Fixed Point Theorems and Applications

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.

List of contents

1 The Banach contraction principle 7.- 2 The Boyd-Wongtheorem 13.- 3 Further extensions of the contraction principle 16.- 4 Weak contractions 23.- 5 Contractions of -type 29.- 6 Sequences of maps  and fixed points 36.- 7 Fixed points of non-expansive maps 39.- 8 The Riesz mean ergodic theorem 42.- 9 The Brouwer fixed point theorem 46.- 10 The Schauder-Tychono fixed point theorem 50.- 11 Further consequences of the Schauder-Tychono theorem 55.- 12 TheMarkov-Kakutani theorem 60.- 13 TheKakutani-Ky Fan theorem 62.- 14 The implicit function theorem 70.- 15 Location of zeros 75.- 16 Ordinary di erential equations in Banach spaces 78.- 17 The Lax-Milgram lemma 89.- 18 An abstract elliptic problem 97.- 19 Semilinear evolution equations 101.- 20 An abstract parabolic problem 108.- 21 The invariant subspace problem 114.- 22 Measure preserving maps on compact Hausdor spaces 118.- 23 Invariant means on semigroups 120.- 24 Haar measures 123.- 25 Game theory 130.- 26 Problems.

About the author

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.

Summary

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.

Additional text

“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)

Report

"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)

Product details

• Authors: Vittorino Pata
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2019
• EAN: 9783030196691
• ISBN: 978-3-0-3019669-1
• No. of pages: 171
• Dimensions: 156 mm x 11 mm x 236 mm
• Weight: 330 g
• Illustrations: XVII, 171 p. 1 illus.
• Series: UNITEXT; La Matematica per il 3+2
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Analysis, B, Topologie, Mathematics and Statistics, Topology, Functional Analysis, Ordinary Differential Equations, Partial Differential Equations, Differential calculus & equations, Differential equations