈ C ∩ B(0, M + 1) : x ∈ T (x)}

.

Proof of Theorem 6.23

311

which is compact. There exist a natural number .p(M, q) and (M,q)

zi

.

∈ ΩM , i = 1, . . . , p(M, q)

such that , (4q)−1 rz(M,q) )}.

(6.254)

r (M,q) = min{(4q)−1 rz(M,q) : i = 1, . . . , p(M, q)}.

(6.255)

p(M,q)

ΩM ⊂ ∪i=1

.

(M,q)

{B 0 (zi

i

Set .

i

Proposition 6.27 Let .M ≥ 1, q ≥ 1 be integers, . ɛ ∈ (0, 1), .c¯ ∈ (0, m−1 ) q −1 < ɛ /8,

(6.256)

δ0 ∈ (0, r (M,q)/4 )

(6.257)

.

let (A4) hold and .

be such the following property holds: (a) for each .x ∈ B(0, M) ∩ C satisfying .||x − Ti (x)|| ≤ δ0 , i ∈ φ(x) there exists z ∈ B(0, M + 1) ∩ C

.

such that φ(x) ⊂ φ(z),

.

||z − x|| ≤ r (M,q) /4, Ti (z) = z, i ∈ φ0 (z)

.

(it exists by Lemma 6.25). Assume that Q is a natural number. Then there exists .δ ∈ (0, δ0 ) such that for Q each finite sequence .{xt }t=0 ⊂ C and each γt,i ≥ c, ¯ t = 0, 1, . . . , Q i = 1, . . . , m,

(6.258)

||x0 || ≤ M, max{||Ti (x0 ) − x0 || : i ∈ φ(x0 )} ≤ δ0 ,

(6.259)

.

satisfying .

312

6 Algorithms Based on Unions of Nonexpansive Maps

for each integer .t = 0, . . . , Q − 1, Σ .

γt,i = 1

(6.260)

γt,i Ti (xt )|| ≤ δ

(6.261)

i∈φ(xt )

and Σ

||xt+1 −

.

i∈φ(xt )

the inequality max{||Ti (xt ) − xt || : i ∈ φ(xt )} ≤ ɛ

.

holds for all .t = 0, . . . , Q. Proof Choose a positive number .δ such that δ < min{δ0 , (8Q)−1 r (M,q) }.

(6.262)

.

Q

Assume that a finite sequence .{xt }t=0 ⊂ C satisfies (6.258), (6.259) and (6.260), (6.261) for each .t ∈ {0, . . . , Q − 1}. Property (a) and Eq. (6.259) imply that there exists z ∈ ΩM

(6.263)

||z − x0 || ≤ r (M,q) /4,

(6.264)

φ(x0 ) ⊂ φ(z), Ti (z) = z, z ∈ φ(z).

(6.265)

.

such that .

.

By (6.263) and (6.264), there exists j ∈ {1, . . . , p(M, q)}

.

such that (M,q)

||z − zj

.

|| < (4q)−1 rz(M,q) .

(6.266)

j

In view of (6.255) and (6.264)–(6.266), (M,q)

||x0 − zj

.

.

(M,q)

|| ≤ ||x0 − z|| + ||z − zj

< r (M,q) /4 + (4q)−1 rz(M,q) < (3q)−1 rz(M,q) . j

j

|| (6.267)

Proof of Theorem 6.23

313

It follows from the choice of .rx , x ∈ ΩM and (6.265) that (M,q)

φ(x0 ) ⊂ φ0 (zj

.

(6.268)

).

Assumption (A2) and Eqs. (6.252), (6.267) and (6.268) imply that for each .i ∈ φ(x0 ), (M,q)

||Ti (x0 ) − zj

.

(M,q)

|| ≤ ||x0 − zj

|| < (3q)−1 rz(M,q) .

(6.269)

j

By (6.258), (6.260), (6.261), (6.269) and the convexity of the norm, (M,q)

||x1 − zj

.

|| ≤ ||x1 −

Σ i∈φ(x0 )

.

(M,q)

γ0,i Ti (x0 ) − zj

||

i∈φ(x0 )

Σ

≤δ+

Σ

γ0,i Ti (x0 )|| + ||

(M,q)

γ0,i ||Ti (x0 ) − zj

||

i∈φ(x0 ) .

≤ δ + (3q)−1 rz(M,q) ).

(6.270)

j

We show that for all .k = 0, . . . , Q, (M,q)

||xk − zj

.

)|| ≤ kδ + (3q)−1 rz(M,q) .

(6.271)

j

(By (6.267) and (6.270) Eq. (6.271) holds for .k = 0, 1.) Assume that .k < Q is a natural number and (6.271) holds. By (6.255), (6.262) and (6.271), (M,q)

||xk − zj

.

.

|| ≤ (3q)−1 rz(M,q) + Qδ j

≤ (3q)−1 rz(M,q) + 8−1 r (M,q) ≤ 2−1 rz(M,q) . j

(6.272)

j

(M,q)

Assumption (A4), the inclusion .zj (6.272) imply that

∈ ΩM , the choice of .rx , x ∈ ΩM and

(M,q)

φ(xk ) ⊂ φ(zj

.

(M,q)

) = φ0 (zj

).

(6.273)

Assumption (A2) and (6.273) imply that for each .i ∈ φ(xk ), (M,q)

||Ti (xk )) − zj

.

(M,q)

|| ≤ ||xk − zj

||.

(6.274)

314

6 Algorithms Based on Unions of Nonexpansive Maps

By (6.258), (6.260), (6.261), (6.274) and the convexity of the norm, (M,q)

||xk+1 − zj

.

Σ

|| ≤ ||xk+1 −

i∈φ(xk )

.

Σ

≤δ+

Σ

γk,i Ti (xk )|| + ||

(M,q)

γk,i Ti (xk ) − zj

||

i∈φ(xk ) (M,q)

γk,i ||Ti (xk ) − zj

||

i∈φ(xk )

.

.

(M,q)

≤ δ + ||xk − zj

||

≤ δ + (3q)−1 rz(M,q) ) + kδ. j

Thus by induction we showed that (6.271) holds for all .k = 1, . . . , Q. By (6.255), (6.262) and (6.271), for all integers .k = 0, . . . , Q, (M,q)

||xk − zj

.

|| ≤ (3q)−1 rz(M,q) + Qδ < (2q)−1 rz(M,q) . j

j

Assumption (A4), the relation above and the choice of .rx , x ∈ ΩM imply that (M,q)

φ(xk ) ⊂ φ(zj

.

(M,q)

) = φ0 (zj

(6.275)

)

and by (A2) and (6.275), for each .i ∈ φ(xk ), (M,q)

||Ti (xk ) − zj

.

(M,q)

|| ≤ ||xk − zj

||.

The equations above and (6.256) imply that for all .k = 0, . . . , Q, and each .i ∈ φ(xk ), (M,q)

||xk − Ti (xk ))|| ≤ ||xk − zj

.

(M,q)

|| + ||zj

− Ti (xk )||

≤ q −1 rz(M,q) < ɛ .

.

j

Proposition 6.27 is proved. Proof of Theorem 6.23 Choose an integer .q ≥ 1 such that q −1 < ɛ /8.

.

(6.276)

Lemma 6.25 implies that there exists δ0 ∈ (0, r (M,q) /4)

.

(6.277)

Proof of Theorem 6.23

315

such that property (a) of Proposition 6.27 holds. Theorem 6.22 implies that there exist .δ1 ∈ (0, δ0 /2) and an integer .Q ≥ 1 such that the following property holds: (b) for each sequence .{xi }∞ i=0 ⊂ C, and each γt,i ≥ c, ¯ t = 0, 1, . . . , , i = 1, . . . , m

.

satisfying for each integer .t = 0, 1, . . . ,, Σ

||xt || ≤ M,

.

γt,i = 1

i∈φ(xt )

and Σ

||xt+1 −

γt,i Ti (xt )|| ≤ δ1

.

i∈φ(xt )

there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q,

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q

.

and .

max{||Ti (xqp ) − xqp || : i ∈ φ(xqp )} ≤ δ0 /2.

(6.278)

Proposition 6.27 implies that there exists .δ ∈ (0, δ1 ) such that the following property holds: Q (c) for each finite sequence .{xt }t=0 ⊂ C and each γt,i ≥ c, ¯ t = 0, . . . , Q − 1, i = 1, . . . , m

.

satisfying ||x0 || ≤ M, max{||Ti (x0 ) − x0 || : i ∈ φ(x0 )} ≤ δ0 ,

.

for each .t ∈ {0, . . . , Q − 1}, Σ .

γt,i = 1,

(6.279)

i∈φ(xt )

||xt+1 −

Σ

.

i∈φ(xt )

γt,i Ti (xt )|| ≤ δ

(6.280)

316

6 Algorithms Based on Unions of Nonexpansive Maps

the inequality .

max{||Ti (xt ) − xt || : i ∈ φ(xt )} ≤ ɛ

holds for all .t = 0, . . . , Q. Assume that .{xi }∞ i=0 ⊂ C satisfies ||xt || ≤ M, t = 0, 1, . . . ,

(6.281)

γt,i ≥ c, ¯ t = 0, 1, . . . , i = 1, . . . , m,

(6.282)

.

.

(6.279) and (6.280) hold for all integers .t ≥ 0. Property (b) and Eqs. (6.279)–(6.282) imply that there exists a strictly increasing sequence of natural numbers .{qi }∞ i=1 such that 1 ≤ q1 ≤ Q

.

for every integer .p ≥ 1, 1 ≤ qp+1 − qp ≤ Q,

.

and (6.278) holds. Let .p ≥ 1 be an integer. Property (c) applied to the sequence {xi+qp }Q i=0 and (6.278)–(6.282) imply that

.

.

max{||Ti (xt ) − xt || : i ∈ φ(xt )} ≤ ɛ

holds for all .t = qp , . . . , qp+1 . Since p is any natural number we conclude that Theorem 6.23 is proved.

The Behavior of Iterates Without Compactness Assumptions In previous sections our results were obtained under assumptions (A1)–(A3). In particular in (A1) we assume that bounded, closed subsets of C are compact. In this section and in the next sections of this chapter we will extend these results for self-mappings of any closed subset of an arbitrary metric space. The results of this section were obtained in [166]. Suppose that .(X, ρ) is a metric space and that .C ⊂ X is its nonempty, closed set. For every point .x ∈ X and every positive number r define B(x, r) = {y ∈ X : ρ(x, y) ≤ r}

.

and for every point .x ∈ C and every nonempty set .B ⊂ C put ρ(x, B) = sup{ρ(x, y) : y ∈ B}.

.

The Behavior of Iterates Without Compactness Assumptions

317

For every operator .S : C → C set Fix(S) = {x ∈ C : S(x) = x}.

.

Assume that .m ≥ 1 is an integer, .Ti : C → C, .i = 1, . . . , m, .c¯ ∈ (0, 1] and that for each .i ∈ {1, . . . , m}, each .z ∈ Fix(Ti ) and each .y ∈ C, ρ(z, y)2 − ρ(z, Ti (y))2 ≥ cρ(y, ¯ Ti (y))2 .

.

(6.283)

Note that (6.283) holds for many nonlinear mappings [163, 165] including projections on closed, convex sets in a Hilbert space. Assume that for each .x ∈ X, φ(x) ⊂ {1, . . . , m}

.

is given. In other words, φ : X → 2{1,...,m} \ {∅}.

.

Define T (x) = {Ti (x) : i ∈ φ(x)}

.

for each .x ∈ C, F¯ (T ) = {z ∈ C : Ti (z) = z, i = 1, . . . , m}

.

(6.284)

and F (T ) = {x ∈ C : x ∈ T (x)}.

.

(6.285)

Assume that F¯ (T ) /= ∅.

.

Fix θ ∈ C.

.

In this section we prove the following result which shows that almost all iterates of our set-valued mappings are approximated solutions of the corresponding fixed point problem. Many results of this type are collected in [163, 165]. Theorem 6.28 Assume that .M > 0, . ɛ ∈ (0, 1], F¯ (T ) ∩ B(θ, M) /= ∅

.

(6.286)

318

6 Algorithms Based on Unions of Nonexpansive Maps

and that Q = L4M 2 c¯−1 ɛ −2 ⎦.

.

(6.287)

Then for each sequence .{xi }∞ i=0 ⊂ C which satisfies ρ(θ, x0 ) ≤ M

(6.288)

xi+1 ∈ T (xi ), i = 0, 1, . . .

(6.289)

.

and .

the inequality Card({i ∈ {0, 1, . . . , } : min{ρ(xi , Tj (xi )) : j ∈ φ(xi )} ≥ ɛ }) ≤ Q

.

holds. Proof By (6.286), there exists z ∈ F¯ (T ) ∩ B(θ, M).

.

(6.290)

In view of (6.284) and (6.290), z = Ti (z), i = 1, . . . , m.

.

(6.291)

Assume that .{xi }∞ i=0 ⊂ C and (6.288) and (6.289) hold. It follows from (6.289) that for each integer .i ≥ 0 there exists j (i) ∈ φ(xi )

(6.292)

xi+1 = Tj (i) (xi ).

(6.293)

.

such that .

Let .i ≥ 0 be an integer. By (6.283) and (6.291)–(6.293), 2 ρ(z, xi )2 ≥ ρ(z, Tj (i) (xi ))2 + cρ(x ¯ i , Tj (i) (xi ))

.

.

2 = ρ(z, xi+1 )2 + cρ(x ¯ i , xi+1 ) .

Let .q ≥ 1 be an integer. It follows from (6.288), (6.290) and (6.294) that 4M 2 ≥ (M + ρ(z, θ ))2 ≥ (ρ(z, θ ) + ρ(θ, x0 ))2

.

.

≥ ρ(z, z0 )2 ≥ ρ(z, x0 )2 − ρ(z, xq )2

(6.294)

The Behavior of Iterates Without Compactness Assumptions

.

=

319

q−1 Σ (ρ(z, xi )2 − ρ(z, xi+1 )2 ) i=0

.

≥ c¯

q−1 Σ

ρ(xi , xi+1 )2 ≥ cCard({i ¯ ∈ {0, . . . , q − 1} : ρ(xi , xi+1 ) ≥ ɛ }) ɛ 2

i=0

and Card({i ∈ {0, . . . , q − 1} : ρ(xi , xi+1 ) ≥ ɛ }) ≤ 4M 2 c¯−1 ɛ −2 .

.

(6.295)

By (6.287) and (6.295), Card({i ∈ {0, 1, . . . } : ρ(xi , xi+1 ) ≥ ɛ }) ≤ L4M 2 c¯−1 ɛ −2 ⎦ = Q.

.

Theorem 6.28 is proved. In our next result taking into account computational errors we show that an approximate fixed point is obtained after a certain number of iterates which depends on a computational error. Theorem 6.29 Assume that .M > 0, . ɛ ∈ (0, 1), F¯ (T ) ∩ B(θ, M) /= ∅,

(6.296)

Q > 8M 2 c¯−1 ɛ −2 ,

(6.297)

δ ≤ (4M + 1)−1 ɛ 2 c/2, ¯

(6.298)

.

an integer Q satisfies .

δ ∈ (0, 1) satisfies

.

.

a sequence .{xi }∞ i=0 ⊂ C satisfies ρ(θ, x0 ) ≤ M

(6.299)

B(xi+1 , δ) ∩ T (xi ) /= ∅.

(6.300)

.

and that for each integer .i ≥ 0, .

Then there exists a nonnegative integer .j ≤ Q − 1 such that B(xj , ɛ ) ∩ T (xj ) /= ∅.

.

(6.301)

320

6 Algorithms Based on Unions of Nonexpansive Maps

Proof Fix z ∈ F¯ (T ) ∩ B(θ, M).

.

(6.302)

Assume that for each nonnegative integer .i ≤ Q − 1, B(xi , ɛ ) ∩ T (xi ) = ∅.

(6.303)

i ∈ {0, . . . , Q − 1}.

(6.304)

.

Let .

In view of (6.300) and (6.304), there exists xˆi+1 ∈ T (xi )

(6.305)

ρ(xi+1 , xˆi+1 ) ≤ δ.

(6.306)

.

such that .

It follows from (6.283), (6.284), (6.302) and (6.305) that 2 ρ(z, xi )2 ≥ ρ(z, xˆi+1 )2 + cρ(x ¯ i , xˆi+1 ) .

.

(6.307)

By (6.303) and (6.305), ρ(xi , xˆi+1 ) > ɛ .

.

(6.308)

Equations (6.307) and (6.308) imply that ρ(z, xi )2 ≥ ρ(z, xˆi+1 )2 + ɛ 2 c. ¯

.

(6.309)

By (6.306), ρ(z, xˆi+1 ) ≥ ρ(z, xi+1 ) − ρ(xˆi+1 , xi+1 ) ≥ ρ(z, xi+1 ) − δ.

.

(6.310)

In view (6.310), ρ(z, xi+1 )2 ≤ (ρ(z, xˆi+1 ) + δ)2

.

.

= δ 2 + ρ(z, xˆi+1 )2 + 2δρ(z, xˆi+1 ).

Equations (6.309) and (6.311) imply that ρ(z, xi )2 ≥ ρ(z, xi+1 )2 − δ 2 − 2δρ(z, xˆi+1 ) + c ɛ ¯ 2

.

(6.311)

The Behavior of Iterates Without Compactness Assumptions

.

≥ ρ(z, xi+1 )2 − δ 2 − 2δρ(z, xi ) + c ɛ ¯ 2.

321

(6.312)

It follows from (6.306) and (6.309) that ρ(z, xi+1 ) ≤ ρ(z, xˆi+1 ) + ρ(xi+1 , xˆi+1 )

.

≤ ρ(z, xˆi+1 ) + δ ≤ ρ(z, xi ) + δ.

.

(6.313)

We show by induction that for all integers .i = 0, . . . , Q − 1, ρ(z, xi ) ≤ 2M,

.

ρ(z, xi )2 ≥ ρ(z, xi+1 )2 + c ɛ ¯ 2 /2.

.

In view of (6.299) and (6.312), ρ(z, x0 ) ≤ ρ(z, θ ) + ρ(θ, x0 ) ≤ 2M.

.

Assume that an integer .i ∈ {0, . . . , Q − 1} and that ρ(z, xi ) ≤ 2M.

.

(6.314)

By (6.298), (6.312) and (6.314), ρ(z, xi )2 ≥ ρ(z, xi+1 )2 − δ 2 − 2δρ(z, xi ) + c ɛ ¯ 2

.

≥ ρ(z, xi+1 )2 − δ 2 − 4δM + c ɛ ¯ 2

.

.

≥ ρ(z, xi+1 )2 − δ(4M + 1) + c ɛ ¯ 2 .

≥ ρ(z, xi+1 )2 + c ɛ ¯ 2 /2.

(6.315)

In view of (6.314) and (6.315), ρ(z, xi+1 ) ≤ ρ(z, xi ) ≤ 2M,

.

ρ(z, xi )2 − ρ(z, xi+1 )2 ≥ c ɛ ¯ 2 /2.

.

Thus by induction we have shown that for all .i = 0, . . . , Q − 1, ρ(z, xi ) ≤ 2M

.

(6.316)

322

6 Algorithms Based on Unions of Nonexpansive Maps

and (6.316) holds. It follows from (6.316) and the relation above that 4M 2 ≥ ρ(z, z0 )2 − ρ(z, xQ )2

.

.

=

Q−1 Σ

(ρ(z, xi )2 − ρ(z, xi+1 )2 ) ≥ Q ɛ 2 c/2 ¯

i=0

and Q ≤ 8M 2 ɛ −2 c¯−1 .

.

This contradicts (6.297). The contradiction we have reached proves Theorem 6.29.

Inexact Iterates Without Compactness Assumptions We use the definitions, notation and assumptions introduced in Sect. “The Behavior of Iterates Without Compactness Assumptions”. The results of this section were obtained in [167]. Assume that for each .x ∈ X, φ(x) ⊂ {1, . . . , m}

.

is nonempty. Set T (x) = {Ti (x) : i ∈ φ(x)}

.

for every .x ∈ C, F¯ (T ) = {z ∈ C : Ti (z) = z, i = 1, . . . , m}

.

and F (T ) = {x ∈ C : x ∈ T (x)}.

.

We assume that F¯ (T ) /= ∅

.

and fix θ ∈ C.

.

Inexact Iterates Without Compactness Assumptions

323

In this section we obtain the following theorem showing that almost all inexact iterates of our set-valued mappings with summable computational errors are approximated solutions of the corresponding fixed point problem. Theorem 6.30 Let M be a positive number, . ɛ ∈ (0, 1], F¯ (T ) ∩ B(θ, M) /= ∅

.

(6.317)

and let a sequence .{ri }∞ i=0 ⊂ (0, 1] satisfy ∞ Σ

ri < ∞.

(6.318)

ρ(θ, x0 ) ≤ M

(6.319)

.

i=0

Assume that .{xi }∞ i=0 ⊂ C satisfies .

and .

inf{ρ(xi+1 , y) : y ∈ T (xi )} ≤ ri , i = 0, 1, . . . ,

(6.320)

Q1 ≥ 1 is an integer such that

.

ri ≤ ɛ /2 for all integers i ≥ Q1

.

(6.321)

and that Q0 = L4 ɛ

.

−2

(4M + 6 2

∞ Σ j =0

rj (2M +

∞ Σ

rj ))⎦ + 1.

(6.322)

j =0

Then Card({i ∈ {0, 1, . . . , } : ρ(xi , xi+1 ) ≥ ɛ /2}) ≤ Q0 ,

.

if an integer .i ≥ Q1 and .ρ(xi , xi+1 ) ≤ ɛ /2, then .

min{ρ(xi , Tj (xi )) : j ∈ φ(xi )} ≤ ɛ

and Card({i ∈ {0, 1, . . . , } : min{ρ(xi , Tj (xi )) : j ∈ φ(xi )} > ɛ }) ≤ Q0 + Q1 .

.

Proof By (6.317), there exists z ∈ F¯ (T ) ∩ B(θ, M).

.

(6.323)

324

6 Algorithms Based on Unions of Nonexpansive Maps

By (6.323), z = Ti (z), i = 1, . . . , m.

(6.324)

.

It follows from (6.319) and (6.323) that ρ(x0 , z) ≤ 2M.

(6.325)

.

Assume that .i ≥ 0 is an integer. It follows from (6.320) that there exists j (i) ∈ φ(xi )

(6.326)

ρ(xi+1 , Tj (i) (xi )) ≤ ri .

(6.327)

.

such that .

Equations (6.283) and (6.324) imply that ρ(z, Tj (i) (xi )) ≤ ρ(z, xi ).

(6.328)

.

In view of (6.327) and (6.328), ρ(xi+1 , z) ≤ ρ(xi+1 , Tj (i) (xi )) + ρ(Tj (i) (xi ), z) ≤ ri + ρ(z, xi ).

.

(6.329)

By (6.325) and (6.329), for every natural number j , ρ(xj , z) ≤ ρ(z, x0 ) +

j −1 Σ

.

k=0

rk ≤ 2M +

∞ Σ

(6.330)

rk .

k=0

By (6.283) and (6.324), 2 ρ(z, xi )2 ≥ ρ(Tj (i) (xi ), z)2 + cρ(T ¯ j (i) (xi ), xi ) .

(6.331)

.

By (6.327), (6.328) and (6.330), |ρ(xi+1 , z)2 − ρ(Tj (i) (xi ), z)2 |

.

.

.

≤ (ρ(xi+1 , z) + ρ(Tj (i) (xi ), z))|ρ(xi+1 , z) − ρ(Tj (i) (xi ), z)|

≤ (ρ(xi+1 , z) + ρ(xi , z))ρ(xi+1 , Tj (i) (xi )) ≤ 2ri (2M +

∞ Σ k=0

rk ).

(6.332)

Inexact Iterates Without Compactness Assumptions

325

By (6.327), (6.328) and (6.330), |ρ(xi+1 , xi )2 − ρ(Tj (i) (xi ), xi )2 |

.

.

≤ (ρ(xi+1 , xi ) + ρ(Tj (i) (xi ), xi ))|ρ(xi+1 , xi ) − ρ(Tj (i) (xi ), xi )| .

≤ (ρ(xi+1 , z) + 2ρ(xi , z) + ρ(z, Tj (i) (xi )))ρ(xi+1 , Tj (i) (xi ))

.

≤ ri (3ρ(xi , z) + ρ(xi+1 , z)) ≤ 4ri (2M +

∞ Σ

(6.333)

rk ).

k=0

In view of (6.331)–(6.333), 2 ρ(z, xi )2 ≥ ρ(Tj (i) (xi ), z)2 + cρ(T ¯ j (i) (xi ), xi )

.

.

≥ ρ(xi+1 , z)2 − 2ri (2M +

∞ Σ

2 rk ) + cρ(x ¯ i+1 , xi ) − 4ri (2M +

k=0

∞ Σ

rk ).

(6.334)

k=0

Let q be a natural number. Equations (6.319), (6.323) and (6.334) imply that 4M 2 ≥ (M + ρ(z, θ ))2 ≥ (ρ(z, θ ) + ρ(θ, x0 ))2

.

.

≥ ρ(z, z0 )2 ≥ ρ(z, x0 )2 − ρ(z, xq )2

.

=

q−1 Σ (ρ(z, xi )2 − ρ(z, xi+1 )2 ) i=0

.

≥ c¯

q−1 Σ

ρ(xi , xi+1 )2 − 6(2M +

∞ Σ

rj )

j =0

i=0

4M + 6(2M +

.

2

∞ Σ j =0

.

≥ c¯

q−1 Σ

rj )

∞ Σ

∞ Σ

rj ,

j =0

rj

j =0

ρ(xi , xi+1 )2

i=0 .

≥ cCard({i ¯ ∈ {0, . . . , q − 1} : ρ(xi , xi+1 ) ≥ ɛ /2}) ɛ 2 /4

326

6 Algorithms Based on Unions of Nonexpansive Maps

and Card({i ∈ {0, . . . , q − 1} : ρ(xi , xi+1 ) ≥ ɛ /2})

.

.

≤ 4 ɛ −2 (4M 2 + 6

∞ Σ

rj (2M +

j =0

∞ Σ

rj )).

j =0

By the equation above and (6.322), Card({i ∈ {0, 1, . . . } : ρ(xi , xi+1 ) ≥ ɛ /2}) ≤ Q0 .

.

(6.335)

Assume that .i ≥ Q1 is an integer and ρ(xi , xi+1 ) ≤ ɛ /2.

.

(6.336)

By (6.321), (6.326), (6.327), ρ(Tj (i) (xi ), xi ) ≤ ρ(xi+1 , xi ) + ρ(Tj (i) (xi ), xi+1 ) ≤ ri + ɛ /2 ≤ ɛ

.

and .

min{ρ(xi , Tj (xi )) : j ∈ φ(xi )} ≤ ɛ .

Together with (6.325) this completes the proof of Theorem 6.30. In our second main result we obtain an extension of Theorem 6.30 under the presence of computational errors tends to zero but not necessarily summable. Theorem 6.31 Assume that .{xi }∞ i=0 ⊂ C satisfies lim inf ρ(θ, xt ) < ∞

(6.337)

lim inf{ρ(xt+1 , y) : y ∈ T (xt )} = 0.

(6.338)

.

t→∞

and .

t→∞

Then there exists a strictly increasing sequence of natural numbers .{tk }∞ k=1 such that .

lim ρ(xtk , xtk +1 ) = 0

k→∞

and .

lim inf{ρ(xtk , ξ ) : ξ ∈ T (xtk )} = 0.

k→∞

(6.339)

Inexact Iterates Without Compactness Assumptions

327

Proof By (6.337), there exists M > lim inf ρ(θ, xt ).

.

t→∞

(6.340)

In view of (6.338), in order to prove the theorem it is sufficient to show the existence of a strictly increasing sequence of natural numbers .{tk }∞ k=1 such that (6.339) holds. Assume the contrary. Then by (6.340), there is a real number . ɛ ∈ (0, 1) and an integer .τ0 ≥ 1 such that ρ(xτ0 , θ ) ≤ M

(6.341)

ρ(xt , xt+1 ) ≥ ɛ .

(6.342)

z ∈ F¯ (T )

(6.343)

δ(1 + ρ(θ, z) + M) ≤ c ɛ ¯ 2 /16.

(6.344)

.

and that for every integer .t ≥ τ0 , .

Fix .

and .δ > 0 such that .

By (6.338) and (6.340), there exists an integer .τ1 ≥ τ0 for which ρ(xτ1 , θ ) ≤ M

(6.345)

inf{ρ(xt+1 , ξ ) : ξ ∈ T (xt )} ≤ δ.

(6.346)

.

and that for every integer .t ≥ τ1 , .

In view of (6.345), ρ(xτ1 , z) ≤ ρ(xτ1 , θ ) + ρ(θ, z) ≤ ρ(θ, z) + M.

.

(6.347)

Let .t ≥ τ1 be an integer.Clearly, there exists .j (t) ∈ φ(xt ) such that ρ(xt+1 , Tj (t) (xt )) = inf{ρ(xt+1 , ξ ) : ξ ∈ T (xt )}.

.

(6.348)

It follows from (6.283), (6.284), (6.342), (6.343), (6.346) and (6.348) that 2 ρ(z, xt )2 ≥ ρ(Tj (t) (xt ), z)2 + cρ(T ¯ j (t) (xt ), xt )

.

.

2 ≥ ρ(Tj (t) (xt ), z)2 + c(ρ(x ¯ t , xt+1 ) − ρ(Tj (t) (xt ), xt+1 )) .

≥ ρ(Tj (t) (xt ), z)2 + c ɛ ¯ 2 /4.

(6.349)

328

6 Algorithms Based on Unions of Nonexpansive Maps

Equations (6.283), (6.284), (6.343), (6.345) and (6.348) imply that |ρ(xt+1 , z)2 − ρ(Tj (t) (xt ), z)2 |

.

.

≤ (ρ(xt+1 , z) + ρ(Tj (t) (xt ), z))|ρ(xt+1 , z) − ρ(Tj (t) (xt ), z)| (2ρ(z, Tj (t) (xt )) + ρ(Tj (t) (xt ), xt+1 ))ρ(xt+1 , Tj (t) (xt ))

.

.

≤ δ(2ρ(xt , z) + δ).

(6.350)

By (6.349) and (6.350), ρ(z, xt )2 ≥ ρ(xt+1 , z)2 − δ(2ρ(xt , z) + δ) + c ɛ ¯ 2 /4.

.

(6.351)

Assume that .t ≥ τ1 is an integer and that ρ(xt , z) ≤ ρ(θ, z) + M.

.

(6.352)

(Note that in view of (6.347), Eq. (6.352) holds for .t = τ1 .) By (6.344), (6.351) and (6.352), ρ(xt+1 , z)2 ≤ ρ(xt , z)2 + 2δ(ρ(θ, z) + M + δ) − c ɛ ¯ 2 /4

.

.

≤ ρ(xt , z)2 + 2δ(ρ(θ, z) + M + 1) − c ɛ ¯ 2 /4 .

≤ ρ(xt , z)2 − c ɛ ¯ 2 /8.

(6.353)

In view of (6.352) and (6.353), ρ(xt+1 , z) ≤ ρ(xt , z) ≤ ρ(θ, z) + M.

.

Thus by induction we showed that for every integer .t ≥ τ1 , ρ(xt , z) ≤ ρ(θ, z) + M

.

and ρ(xt+1 , z)2 ≤ ρ(xt , z)2 − c ɛ ¯ 2 /8

.

and .ρ(xt+1 , z)2 → −∞ as .t → ∞. The contradiction we have reached completes the proof of Theorem 6.31.

Extensions

329

Extensions In this section we prove an extension of the result of Sect. “The Behavior of Iterates Without Compactness Assumptions” in the case when the common fixed problem has only an approximated solution. We use the notation, definitions and assumptions introduced in Sect. “The Behavior of Iterates Without Compactness Assumptions”. The results of this section were obtained in [170]. Assume that φ : X → 2{1,...,m} \ {∅}.

.

Define T (x) = {Ti (x) : i ∈ φ(x)}

.

for each .x ∈ C and F (T ) = {x ∈ C : x ∈ T (x)}.

.

Fix θ ∈ C.

.

We prove the following theorem under the presence of computational errors. This theorem shows that after a certain number of iterates we obtain an approximate solution of our fixed point problem. The number of iterates depends on the computational error. Theorem 6.32 Let .M > 0, . ɛ ∈ (0, 1], γ ∈ (0, (18)−1 (4M + 4)−1 ɛ 2 c), ¯

(6.354)

z ∈ B(θ, M)

(6.355)

B(z, γ ) ∩ Fix(Ti ) /= ∅, i = 1, . . . , m,

(6.356)

Q = L8 ɛ −2 M 2 c¯−1 ⎦ + 1

(6.357)

.

.

satisfy .

.

and .δ ∈ (0, γ ). Assume that .{xk }∞ k=0 ⊂ C, ρ(θ, x0 ) ≤ M

.

(6.358)

330

6 Algorithms Based on Unions of Nonexpansive Maps

and that B(xi+1 , δ) ∩ T (xi ) = / ∅, i = 0, 1, . . . .

.

(6.359)

Then there is a nonnegative integer .j < Q for which B(xj , ɛ ) ∩ T (xj ) /= ∅.

.

(6.360)

In the theorem above we assume the existence of the point z satisfying (6.356) which means that z is an approximate fixed point for all the mappings .Ti , .i = 1, . . . , m. This result has prototype in Sect. “The Behavior of Iterates Without Compactness Assumptions”, which was obtained under the assumption that z is a common fixed point for all .Tk , .k = 1, . . . , m. Proof Assume that for every nonnegative integer .j < Q relation (6.360) is not true. Then for every nonnegative integer .i < Q, B(xi , ɛ ) ∩ T (xi ) = ∅.

(6.361)

M0 = 2M + 1.

(6.362)

.

Set .

By (6.356), for every .k ∈ {1, . . . , m}, there is .

zi ∈ Fix(Ti )

(6.363)

ρ(z, zi ) ≤ γ .

(6.364)

ρ(x0 , z) ≤ 2M.

(6.365)

such that .

By (6.355) and (6.359), .

Let .i ∈ [0, Q − 1] be an integer. By (6.359), there is ˆ xi+1 ∈ T (xi )

(6.366)

ρ(xi+1 , ˆ xi+1 ) ≤ δ.

(6.367)

.

for which .

Equations (6.283) and (6.366) imply that there is an integer .j ∈ [1, m] for which ˆ xi+1 = Tj (xi ).

.

(6.368)

Extensions

331

It follows from (6.283) and (6.368) that ρ(zj , xi )2 ≥ ρ(zj , ˆ xi+1 )2 + cρ(x ¯ xi+1 )2 . i,ˆ

.

(6.369)

By (6.361) and (6.368), ρ(xi , ˆ xi+1 ) > ɛ .

(6.370)

ρ(zj , xi )2 ≥ ρ(zj , ˆ xi+1 )2 + c ɛ ¯ 2.

(6.371)

ρ(z, xi ) ≤ M0 .

(6.372)

.

In view of (6.369) and (6.370), .

Assume that .

(In view of (6.362) and (6.365), Eq. (6.372) holds for .i = 0.) Equations (6.364) and (6.372) imply that ρ(zj , xi ) ≤ ρ(zj , z) + ρ(z, xi ) ≤ M0 + γ .

.

(6.373)

It follows from (6.354), (6.362), (6.371) and (6.373) that ρ(zj , ˆ xi+1 )2 ≤ ρ(zj , xi )2 − ɛ 2 c¯ ≤ (M0 + γ )2 − ɛ 2 c¯

.

.

= M02 + γ (γ + 2M0 ) − ɛ 2 c¯ ≤ M02 + γ (1 + 2M0 ) − ɛ 2 c¯ .

≤ M02 − 7γ (1 + 2M0 ) ≤ (M0 − 2γ )2

and ρ(zj , ˆ xi+1 ) ≤ M0 − 2γ .

.

(6.374)

By (6.364) and (6.374), ρ(z, xi+1 ) ≤ ρ(z, zj ) + ρ(zj , ˆ xt+1 ) + ρ(ˆ xi+1 , xi+1 )

.

.

≤ M0 − 2γ + γ + γ ≤ M0

and ρ(z, xi+1 ) ≤ M0 .

.

(6.375)

332

6 Algorithms Based on Unions of Nonexpansive Maps

By (6.371), ρ(zj , ˆ xi+1 )2 ≤ ρ(zj , xi )2 − ɛ 2 c. ¯

.

(6.376)

Equations (6.364) and (6.372) imply that |ρ(xi , z)2 − ρ(xi , zj )2 |

.

.

.

≤ (ρ(xi , z) + ρ(xi , zj ))|ρ(xi , z) − ρ(xi , zj )|

≤ (ρ(xi , z) + ρ(xi , z) + γ )ρ(zj , z) ≤ γ (2M0 + 1).

(6.377)

It follows from (6.364), (6.367) and (6.375) that |ρ(xi+1 , z)2 − ρ(ˆ xi+1 , zj )2 |

.

.

.

≤ (ρ(xi+1 , z) + ρ(ˆ xi+1 , zj ))|ρ(xi+1 , z) − ρ(ˆ xi+1 , zj )|

≤ (2M0 + γ + δ)(ρ(zj , z) + ρ(ˆ xi+1 , xi+1 )) ≤ (2M0 + 2)(γ + δ).

(6.378)

By (6.354), (6.362), (6.371) and (6.378), ρ(xi+1 , z)2 ≤ ρ(zj , ˆ xi+1 )2 + 2γ (2M0 + 2)

.

.

.

≤ ρ(zj , xi )2 − c ɛ ¯ 2 + 2γ (2M0 + 2)

≤ ρ(xi , z)2 − ɛ 2 c¯ + γ (2M0 + 1) + 2γ (2M0 + 2) ρ(xi , z)2 − ɛ 2 c¯ + 3γ (2M0 + 1)

.

.

≤ ρ(xi , z)2 − ɛ 2 c/2. ¯

(6.379)

Thus we have shown by induction that (6.372) and (6.379) hold for .i = 0, . . . , Q−1. By (6.365) and (6.379), 4M 2 ≥ ρ(z, x0 ))2

.

.

.

=

Q−1 Σ

≥ ρ(z, x0 )2 − ρ(z, xQ )2

(ρ(z, xi )2 − ρ(z, xi+1 )2 ) ≥ Qc ɛ ¯ 2 /2

i=0

and Q ≤ 8M 2 c¯−1 ɛ −2 .

.

Extensions

333

This contradicts (6.357). The contradiction we have reached proves Theorem 6.32. Lemma 6.33 Assume that .M0 > 0, z ∈ B(θ, M0 ),

(6.380)

B(z, 1) ∩ Fix(Ti ) /= ∅, i = 1, . . . , m,

(6.381)

x0 ∈ B(θ, M0 ),

(6.382)

B(x1 , 1) ∩ T (x0 ) /= ∅.

(6.383)

.

.

.

x1 ∈ C and that

.

.

Then ρ(x1 , θ ) ≤ 3M0 + 3.

.

Proof Clearly, there is an integer .j ∈ [1, m] for which ρ(x1 , Tj (x0 )) ≤ 1.

(6.384)

.

zj ∈ Fix(Tj )

(6.385)

ρ(z, zj ) ≤ 1.

(6.386)

.

By (6.381), there is

for which .

Equations (6.283), (6.380), (6.382) and (6.384)–(6.386) imply that ρ(x1 , θ ) ≤ ρ(θ, Tj (x0 )) + ρ(Tj (x0 ), x1 )

.

.

≤ 1 + ρ(θ, z) + ρ(z, zj ) + ρ(zj , Tj (x0 )) .

.

≤ 1 + M0 + 1 + ρ(zj , x0 )

≤ 2 + M0 + ρ(θ, x0 ) + ρ(θ, z) + ρ(z, zj ) .

Lemma 6.33 is proved.

≤ 3 + 3M0 .

334

6 Algorithms Based on Unions of Nonexpansive Maps

Theorem 6.34 Let .M > 0, . ɛ ∈ (0, 1], γ ∈ (0, (18)−1 (12M + 12)−1 ɛ 2 c), ¯

.

z ∈ B(θ, M)

.

satisfy B(z, γ ) ∩ Fix(Ti ) = / ∅, i = 1, . . . , m,

.

Q = L8 ɛ −2 (3M + 3)2 c¯−1 ⎦ + 1

.

and .δ ∈ (0, γ ). Assume that .{xk }∞ k=0 ⊂ C, ρ(θ, x0 ) ≤ M

.

and that B(xk+1 , δ) ∩ T (xk ) = / ∅, k = 0, 1, . . . .

.

Then there is .j ∈ {1, . . . , Q} for which B(xj , ɛ ) ∩ T (xj ) /= ∅.

.

Proof Lemma 6.33 implies that ρ(x1 , θ ) ≤ 3M + 3.

.

Theorem 6.32 applied to the sequence .{xi+1 }∞ i=0 implies our result. Theorem 6.35 Let .M > 0, . ɛ ∈ (0, 1], {ξ ∈ C : B(ξ, ɛ ) ∩ T (ξ ) /= ∅} ⊂ B(θ, M),

.

γ ∈ (0, (18)−1 (12M + 12)−1 ɛ 2 c), ¯

.

z ∈ B(θ, M)

.

satisfy B(z, γ ) ∩ Fix(Ti ) = / ∅, i = 1, . . . , m,

.

Q = L8 ɛ −2 (3M + 3)2 c¯−1 ⎦ + 1

.

and .δ ∈ (0, γ ). Assume that .{xk }∞ k=0 ⊂ C, ρ(θ, x0 ) ≤ M

.

(6.387)

Extensions

335

and that B(xk+1 , δ) ∩ T (xk ) = / ∅, k = 0, 1, . . . .

.

Then there exists a strictly increasing sequence of natural numbers .{qj }∞ j =1 such that 1 ≤ q0 ≤ Q

(6.388)

qj +1 − qj ≤ Q

(6.389)

B(xqj , ɛ ) ∩ T (xqj ) /= ∅.

(6.390)

.

and for each integer .j ≥ 0, .

.

Proof Theorem 6.34 implies that there exists .q0 ∈ {1, . . . , Q} for which B(xq0 , ɛ ) ∩ T (xq0 ) /= ∅.

.

Assume that .p ∈ {0, 1, . . . }, .qj , .j = 0, . . . , p are natural numbers such that for any integer j satisfying .0 ≤ j < p, (6.389) holds and that (6.390) is true for all .j = 0, . . . , p. Set yi = xi+qp , i = 0, 1, . . . .

.

By (6.387) and (6.390), ρ(θ, y0 ) ≤ M.

.

Clearly, all the assumptions of Theorem 6.34 holds with .xi = yi , .i = 0, 1, . . . and Theorem 6.34 implies that there is .j ∈ {1, . . . , Q} for which B(yj , ɛ ) ∩ T (yj ) /= ∅.

.

Set qp+1 = qp + j.

.

Clearly, B(xqp+1 , ɛ ) ∩ T (xqp+1 ) /= ∅.

.

Thus by induction we constructed the sequence of natural numbers .{qj }∞ j =1 and proved Theorem 6.35.

Chapter 7

Inconsistent Convex Feasibility Problems

In this chapter we study the method of cyclic projections for inconsistent convex feasibility problems in a Hilbert space under the presence of computational errors. We show that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterations one needs for this.

Preliminaries The convex feasibility problem is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the convex feasibility problem is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent feasibility problems exists and is developed in the following two directions (See [2, 18, 19, 52, 57, 62, 67, 70, 74, 101, 123] and the references mentioned therein.) One direction is to define other solution concepts which can be used for inconsistent feasibility problems such as proximity function minimization, wherein a proximity function measures in some way the total violation of all constraints. The second direction analyzed the behavior of algorithms which are used to solve a consistent feasibility problems, when they are applied to inconsistent problems. In this chapter the second direction is chosen. The results presented here were obtained in [168]. Let .(X, ) be a real Hilbert space equipped with an inner product . which induces a complete norm .|| · || in X. For each nonempty convex closed set .C ⊂ X and each .x ∈ X, there exists a unique point .PC (x) ∈ C such that ||PC (x) − x|| = inf{||y − x|| : y ∈ X}.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_7

337

338

7 Inconsistent Convex Feasibility Problems

The mapping .PC is called the metric projection on C. For each .x ∈ X and each .r > 0, set B(x, r) = {y ∈ X : ||x − y|| ≤ r}.

.

For each .z ∈ R 1 , set Lz⎦ = max{j : j is an integer and j ≤ z}.

.

Denote by I the identity operator in X. For each mapping .S : X → X, define S 0 = I , .S 1 = S and .S i+1 = S ◦ S i for all integers .i ≥ 0. A mapping .T : X → X is firmly nonexpansive [19, 79] if for all .x, y ∈ X,

.

||T (x) − T (y)||2 + ||(I − T )(x) − (I − T )(y)||2 ≤ ||x − y||2 .

.

(7.1)

We will use the following well-known fact. Proposition 7.1 ([19, 79]) Assume that .C ⊂ X is a nonempty closed and convex set. Then the mapping .PC is firmly nonexpansive. Assume that .Ci , .i = 1, . . . , m, where .m ≥ 2 is an integer, are closed and convex sets and .Pi = PCi , .i = 1, . . . , m are the corresponding metric projections. We consider the case when .

∩m i=1 Ci = ∅.

If .m = 2, then the solution of our inconsistent feasibility problem is a pair of points .xi ∈ Ci , .i = 1, 2 such that ||x1 − x2 || = inf{||y1 − y2 || : yi ∈ Ci , i = 1, 2}.

.

This condition is equivalent to one in the general case below for .m = 2. In the general case [62, 94] the solution is a finite sequence .x¯i ∈ Ci , .i = 1, . . . , m such that Pi+1 (x¯i ) = x¯i+1 , i = 1, . . . , m − 1, P1 (x¯m ) = x¯1 .

.

It turns out that under certain mild assumptions, the cyclic projections method converges to the solution of our inconsistent problem. More precisely, for each integer .n ≥ 0, set i(n) = n(mod m) + 1

.

and for a given .x0 ∈ X, define xn+1 = Pi(n) (xn ) for all integers n ≥ 0.

.

(7.2)

Main Results

339

It was shown in [94] (see also Theorem 2.4 of [62]) that if at least one of the sets .Ci , i = 1, . . . , m is bounded, then there exists a finite sequence .x¯i ∈ Ci , .i = 1, . . . , m such that

.

Pi+1 (x¯i ) = x¯i+1 , i = 1, . . . , m − 1, P1 (x¯m ) = x¯1

.

and xkm+i+1 − xkm+i → x¯i+1 − x¯i , i = 1, . . . , m − 1, xkm+1 → x¯1

.

weakly in X, as .k → ∞. In this chapter based on [168] we study the cyclic projections algorithm taking into account computational errors which are always present 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. Clearly, in practice it is sufficient to find a good approximate solution instead of constructing a minimizing sequence. On the other hand in practice computations induce numerical errors and if one uses methods in order to solve feasibility problems, these methods usually provide only approximate solutions of the problems. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterations one needs for this. A finite sequence .{x¯i }m i=1 ⊂ X is considered as an approximate solution of our problems if Pi+1 (x¯i ) = x¯i+1 , i = 1, . . . , m − 1, ||P1 (x¯m ) − x¯1 || ≤ ɛ ,

.

where .ɛ > 0 is a small number. Under the presence of a computational error .δ, a finite sequence .{x¯i }m i=1 ⊂ X is an approximate solution of our problems if ||Pi+1 (x¯i ) − x¯i+1 || ≤ δ, i = 1, . . . , m − 1, ||P1 (x¯m ) − x¯1 || ≤ ɛ ,

.

where the computational error .δ is essentially smaller than .ɛ . As a matter of fact, .ɛ depends on .δ and mappings .Pi , .i = 1, . . . , m.

Main Results Recall that .(X, ) is a real Hilbert space equipped with an inner product . which induces a complete norm .|| · || in X and that .Ci , .i = 1, . . . , m, where .m ≥ 2 is an integer, are closed and convex sets and .Pi = PCi , .i = 1, . . . , m are the corresponding projections. We continue to assume that .

∩m i=1 Ci = ∅.

340

7 Inconsistent Convex Feasibility Problems

Assume that x¯i ∈ Ci , i = 1, . . . , m,

.

x¯i+1 = Pi+1 (x¯i ), i = 1, . . . , m − 1, x¯1 = P1 (x¯m ).

.

(7.3)

We mentioned in Sect. “Preliminaries” that such finite sequence exists if at least one of the sets .Ci , .i = 1, . . . , m is bounded. Set Pm+1 = P1 , x¯m+1 = x¯1 .

.

(7.4)

The following theor is our first main result. Theorem 7.2 Let .M > 0, .ɛ ∈ (0, 1], ||x¯i || ≤ M, i = 1, . . . , m,

.

(7.5)

a nonnegative number .δ satisfy δ < (8m)−1 (2M + 2)−1 ɛ 2

(7.6)

n0 = L8(2M + 1)2 ɛ −2 ⎦ + 1.

(7.7)

.

and .

Assume that a sequence .{xk }∞ k=0 ⊂ X satisfies x0 ∈ B(0, M)

(7.8)

xk+1 ∈ B(Pi(k) (xk ), δ).

(7.9)

.

and that for each integer .k ≥ 0, .

Then the following assertions hold. 1. There exists an integer .k ∈ {0, . . . , n0 } such that for all .i = 1, . . . , m, ||xkm+i+1 − xkm+i − (x¯i+1 − x¯i )|| ≤ ɛ ,

.

if an integer .j ≥ 0 satisfies .j < k, then .

max{||xj m+i+1 − xj m+i − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ

and ||xj m+1 − x¯1 || ≤ 2M + 1, j = 0, . . . , k.

.

Main Results

341

Moreover, ||xkm+1 − x(k+1)m+1 || ≤ ɛ m.

.

2. Assume that 16ɛ m(4M + 2) ≤ 1,

.

k ∈ {0, . . . , n0 },

.

||xkm+1 − x(k+1)m+1 || ≤ ɛ m

.

(7.10)

and that for all integers j satisfying .0 ≤ j < k, ||xj m+1 − x(j +1)m+1 || > ɛ m.

.

(7.11)

Then ||xkm+1 − x¯1 || ≤ 2M + 1

.

and for all .i = 1, . . . , m, ||xkm+i+1 − xkm+i − (x¯i+1 − x¯i )|| ≤ 4(ɛ m(4M + 2))1/2 .

.

Theorem 7.2 can be used in a very easy way. We need just to find the smallest integer .k ∈ {0, . . . , n0 } for which ||xkm+1 − x(k+1)m+1 || ≤ ɛ m.

.

(k+1)m

Then .{xi }i=km+1 is our approximate solution. Theorem 7.2 is proved in Sect. “Proof of Theorem 7.2”. The next theor is related to the case of exact iterations. It is proved in Sect. “Proof of Theorem 7.3”. Theorem 7.3 Let .M > 1, .ɛ ∈ (0, 1] satisfy 16ɛ m(4M + 2) ≤ 1,

.

||x¯i || ≤ M, i = 1, . . . , m

.

and n0 = L8(2M + 1)2 ɛ −2 ⎦ + 1.

.

Assume that x0 ∈ B(0, M),

.

(7.12)

342

7 Inconsistent Convex Feasibility Problems

for each integer .k ≥ 0, xk+1 = Pi(k) (xk )

.

(7.13)

and that an integer .k ∈ {0, . . . , n0 } is as guaranteed by Theorem 7.2 with .δ = 0. Then for each integer .j ≥ k, ||xj m+1 − P1 Pm · · · P2 (xj m+1 )|| = ||xj m+1 − x(j +1)m+1 || ≤ ɛ m

.

and for all .i = 1, . . . , m, ||xj m+i+1 − xj m+i − (x¯i+1 − x¯i )|| ≤ 4(ɛ m(4M + 2))1/2 .

.

In Theorem 7.3 we use Theorem 7.2 with .δ = 0 because only exact iterations are considered. Theorem 7.3 easily implies the following result. Theorem 7.4 Assume that for each .γ > 0 there exists .ɛ > 0 such that if .x ∈ X satisfies ||x − P1 Pm · · · P2 (x)|| ≤ ɛ ,

.

then .||x − x¯1 || ≤ γ . Let .M > 0, .γ > 0. Then there exists an integer .k ≥ 0 such that for each sequence ∞ ⊂ X satisfying .{xi } i=0 x0 ∈ B(0, M),

.

xp+1 = Pi(p) (xp ) for each integer p ≥ 0,

.

each integer .j ≥ k and each .i ∈ {1, . . . , m}, ||xj m+i − x¯i || ≤ γ , i = 1, . . . , m.

.

The last result of this section easily follows from Theorem 7.4 and Theorem 2.65 in [132] (obtained in [35] and applied here for the mapping .P1 Pm · · · P2 with the fixed point .x¯1 ). Theorem 7.5 Assume that for each .γ > 0 there exists .ɛ > 0 such that if .x ∈ X satisfies ||x − P1 Pm · · · P2 (x)|| ≤ ɛ ,

.

then .||x − x¯1 || ≤ γ .

An Auxiliary Result

343

Let .M > 0, .γ > 0. Then there exist an integer .k ≥ 0 and .δ > 0 such that for each sequence .{xi }∞ i=0 ⊂ X satisfying x0 ∈ B(0, M),

.

xp+1 ∈ B(Pi(p) (xp ), δ) for each integer p ≥ 0,

.

each integer .j ≥ k and each .i ∈ {1, . . . , m}, ||xj m+i − x¯i || ≤ γ , i = 1, . . . , m.

.

An Auxiliary Result Lemma 7.6 Let M > 0, ɛ ∈ (0, 1]

.

(7.14)

and a nonnegative number δ satisfy δ < (8m)−1 (4M + 4)−1 ɛ 2 .

.

(7.15)

m+1 Assume that a finite sequence {yi }i=1 ⊂ X satisfies .

||y1 − x¯1 || ≤ 2M + 1,

(7.16)

yi+1 ∈ B(Pi+1 (yi ), δ),

(7.17)

ym+1 ∈ B(P1 (ym ), δ).

(7.18)

for each i ∈ {1, . . . , m − 1}, .

.

Then the following assertions hold. 1. Assume that .

max{||yi+1 − yi − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ .

Then ||ym+1 − x¯1 ||2 ≤ ||y1 − x¯1 ||2 − ɛ 2 /8

.

(7.19)

344

7 Inconsistent Convex Feasibility Problems

and ||ym+1 − x¯1 || ≤ ||y1 − x¯1 || − (4M + 2)−1 ɛ 2 /8.

.

2. If .

max{||yi+1 − yi − (x¯i+1 − x¯i )|| : i = 1, . . . , m} ≤ ɛ ,

(7.20)

then ||y1 − ym+1 || ≤ ɛ m.

.

3. Let ɛ 1 = 4(ɛ m(4M + 2))1/2 ≤ 1,

.

8δm(2M + 2) < ɛ 12 .

.

If ||y1 − ym+1 || ≤ ɛ m,

.

(7.21)

then .

max{||yi+1 − yi − (x¯i+1 − x¯i )|| : i = 1, . . . , m} ≤ 4(ɛ m(4M + 2))1/2 .

Proof Set ξ1 = y1

(7.22)

ξi+1 = Pi+1 (ξi ).

(7.23)

ξm+1 = P1 (ξm ).

(7.24)

.

and for each i ∈ {1, . . . , m}, .

In view of (7.4) and (7.23), .

By (7.17), (7.18) and (7.22)–(7.24), ||y2 − ξ2 || = ||y2 − P2 (y1 )|| ≤ δ.

.

(7.25)

We show that for all i = 1, . . . , m + 1, ||yi − ξi || ≤ δ(i − 1).

.

(7.26)

An Auxiliary Result

345

Equations (7.22) and (7.25) imply that (7.26) holds for i = 1, 2. Assume that k ∈ {1, . . . , m} is an integer and (7.26) holds for all i = 1, . . . , k. Proposition 7.1, (7.1), (7.17), (7.18), (7.23) and (7.26) imply that ||yk+1 − ξk+1 || = ||yk+1 − Pk+1 (ξk )||

.

.

≤ ||yk+1 − Pk+1 (yk )|| + ||Pk+1 (yk ) − Pk+1 (ξk )|| ≤ δ + ||yk − ξk || ≤ kδ.

.

Thus the assumption made for k also holds for k + 1. Therefore by induction we showed that (7.26) holds for all i = 1, . . . , m + 1. Proposition 7.1, (7.1), (7.2)–(7.4) and (7.23) imply that for each i ∈ {1, . . . , m}, ||ξi+1 − x¯i+1 ||2 + ||ξi − ξi+1 − (x¯i − x¯i+1 )||2

.

.

= ||Pi+1 (ξi ) − Pi+1 (x¯i )||2 + ||ξi − Pi+1 (ξi ) − (x¯i − Pi+1 (x¯i ))||2 ≤ ||ξi − x¯i ||2 .

(7.27)

||ξi+1 − x¯i+1 || ≤ ||ξi − x¯i ||.

(7.28)

.

By (7.17), for each i ∈ {1, . . . , m}, .

Equation (7.10) implies that there exists j ∈ {1, . . . , m} such that ||yj +1 − yj − (x¯j +1 − x¯j )|| > ɛ .

.

(7.29)

It follows from (7.15) and (7.26) that for i = 1, . . . , m + 1, ||yi − ξi || ≤ δm ≤ ɛ /4.

.

(7.30)

In view of (7.29) and (7.30), ||ξj +1 − ξj − (x¯j +1 − x¯j )||

.

.

.

≥ ||yj +1 − yj − (x¯j +1 − x¯j )||

− ||yj +1 − yj − (ξj +1 − ξj )|| ≥ ɛ − ɛ /2.

(7.31)

By (7.27) and (7.31), ||ξj +1 − x¯j +1 ||2 + 4−1 ɛ 2 ≤ ||ξj − x¯j ||2 .

.

(7.32)

346

7 Inconsistent Convex Feasibility Problems

It follows from (7.27) and (7.32) that ||ξ1 − x¯1 ||2 − ||ξm+1 − x¯1 ||2

.

=

.

m Σ

(||ξi − x¯i ||2 − ||ξi+1 − x¯i+1 ||2 )

i=1 .

≥ ||ξj − x¯j ||2 − ||ξj +1 − x¯j +1 ||2 ≥ 4−1 ɛ 2

and ||ξm+1 − x¯1 ||2 ≤ ||ξ1 − x¯1 ||2 − 4−1 ɛ 2 .

.

(7.33)

By (7.15), (7.16), (7.22), (7.27) and (7.30), ||||ym+1 − x¯1 ||2 − ||ξm+1 − x¯1 ||2 ||

.

.

= ||||ym+1 − x¯1 || − ||ξm+1 − x¯1 ||||(||ym+1 − x¯1 || + ||ξm+1 − x¯1 ||) .

.

≤ ||ym+1 − ξm+1 ||(2||ξm+1 − x¯1 || + ||ym+1 − ξm+1 ||)

≤ δm(2||ξ1 − x¯1 || + δm) ≤ δm(4M + 2 + δm) ≤ δm(4M + 3).

(7.34)

Equations (7.15), (7.22), (7.33) and (7.34) imply that ||ym+1 − x¯1 ||2 ≤ ||ξm+1 − x¯1 ||2 + δm(4M + 3)

.

.

≤ ||ξ1 − x¯1 ||2 − 4−1 ɛ 2 + δm(4M + 3)

.

= ||y1 − x¯1 ||2 − 4−1 ɛ 2 + δm(4M + 3) .

≤ ||y1 − x¯1 ||2 − 8−1 ɛ 2 .

By (7.16) and (7.35), ɛ 2 /8 ≤ ||y1 − x¯1 ||2 − ||ym+1 − x1 ||2

.

.

= (||y1 − x¯1 || − ||ym+1 − x1 ||)(||y1 − x¯1 || + ||ym+1 − x1 ||) .

≤ 2(||y1 − x¯1 || − ||ym+1 − x1 ||)||y1 − x¯1 || .

≤ (4M + 2)(||y1 − x¯1 || − ||ym+1 − x1 ||)

(7.35)

Proof of Theorem 7.2

347

and ||y1 − x¯1 || − ||ym+1 − x1 || ≥ 8−1 ɛ 2 (4M + 2)−1 .

.

Assertion 1 is proved. Now we prove Assertion 2. Assume that (7.20) holds. In view of (7.4), .

m Σ (x¯i+1 − x¯i ) = x¯m+1 − x¯1 = 0.

(7.36)

i=1

By (7.20) and (7.36), ||ym+1 − y1 || = ||

.

m Σ (yi+1 − yi )|| ≤ ɛ m. i=1

Assertion 2 is proved. Let us prove Assertion 3. Recall that ɛ 1 = (16ɛ m(4M + 2))1/2 .

.

(7.37)

Let (7.21) hold. Assume to the contrary that max{||yi+1 − yi − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ 1 .

.

Applying Assertion 1 with ɛ = ɛ 1 we obtain that ||ym+1 − x¯1 || ≤ ||y1 − x¯1 || − (4M + 2)−1 ɛ 2 /8.

.

(7.38)

Equation (7.38) implies that ||y1 − ym+1 || ≥ 8−1 ɛ 12 (4M + 2)−1 > ɛ m.

.

(7.39)

Clearly, (7.39) contradicts (7.21). The contradiction we have reached proves that for all i = 1, . . . , m, ||yi+1 − yi − (x¯i+1 − x¯i )|| ≤ ɛ 1 = 4(ɛ m(4M + 2))1/2 .

.

Lemma 7.6 is proved.

Proof of Theorem 7.2 By (7.5) and (7.8), ||x0 − x¯1 || < 2M.

.

(7.40)

348

7 Inconsistent Convex Feasibility Problems

Proposition 7.1, (7.1) and (7.40) imply that ||P1 (x0 ) − x¯1 || ≤ ||x0 − x¯1 || ≤ 2M.

.

(7.41)

In view of (7.6), (7.9) and (7.41), ||x1 − x¯1 || ≤ ||x1 − P1 (x0 )|| + ||P1 (x0 ) − x¯1 || ≤ 2M + 1.

.

(7.42)

Let us prove Assertion 1. If ||xi+1 − xi − (x¯i+1 − x¯i )|| ≤ ɛ , i = 1, . . . , m,

.

(7.43)

then in view of (7.43) and (7.4), ||x1 − xm+1 || = ||

m Σ

.

(xi+1 − xi )||

i=1

.

≤ mɛ + ||

m Σ

(x¯i+1 − x¯i )|| = mɛ

i=1

and Assertion 1 holds. Now assume that .

max{||xi+1 − xi − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ .

(7.44)

Assume that k is a natural number and that for all integers .j = 0, . . . , k − 1, .

max{||xj m+i+1 − xj m+i − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ .

(7.45)

(Note that by (7.44) our assumption holds for .k = 1.) Assume that .j ∈ {0, . . . , k − 1} and that ||xj m+1 − x¯1 || ≤ 2M + 1.

.

(7.46)

(In view of (7.42), this assumption holds for .j = 0.) Set yi = xj m+i , i = 1, . . . , m + 1.

(7.47)

||y1 − x¯1 || ≤ 2M + 1.

(7.48)

.

By (7.46) and (7.47), .

Proof of Theorem 7.2

349

It follows from (7.4), (7.9) and (7.47) that for all .i = 1, . . . , m, ||yi+1 − Pi+1 (yi )|| = ||xj m+i+1 − Pi+1 (xj m+i )|| ≤ δ.

.

(7.49)

In view of (7.15), (7.45) and (7.47), .

max{||yi+1 − yi − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ .

(7.50)

Assertion 1 of Lemma 7.6 and (7.48)–(7.50) imply that ||ym+1 − x¯1 ||2 ≤ ||y1 − x¯1 ||2 − ɛ 2 /8,

.

||ym+1 − x¯1 || ≤ ||y1 − x¯1 || − (ɛ 2 /8)(4M + 2)−1 .

.

(7.51) (7.52)

By (7.47), (7.51) and (7.52), ||x(j +1)m+1 − x¯1 ||2 ≤ ||xj m+1 − x¯1 ||2 − ɛ 2 /8,

.

(7.53)

and ||x(j +1)m+1 − x¯1 || ≤ ||xj m+1 − x¯1 || − (ɛ 2 /8)(4M + 2)−1 .

.

(7.54)

In view of (7.46) and (7.54), ||x(j +1)m+1 − x¯1 || ≤ 2M + 1.

.

Hence the assumption made for j (see (7.46)) also holds for .j +1. Thus by induction we showed that ||xj m+1 − x¯1 || ≤ 2M + 1, j = 0, . . . , k

.

and that for all .j = 0, . . . , k − 1 (see (7.53)) ||x(j +1)m+1 − x¯1 ||2 ≤ ||xj m+1 − x¯1 ||2 − ɛ 2 /8.

.

Together with (7.42) this implies that (2M + 1)2 ≥ ||x1 − x¯1 ||2 ≥ ||x1 − x¯1 ||2 − ||xkm+1 − x¯1 ||2

.

.

=

k−1 Σ (||xj m+1 − x¯1 ||2 − ||x(j +1)m+1 − x¯1 ||2 ) ≥ 8−1 ɛ 2 k, j =0

k ≤ 8(2M + 1)2 ɛ −2

.

350

7 Inconsistent Convex Feasibility Problems

and k ≤ L8(2M + 1)2 ɛ −2 ⎦ < n0 .

.

This implies that there exists an integer .k ∈ {1, . . . , n0 } such that ||xkm+i+1 − xkm+i − (x¯i+1 − x¯i )|| ≤ ɛ , i = 1, . . . , m

.

(7.55)

and if an integer j satisfies .0 ≤ j < k, then .

max{||xj m+i+1 − xj m+i − (x¯i+1 − x¯i )|| : i = 1, . . . , m} > ɛ

and ||xj m+1 − x¯1 || ≤ 2M + 1, j = 0, . . . , k.

.

By (7.4) and (7.55), ||xkm+1 − x(k+1)m+1 || = ||

.

m Σ (xkm+i − xkm+i+1 )|| i=1

≤ ɛ m + ||

.

m Σ

(x¯i − x¯i+1 )|| = ɛ m.

i=1

Thus Assertion 1 is proved. Let us prove Assertion 2. By (7.11) and the equation

.

m Σ (x¯i+1 − x¯i ) = x¯m+1 − x¯1 = 0 i=1

for all integers j satisfying .0 ≤ j ≤ k, .

max{||xj m+i+1 − xj m+i − (x¯i − x¯i+1 )|| : i = 1, . . . , m} > ɛ .

If an integer j satisfies .0 ≤ j ≤ k and ||xj m+1 − x¯1 || ≤ 2M + 1,

.

then we apply Lemma 7.6 to yi = xj m+i , i = 1, . . . , m + 1

.

and obtain that ||x(j +1)m+1 − x¯1 ||2 ≤ ||xj m+1 − x¯1 ||2 − ɛ 2 /8.

.

(7.56)

Proof of Theorem 7.3

351

Thus by induction and (7.42) we showed that ||xkm+1 − x¯1 || ≤ ||x1 − x¯1 || ≤ 2M + 1.

.

Now Assertion 2 follows from Lemma 7.6 applied with yi = xkm+i , i = 1, . . . , m + 1.

.

Proof of Theorem 7.3 Since k is guaranteed by Theorem 7.2 we have ||xkm+1 − x(k+1)m+1 || ≤ ɛ m.

.

(7.57)

Proposition 7.1, (7.1), (7.4) and (7.13) imply that for each integer .j ≥ 0, ||x(j +1)m+1 − x¯1 || = ||

m+1 | |

.

Pi (xj m+1 ) − x¯1 ||

i=2 .

= ||P1 Pm · · · P2 (xj m+1 ) − P1 Pm · · · P2 (x¯1 )|| ≤ ||xj m+1 − x¯1 ||

and ||xj m+1 − x¯1 || ≤ ||x1 − x¯1 || = ||P1 (x0 ) − x¯1 ||

.

.

≤ ||x0 − x¯1 || ≤ 2M.

Proposition 7.1, (7.1), (7.13) and (7.57) imply that for all integers .j ≥ k + 1, ||xj m+1 − x(j +1)m+1 ||

.

.

= ||(Pm+1 · · · P2 )j −k (xkm+1 ) − (Pm+1 · · · P2 )j −k (x(k+1)m+1 )|| .

≤ ||xkm+1 − x(k+1)m+1 || ≤ ɛ m.

Let .j ≥ k + 1 be an integer. Now we apply Lemma 7.6 with .yi = xj m+i , .i = 1, . . . , m + 1 and obtain ||xj m+i+1 − xj m+i − (x¯i+1 − x¯i )|| ≤ 4(ɛ m(4M + 2))1/2 ,

.

i = 1, . . . , m.

.

Chapter 8

Split Common Fixed Point Problems

In this chapter we study split common fixed point problems in a Hilbert space under the presence of computational errors. We show that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterations one needs for this.

Preliminaries Assume that for .i = 1, 2, .(Hi , ) be Hilbert spaces equipped with inner products which induce the norm ||x|| = 1/2 , x ∈ Hi , i = 1, 2.

.

For each linear bounded operator .A : H1 → H2 denote by .A∗ : H2 → H1 its dual operator. Assume that .U : H1 → H1 , .T : H2 → H2 are mappings .A : H1 → H2 are linear bounded operators. Recall that for each .S : X → X, where X is a nonempty set, .Fix(S) = {x ∈ X : S(x) = x}. Consider the problem Find x ∗ ∈ Fix(U ) such that Ax ∗ ∈ Fix(T ).

.

Set F = {x ∈ Fix(U ); Ax ∈ Fix(T )}.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0_8

353

354

8 Split Common Fixed Point Problems

This split common fixed point problem was introduced and studied in [59]. In this chapter we use the algorithm of [59] taking into account computational errors produced by our computer system. Let .(H, ) be Hilbert spaces equipped with the inner product .. Denote by .I : H → H the identity operator. For each .x, y ∈ H define H (x, y) = {u ∈ H : ≤ 0}.

.

(8.1)

An operator .T : H → H is called a directed operator [59] if Fix(T ) ⊂ H (x, T (x)) for all x ∈ H

.

(8.2)

or equivalently if .g ∈ Fix(T ), then ≤ 0 for all x ∈ H.

.

(8.3)

Clearly, if .T : H → H is a directed operator and .Fix(T ) /= ∅, then Fix(T ) = ∩x∈X H (x, T (x))

.

and it is a closed, convex set and for each .λ ∈ [0, 1] the mapping .I + λ(T − I ) is also directed. For each .x ∈ Hi , .i = 1, 2 and each .r > 0 set B(x, r) = {y ∈ Hi : ||x − y|| ≤ r}.

.

Proposition 8.1 ([147]) Let .T : H → H be a mapping. Then the following statements are equivalent. (i) T is directed; (ii) .||x − T (x)||2 ≤ for each .z ∈ Fix(T ) and each .x ∈ H ; (iii) .||z − T (z)||2 ≤ ||x − z||2 − ||x − T (x)||2 for each .z ∈ Fix(T ) and each .x ∈ H . Assume that .T , U are directed mappings and that γ ∈ (0, 2||A||−2 ).

.

(8.4)

The following algorithm was studied in [59]. Initialization: select an arbitrary .x0 ∈ H1 . Iterative step: for an integer .k ≥ 0, given a current iteration point .xk ∈ H1 calculate the next iteration point xk+1 = U (xk + γ A∗ (T − I )(Axk )).

.

The Main Result

355

Now we consider this algorithm under the presence of computational errors. Assume that δU , δT , δA ∈ [0, 1].

.

(8.5)

Initialization: select an arbitrary .x0 ∈ H1 . Iterative step: for an integer .k ≥ 0, given a current iteration point .xk ∈ H1 calculate .zk ∈ H2 such that ||zk − (T − I )A(xk )|| ≤ δT ,

.

yk ∈ H1 such that

.

||yk − γ A∗ zk || ≤ δA

.

and .xk+1 ∈ H1 such that ||xk+1 − U (xk + yk )|| ≤ δU .

.

Here .δT is a computational error when we calculate .(T − I )A(xk ), .δA is a computational error when we calculate .γ A∗ zk and .δU is a computational error when we calculate .U (xk + yk ).

The Main Result We prove the following result which shows that our algorithm generate a good approximate solution, if computational errors are bounded from above by a small positive constant. For known computational errors, it shows what approximate solution can be obtained and how many iterations one needs for this. Theorem 8.2 Assume that .M ≥ 1, {x ∈ H1 : U (x) = x, T (Ax) = Ax} ∩ B(0, M) /= ∅,

.

a positive number .ɛ satisfies ɛ ≥ max{4δT , 4δU , 4δA , (32(min{1, γ (2 − γ ||A||2 )−1 })1/2

.

.

× [4M(2δU + δA + γ ||A||δT )(γ ||A||δT + δA )2 + δU2 + 2δU (γ ||A||δT + δA )]1/2 }, Δ = 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )},

.

356

8 Split Common Fixed Point Problems

∞ ∞ x0 ∈ H1 ∩ B(0, M), .{xk }∞ k=1 ⊂ H1 , .{yk }k=0 ⊂ H1 , .{zk }k=0 ⊂ H2 , for each integer .k ≥ 0, .

||zk − (T − I )(Axk )|| ≤ δT ,

.

||yk − γ A∗ zk || ≤ δA ,

.

||xk+1 − U (xk + yk )|| ≤ δU .

.

Then there exists a nonnegative integer .n ≤ 4M 2 Δ−1 such that for each integer .k ∈ {0, . . . , n} \ {n}, .

.

max{||xk + yk − xk+1 ||, ||zk ||} ≥ ɛ /2, max{||xn + yn − xn+1 ||, ||zn ||} < ɛ /2,

(8.6)

||xk || ≤ 3M, k = 0, . . . , n

.

If .n ≥ 0 is an integer and (8.6) holds, then ||yn || ≤ 2−1 ɛ (1 + γ ||A||),

.

||T (Axn ) − Axn || < ɛ ,

.

||U (xn + yn ) − (xn + yn )|| < ɛ .

.

By Theorem 8.2, there is a nonnegative integer .n ≤ 4M 2 Δ−1 such that ||T (Axn ) − Axn || < ɛ

.

and there is .ξ ∈ B(xn , 2−1 ɛ (1 + γ ||A||)) such that ||U (ξ ) − ξ || < ɛ .

.

This .xn is an approximate solution of our split common fixed point problem.

Proof of Theorem 8.2 Theorem 8.2 easily follows from the next lemma. ∞ ∞ Lemma 8.3 Assume that .x0 ∈ H1 , .{xk }∞ k=1 ⊂ H1 , .{yk }k=0 ⊂ H1 , .{zk }k=0 ⊂ H2 , for each integer .k ≥ 0,

||zk − (T − I )(Axk )|| ≤ δT ,

.

(8.7)

Proof of Theorem 8.2

357

||yk − γ A∗ zk || ≤ δA ,

(8.8)

||xk+1 − U (xk + yk )|| ≤ δU

(8.9)

z ∈ Fix(U ), Az ∈ Fix(T ).

(8.10)

.

.

and .

Then the following assertions hold. 1. For each integer .k ≥ 0, ||xk+1 − z||2 ≤ ||xk − z||2 − ||xk + yk − U (xk + yk )||2

.

.

− γ (2 − γ ||A||2 )||(T − I )(Axk )||2 + 2||xk − z||(γ ||A||δT + δA + 2δU ) .

2 + (γ ||A||δT + δA )2 + 2δU (γ ||A||δT + δA ) + δU .

2. Let .M ≥ 1, ɛ > 0, δA , δT , δU ≤ ɛ /4,

.

2 4M(2δU + δA + γ ||A||δT ) + (γ ||A||δT + δA )2 + δU + 2δU (γ ||A||δT + δA )

.

≤ 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )},

(8.11)

z, x0 ∈ B(0, M),

(8.12)

Δ = 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )},

(8.13)

.

.

.

n be a natural number and for each integer .k ∈ {0, . . . , n − 1}, .

max{||xk + yk − xk+1 ||, ||zk ||} ≥ ɛ /2.

Then ||xi − z|| ≤ 2M, i = 0, . . . , n

.

and n ≤ 4M 2 Δ−1 .

.

3. Assume that the assumptions of Assertion 2 hold, .m ≥ 0 is an integer, .

max{||xm + ym − xm+1 ||, ||zm ||} < ɛ /2.

(8.14)

358

8 Split Common Fixed Point Problems

Then ||T (Axm ) − A(xm )|| < ɛ ,

.

||ym || ≤ δA + γ ||A||ɛ /2 ≤ 2−1 ɛ (1 + γ ||A||),

.

||U (xm + ym ) − (xm + ym )|| < ɛ .

.

Proof Let .k ≥ 0 be an integer. By (8.9) and the properties of the inner product, ||xk+1 − z||2 = ||xk+1 − U (xk + yk ) + U (xk + yk ) − z||2

.

.

.

≤ ||xk+1 − U (xk + yk )||2

+ 2||xk+1 − U (xk + yk )||||U (xk + yk ) − z|| + ||U (xk + yk ) − z||2 .

≤ δU2 + 2δU ||U (xk + yk ) − z|| + ||U (xk + yk ) − z||2 .

(8.15)

In view of (8.10) and Proposition 8.1, ||U (xk + yk ) − z|| ≤ ||xk + yk − z||.

.

(8.16)

Equations (8.15) and (8.16) imply that ||xk+1 − z||2 ≤ δU2 + 2δU ||xk + yk − z|| + ||U (xk + yk ) − z||2 .

.

(8.17)

By (8.8), ||xk + yk − z|| ≤ ||xk − z + γ A∗ zk − γ A∗ zk + yk ||

.

.

≤ ||xk − z + γ A∗ zk || + δA .

(8.18)

In view of (8.7), ||xk − z + γ A∗ zk ||

.

.

≤ ||xk − z + γ A∗ (T − I )(Axk ) − γ A∗ (T − I )(Axk ) + γ A∗ zk || .

≤ ||xk − z + γ A∗ (T − I )(Axk )|| + γ ||A||δT .

(8.19)

Equations (8.18) and (8.19) imply that ||xk + yk − z|| ≤ ||xk − z + γ A∗ (T − I )(Axk )|| + γ ||A||δT + δA .

.

(8.20)

Proof of Theorem 8.2

359

It follows from (8.17) and (8.20) that ||xk+1 − z||2

.

.

2 ≤ δU + 2δU ||xk − z + γ A∗ (T − I )(Axk )|| + 2δU (γ ||A||δT + δA ) .

.

+ ||xk − z + γ A∗ (T − I )(Axk )||2 + (γ ||A||δT + δA )2

+ 2(γ ||A||δT + δA )||xk − z + γ A∗ (T − I )(Axk )||.

(8.21)

In view of (8.20), ||xk + yk − z||2 ≤ (||xk − z + γ A∗ (T − I )(Axk )|| + γ ||A||δT + δA )2

.

.

.

≤ ||xk − z + γ A∗ (T − I )(Axk )||2

+ 2||xk − z + γ A∗ (T − I )(Axk )||(γ ||A||δT + δA ) + (γ ||A||δT + δA )2 .

(8.22)

It is easy to see that ||xk − z + γ A∗ (T − I )(Axk )||2 = ||xk − z||2 + γ 2 ||A∗ (T − I )(Axk )||2

.

+ 2γ

.

.

= ||xk − z||2 + γ 2 + 2γ

.

(8.23)

and γ 2 ≤ γ 2 ||A||2 ||(T − I )(Axk )||2 .

.

(8.24)

By (8.3) and (8.10), 2γ

.

.

= 2γ .

= 2γ − ||(T − I )(Axk )||2 .

≤ −2γ ||(T − I )(Axk )||2 .

It follows from (8.23)–(8.25) that ||xk − z + γ A∗ (T − I )(Axk )||2

.

(8.25)

360

8 Split Common Fixed Point Problems

.

≤ ||xk − z||2 + γ 2 ||A||2 ||(T − I )(Axk )||2 − 2γ ||(T − I )(Axk )||2 ≤ ||xk − z||2 + γ (γ ||A||2 − 2)||(T − I )(Axk )||2 .

.

(8.26)

Equations (8.4) and (8.26) imply that ||xk − z + γ A∗ (T − I )(Axk )|| ≤ ||xk − z||.

.

(8.27)

By (8.20) and (8.27), ||xk + yk − z|| ≤ ||xk − z|| + γ ||A||δT + δA .

.

(8.28)

In view of (8.21), (8.26) and (8.27), ||xk+1 − z||2

.

.

≤ δU2 + 2δU ||xk − z||2 + 2δU (γ ||A||δT + δA )

.

+ (γ ||A||δT + δA )2 + 2(γ ||A||δT + δA )||xk − z||

.

+ ||xk − z||2 − γ (2 − γ ||A||2 )||(T − I )(Axk )||2 .

Proposition 8.1 (iii) and Eqs. (8.9), (8.10), (8.26) and (8.28) imply that ||xk+1 − z||2 = ||xk+1 − U (xk + yk ) + U (xk + yk ) − z||2

.

.

≤ δU2 + 2δU ||U (xk + yk ) − z|| + ||U (xk + yk ) − z||2 ≤ δU2 + 2δU ||xk + yk − z|| + ||U (xk + yk ) − z||2

.

.

.

+ ||xk + yk − z||2 − ||xk + yk − U (xk + yk )||2 .

.

.

≤ δU2 + 2δU (||xk − z|| + γ ||A||δT + δA )

≤ δU2 + 2δU (||xk − z|| + γ ||A||δT + δA )

+ ||xk − z||2 + γ (γ ||A||2 − 2)||(T − I )(Axk )||2

+ 2||xk − z||(γ ||A||δA + δA ) + (γ ||A||δT + δA )2 − ||xk + yk − U (xk + yk )||2 .

≤ ||xk − z||2 − ||xk + yk − U (xk + yk )||2 .

− γ (2 − γ ||A||2 )||(T − I )(Axk )||2

Proof of Theorem 8.2

361

+ 2||xk − z||(γ ||A||δT + δA + 2δU ) + (γ ||A||δT + δA )2

.

.

2 + δU + 2δU (γ ||A||δT + δA ).

Assertion 1 is proved. Let us prove Assertion 2. We show that for all .k = 0, . . . , n, ||xk − z|| ≤ 2M.

.

(8.29)

(In view of (8.12), Eq. (8.29) holds for .k = 0.) Assume that .p ∈ [0, n) is an integer and ||xp − z|| ≤ 2M.

.

(8.30)

Assertion 1 implies that ||xp+1 − z||2 ≤ ||xp − z||2 − ||xp + yp − U (xp + yp )||2

.

.

.

− γ (2 − γ ||A||2 )||(T − I )(Axp )||2

+2||xp −z||(γ ||A||δT +δA +2δU )+(γ ||A||δT +δA )2 +2δU (γ ||A||δT +δA ).

(8.31)

There are two cases: ||xp + yp − xp+1 || ≥ ɛ /2;

(8.32)

||zp || ≥ ɛ /2.

(8.33)

.

.

Assume that (8.32) holds. Then by (8.9), (8.11) and (8.32), ||xp + yp − U (xp + yp )||

.

.

≥ ||xp + yp − xp+1 || − ||U (xp + yp ) − xp+1 || .

≥ ɛ /2 − δU ≥ ɛ /4.

This implies that ||xp + yp − U (xp + yp )||2 + γ (2 − γ ||A||2 )||(T − I )(Axp )||2 ≥ 16−1 ɛ 2 .

.

Assume that (8.33) holds. Then by (8.7), (8.11) and (8.33), ||(T − I )(Axp )|| ≥ ||zp || − ||zp − (T − I )(Axp )|| ≥ ɛ /2 − δT ≥ ɛ /4.

.

362

8 Split Common Fixed Point Problems

This implies that ||xp + yp − U (xp + yp )||2 + γ (2 − γ ||A||2 )||(T − I )(Axp )||2

.

.

≥ 16−1 ɛ 2 γ (2 − γ ||A||2 ).

Therefore in both cases ||xp + yp − U (xp + yp )||2 + γ (2 − γ ||A||2 )||(T − I )(Axp )||2

.

.

≥ 16−1 ɛ 2 min{1, γ (2 − γ ||A||2 )}.

(8.34)

By (8.11), (8.30), (8.31) and (8.34), ||xp+1 − z||2 ≤ ||xp − z||2 − 16−1 ɛ 2 min{1, γ (2 − γ ||A||2 )} + δU2

.

+ 2||xp − z||(γ ||A||δT + δA + 2δU ) + (γ ||A||δT + δA )2 + 2δU (γ ||A||δT + δA )

.

.

.

≤ |xp − z||2 − 16−1 ɛ 2 min{1, γ (2 − γ ||A||2 )}

+ 4M(γ ||A||δT + δA + 2δU ) + (γ ||A||δT + δA )2 + δU2 + 2δU (γ ||A||δT + δA ) .

≤ ||xp − z||2 − 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )}.

(8.35)

In view of (8.30) and (8.35), ||xp+1 − z|| ≤ ||T (xp ) − z|| ≤ 2M,

.

||xp+1 − z||2 ≤ ||xp − z||2 − 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )}.

.

Thus by induction we showed that ||xp − z|| ≤ 2M, p = 0, . . . , n,

.

and that for all .p = 0, . . . , n − 1, ||xp+1 − z||2 ≤ ||xp − z||2 − 32−1 ɛ 2 min{1, γ (2 − γ ||A||2 )}.

.

It follows from (8.13), (8.36) and the relation above that 4M 2 ≥ ||x0 − z||2 ≥ ||x0 − z||2 − ||xn − z||2

.

.

=

n−1 Σ p=0

(||xp − z||2 − ||xp+1 − z||2 ) ≥ nΔ

(8.36)

The Split Common Fixed Point Problem with Several Maps

363

and n ≤ 4M 2 Δ−1 .

.

Assertion 2 is proved. Let us prove Assertion 3. By our assumptions and (8.7) and (8.11), ||(T − I )(Axm )|| ≤ ||zm || + δT < ɛ /2 + δT < ɛ ,

.

||xm + ym − U (xm + ym )|| ≤ ||xm+1 − (xm + ym )||

.

.

+ ||U (xm + ym ) − xm+1 || ≤ ɛ /2 + δU < ɛ .

By (8.8) and the assumptions of Assertion 3, ||ym || ≤ δA + γ ||A||||zm || ≤ δA + γ ||A||ɛ /2.

.

U ∩

Assertion 3 is proved.

The Split Common Fixed Point Problem with Several Maps Assume that .m1 , m2 are natural numbers, Ui : H1 → H1 , i = 1, . . . , m1 , Ti : H2 → H2 , i = 1, . . . , m2

.

(8.37)

are directed operators, .A : H1 → H2 is a bounded linear operator. For .i = 1, . . . , m1 set Fix(Ui ) = {z ∈ H1 : Ui (z) = z},

(8.38)

Fix(Ti ) = {z ∈ H2 : Ti (z) = z}.

(8.39)

m2 ∗ 1 x ∗ ∈ ∩m i=1 Fix(Ui ) such that Ax ∈ ∩i=1 Fix(Ti ).

(8.40)

.

for .i = 1, . . . , m2 set .

Our goal is to find .

Consider the space .H3 = H1m1 × H2m2 which consists of vectors x = (x1 , . . . , xm1 , xm1 +1 , . . . , xm1 +m2 )

.

364

8 Split Common Fixed Point Problems

such that .xi ∈ H1 , i = 1, . . . , m1 , .xi ∈ H2 , i = m1 + 1, . . . , m1 + m2 . Let αi > 0, i = 1, . . . , m1 + m2 .

.

Consider a mapping .T˜ : H3 → H3 defined by T˜ (x1 , . . . , xm1 , xm1 +1 , . . . , xm1 +m2 )

.

−1/2

1/2

= (α1 U1 (α1

.

−1/2

1/2

x1 ), . . . , αi Ui (αi −1/2

1/2

1/2

−1/2

1/2

αm1 +1 T1 (αm1 +1 xm1 +1 ), . . . , αm1 +i Ti (α1

.

−1/2

1/2

αm1 +m2 Tm2 (α1

.

−1/2

xi ), . . . , αm1 Um1 (α1

xm1 ),

xm1 +i ), . . . ,

xm1 +m2 )).

(8.41)

Set Q=

m1 | |

.

1/2

αi Fix(Ui ) ×

i=1

m2 | |

1/2

αi+m1 Fix(Ti ).

(8.42)

i=1

Define an operator .A˜ : H1 → H3 by 1/2 1/2 1/2 ˜ = (α 1/2 x, . . . , αm Ax 1 x, αm1 +1 Ax, . . . , αm1 +m2 Ax), x ∈ H1 . 1

.

(8.43)

Let .I : H1 → H1 be an identity operator. Clearly, −1 m1 1 ˜ x ∈ ∩m i=1 Fix(Ui ) ∩ A (∩i=1 Fix(Ti )) if and only if Ax ∈ Q.

.

(8.44)

For each .x ∈ H1 , ˜ =( ||Ax||

m1 Σ

.

αi ||x||2 +

m2 Σ

i=1

.

≤ ||x||(

m1 Σ

αm1 +i ||Ax||2 )1/2

i=1

αi +

m2 Σ

i=1

i=1

m1 Σ

m2 Σ

αm1 +i ||A||2 )1/2

and ˜ ≤( .||A||

i=1

αi +

αm1 +i ||A||2 )1/2 .

i=1

It is not difficult to see that .x ∈ H1 satisfies Ui (x) = x, i = 1, . . . , m1 , Ti (Ax) = Ax, i = m1 + 1, . . . , m1 + m2

.

(8.45)

The Split Common Fixed Point Problem with Several Maps

365

if and only if ˜ = Ax. ˜ T˜ (Ax)

(8.46)

.

It is easy to see that .T˜ is a directed operator. Set .U˜ = I : H1 → H1 - the identity operator. Now we consider the equivalent problem ˜ = Ax. ˜ Find x ∈ H1 such that T˜ (Ax)

.

and use the following algorithm of [59]. Initialization: select an arbitrary .x0 ∈ H1 . Iterative step: for an integer .k ≥ 0, given a current iteration point .xk ∈ H1 calculate the next iteration point ˜ k) xk+1 = xk + γ A˜ ∗ (T˜ − I )(Ax

.

(here I is the identity operator in .H3 ). Now we consider this algorithm under the presence of computational errors. First, we calculate .A˜ ∗ : H3 → H1 . Let .x ∈ H1 , .l ∈ H3 . Then 1/2 1/2 1/2 1/2 = = 0, .i = 1, . . . , m2 , .δU,i > 0, .i = 1, . . . , m1 , .δA , δU > 0, .γ > 0, .αi > 0, .i = 1, . . . , m1 + m2 . Initialization: select an arbitrary .x0 ∈ H1 . Iterative step: for an integer .k ≥ 0, given a current iteration point .xk ∈ H1 for .i = 1, . . . , m1 find .yk,i ∈ H1 satisfying ||Ui (xk ) − xk − yk,i || ≤ δU,i ,

.

for .i = 1, . . . , m2 , define .zk,i+m1 ∈ H2 satisfying ||Ti (Axk ) − Axk − zk,m1 +i || ≤ δT ,i

.

and .yk,i+m1 ∈ H1 satisfying ||yk,i+m1 − A∗ (zk,m1 +i )|| ≤ δA

.

and xk+1 ∈ xk + γ

mΣ 1 +m2

.

αi yk,i + B(0, δU ).

i=1

Set δ˜T = (

m1 Σ

.

m2 Σ

2 αU,i +

αi+m1 δT2 ,i )1/2 ,

(8.49)

αi2 )1/2 .

(8.50)

+αm1 +i ||A||2 ).

(8.51)

i=m1 +1

i=1

δ˜A = γ δA (

mΣ 1 +m2

.

i=m1 +1

Assume that 0 < γ < 2(

m1 Σ

.

i=1

αi +

m2 Σ i=1

The Split Common Fixed Point Problem with Several Maps

367

In the next section we prove the following result. Theorem 8.4 Assume that .M ≥ 1, {z ∈ H1 : Ui (z) = z, i = 1, . . . , m,

.

Ti (Az) = Az, i = 1, . . . , m2 } ∩ B(0, M) /= ∅,

.

(8.52)

{xk }∞ k=0 ⊂ H1 , yk,i ∈ H1 , k = 0, 1, . . . , i = 1, . . . , m1 , .zl.m1 +i ∈ H2 , k = 0, 1, . . . , i = 1, . . . , m2 ,

.

yk,i+m1 ∈ H1 , k = 0, 1, . . . , ; i = 1, . . . , m2 ,

.

||x0 || ≤ M,

(8.53)

.

for every integer .k ≥ 0 and every integer .i ∈ {1, . . . , m1 }, ||Ui (xk ) − xk − yk,i || ≤ δU,i ,

.

(8.54)

for every integer .k ≥ 0 and every integer .i ∈ {1, . . . , m2 }, ||Ti (Axk ) − Axk − zk,m1 +i || ≤ δT ,i ,

(8.55)

||yk,i+m1 − A∗ (zk,m1 +i )|| ≤ δA

(8.56)

.

.

and ||xk+1 − xk − γ

mΣ 1 +m2

.

αi yk,i || ≤ δU .

(8.57)

i=1

Let δ˜T ∈ (0, 1], δ∗ = δ˜T γ (

m1 Σ

.

αi +

i=1

m2 Σ

αi+m1 ||A||2 )1/2 + δ˜A ,

i=1

ɛ ≥ max{4δUi , i = 1, . . . , m, 4δTi , i = 1, . . . , m2 ,

.

[(γ∗ + δU )2 + 32(2δU + δ∗ )(2M + 2) min{αi : i = 1, . . . , m1 + m2 }−1 γ −1 (2

.

.

− γ(

m1 Σ i=1

αi +

m2 Σ i=1

αi+m1 ||A||2 )−1 ]1/2 }.

368

8 Split Common Fixed Point Problems

Then there exists a nonnegative integer n such that n ≤ 642 M 2 γ −1 ɛ −2 min{αi : i = 1, . . . , m1 + m2 }−1 (2 − γ (

m1 Σ

.

αi

i=1

.

+

m2 Σ

αi+m1 ||A||2 ))−1

i=1

such that for each integer .k ∈ {0, . . . , n}, ||xk || ≤ 3M,

.

for each integer .k ∈ {0, . . . , n − 1}, .

max{||yk,i || : i = 1, . . . , m1 , ||zk,m1 +i || : i = 1, . . . , m2 } ≥ ɛ /2,

.

max{||yn,i || : i = 1, . . . , m1 , ||zn,m1 +i || : i = 1, . . . , m2 } < ɛ /2.

Moreover for every integer .k ≥ 0 for which .

max{||yk,i || : i = 1, . . . , m1 , ||zk,m1 +i || : i = 1, . . . , m2 } < ɛ /2

the inequality ||Ui (xk ) − xk || ≤ ɛ , i = 1, . . . , m1 , ||Ti (Axk ) − Axk || ≤ ɛ .

.

By Theorem 8.4, there exists a nonnegative integer k ≤ 642 M 2 γ −1 ɛ −2 min{αi : i = 1, . . . , m1 + m2 }−1 (2 − γ (

m1 Σ

.

αi

i=1

.

+

m2 Σ

αi+m1 ||A||2 ))−1

i=1

such that ||Ui (xk ) − xk || ≤ ɛ , i = 1, . . . , m1 , ||Ti (Axk ) − Axk || ≤ ɛ .

.

This .xk is an appropriate solution of our split common fixed point problem.

Proof of Theorem 8.4

369

Proof of Theorem 8.4 Theorem follows easily from the next lemma. Lemma 8.5 z ∈ H1 : Ui (z) = z, i = 1, . . . , m, Ti (Az) = Az, i = 1, . . . , m2 ,

.

(8.58)

{xk }∞ k=0 ⊂ H1 , yk,i ∈ H1 , k = 0, 1, . . . , i = 1, . . . , m1 ,

.

.

zk,m1 +i ∈ H2 , k = 0, 1, . . . , i = 1, . . . , m2 ,

(8.59)

yk,i+m1 ∈ H1 , k = 0, 1, . . . , ; i = 1, . . . , m2 ,

(8.60)

.

for every integer .k ≥ 0 and every integer .i ∈ {1, . . . , m1 }, ||Ui (xk ) − xk − yk,i || ≤ δU,i ,

.

(8.61)

for every integer .k ≥ 0 and every integer .i ∈ {1, . . . , m2 }, ||Ti (Axk ) − Axk − zk,m1 +i || ≤ δT ,i ,

(8.62)

||yk,i+m1 − A∗ (zk,m1 +i )|| ≤ δA

(8.63)

.

.

and ||xk+1 − xk − γ

mΣ 1 +m2

.

αi yk,i || ≤ δ0 .

(8.64)

i=1

Then the following assertions hold. 1. For each integer .k ≥ 0, ||xk+1 − z||2 ≤ ||xk − z||2

.

.

m1 m2 Σ Σ − γ (2 − γ ( αi + αi+m1 ||A||2 )) i=1

.

m1 m2 Σ Σ ×( αi ||Ui (xk ) − xk ||2 + αi+m1 ||Ti (Axk ) − Axk ||2 ) i=1

i=1 .

.

i=1

˜ δ˜T + δ˜A + 2δU ) + 2||xk − z||(γ ||A||

2 ˜ δ˜T + δ˜A )2 + δU ˜ δ˜T + δ˜A ). + (γ ||A|| + 2δU ((γ ||A||

(8.65)

370

8 Split Common Fixed Point Problems

2. Let .M > 0, .0 < δ˜T ≤ 1, δ∗ = δ˜T γ (

m1 Σ

.

αi +

i=1

m2 Σ

αi+m1 ||A||2 )1/2 + δ˜A ,

i=1

ɛ ≥ max{4δUi , i = 1, . . . , m, 4δTi , i = 1, . . . , m2 },

.

[(δ∗ + δU )2 + 32(2δU + δ∗ )(2M + 2) min{αi : i = 1, . . . , m1 + m2 }−1 γ −1 (2

.

.

− γ(

m1 Σ

αi +

i=1

m2 Σ

αi+m1 ||A||2 ))−1 ]1/2 },

i=1

n ≥ 1 is an integer,

.

z, x0 ∈ B(0, M)

.

and for all integers .k = 0, . . . , n − 1, .

max{||yk,i || : i = 1, . . . , m1 , ||zk,m1 +i || : i = 1, . . . , m2 } ≥ ɛ /2

(8.66)

Then n ≤ 642 M 2 γ −1 ɛ −2 min{αi :

.

.

i = 1, . . . , m1 + m2 }−1 (2 − γ (

m1 Σ i=1

αi +

m2 Σ

αi+m1 ||A||2 )−1 ),

i=1

||xi − z|| ≤ 2M, i = 0, . . . , n.

.

3. Let .k ≥ 0 be an integer, .

max{||yk,i || : i = 1, . . . , m1 , ||zk,m1 +i || : i = 1, . . . , m2 } < ɛ /2.

Then ||Ui (xk ) − xk || ≤ ɛ , i = 1, . . . , m1 , ||Ti (Axk ) − Axk || ≤ ||zk,m1 +i || + δT ,i < ɛ .

.

Proof Let .k ≥ 0 be an integer. Set (see (8.48)) ˜ k) x˜k+1 = xk + γ A˜ ∗ (T˜ − I )(Ax

.

.

= xk + γ (

m1 Σ i=1

αi (Ui (xk ) − xk ) +

m2 Σ i=1

αi+m1 A∗ (Ti (Axk ) − Axk )).

(8.67)

Proof of Theorem 8.4

371

Set .

1/2

1/2

1/2

z˜ k = (α1 yk,1 . . . , αm1 yk,m1 , αm1 +1 zk,m1 +1 , . . . , αm1 +m2 zk,m1 +m2 ).

(8.68)

By (8.41), (8.43), (8.61) and (8.62), ˜ k ) = (α U1 (xk ) − α xk , . . . , αm1 Um1 (xk ) − αm1 xk , (T˜ − I )(Ax 1 1 1/2

.

1/2

1/2

1/2

1/2

1/2

1/2

1/2

αm1 +1 T1 (Axk ) − αm1 +1 (Axk ), . . . , αm1 +m2 Tm1 (Axk ) − αm1 +m2 (Axk )).

.

(8.69)

It follows from (8.49), (8.68) and (8.69) that m1 m2 Σ Σ 2 ˜ k )|| ≤ ( ||˜zk − (T˜ − I )Ax αi δU,i + αi+m1 δT2 ,i )1/2 ≤ δ˜T .

.

i=1

(8.70)

i=1

In view of (8.47) and (8.68), A˜ ∗ (˜zk ) =

m1 Σ

.

1/2 1/2

αi αi yk,i +

mΣ 1 +m2

αi A∗ zk,i

i=m2 +1

i=1

and γ A˜ ∗ (˜zk ) = γ

m1 Σ

.

αi yk,i + γ

mΣ 1 +m2

αi A∗ zk,i .

(8.71)

i=m2 +1

i=1

Set y˜k = γ

m1 Σ

.

αi yk,i + γ

mΣ 1 +m2

αi yk,i .

(8.72)

αi2 )1/2 = δA˜ .

(8.73)

i=m2 +1

i=1

In view of (8.50), (8.63), (8.71) and (8.72), ||y˜k − γ A˜ ∗ z˜ k || ≤ γ δA (

mΣ 1 +m2

.

i=m1 +1

By (8.62) and (8.64), ||xk+1 − xk − y˜k || ≤ δU .

.

(8.74)

372

8 Split Common Fixed Point Problems

It follows from (8.64), (8.65), (8.68), (8.70), (8.72), (8.73) and Lemma 8.3 applied ˜ .T = T˜ , .zk = z˜ k , .y = y˜k that with .δA = δ˜A , .δT = δ˜T , .U = I , .A = A, ||xk+1 − z||2 ≤ ||xk − z||2

.

.

.

˜ 2 ||)(T˜ − I )(Ax ˜ k )||2 − γ (2 − γ (||A||

.

˜ δ˜T + δ˜A + 2δU ) + 2||xk − z||(γ ||A||

˜ δ˜T + δ˜A )2 + δU2 + 2δU (γ ||A|| ˜ δ˜T + δ˜A ). + (γ ||A||

(8.75)

In view of (8.69), ˜ k )||2 = ( ||(T˜ − I )(Ax

m1 Σ

αi ||Ui (xk ) − xk ||2 +

.

i=1

m2 Σ

αm1 +i ||Ti (Axk ) − Axk ||2 ).

i=1

(8.76) By (8.45), (8.75) and (8.76), ||xk+1 − z||2 ≤ ||xk − z||2

.

.

m1 m2 Σ Σ − γ (2 − γ ( αi + αi+m1 ||A||2 )) i=1

.

×(

m1 Σ

αi ||Ui (xk ) − xk ||2 +

i=1

m2 Σ

αi+m1 ||Ti (Axk ) − Axk ||2 )

i=1 .

.

i=1

˜ δ˜T + δ˜A + 2δU ) + 2||xk − z||(γ ||A||

˜ δ˜T + δ˜A )2 + δU2 + 2δU (γ ||A|| ˜ δ˜T + δ˜A ). + (γ ||A||

Assertion 1 is proved. Let us prove Assertion 2. Assume that .p ∈ [0, n) is an integer and ||xp − z|| ≤ 2M.

.

(8.77)

(By the relations .||z||, ||x0 || ≤ M, our assumption holds for .p = 0.) Assertion 1 and (8.45), (8.64), (8.65) imply that ||xp+1 − z||2 ≤ ||xp − z||2

.

.

m1 m2 Σ Σ − γ (2 − γ ( αi + αi+m1 ||A||2 )) i=1

i=1

Proof of Theorem 8.4

.

×(

373

m1 Σ

||Ui (xp ) − xp ||2 +

i=1 .

m2 Σ

αi+m1 ||Ti (Axp ) − Axp ||2 )

i=1

2 + 2||xp − z||(δ∗ + 2δU ) + δ∗2 + δU + 2δU δ∗ .

(8.78)

In view of (8.66), at least one of the following cases holds: (i) there is .j ∈ {1, . . . , m1 } such that .||yk,j || ≥ ɛ /2; (ii) there is .j ∈ {1, . . . , m2 } such that .||zk,m1 +j || ≥ ɛ /2. Assume that case (i) holds and j ∈ {1, . . . , m1 }, ||yk,j || ≥ ɛ /2.

.

(8.79)

By (8.61), (8.65) and (8.79), ||Uj (xp ) − xp || ≥ ||yp,j || − δU,j ≥ ɛ /2 − δU,j ≥ ɛ /4

.

and m1 Σ .

αi ||Ui (xp ) − xp ||2 +

i=1

m2 Σ

αi+m1 ||Ti (Axp ) − Axp ||2

i=1

≥ 16−1 ɛ 2 min{αi : i = 1, . . . , m1 }.

.

(8.80)

Assume that case (ii) holds and j ∈ {1, . . . , m2 }, ||zp,m1 +j || ≥ ɛ /2.

.

By (8.62), (8.64), (8.65) and (8.81), ||Tj (Axp ) − Axp || ≥ ||zp,m1 +j || − δT ,j ≥ ɛ /2 − δT ,j ≥ ɛ /4

.

and m1 Σ .

i=1 .

αi ||Ui (xp ) − xp ||2 +

m2 Σ

αi+m1 ||Ti (Axp ) − Axp ||2

i=1

≥ 16−1 αj +m1 ɛ 2 ≥ 16−1 ɛ 2 min{αj +m1 : j = 1, . . . , m2 }.

(8.81)

374

8 Split Common Fixed Point Problems

Together with (8.80) this implies that in the both cases m1 Σ .

αi ||Ui (xp ) − xp ||2 +

i=1

m2 Σ

αi+m1 ||Ti (Axp ) − Axp ||2

i=1 .

≥ 16−1 ɛ 2 min{αj : j = 1, . . . , m1 + m2 }.

(8.82)

It follows from (8.64), (8.65), (8.78) and (8.82) that ||xp+1 − z||2 ≤ ||xp − z||2

.

.

m1 m2 Σ Σ − 16−1 γ (2 − γ ( αi + αi+m1 ||A||2 ))ɛ 2 min{αj : j = 1, . . . , m1 + m2 } i=1 .

i=1

+ (2M + 2)(δ∗ + 2δU ) + δ∗2 + δU2 + 2δU δ∗ .

.

≤ ||xp − z||2

m1 m2 Σ Σ − 32−1 γ (2 − γ ( αi + αi+m1 ||A||2 ))ɛ 2 min{αj : j = 1, . . . , m1 + m2 }. i=1

i=1

(8.83) By (8.77) and (8.83), ||xp+1 − z|| ≤ ||xp − z|| ≤ 2M.

.

Thus by induction we showed that for all .p = 0, . . . , n, ||xp − z|| ≤ 2M

.

(8.84)

and that for all .p = 0, . . . , n−1. Equation (8.83) holds. In view of (8.83) and (8.84), 4M 2 ≥ ||x0 − z||2 ≥ ||x0 − z||2 − ||xn − z||2

.

.

=

n−1 Σ

(||xp − z||2 − ||xp+1 − z||2 )

p=0

.

≥ 32−1 nγ (2 − γ (

m1 Σ i=1

αi +

m2 Σ i=1

αi+m1 ||A||2 ))ɛ 2 min{αj : j = 1, . . . , m1 + m2 }

Proof of Theorem 8.4

375

and n ≤ 128M 2 γ −1 ɛ −2 min{αi :

.

.

m1 m2 Σ Σ i = 1, . . . , m1 + m2 }−1 (2 − γ ( αi + αi+m1 ||A||2 ))−1 . i=1

i=1

Assertion 2 is proved. Let us prove Assertion 3. By (8.61), (8.63) and (8.64), for .i = 1, . . . , m1 , ||Ui (xk ) − xk − yk,i || ≤ δU,i ,

.

||Ui (xk ) − xk || ≤ ||yk,i || + δU,i ≤ ɛ

.

and for .i = 1, . . . , m2 , ||Ti (Axk ) − Axk − zk,m1 +i || ≤ δT ,i

.

||Ti (Axk ) − Axk || ≤ ||zk,m1 +i || + δT ,i ≤ ɛ .

.

This completes the proof of Assertion 3 and Lemma 8.5 itself.

U ∩

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Index

A Almost star-shaped feasibility problem, 76–83 Approximate solution, 3

Firmly nonexpansive mapping, 18 Fixed point, 12 Fixed point problem, 12

B Boundary regularity property (BRP), 4 Bounded subset, 15

H Hilbert space, 1

C Cardinality, 1 Cimmino algorithm, 3 Closed set, 2, 10 Common fixed point problem, 10 Compact set, 11 Complete norm, 17 Continuous operator, 11 Convex feasibility problem, 2 Convex set, 2 Cyclic projections method, 18

I Identity operator, 22 Inconsistent convex feasibility problem, 17–21 Index vector, 138 Inner product, 1 Iterative method, 3

K Krasnosel’ski–Mann iteration, 10

L Linear bounded operator, 21 D Directed operator, 22 Dual operator, 21 Dynamic string-averaging algorithm, 3

M Metric space, 10

F Feasibility problem, 2 Finite-dimensional space, 29

N Nonexpansive operator, 2 Norm, 1

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. J. Zaslavski, Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications 210, https://doi.org/10.1007/978-3-031-50879-0

385

386 P Paracontraction, 11 Projection, 15

Q Quasi–nonexpansive operator, 10

R Remotest set control, 179

Index S Set-valued mappings, 12 Split common fixed point problem, 21–24 Star-shaped feasibility problem, 1–10 Strict contraction, 2 Strong boundary regularity property, 33 Structured set-valued operator, 10