Nonlinear functional analysis. Fixed point theory and its applications.

*(English)*Zbl 0997.47002
Yokohama: Yokohama Publishers. iv, 276 p. (2000).

There exists a very large literature on fixed point theory, being one of the most important topics in nonlinear analysis and its applications. Standard references which appeared in the last decade are, e.g., the monographs of K. Goebel and W. A. Kirk [“Topics in metric fixed point theory” (1990; Zbl 0708.47031)], E. Zeidler [“Nonlinear functional analysis and its applications. I” (1993; Zbl 0794.47033)], J. M. Ayerbe Toledano, T. Domínguez Benavides, and G. López Acedo [“Measures of noncompactness in metric fixed point theory” (1997; Zbl 0885.47021)], S. Singh, B. Watson and P. Srivastava [“Fixed point theory and best approximation; the KKM-map prinziple (1997; Zbl 0901.47039)], L. Górniewicz [“Topological fixed point theory of multivalued mappings” (1999; Zbl 0937.55001)], R. P. Agarwal, M. Mechan and D. O’Regan [“Fixed point theory and applications” (2001; Zbl 0960.54027)], or W. A. Kirk and B. Sims [“Handbook of metric fixed point theory” (2001; Zbl 0970.54001)]. Most of these monographs are concerned with some more or less special aspects of metric or topological fixed point theory.

Inspite of its general title, the book under review is still more specialized. In fact, it is almost entirely devoted to (geo)metric properties of nonexpansive, accretive and similar maps, as well as to applications to resolvent operators, nonlinear ergodic theory, and minimax theorems. This clearly reflects the personal interests of the author and his Japanese co-authors, which is of course legitimate, but does not justify the misleading title. As a result, the book may be of interest for specialists in the field, while those who want to get in touch with fixed point theory for the first time are recommended to consult one of the monographs cited above.

Inspite of its general title, the book under review is still more specialized. In fact, it is almost entirely devoted to (geo)metric properties of nonexpansive, accretive and similar maps, as well as to applications to resolvent operators, nonlinear ergodic theory, and minimax theorems. This clearly reflects the personal interests of the author and his Japanese co-authors, which is of course legitimate, but does not justify the misleading title. As a result, the book may be of interest for specialists in the field, while those who want to get in touch with fixed point theory for the first time are recommended to consult one of the monographs cited above.

Reviewer: Jürgen Appell (Würzburg)

##### MSC:

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

47H10 | Fixed-point theorems |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

47H20 | Semigroups of nonlinear operators |