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**Optima and equilibria. An introduction to nonlinear analysis. Transl. from the French by Stephen Wilson.
2nd ed.**
*(English)*
Zbl 0930.91001

Graduate Texts in Mathematics. 140. Berlin: Springer. xviii, 430 p. (1998).

The author presents in this textbook the basic principles of nonlinear analysis and demonstrates the application of these tools in mathematical optimization, game theory and mathematical economics. Conversely, with numerous historical reflections he shows that the progress in nonlinear analysis and the constructions of mathematical techniques are very influenced by the development of economic equilibrium theory and game theory.

The book is divided in two parts. Part I (226 pages) describes the mentioned theory. Part II (166 pages) contains exercises and problem statements with solutions. An appendix (20 pages) at the end of the book gives a short summary of the presented results.

In detail, section 1 contains the basic tools of mathematical optimization: topological properties of sets and functions, existence theorems of minimizers, extremal principles and fixed-point theorems. The next three sections are devoted to the foundations of convex analysis. The author presents the most important results about convex sets and convex functions, first of all the separation theorems, the proximal theorem, the characterization of conjugate functions and the calculus rules for the subdifferential mapping. The last results are used in section 5 for the characterization of the solution of convex minimization problems by differential inclusions (generalized Fermat’s rules). To extend these principles to nonconvex problems, the author introduces in section 6 the notion of generalized gradients (Clarke’s subdifferential). The relations between subdifferential and directional derivatives and the duality between tangent cone and normal cone are essential parts of this section.

The sections 7 and 8 give a general description (definitions, different strategies, equilibrium) of two-person games and two-person zero-sum games respectively. In this connection the authors prove Browers Fixed-point Theorem, the Minimax Theorem of von Neumann and the Ky Fan Inequality. Section 9 is devoted to the solutions of nonlinear equations and nonlinear inclusions. Using Ky Fan’s Inequality, special existence theorems are proved but also some of the most important theorems of the topic (e.g. Kakutani’s Fixed-point Theorem, the Leray-Schauder Theorem, Shapleys Theorem, the Debreu-Gale-Nikkaïdo Theorem and the Viability Theorem).

Section 10 gives an introduction to the theory of economic equilibrium. The authors provide two explanations to the role of prices in a decentralized mechanism: by means of the Walrasian equilibrium definition and by a more dynamic (viable) model. The discussion of the von Neumann growth model including the proof of the Perron-Frobenius Theorem is contained in section 11. Finally, the sections 12 and 13 give some modifications of the former theory: the extension of special results of two-person games to \(n\)-person games, the discussion of cooperative games and the consideration of fuzzy concepts in game theory.

In part II (sections 14-16) the author has formulated 175 exercises and 48 special problems with solutions. These exercises and problems are grouped accordingly the former sections and follow the order of part I.

The book is written in a very well understandable style. Especially, the motivations given at the beginning of all sections combined with historical remarks are advantageous for the reader. The book is written for second-level students in Mathematics and Mathematical Economics which have the basic knowledge of analysis and linear algebra. Beside it, it should be used also as reference-book for all scientists who have to deal with analytic problems, economic equilibrium models and game theory.

The book is divided in two parts. Part I (226 pages) describes the mentioned theory. Part II (166 pages) contains exercises and problem statements with solutions. An appendix (20 pages) at the end of the book gives a short summary of the presented results.

In detail, section 1 contains the basic tools of mathematical optimization: topological properties of sets and functions, existence theorems of minimizers, extremal principles and fixed-point theorems. The next three sections are devoted to the foundations of convex analysis. The author presents the most important results about convex sets and convex functions, first of all the separation theorems, the proximal theorem, the characterization of conjugate functions and the calculus rules for the subdifferential mapping. The last results are used in section 5 for the characterization of the solution of convex minimization problems by differential inclusions (generalized Fermat’s rules). To extend these principles to nonconvex problems, the author introduces in section 6 the notion of generalized gradients (Clarke’s subdifferential). The relations between subdifferential and directional derivatives and the duality between tangent cone and normal cone are essential parts of this section.

The sections 7 and 8 give a general description (definitions, different strategies, equilibrium) of two-person games and two-person zero-sum games respectively. In this connection the authors prove Browers Fixed-point Theorem, the Minimax Theorem of von Neumann and the Ky Fan Inequality. Section 9 is devoted to the solutions of nonlinear equations and nonlinear inclusions. Using Ky Fan’s Inequality, special existence theorems are proved but also some of the most important theorems of the topic (e.g. Kakutani’s Fixed-point Theorem, the Leray-Schauder Theorem, Shapleys Theorem, the Debreu-Gale-Nikkaïdo Theorem and the Viability Theorem).

Section 10 gives an introduction to the theory of economic equilibrium. The authors provide two explanations to the role of prices in a decentralized mechanism: by means of the Walrasian equilibrium definition and by a more dynamic (viable) model. The discussion of the von Neumann growth model including the proof of the Perron-Frobenius Theorem is contained in section 11. Finally, the sections 12 and 13 give some modifications of the former theory: the extension of special results of two-person games to \(n\)-person games, the discussion of cooperative games and the consideration of fuzzy concepts in game theory.

In part II (sections 14-16) the author has formulated 175 exercises and 48 special problems with solutions. These exercises and problems are grouped accordingly the former sections and follow the order of part I.

The book is written in a very well understandable style. Especially, the motivations given at the beginning of all sections combined with historical remarks are advantageous for the reader. The book is written for second-level students in Mathematics and Mathematical Economics which have the basic knowledge of analysis and linear algebra. Beside it, it should be used also as reference-book for all scientists who have to deal with analytic problems, economic equilibrium models and game theory.

Reviewer: J.Thierfelder (Ilmenau)

### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

91B50 | General equilibrium theory |

49J52 | Nonsmooth analysis |

91B62 | Economic growth models |

91A05 | 2-person games |

91A12 | Cooperative games |