Fixed point theory and applications.

*(English)*Zbl 0960.54027
Cambridge Tracts in Mathematics. 141. Cambridge: Cambridge University Press. x, 170 p. (2001).

The book can be considered as a handbook containing an introduction from the metric and topological fixed-point theory both for single-valued and multivalued mappings.

The first three chapters are devoted to nonexpansive and contractive mappings. The case of contractive multivalued mappings is considered in chapter 9. In chapters 4-8 results connected with the Brouwer and the Schauder fixed-point theorems are studied first for compact and later for condensing mappings. Chapters 10 and 11 are devoted to multivalued mappings, mainly to the class of so-called KKM-mappings (Knaster-Kuratowski-Mazurkiewicz). Note that these two chapters (10,11) are unconnected with the rest of the book. The last chapter is devoted to a very short introduction to topological degree theory. The definition based on the Sard theorem.

The book is written in a very nice and clear style and is also agreeable from the editorial point of view. The presentation of the material is very good but it is not far going and is concentrated only on very elementary facts.

Summing up all of the above I would like to say that this book is a very good one for students for the first looking on the fixed point theory (both metric and topological).

The first three chapters are devoted to nonexpansive and contractive mappings. The case of contractive multivalued mappings is considered in chapter 9. In chapters 4-8 results connected with the Brouwer and the Schauder fixed-point theorems are studied first for compact and later for condensing mappings. Chapters 10 and 11 are devoted to multivalued mappings, mainly to the class of so-called KKM-mappings (Knaster-Kuratowski-Mazurkiewicz). Note that these two chapters (10,11) are unconnected with the rest of the book. The last chapter is devoted to a very short introduction to topological degree theory. The definition based on the Sard theorem.

The book is written in a very nice and clear style and is also agreeable from the editorial point of view. The presentation of the material is very good but it is not far going and is concentrated only on very elementary facts.

Summing up all of the above I would like to say that this book is a very good one for students for the first looking on the fixed point theory (both metric and topological).

Reviewer: Lech Górniewicz (Toruń)