Fixed Point Theorems and Applications

1. Front Matter

   Pages i-xvii

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2. Fixed Point Theorems

   1. Front Matter

      Pages 1-1

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   2. The Banach Contraction Principle

      • Vittorino Pata

      Pages 3-8

   3. The Boyd-Wong Theorem

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      Pages 9-11

   4. Further Extensions of the Contraction Principle

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      Pages 13-19

   5. Weak Contractions

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      Pages 21-26

   6. Contractions of \(\varepsilon \)-Type

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      Pages 27-34

   7. Sequences of Maps and Fixed Points

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      Pages 35-37

   8. Fixed Points of Non-expansive Maps

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      Pages 39-41

   9. The Riesz Mean Ergodic Theorem

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      Pages 43-46

   10. The Brouwer Fixed Point Theorem

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       Pages 47-51

   11. The Schauder-Tychonoff Fixed Point Theorem

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       Pages 53-57

   12. Further Consequences of the Schauder-Tychonoff Theorem

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       Pages 59-63

   13. The Markov-Kakutani Theorem

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       Pages 65-66

   14. The Kakutani-Ky Fan Theorem

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       Pages 67-72

3. Some Applications

   1. Front Matter

      Pages 2-73

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   2. The Implicit Function Theorem

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      Pages 75-80

   3. Location of Zeros

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      Pages 81-83

   4. Ordinary Differential Equations in Banach Spaces

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      Pages 85-95

   5. The Lax-Milgram Lemma

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      Pages 97-104

This book addresses fixed point theory, a fascinating and far-reaching field with applications in several areas of mathematics. The content is divided into two main parts. The first, which is more theoretical, develops the main abstract theorems on the existence and uniqueness of fixed points of maps. In turn, the second part focuses on applications, covering a large variety of significant results ranging from ordinary differential equations in Banach spaces, to partial differential equations, operator theory, functional analysis, measure theory, and game theory. A final section containing 50 problems, many of which include helpful hints, rounds out the coverage. Intended for Master’s and PhD students in Mathematics or, more generally, mathematically oriented subjects, the book is designed to be largely self-contained, although some mathematical background is needed: readers should be familiar with measure theory, Banach and Hilbert spaces, locally convex topological vector spaces and, in general, with linear functional analysis.

• Fixed point theory
• Functional analysis
• Partial differentia equation
• Ordinary differential equations
• Operator theory
• partial differential equations

“This book is certainly one of the best. It should be quite useful to anybody interested in the theory, methods, and applications of nonlinear functional analysis, in particular to students during their masters studies (il maledetto 3+2, una grossa sciocchezza politica che fa soffrire tanti studenti).” (Jürgen Appell, zbMATH 1448.47001, 2020)

“This is a good reference book on the applications of fixed point theory to other fields in mathematics, optimization theory, economics, engineering, and image science. … I must point out that this book lets researchers get their required knowledge on fixed points in a short time.” (Lai-Jiu Lin, Mathematical Reviews, May, 2020)

“It could be used as a textbook for a somewhat unusual functional analysis course, or for a real analysis coursethat introduces functional analysis. It would also be good preparation for a course in nonlinear partial differential equations or control theory.” (John D. Cook, MAA Reviews, October 6, 2019)

• Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

  Vittorino Pata

Vittorino Pata completed his PhD in Pure Mathematics at Indiana University, USA in December 1994. Since 2000, he has been a Professor of Mathematical Analysis at the Politecnico di Milano, Italy. His research interests touch several areas of pure and applied mathematics, including ordinary and partial differential equations (with particular emphasis on the asymptotic behavior of solutions), infinite-dimensional dynamical systems, real and functional analysis, operator theory, and noncommutative probability.

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