Fixed Point Theory
This monograph provides an immense text (690 pages!) on classical topics in fixed point theory that lie on the borderline of topology and nonlinear functional analysis. The book grew from the book “Fixed point theory, Vol. I”, published by the same authors in 1982 and was finished after the death of Jim Dugundji in 1985. The book starts with an introduction and elementary fixed point theorems (e.g., Banach contraction principle and its extensions, the Knaster-Tarski theorem, applications to the geometry of Banach spaces, to theory of critical points and to integral and differential equations, the Markoff-Kakutani theorem, theorems on nonexpansive maps in Hilbert spaces). The next chapter describes the Borsuk theorems and topological transversality. It contains paragraphs on the Brouwer and Borsuk theorems, on fixed points for compact maps and on further applications (the antipodal theorem, the Schauder theorem, the invariance subspace problem, absolute retracts, the Ryll-Nardzewski theorem). The next chapter deals with homology and fixed points (simplicial homology, the Lefschetz-Hopf theorem, the Brouwer degree). The monograph emphasizes developments related to the Leray-Schauder theory and consecutive chapters are dedicated to this subject. The final chapter is devoted to selected topics (finite-codimensional Čech cohomology, Vietoris fractions and coincidence theory).
Each chapter is accompanied by “Miscellaneous results and examples” in the form of exercises, which give further applications and extensions of the theory. They are followed by “Notes and comments” containing references to literature and providing some additional information. The book is equipped with a comprehensive bibliography (30 pages), list of standard symbols and indices of names and terms. The book is well understandable and requires only a basic knowledge of topology and functional analysis; moreover, the necessary background material is collected in an appendix. It can be warmly recommended to a broad spectrum of readers - to graduate students, experts and everybody who wishes to become acquainted with the basic elements and deeper properties of this part of functional analysis