##
**Iterative approximation of fixed points.**
*(English)*
Zbl 1036.47037

Baia Mare: Efemeride (ISBN 973-85243-6-9/pbk). xii, 283 p. (2002).

As the title suggests, this book is concerned not only with examining methods of iteration to obtain fixed points, but also with the important question of the rate of convergence and error analysis.

Chapter 1 provides some background for metrical fixed point theory, including a number of examples of functional, differential, and integral equations which can be formulated as fixed point problems.

Chapter 2 deals with Picard iteration, or function iteration, to determine fixed points of various classes of functions, including \(\varphi\)-contractions.

Let \(X\) be a Banach space, \(T\) a selfmap of \(X\). Specialized Mann iteration is defined by \(x_0 \in X, x_{n+1} = (1 - \alpha_n)x_n + \alpha_nTx_n\), \(n \geq 0\), where \(\alpha_0 = 1, \alpha_n \in [0, 1]\) for \(n > 0\), and \(\sum \alpha_n = \infty\) [W. R. Mann, Proc. Am. Math. Soc. 4, 506–510 (1953; Zbl 0050.11603)]. The special case in which each \(\alpha_n = \lambda\), \(0 < \lambda < 1\), was defined by M. A. Krasnosel’skij [Usp. Mat. Nauk 10, 123–127 (1955; Zbl 0064.12002)]. Chapter 3 deals with the weak and strong convergence of Krasnosel’skij iteration for nonexpansive operators in Hilbert space. Convergence rates and error estimates are obtained for strictly \(\phi\)-contractive operators as well as generalized pseudocontractive and Lipschitzian operators. For this latter class, the author has also determined the value of \(\lambda\) which yields the fastest iteration.

Chapter 4 contains theorems showing that Mann iteration converges for nonexpansive, quasi-nonexpansive, and Lipschitzian strictly pseudocontractive operators, with appropriate restrictions on \(\{\alpha_n\}\) and the space \(X\).

Attempts by the authors to use the same proof techniques on Lipschitzian pseudocontractive maps failed. In [Proc. Am. Math. Soc. 44, 147–150 (1974; Zbl 0286.47036)], S. Ishikawa developed an iteration scheme which provided strong convergence to a fixed point for a Lipschitzian pseudocontractive selfmap \(T\) of a convex compact subset of a Hilbert space. T. L. Hicks and J. D. Kubicek [J. Math. Anal. Appl. 59, 498–504 (1977; Zbl 0361.65057)] provided an example to show that Mann iteration need not converge if \(T\) is not Lipschitzian. In [Proc. Am. Math. Soc. 129, 2359–2363 (2001; Zbl 0972.47062)], C. E. Chidume and S. A. Mutangadura provided an example of a Lipschitzian pseudocontractive map for which no Mann iteration converges.

The scheme developed by Ishikawa is defined by \[ x_0 \in X, x_{n+1} = (1 - \alpha_n)x_n + \alpha_nTy_n, \quad y_n = (1 - \beta_n)x_n + \beta_nTx_n, n \geq 0, \] where (i\(^{\prime}\)) \(0 \leq \alpha_n \leq \beta_n \leq 1,\) (ii) \(\lim \beta_n = 0\), and (iii) \(\sum \nolimits \alpha_n\beta_n = \infty\). As a result of condition (i\(^{\prime}\)), the definition of Ishikawa does not reduce to that of Mann by setting \(\beta_n = 0\). The reviewer [J. Math. Anal. Appl. 56, 741–750 (1976; Zbl 0353.47029)] observed that, for certain other contractive conditions, one could replace (i\(^{\prime}\)) with (i) \(0 \leq \alpha_n\), \(\beta_n \leq 1\). Then each theorem proved for an Ishikawa iteration satisfying (i) would automatically include the corresponding result for Mann iteration as a special case. Since that time, many such papers have been written by a number of authors. Chapter 5 treats Ishikawa iteration.

In Chapter 6, there is a treatment of modified Mann and Ishikawa iterations, ergodic fixed point iteration, and iterations with errors.

Let \((X, d)\) be a metric space, \(T\) a selfmap of \(X, x_0 \in X\), and let (1) \(x_{n+1} = f(T, x_n)\) represent some iteration process that depends on \(T\) and \(x_n\), and suppose that \(\lim x_n = p\), \(p\) a fixed point of \(T\). Let \(\{y_n\} \subset X\) and set \(\varepsilon_n = d(y_{n+1}, f(T, y_n))\). Then (1) is said to be \(T\)-stable if \(\lim \varepsilon_n = 0\) implies \(\lim y_n = p\). The iteration procedures is said to be almost \(T\)-stable if, in addition, \(\sum \nolimits \epsilon_n < \infty\). A treatment of these topics, along with weak stability, continuous dependence on fixed points, and sequences of maps possessing unique fixed points, constitutes Chapter 7.

Chapter 8 provides a survey of some theorems on Mann and Ishikawa iterations (using (i)) applied to certain operators, including Lipschitz strongly accretive and \(\phi\)-strongly accretive operators.

The topics of Chapter 9, which deals with error analysis of fixed point iteration procedures, are the rate of convergence of iterative processes and the theoretical and empirical comparison of some fixed point iteration procedures.

The bibliography contains over 1,000 references, and there is an in-depth treatment of about 400 of them in the text. There are no exercises in the text, but a number of results are stated without proof, providing a source of problems for the interested reader. Each chapter concludes with a section-by-section summary which lists additional references as well as pertinent historical comments.

This book is an excellent introduction to various aspects of fixed point theory. The author has provided a model exposition to accompany the mathematics. This book not only qualifies as an excellent text for graduate students, but provides an exemplary treatment of the subject for present and future researchers in this field.

Chapter 1 provides some background for metrical fixed point theory, including a number of examples of functional, differential, and integral equations which can be formulated as fixed point problems.

Chapter 2 deals with Picard iteration, or function iteration, to determine fixed points of various classes of functions, including \(\varphi\)-contractions.

Let \(X\) be a Banach space, \(T\) a selfmap of \(X\). Specialized Mann iteration is defined by \(x_0 \in X, x_{n+1} = (1 - \alpha_n)x_n + \alpha_nTx_n\), \(n \geq 0\), where \(\alpha_0 = 1, \alpha_n \in [0, 1]\) for \(n > 0\), and \(\sum \alpha_n = \infty\) [W. R. Mann, Proc. Am. Math. Soc. 4, 506–510 (1953; Zbl 0050.11603)]. The special case in which each \(\alpha_n = \lambda\), \(0 < \lambda < 1\), was defined by M. A. Krasnosel’skij [Usp. Mat. Nauk 10, 123–127 (1955; Zbl 0064.12002)]. Chapter 3 deals with the weak and strong convergence of Krasnosel’skij iteration for nonexpansive operators in Hilbert space. Convergence rates and error estimates are obtained for strictly \(\phi\)-contractive operators as well as generalized pseudocontractive and Lipschitzian operators. For this latter class, the author has also determined the value of \(\lambda\) which yields the fastest iteration.

Chapter 4 contains theorems showing that Mann iteration converges for nonexpansive, quasi-nonexpansive, and Lipschitzian strictly pseudocontractive operators, with appropriate restrictions on \(\{\alpha_n\}\) and the space \(X\).

Attempts by the authors to use the same proof techniques on Lipschitzian pseudocontractive maps failed. In [Proc. Am. Math. Soc. 44, 147–150 (1974; Zbl 0286.47036)], S. Ishikawa developed an iteration scheme which provided strong convergence to a fixed point for a Lipschitzian pseudocontractive selfmap \(T\) of a convex compact subset of a Hilbert space. T. L. Hicks and J. D. Kubicek [J. Math. Anal. Appl. 59, 498–504 (1977; Zbl 0361.65057)] provided an example to show that Mann iteration need not converge if \(T\) is not Lipschitzian. In [Proc. Am. Math. Soc. 129, 2359–2363 (2001; Zbl 0972.47062)], C. E. Chidume and S. A. Mutangadura provided an example of a Lipschitzian pseudocontractive map for which no Mann iteration converges.

The scheme developed by Ishikawa is defined by \[ x_0 \in X, x_{n+1} = (1 - \alpha_n)x_n + \alpha_nTy_n, \quad y_n = (1 - \beta_n)x_n + \beta_nTx_n, n \geq 0, \] where (i\(^{\prime}\)) \(0 \leq \alpha_n \leq \beta_n \leq 1,\) (ii) \(\lim \beta_n = 0\), and (iii) \(\sum \nolimits \alpha_n\beta_n = \infty\). As a result of condition (i\(^{\prime}\)), the definition of Ishikawa does not reduce to that of Mann by setting \(\beta_n = 0\). The reviewer [J. Math. Anal. Appl. 56, 741–750 (1976; Zbl 0353.47029)] observed that, for certain other contractive conditions, one could replace (i\(^{\prime}\)) with (i) \(0 \leq \alpha_n\), \(\beta_n \leq 1\). Then each theorem proved for an Ishikawa iteration satisfying (i) would automatically include the corresponding result for Mann iteration as a special case. Since that time, many such papers have been written by a number of authors. Chapter 5 treats Ishikawa iteration.

In Chapter 6, there is a treatment of modified Mann and Ishikawa iterations, ergodic fixed point iteration, and iterations with errors.

Let \((X, d)\) be a metric space, \(T\) a selfmap of \(X, x_0 \in X\), and let (1) \(x_{n+1} = f(T, x_n)\) represent some iteration process that depends on \(T\) and \(x_n\), and suppose that \(\lim x_n = p\), \(p\) a fixed point of \(T\). Let \(\{y_n\} \subset X\) and set \(\varepsilon_n = d(y_{n+1}, f(T, y_n))\). Then (1) is said to be \(T\)-stable if \(\lim \varepsilon_n = 0\) implies \(\lim y_n = p\). The iteration procedures is said to be almost \(T\)-stable if, in addition, \(\sum \nolimits \epsilon_n < \infty\). A treatment of these topics, along with weak stability, continuous dependence on fixed points, and sequences of maps possessing unique fixed points, constitutes Chapter 7.

Chapter 8 provides a survey of some theorems on Mann and Ishikawa iterations (using (i)) applied to certain operators, including Lipschitz strongly accretive and \(\phi\)-strongly accretive operators.

The topics of Chapter 9, which deals with error analysis of fixed point iteration procedures, are the rate of convergence of iterative processes and the theoretical and empirical comparison of some fixed point iteration procedures.

The bibliography contains over 1,000 references, and there is an in-depth treatment of about 400 of them in the text. There are no exercises in the text, but a number of results are stated without proof, providing a source of problems for the interested reader. Each chapter concludes with a section-by-section summary which lists additional references as well as pertinent historical comments.

This book is an excellent introduction to various aspects of fixed point theory. The author has provided a model exposition to accompany the mathematics. This book not only qualifies as an excellent text for graduate students, but provides an exemplary treatment of the subject for present and future researchers in this field.

Reviewer: Billy E. Rhoades (Bloomington)

### MSC:

47H10 | Fixed-point theorems |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47J25 | Iterative procedures involving nonlinear operators |