Fixed Points (MATHEMATICAL WORLD) by Yu. A. Shashkin (PDF)

Description

The theory of fixed points finds its roots in the work of Poincare, Brouwer, and Sperner and makes extensive use of such topological notions as continuity, compactness, homotopy, and the degree of a mapping. Fixed point theorems have numerous applications in mathematics; most of the theorems ensuring the existence of solutions for differential, integral, operator, or other equations can be reduced to fixed point theorems. In addition, these theorems are used in such areas as mathematical economics and game theory. This book presents a readable exposition of fixed point theory. The author focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. Another aim of the book is to show how fixed point theory uses combinatorial ideas related to decomposition (triangulation) of figures into distinct parts called faces (simplexes), which adjoin each other in a regular fashion. All necessary background concepts – such as continuity, compactness, degree of a map, and so on – are explained, making the book accessible even to students at the high school level. In addition, the book contains exercises and descriptions of applications.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a short booklet on the Fixed Point Theory. It focuses on the Brouwer’s Fixed Point Theorem and on Newton method of approximation (albeit the name of Newton is not present). It is only a very short introduction to this theory and it presupposes (almost) no knowledge in mathematics. For this reason (I suppose) the author don’t introduce the Picard-Banach Theorem. For this it is necessary much more space. At any rate, I strongly suggest it. Finally, look at the references: there are many interesting and easy to read books. I adore the cover!

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